Foundational Problems in Quantum Mechanics

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*Quantum Mechanics*

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883-907

**44**

Singapore; 2001

**Chapter 4**

**Abstract**

relations.

**1. Introduction**

defined by

**47**

where *A*<sup>0</sup> ¼ *A* � *Eρ*ð Þ *A I*.

skew information *Iρ*ð Þ *A* is defined by

*<sup>I</sup>ρ*ð Þ¼ *<sup>A</sup>* <sup>1</sup> 2

information *I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* is defined by

*Tr i ρ*<sup>1</sup>*=*<sup>2</sup>

*Kenjiro Yanagi*

Uncertainty Relations

Uncertainty relations are inequalities representing the impossibility of simultaneous measurement in quantum mechanics. The most well-known uncertainty relations were presented by Heisenberg and Schrödinger. In this chapter, we generalize and extend them to produce several types of uncertainty

**Keywords:** trace inequality, variance, covariance, skew information, metric adjusted skew information, noncommutativity, observable, operator inequality

Let *Mn*ð Þ (resp. *Mn*,*sa*ð Þ ) be the set of all *n* � *n* complex matrices (resp. all *n* � *n* self-adjoint matrices), endowed with the Hilbert-Schmidt scalar product h i *<sup>A</sup>*, *<sup>B</sup>* <sup>¼</sup> *Tr A*<sup>∗</sup> ½ � *<sup>B</sup>* . Let *Mn*,þð Þ be the set of strictly positive elements of *Mn*ð Þ and

*Mn*,þ,1ð Þ¼ f g *ρ*∈ *Mn*ð Þj *Tr*½ �¼ *ρ* 1, *ρ* >0 . If not otherwise specified, hereafter, we address the case of faithful states, that is *ρ*> 0. It is known that the expectation of an

*Eρ*ð Þ¼ *A Tr*½ � *ρA* ,

*<sup>V</sup>ρ*ð Þ¼ *<sup>A</sup> Tr <sup>ρ</sup> <sup>A</sup>* � *<sup>E</sup>ρ*ð Þ *<sup>A</sup> <sup>I</sup>* � �<sup>2</sup> h i <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup> � � � *<sup>E</sup>ρ*ð Þ *<sup>A</sup>* <sup>2</sup> <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup>

In Section 2, we introduce the Heisenberg and Schrödinger uncertainty relations. In Section 3, we present uncertainty relations with respect to the Wigner-Yanase

� � h i <sup>2</sup> � � <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup> � � � *Tr <sup>ρ</sup>*<sup>1</sup>*=*<sup>2</sup>

where ½ �¼ *X*, *Y XY* � *YX*. Furthermore, the Wigner-Yanase-Dyson skew

0 � �,

*Aρ*<sup>1</sup>*=*<sup>2</sup> *A* h i,

and the variance of an observable *A* ∈ *Mn*,*sa*ð Þ in state *ρ*∈ *Mn*,þ,1ð Þ is

and Wigner-Yanase-Dyson skew information. To represent the degree of noncommutativity between *ρ*∈ *Mn*,þ,1ð Þ and *A* ∈ *Mn*,*sa*ð Þ , the Wigner-Yanase

, *A*

*Mn*,þ,1ð Þ ⊂ *Mn*,þð Þ be the set of strictly positive density matrices, that is

observable *A* ∈ *Mn*,*sa*ð Þ in state *ρ*∈ *Mn*,þ,1ð Þ is defined by

## **Chapter 4** Uncertainty Relations

*Kenjiro Yanagi*

#### **Abstract**

Uncertainty relations are inequalities representing the impossibility of simultaneous measurement in quantum mechanics. The most well-known uncertainty relations were presented by Heisenberg and Schrödinger. In this chapter, we generalize and extend them to produce several types of uncertainty relations.

**Keywords:** trace inequality, variance, covariance, skew information, metric adjusted skew information, noncommutativity, observable, operator inequality

#### **1. Introduction**

Let *Mn*ð Þ (resp. *Mn*,*sa*ð Þ ) be the set of all *n* � *n* complex matrices (resp. all *n* � *n* self-adjoint matrices), endowed with the Hilbert-Schmidt scalar product h i *<sup>A</sup>*, *<sup>B</sup>* <sup>¼</sup> *Tr A*<sup>∗</sup> ½ � *<sup>B</sup>* . Let *Mn*,þð Þ be the set of strictly positive elements of *Mn*ð Þ and *Mn*,þ,1ð Þ ⊂ *Mn*,þð Þ be the set of strictly positive density matrices, that is *Mn*,þ,1ð Þ¼ f g *ρ*∈ *Mn*ð Þj *Tr*½ �¼ *ρ* 1, *ρ* >0 . If not otherwise specified, hereafter, we address the case of faithful states, that is *ρ*> 0. It is known that the expectation of an observable *A* ∈ *Mn*,*sa*ð Þ in state *ρ*∈ *Mn*,þ,1ð Þ is defined by

$$E\_{\rho}(\mathcal{A}) = Tr[\rho \mathcal{A}],$$

and the variance of an observable *A* ∈ *Mn*,*sa*ð Þ in state *ρ*∈ *Mn*,þ,1ð Þ is defined by

$$V\_{\rho}(A) = \operatorname{Tr}\left[\rho\left(A - E\_{\rho}(A)I\right)^{2}\right] = \operatorname{Tr}\left[\rho A^{2}\right] - E\_{\rho}(A)^{2} = \operatorname{Tr}\left[\rho A\_{0}^{2}\right],$$

where *A*<sup>0</sup> ¼ *A* � *Eρ*ð Þ *A I*.

In Section 2, we introduce the Heisenberg and Schrödinger uncertainty relations. In Section 3, we present uncertainty relations with respect to the Wigner-Yanase and Wigner-Yanase-Dyson skew information. To represent the degree of noncommutativity between *ρ*∈ *Mn*,þ,1ð Þ and *A* ∈ *Mn*,*sa*ð Þ , the Wigner-Yanase skew information *Iρ*ð Þ *A* is defined by

$$I\_{\rho}(A) = \frac{1}{2}Tr\left[\left(i\left[\rho^{1/2}, A\right]\right)^2\right] = Tr\left[\rho A^2\right] - Tr\left[\rho^{1/2} A \rho^{1/2} A\right],$$

where ½ �¼ *X*, *Y XY* � *YX*. Furthermore, the Wigner-Yanase-Dyson skew information *I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* is defined by

$$I\_{\rho,a}(A) = \frac{1}{2} \operatorname{Tr} \left[ (i[\rho^a, A]) \left( i[\rho^{1-a}, A] \right) \right] = \operatorname{Tr} \left[ \rho A^2 \right] - \operatorname{Tr} \left[ \rho^a A \rho^{1-a} A \right], \ (a \in [0, 1]).$$

Since *Tr*½ �¼ *ρ*½ � *A*0, *B*<sup>0</sup> *Tr*½ � *ρ*½ � *A*, *B* , we obtain

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

*Vρ*ð Þ� *A Vρ*ð Þ� *B* ∣ Re f g *Tr*½ � *ρA*0*B*<sup>0</sup>

*Tr i ρ*<sup>1</sup>*=*<sup>2</sup>

*Tr ρ*f g *A*0, *B*<sup>0</sup>

uncertainty excluding the classical mixture is defined as

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Iρ*ð Þ� *A Jρ*ð Þ *A*

Luo [8] then derived the uncertainty relation of *Uρ*ð Þ *A* .

*Uρ*ð Þ� *A Uρ*ð Þ *B* ≥

Theorem 1.3. For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

Inequality (3) is a refinement of (1) in terms of (2).

*Tr i ρα* ð Þ ½ � , *<sup>A</sup>*<sup>0</sup> *<sup>i</sup> <sup>ρ</sup>*<sup>1</sup>�*<sup>α</sup>*, *<sup>A</sup>*<sup>0</sup>

*Tr <sup>ρ</sup><sup>α</sup>* f g , *<sup>A</sup>*<sup>0</sup> *<sup>ρ</sup>*<sup>1</sup>�*<sup>α</sup>*, *<sup>A</sup>*<sup>0</sup>

� � � � � ¼ <sup>1</sup>

*Tr i <sup>ρ</sup><sup>α</sup>* ½ Þ ½ � , *<sup>A</sup>*<sup>0</sup> *<sup>i</sup> <sup>ρ</sup>*<sup>1</sup>�*<sup>α</sup>*, *<sup>A</sup>*<sup>0</sup>

**3.2 Wigner-Yanase-Dyson skew information**

skew information as follows:

*<sup>J</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ¼ *<sup>A</sup>* <sup>1</sup>

1 2 2

2

*<sup>I</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ¼ *<sup>A</sup>* <sup>1</sup>

We also define

We note that

**49**

, *A*<sup>0</sup> � � h i <sup>2</sup> � � <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup>

<sup>2</sup> h i <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup>

where f g *A*, *B* ¼ *AB* þ *BA*. The quantity *Uρ*ð Þ *A* representing a quantum

¼

q

1 4

Here, we introduce a one-parameter inequality extended from (3). For 0≤*α*≤ 1, *A*, *B*∈ *Mn*,*sa*ð Þ and *ρ*∈ *Mn*,þ,1ð Þ , we define the Wigner-Yanase-Dyson

� � � � � � <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup>

� <sup>h</sup> �

**3.1 Wigner-Yanase skew information**

*<sup>I</sup>ρ*ð Þ¼ *<sup>A</sup>* <sup>1</sup> 2

> *<sup>J</sup>ρ*ð Þ¼ *<sup>A</sup>* <sup>1</sup> 2

*Uρ*ð Þ¼ *A*

We note the following relation:

are defined as

*Uncertainty Relations*

**3. Uncertainty relation for Wigner-Yanase-Dyson skew information**

To represent the degree of noncommutativity between *ρ* ∈ *Mn*,þ,1ð Þ and *A* ∈ *Mn*,*sa*ð Þ , the Wigner-Yanase skew information *Iρ*ð Þ *A* and related quantity *Jρ*ð Þ *A*

h � � � �

> 0 � � � *Tr <sup>ρ</sup>*<sup>1</sup>*=*<sup>2</sup>

0 � � <sup>þ</sup> *Tr <sup>ρ</sup>*<sup>1</sup>*=*<sup>2</sup>

*Tr ρ*½*A*, *B*��j

2

*:* □

*A*0*ρ*<sup>1</sup>*=*<sup>2</sup> *A*<sup>0</sup>

*:*

*A*0*ρ*<sup>1</sup>*=*<sup>2</sup> *A*<sup>0</sup> h i,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>V</sup>ρ*ð Þ *<sup>A</sup>* <sup>2</sup> � *<sup>V</sup>ρ*ð Þ� *<sup>A</sup> <sup>I</sup>ρ*ð Þ *<sup>A</sup>* � �<sup>2</sup>

0≤*Iρ*ð Þ *A* ≤ *Uρ*ð Þ *A* ≤*Vρ*ð Þ *A :* (2)

2 *:*

0

<sup>2</sup> *Re i ρα* ð Þ ½ � , *<sup>A</sup> <sup>i</sup> <sup>ρ</sup>*<sup>1</sup>�*<sup>α</sup>*, *<sup>A</sup>* � � � � � � ;

� � � � <sup>¼</sup> *Tr*½ �þ *<sup>ρ</sup>A*<sup>2</sup> *Tr <sup>ρ</sup><sup>α</sup>A*0*ρ*<sup>1</sup>�*<sup>α</sup>A*<sup>0</sup>

� (3)

� � � *Tr <sup>ρ</sup><sup>α</sup>A*0*ρ*<sup>1</sup>�*<sup>α</sup>A*<sup>0</sup>

� �*:*

� �*:*

*Tr ρ*½*A*, *B*��j

h i*:*

The convexity of *I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* with respect to *ρ* was famously demonstrated by Lieb [1], and the relationship between the Wigner-Yanase skew information and the uncertainty relation was originally developed by Luo and Zhang [2]. Subsequently, the relationship between the Wigner-Yanase-Dyson skew information and the uncertainty relation was provided by Kosaki [3] and Yanagi-Furuichi-Kuriyama [4]. In Section 4, we discuss the metric adjusted skew information defined by Hansen [5], which is an extension of the Wigner-Yanase-Dyson skew information. The relationship between metric adjusted skew information and the uncertainty relation was provided by Yanagi [6] and generalized by Yanagi-Furuichi-Kuriyama [7] for generalized metric adjusted skew information and the generalized metric adjusted correlation measure. In Sections 5 and 6, we provide non-Hermitian extensions of Heisenberg-type and Schrödinger-type uncertainty relations related to generalized quasi-metric adjusted skew information and the generalized quasimetric adjusted correlation measure. As a result, we obtain results for non-Hermitian uncertainty relations provided by Dou and Du as corollaries of our results. Finally, in Section 7, we present the sum types of uncertainty relations.

#### **2. Heisenberg and Schrödinger uncertainty relations**

Theorem 1.1 (Heisenberg uncertainty relation). For *A*, *B*∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

$$\left| V\_{\rho}(A)V\_{\rho}(B) \ge \frac{1}{4} \right| Tr\left[ \rho[A,B] \right]|^2,\tag{1}$$

where ½ �¼ *A*, *B AB* � *BA* is the commutator.

Theorem 1.2 (Schrödinger uncertainty relation). For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

$$\left| V\_{\rho}(A)V\_{\rho}(B) - \left| \operatorname{Re} \left\{ Tr[\rho A\_0 B\_0] \right\} \right| \right|^2 \geq \frac{1}{4} Tr \left[ \rho [A, B] \right] |^2 A$$

Proof of Theorem 1.2. By the Schwarz inequality

$$\begin{aligned} \left| Tr[\rho A\_0 B\_0] \right|^2 &= \left| Tr\left[ \left( \rho^{1/2} B\_0 \right)^\* \left( \rho^{1/2} A\_0 \right) \right] \right|^2 \\ &\leq Tr\left[ \left( \rho^{1/2} B\_0 \right)^\* \left( \rho^{1/2} B\_0 \right) \right] \cdot Tr\left[ \left( \rho^{1/2} A\_0 \right)^\* \left( \rho^{1/2} A\_0 \right) \right] \right] \\ &= Tr\left[ \rho A\_0^2 \right] \cdot Tr\left[ \rho B\_0^2 \right] = V\_\rho(A) \cdot V\_\rho(B). \end{aligned}$$

Since

$$\begin{aligned} \operatorname{Tr}[\rho[A\_0, B\_0]] &= \operatorname{Tr}[\rho A\_0 B\_0] - \operatorname{Tr}[\rho B\_0 A\_0] = \operatorname{Tr}[\rho A\_0 B\_0] - \overline{\operatorname{Tr}[A\_0 B\_0 \rho]} \\ &= \operatorname{Tr}[\rho A\_0 B\_0] - \overline{\operatorname{Tr}[\rho A\_0 B\_0]} = 2i \operatorname{Im}\{\operatorname{Tr}[\rho A\_0 B\_0] \}, \end{aligned}$$

we have

$$\begin{aligned} \left| Tr[\rho A\_0 B\_0] \right|^2 &= \left( \operatorname{Re} \left\{ Tr[\rho A\_0 B\_0] \right\} \right)^2 + \left( \operatorname{Im} \left\{ Tr[\rho A\_0 B\_0] \right\} \right)^2 \\ &= \left( \operatorname{Re} \left\{ Tr[\rho A\_0 B\_0] \right\} \right)^2 + \frac{1}{4} |Tr[\rho [A\_0, B\_0]]|^2. \end{aligned}$$

**48**

*<sup>I</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ¼ *<sup>A</sup>* <sup>1</sup>

*Quantum Mechanics*

*ρ*∈ *Mn*,þ,1ð Þ ,

*ρ*∈ *Mn*,þ,1ð Þ ,

Since

we have

**48**

j j *Tr*½ � *ρA*0*B*<sup>0</sup>

j j *Tr*½ � *ρA*0*B*<sup>0</sup>

2

*Tr i ρα* ð Þ ½ � , *<sup>A</sup> <sup>i</sup> <sup>ρ</sup>*1�*<sup>α</sup>*, *<sup>A</sup>* � � � � � � <sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup> � � � *Tr <sup>ρ</sup><sup>α</sup>Aρ*1�*<sup>α</sup><sup>A</sup>* � �, ð Þ *<sup>α</sup>* <sup>∈</sup>½ � 0, 1 *:*

The convexity of *I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* with respect to *ρ* was famously demonstrated by Lieb [1], and the relationship between the Wigner-Yanase skew information and the uncertainty relation was originally developed by Luo and Zhang [2]. Subsequently, the relationship between the Wigner-Yanase-Dyson skew information and the uncertainty relation was provided by Kosaki [3] and Yanagi-Furuichi-Kuriyama [4]. In Section 4, we discuss the metric adjusted skew information defined by Hansen [5], which is an extension of the Wigner-Yanase-Dyson skew information. The relationship between metric adjusted skew information and the uncertainty relation was provided by Yanagi [6] and generalized by Yanagi-Furuichi-Kuriyama [7] for generalized metric adjusted skew information and the generalized metric adjusted correlation measure. In Sections 5 and 6, we provide non-Hermitian extensions of Heisenberg-type and Schrödinger-type uncertainty relations related to generalized quasi-metric adjusted skew information and the generalized quasimetric adjusted correlation measure. As a result, we obtain results for non-Hermitian uncertainty relations provided by Dou and Du as corollaries of our results. Finally, in Section 7, we present the sum types of uncertainty relations.

**2. Heisenberg and Schrödinger uncertainty relations**

where ½ �¼ *A*, *B AB* � *BA* is the commutator.

Theorem 1.1 (Heisenberg uncertainty relation). For *A*, *B*∈ *Mn*,*sa*ð Þ ,

Theorem 1.2 (Schrödinger uncertainty relation). For *A*, *B* ∈ *Mn*,*sa*ð Þ ,

1 4 � <sup>h</sup> �

*Tr ρ*½*A*, *B*��j

> � � � 2

� � <sup>¼</sup> *<sup>V</sup>ρ*ð Þ� *<sup>A</sup> <sup>V</sup>ρ*ð Þ *<sup>B</sup> :*

� *Tr <sup>ρ</sup>*<sup>1</sup>*=*<sup>2</sup>

h � � � �

2 ,

� (1)

*Tr ρ*½*A*, *B*��j

*A*<sup>0</sup> � �<sup>∗</sup>

<sup>2</sup> <sup>þ</sup> ð Þ Imf g *Tr*½ � *<sup>ρ</sup>A*0*B*<sup>0</sup>

<sup>4</sup> j j *Tr*½ � *<sup>ρ</sup>*½ � *<sup>A</sup>*0, *<sup>B</sup>*<sup>0</sup>

h i � �

2 *:*

> *ρ*<sup>1</sup>*=*<sup>2</sup> *A*<sup>0</sup>

> > 2

2 *:*

*Vρ*ð Þ *A Vρ*ð Þ *B* ≥

*Vρ*ð Þ *A Vρ*ð Þ� *B* ∣ Re f g *Tr*½ � *ρA*0*B*<sup>0</sup>

*B*0 � �<sup>∗</sup>

*B*0 � �<sup>∗</sup>

<sup>2</sup> <sup>¼</sup> ð Þ Re f g *Tr*½ � *<sup>ρ</sup>A*0*B*<sup>0</sup>

¼ ð Þ Re f g *Tr*½ � *ρA*0*B*<sup>0</sup>

� h i � � �

h i � �

*ρ*<sup>1</sup>*=*<sup>2</sup> *A*<sup>0</sup>

*ρ*<sup>1</sup>*=*<sup>2</sup> *B*0

0

*Tr*½ �¼ *ρ*½ � *A*0, *B*<sup>0</sup> *Tr*½ �� *ρA*0*B*<sup>0</sup> *Tr*½ �¼ *ρB*0*A*<sup>0</sup> *Tr*½ �� *ρA*0*B*<sup>0</sup> *Tr A*½ � <sup>0</sup>*B*0*ρ*

¼ *Tr*½ �� *ρA*0*B*<sup>0</sup> *Tr*½ �¼ *ρA*0*B*<sup>0</sup> 2*i*Imf g *Tr*½ � *ρA*0*B*<sup>0</sup> ,

2 þ 1

Proof of Theorem 1.2. By the Schwarz inequality

<sup>2</sup> <sup>¼</sup> *Tr <sup>ρ</sup>*<sup>1</sup>*=*<sup>2</sup>

≤*Tr ρ*<sup>1</sup>*=*<sup>2</sup>

<sup>¼</sup> *Tr <sup>ρ</sup>A*<sup>2</sup> 0 � � � *Tr <sup>ρ</sup>B*<sup>2</sup>

�

Since *Tr*½ �¼ *ρ*½ � *A*0, *B*<sup>0</sup> *Tr*½ � *ρ*½ � *A*, *B* , we obtain

$$\left| \left( V\_{\rho}(A) \cdot V\_{\rho}(B) - |\operatorname{Re} \left\{ Tr[\rho A\_0 B\_0] \right\} | \right) \right|^2 \geq \frac{1}{4} \operatorname{Tr} \left[ \rho [A, B] | \right]^2. \tag{7}$$

#### **3. Uncertainty relation for Wigner-Yanase-Dyson skew information**

#### **3.1 Wigner-Yanase skew information**

To represent the degree of noncommutativity between *ρ* ∈ *Mn*,þ,1ð Þ and *A* ∈ *Mn*,*sa*ð Þ , the Wigner-Yanase skew information *Iρ*ð Þ *A* and related quantity *Jρ*ð Þ *A* are defined as

$$I\_{\rho}(A) = \frac{1}{2}Tr\left[\left(i\left[\rho^{1/2}, A\_0\right]\right)^2\right] = Tr\left[\rho A\_0^2\right] - Tr\left[\rho^{1/2}A\_0\rho^{1/2}A\_0\right].$$

$$J\_{\rho}(A) = \frac{1}{2}Tr\left[\rho\left\{A\_0, B\_0\right\}^2\right] = Tr\left[\rho A\_0^2\right] + Tr\left[\rho^{1/2}A\_0\rho^{1/2}A\_0\right],$$

where f g *A*, *B* ¼ *AB* þ *BA*. The quantity *Uρ*ð Þ *A* representing a quantum uncertainty excluding the classical mixture is defined as

$$U\_{\rho}(\mathbf{A}) = \sqrt{I\_{\rho}(\mathbf{A}) \cdot J\_{\rho}(\mathbf{A})} = \sqrt{V\_{\rho}(\mathbf{A})^2 - \left(V\_{\rho}(\mathbf{A}) - I\_{\rho}(\mathbf{A})\right)^2}.$$

We note the following relation:

$$0 \le I\_{\rho}(A) \le U\_{\rho}(A) \le V\_{\rho}(A). \tag{2}$$

Luo [8] then derived the uncertainty relation of *Uρ*ð Þ *A* . Theorem 1.3. For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

$$\left| U\_{\rho}(A) \cdot U\_{\rho}(B) \geq \frac{1}{4} \right| \text{Tr} \left[ \rho[A, B] \right] |^{2}. \tag{3}$$

Inequality (3) is a refinement of (1) in terms of (2).

#### **3.2 Wigner-Yanase-Dyson skew information**

Here, we introduce a one-parameter inequality extended from (3). For 0≤*α*≤ 1, *A*, *B*∈ *Mn*,*sa*ð Þ and *ρ*∈ *Mn*,þ,1ð Þ , we define the Wigner-Yanase-Dyson skew information as follows:

$$I\_{\rho,a}(A) = \frac{1}{2} \operatorname{Tr} \left[ (i[\rho^a, A\_0]) \left( i[\rho^{1-a}, A\_0] \right) \right] = \operatorname{Tr} \left[ \rho A\_0^2 \right] - \operatorname{Tr} \left[ \rho^a A\_0 \rho^{1-a} A\_0 \right].$$

We also define

$$J\_{\rho,a}(A) = \frac{1}{2} \operatorname{Tr} \left[ \{ \rho^a, A\_0 \} \{ \rho^{1-a}, A\_0 \} \right] \\ = \operatorname{Tr} [\rho A\_2] + \operatorname{Tr} \left[ \rho^a A\_0 \rho^{1-a} A\_0 \right].$$

We note that

$$\frac{1}{2}\operatorname{Tr}[\dot{\iota}[\rho^a, A\_0])\left(\dot{\iota}[\rho^{1-a}, A\_0]\right)\big] = \frac{1}{2}\operatorname{Re}\left[\left(\dot{\iota}[\rho^a, A]\right)\left(\dot{\iota}[\rho^{1-a}, A]\right)\right];$$

however, we have

$$\frac{1}{2}Tr\left[\left\{\rho^a, A\_0\right\}\left\{\rho^{1-a}, A\_0\right\}\right] \neq \frac{1}{2}Tr\left[\left\{\rho^a, A\right\}\left\{\rho^{1-a}, A\right\}\right].$$

We then have the following inequalities:

$$I\_{\rho,a}(A) \le I\_{\rho}(A) \le I\_{\rho}(A) \le I\_{\rho,a}(A),\tag{4}$$

*Tr*½ �¼ *<sup>ρ</sup>*½ � *<sup>A</sup>*, *<sup>B</sup> Tr*½ �¼ *<sup>ρ</sup>*½ � *<sup>A</sup>*0, *<sup>B</sup>*<sup>0</sup> <sup>2</sup>*iImTr*½ �¼ *<sup>ρ</sup>A*0*B*<sup>0</sup> <sup>2</sup>*iIm*<sup>X</sup>

∣*Tr*½ � *ρ*½ � *A*, *B* ∣ ¼ 2∣

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

We then have

*Uncertainty Relations*

¼ 2*i* X *i*< *j*

*λ<sup>i</sup>* � *λ <sup>j</sup>*

� *λ <sup>j</sup>*k*Im ϕi*j*A*0j*ϕ<sup>j</sup>*

*i*<*j*

*i*< *j*

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*λ<sup>i</sup>* þ *λ <sup>j</sup>* � �<sup>2</sup> � *<sup>λ</sup><sup>α</sup>*

*<sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* � *<sup>λ</sup><sup>α</sup>*

*<sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup><sup>α</sup>*

X *i*<*j*

j j *Tr*½ � *<sup>ρ</sup>*½ � *<sup>A</sup>*, *<sup>B</sup>* <sup>2</sup> <sup>≤</sup> <sup>4</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup> *i* <*j*

≤ X *i* <*j*

≤ X *i* <*j*

≤ X *i* <*j*

� X *i* <*j*

Then, we have

We also have

When *<sup>α</sup>* <sup>¼</sup> <sup>1</sup>

**measure**

**51**

2

**4.1 Operator monotone function**

By (7) and the Schwarz inequality,

*<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* j j *Tr*½*ρ*½*A*, *<sup>B</sup>*�� <sup>2</sup> <sup>≤</sup> <sup>4</sup>*α*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>X</sup>

*λ<sup>i</sup>* � *λ <sup>j</sup>*

� �*Im <sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>*

D E *<sup>ϕ</sup>j*j*B*0j*ϕ<sup>i</sup>*

j*λ<sup>i</sup>* � *λ <sup>j</sup>*k*Im ϕi*j*A*0j*ϕ<sup>j</sup>*

j*λ<sup>i</sup>* � *λ <sup>j</sup>*k*Im ϕi*j*A*0j*ϕ<sup>j</sup>* D Eh*ϕj*j*B*0j*ϕi*ij ( )<sup>2</sup>

( )<sup>2</sup>

<sup>j</sup>*B*0j*ϕi*ij ( )<sup>2</sup>

� �<sup>2</sup> � �<sup>1</sup>*=*<sup>2</sup>

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* � *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

� �∣h*ϕi*∣*A*<sup>0</sup> *<sup>ϕ</sup><sup>j</sup>*

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

*<sup>I</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ *<sup>A</sup> <sup>J</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ *<sup>B</sup>* <sup>≥</sup>*α*ð Þ <sup>1</sup> � *<sup>α</sup>* j j *Tr*½*ρ*½*A*, *<sup>B</sup>*�� <sup>2</sup>

*I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *B J<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* ≥ *α*ð Þ 1 � *α Tr*½*ρ*½*A*, *B*��j

**4. Metric adjusted skew information and metric adjusted correlation**

A function *f* : ð Þ! 0, þ∞ is considered operator monotone if, for any *n*∈ℕ, and *A*, *B* ∈ *Mn* such that 0≤ *A* ≤*B*, the inequalities 0≤*f A*ð Þ≤ *f B*ð Þ hold. An operator

, we obtain the result in Theorem 1.3.

Thus, we have the final result, (6). □

� �∣h*ϕi*∣*B*<sup>0</sup> *<sup>ϕ</sup>j*<sup>i</sup>

*<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>p</sup> <sup>j</sup>*λ<sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*k*Im <sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>*

*<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>p</sup> <sup>j</sup>*λ<sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*<sup>k</sup> *<sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>*

� �*Im <sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>*

*i* <*j*

D E *<sup>ϕ</sup>j*j*B*0j*ϕ<sup>i</sup>*

D E∣*:*

D Eh*ϕj*j*B*0j*ϕi*ij ( )<sup>2</sup>

*λ<sup>i</sup>* � *λ <sup>j</sup>*

D E *<sup>ϕ</sup>j*j*B*0j*ϕ<sup>i</sup>*

� � *<sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>*

D E,

D E∣ ≤ <sup>2</sup>

D Eh*ϕj*j*B*0j*ϕi*<sup>i</sup>

D Eh*ϕ<sup>j</sup>*

*j*

� � � � � � 2 *:*

i � � � � � � 2

*:*

2

*:* �

<sup>j</sup>*B*0j*ϕi*ij ( )<sup>2</sup>

jh*ϕi*j*A*0j*ϕ<sup>j</sup>*

ikh*ϕ<sup>j</sup>*

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

*j*

*j*

�

D E *<sup>ϕ</sup><sup>j</sup>*

X *i*<*j* ∣*λi*

*:*

j*B*0j*ϕ<sup>i</sup>* D E

because *Tr <sup>ρ</sup>*1*=*2*Aρ*1*=*2*<sup>A</sup>* � �<sup>≤</sup> *Tr <sup>ρ</sup><sup>α</sup>Aρ*1�*<sup>α</sup>* ½ � *<sup>A</sup>* . We define

$$U\_{\rho,a}(A) = \sqrt{I\_{\rho,a}(A) \cdot I\_{\rho,a}(A)} = \sqrt{V\_{\rho}(A)^2 - \left(V\_{\rho}(A) - I\_{\rho,a}(A)\right)}.\tag{5}$$

From (2), (4), and (5), we have

$$0 \le I\_{\rho,a}(A) \le I\_{\rho}(A) \le U\_{\rho}(A).$$

and

$$0 \le I\_{\rho,a}(A) \le U\_{\rho,a}(A) \le U\_{\rho}(A).$$

We provide the following uncertainty relation with respect to *Uρ*,*<sup>α</sup>*ð Þ *A* as a direct generalization of (3).

Theorem 1.4 ([9]). For *A*, *B*∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

$$\left| U\_{\rho,a}(A) \cdot U\_{\rho,a}(B) \ge a(1-a) \right| \text{Tr} \left[ \rho[A,B] \right] \text{|}^2. \tag{6}$$

Proof of Theorem 1.4. By spectral decomposition, there exists an orthonormal basis j*ϕ*1i, j*ϕ*2i, … , j*ϕ<sup>n</sup>* f gi consisting of eigenvectors of *ρ*. Let *λ*1, *λ*2, … , *λ<sup>n</sup>* be the corresponding eigenvalues, where P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*λ<sup>i</sup>* <sup>¼</sup> 1 and *<sup>λ</sup><sup>i</sup>* <sup>≥</sup>0. Thus *<sup>ρ</sup>* has a spectral representation *<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*λi*∣*ϕi*ih*ϕi*∣. We can obtain the following representations of *Iρ*,*<sup>α</sup>*ð Þ *A* and *Jρ*,*<sup>α</sup>*ð Þ *A* :

$$I\_{\rho,a}(A) = \sum\_{i
$$J\_{\rho,a}(A) \ge \sum\_{i$$
$$

Since 1ð Þ � <sup>2</sup>*<sup>α</sup>* <sup>2</sup> ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>2</sup> � *<sup>t</sup> <sup>α</sup>* � *<sup>t</sup>* <sup>1</sup>�*<sup>α</sup>* ð Þ<sup>2</sup> ≥0 for any *t*>0 and 0 ≤*α*≤ 1, we define *<sup>t</sup>* <sup>¼</sup> *<sup>λ</sup><sup>i</sup> λ j* and have

$$(\mathbf{1} - 2a)^2 \left(\frac{\lambda\_i}{\lambda\_j} - \mathbf{1}\right)^2 - \left(\left(\frac{\lambda\_i}{\lambda\_j}\right)^a - \left(\frac{\lambda\_i}{\lambda\_j}\right)^{1-a}\right)^2 \ge \mathbf{0}.$$

Then,

$$\left(\left(\lambda\_i + \lambda\_j\right)^2 - \left(\lambda\_i^a \lambda\_j^{1-a} + \lambda\_i^{1-a} \lambda\_j^a\right)^2\right) \ge 4a(1-a)\left(\lambda\_i - \lambda\_j\right)^2. \tag{7}$$

Since

*Uncertainty Relations DOI: http://dx.doi.org/10.5772/intechopen.92137*

$$\begin{split} \operatorname{Tr}[\rho[A,B]] &= \operatorname{Tr}[\rho[A\_0,B\_0]] = 2i\operatorname{Im}\operatorname{Tr}[\rho A\_0 B\_0] = 2i\operatorname{Im}\sum\_{i \left<\phi\_j |B\_0|\phi\_i\right> \\ &= 2i\sum\_{i \left<\phi\_j |B\_0|\phi\_i\right>, \\ |\operatorname{Tr}[\rho[A,B]]| &= 2|\sum\_{i \left<\phi\_j |B\_0|\phi\_i\right>| \le 2\sum\_{i \left<\phi\_j |B\_0|\phi\_i\right>|. \end{split}$$

We then have

however, we have

*Quantum Mechanics*

1 2

*U<sup>ρ</sup>*,*<sup>α</sup>*ð Þ¼ *A*

and

generalization of (3).

representation *<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

Since 1ð Þ � <sup>2</sup>*<sup>α</sup>* <sup>2</sup>

and have

*<sup>t</sup>* <sup>¼</sup> *<sup>λ</sup><sup>i</sup> λ j*

Then,

Since

**50**

*Iρ*,*<sup>α</sup>*ð Þ *A* and *Jρ*,*<sup>α</sup>*ð Þ *A* :

From (2), (4), and (5), we have

corresponding eigenvalues, where P*<sup>n</sup>*

*<sup>I</sup>ρ*,*<sup>α</sup>*ð Þ¼ *<sup>A</sup>* <sup>X</sup>

*J<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* ≥

*λ<sup>i</sup>* þ *λ <sup>j</sup>* � �<sup>2</sup> � *<sup>λ</sup><sup>α</sup>*

*i* <*j*

X *i*<*j*

ð Þ *<sup>t</sup>* � <sup>1</sup> <sup>2</sup> � *<sup>t</sup>*

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>α</sup>* <sup>2</sup> *<sup>λ</sup><sup>i</sup>*

*<sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* � *λα*

*<sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* <sup>þ</sup> *λα*

*<sup>α</sup>* � *<sup>t</sup>* <sup>1</sup>�*<sup>α</sup>* ð Þ<sup>2</sup>

*λ j* � 1 � �<sup>2</sup>

*Tr <sup>ρ</sup><sup>α</sup>* f g , *<sup>A</sup>*<sup>0</sup> *<sup>ρ</sup>*1�*<sup>α</sup>*, *<sup>A</sup>*<sup>0</sup> � � � � 6¼ <sup>1</sup>

because *Tr <sup>ρ</sup>*1*=*2*Aρ*1*=*2*<sup>A</sup>* � �<sup>≤</sup> *Tr <sup>ρ</sup><sup>α</sup>Aρ*1�*<sup>α</sup>* ½ � *<sup>A</sup>* . We define

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ� *A J<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A*

Theorem 1.4 ([9]). For *A*, *B*∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

We then have the following inequalities:

q

2

¼

0≤ *Iρ*,*<sup>α</sup>*ð Þ *A* ≤*Iρ*ð Þ *A* ≤ *Uρ*ð Þ *A*

0≤*Iρ*,*<sup>α</sup>*ð Þ *A* ≤ *Uρ*,*<sup>α</sup>*ð Þ *A* ≤ *Uρ*ð Þ *A :*

*Uρ*,*<sup>α</sup>*ð Þ� *A Uρ*,*<sup>α</sup>*ð Þ *B* ≥*α*ð Þ 1 � *α Tr ρ*½*A*, *B*��j

Proof of Theorem 1.4. By spectral decomposition, there exists an orthonormal basis j*ϕ*1i, j*ϕ*2i, … , j*ϕ<sup>n</sup>* f gi consisting of eigenvectors of *ρ*. Let *λ*1, *λ*2, … , *λ<sup>n</sup>* be the

� �

� �

� *<sup>λ</sup><sup>i</sup> λ j* � �*<sup>α</sup>*

*j*

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

� �<sup>2</sup>

We provide the following uncertainty relation with respect to *Uρ*,*<sup>α</sup>*ð Þ *A* as a direct

� <sup>h</sup> �

*<sup>i</sup>*¼<sup>1</sup>*λi*∣*ϕi*ih*ϕi*∣. We can obtain the following representations of

*j*

*j*

� *<sup>λ</sup><sup>i</sup> λ j*

≥4*α*ð Þ 1 � *α λ<sup>i</sup>* � *λ <sup>j</sup>*

� �<sup>1</sup>�*<sup>α</sup>* !<sup>2</sup>

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* � *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

*<sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup>*

*Tr ρα* f g , *<sup>A</sup> <sup>ρ</sup>*1�*<sup>α</sup>*, *<sup>A</sup>* � � � � *:*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>V</sup>ρ*ð Þ *<sup>A</sup>* <sup>2</sup> � *<sup>V</sup>ρ*ð Þ� *<sup>A</sup> <sup>I</sup><sup>ρ</sup>*,*<sup>α</sup>*ð Þ *<sup>A</sup> :* <sup>q</sup> � (5)

> 2 *:*

*<sup>i</sup>*¼<sup>1</sup>*λ<sup>i</sup>* <sup>¼</sup> 1 and *<sup>λ</sup><sup>i</sup>* <sup>≥</sup>0. Thus *<sup>ρ</sup>* has a spectral

∣h*ϕi*∣*A*<sup>0</sup> *ϕ<sup>j</sup>*

∣h*ϕi*∣*A*<sup>0</sup> *ϕ<sup>j</sup>*

≥0 for any *t*>0 and 0 ≤*α*≤ 1, we define

i � � �

i � � �

≥0*:*

� �<sup>2</sup>

*:* (7)

� � � 2 *:*

� � � 2 *:*

� (6)

*I<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* ≤*Iρ*ð Þ *A* ≤ *Jρ*ð Þ *A* ≤*J<sup>ρ</sup>*,*<sup>α</sup>*ð Þ *A* , (4)

$$\left| Tr[\rho[A,B]] \right|^2 \le 4 \left\{ \sum\_{i$$

By (7) and the Schwarz inequality,

*<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* j j *Tr*½*ρ*½*A*, *<sup>B</sup>*�� <sup>2</sup> <sup>≤</sup> <sup>4</sup>*α*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>X</sup> *i*< *j* j*λ<sup>i</sup>* � *λ <sup>j</sup>*k*Im ϕi*j*A*0j*ϕ<sup>j</sup>* D Eh*ϕj*j*B*0j*ϕi*ij ( )<sup>2</sup> <sup>¼</sup> <sup>X</sup> *i* <*j* 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>p</sup> <sup>j</sup>*λ<sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*k*Im <sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>* D Eh*ϕj*j*B*0j*ϕi*<sup>i</sup> ( )<sup>2</sup> ≤ X *i* <*j* 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* <sup>p</sup> <sup>j</sup>*λ<sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*<sup>k</sup> *<sup>ϕ</sup>i*j*A*0j*ϕ<sup>j</sup>* D Eh*ϕj*j*B*0j*ϕi*ij ( )<sup>2</sup> ≤ X *i* <*j λ<sup>i</sup>* þ *λ <sup>j</sup>* � �<sup>2</sup> � *<sup>λ</sup><sup>α</sup> <sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup> j* � �<sup>2</sup> � �<sup>1</sup>*=*<sup>2</sup> jh*ϕi*j*A*0j*ϕ<sup>j</sup>* ikh*ϕ<sup>j</sup>* <sup>j</sup>*B*0j*ϕi*ij ( )<sup>2</sup> ≤ X *i* <*j <sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* � *<sup>λ</sup><sup>α</sup> <sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* � *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup> j* � �∣h*ϕi*∣*A*<sup>0</sup> *<sup>ϕ</sup><sup>j</sup>* i � � � � � � 2 � X *i* <*j <sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>λ</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup><sup>α</sup> <sup>i</sup> <sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>i</sup> <sup>λ</sup><sup>α</sup> j* � �∣h*ϕi*∣*B*<sup>0</sup> *<sup>ϕ</sup>j*<sup>i</sup> � � � � � � 2 *:*

Then, we have

$$I\_{\rho,a}(A)I\_{\rho,a}(B) \ge a(1-a)|Tr[\rho[A,B]]|^2.$$

We also have

$$I\_{\rho,a}(B)I\_{\rho,a}(A) \ge \alpha(\mathbf{1} - a) \left| Tr[\rho[A,B]] \right|^2.$$

Thus, we have the final result, (6). □ When *<sup>α</sup>* <sup>¼</sup> <sup>1</sup> 2 , we obtain the result in Theorem 1.3.

#### **4. Metric adjusted skew information and metric adjusted correlation measure**

#### **4.1 Operator monotone function**

A function *f* : ð Þ! 0, þ∞ is considered operator monotone if, for any *n*∈ℕ, and *A*, *B* ∈ *Mn* such that 0≤ *A* ≤*B*, the inequalities 0≤*f A*ð Þ≤ *f B*ð Þ hold. An operator monotone function is said to be symmetric if *f x*ð Þ¼ *xf x*�<sup>1</sup> ð Þ and normalized if *f*ð Þ¼ 1 1.

*I f*

*Cf*

*U f <sup>ρ</sup>* ð Þ¼ *A*

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

Proposition 1. The following holds:

0

*<sup>ρ</sup>* ð Þ¼ *<sup>A</sup>*<sup>0</sup> *Tr <sup>ρ</sup>A*<sup>2</sup>

Quantity *I <sup>f</sup>*

*Uncertainty Relations*

*<sup>ρ</sup>* ð Þ¼ *<sup>A</sup> <sup>I</sup><sup>f</sup>*

*<sup>ρ</sup>* ð Þ¼ *A*

then it holds that

where *A*, *B*∈ *Mn*,*sa*ð Þ .

*<sup>ρ</sup>* ð Þ¼ *<sup>A</sup> Tr <sup>ρ</sup>A*<sup>2</sup>

*<sup>ρ</sup>* ð Þ *<sup>A</sup>* <sup>≤</sup> *<sup>U</sup> <sup>f</sup>*

*If <sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>J</sup> <sup>f</sup>*

Theorem 1.6 ([6]). For *f* ∈*F<sup>r</sup>*

q

1.*I<sup>f</sup>*

2.*J <sup>f</sup>*

3.0≤ *I<sup>f</sup>*

4.*U <sup>f</sup>*

Since

we have

*x* þ *y* 2 � �<sup>2</sup>

**53**

*<sup>m</sup>*<sup>~</sup>*f*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>y</sup>*~*<sup>f</sup> <sup>x</sup>*

� *m*<sup>~</sup>*f*ð Þ *x*, *y*

*<sup>ρ</sup>* ð Þ¼ *<sup>A</sup> <sup>f</sup>*ð Þ <sup>0</sup>

r

h i *A*, *B <sup>ρ</sup>*,*<sup>f</sup>* is referred to as the metric adjusted correlation measure.

� � � *Tr m*~*<sup>f</sup> <sup>L</sup>ρ*, *<sup>R</sup><sup>ρ</sup>*

� �ð Þ� *<sup>A</sup>*<sup>0</sup> *<sup>A</sup>*<sup>0</sup>

0

� � <sup>þ</sup> *Tr m*<sup>~</sup>*<sup>f</sup> <sup>L</sup>ρ*, *<sup>R</sup><sup>ρ</sup>*

*<sup>ρ</sup>* ð Þ *A*

.

*op*, if

*x* þ 1

Lemma 1. If (8) holds, then the following inequality is satisfied:

� *m*<sup>~</sup>*f*ð Þ *x*, *y*

*<sup>ρ</sup>* ð Þ *A* ≤*Vρ*ð Þ *A* .

*U f*

To prove Theorem 1.6, several lemmas are used.

*x* þ *y* 2 � �<sup>2</sup>

*x* þ *y*

*x y*

<sup>¼</sup> *<sup>f</sup>*ð Þ <sup>0</sup> ð Þ *<sup>x</sup>* � *<sup>y</sup>*

2*m <sup>f</sup>*ð Þ *x*, *y*

<sup>þ</sup> <sup>1</sup> � *<sup>x</sup>*

<sup>2</sup> � *<sup>m</sup>*<sup>~</sup>*f*ð Þ *<sup>x</sup>*, *<sup>y</sup>* n o *x* þ *y*

2

*<sup>y</sup>* � <sup>1</sup> � �<sup>2</sup> *<sup>f</sup>*ð Þ <sup>0</sup>

*x* þ *y*

( )

Proof of Lemma 1. By (8), we have

*y* � �

¼ *y* 2

<sup>2</sup> <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*

*<sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>U</sup> <sup>f</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>ρ</sup>* ð Þ¼ *A Tr m <sup>f</sup> Lρ*, *R<sup>ρ</sup>*

<sup>2</sup> h i *<sup>i</sup>*½ � *<sup>ρ</sup>*, *<sup>A</sup>* , *<sup>i</sup>*½ � *<sup>ρ</sup>*, *<sup>A</sup> <sup>ρ</sup>*,*<sup>f</sup>* ,

� �ð Þ� *<sup>A</sup> <sup>A</sup>* � �,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>V</sup>ρ*ð Þ *<sup>A</sup>* <sup>2</sup> � *<sup>V</sup>ρ*ð Þ� *<sup>A</sup> <sup>I</sup> <sup>f</sup>*

*<sup>ρ</sup>* ð Þ *A* is referred to as the metric adjusted skew information, and

� � <sup>¼</sup> *<sup>V</sup>ρ*ð Þþ *<sup>A</sup> <sup>C</sup>*~*<sup>f</sup>*

*<sup>ρ</sup>* ð Þ *B* ≥*f*ð Þ 0 *Tr*ð*ρ*½*A*, *B*�Þj

� �ð Þ� *<sup>A</sup>*<sup>0</sup> *<sup>A</sup>*<sup>0</sup>

*<sup>ρ</sup>* ð Þ *A*

� � <sup>¼</sup> *<sup>V</sup>ρ*ð Þ� *<sup>A</sup> <sup>C</sup>*~*<sup>f</sup>*

<sup>2</sup> <sup>þ</sup> <sup>~</sup>*f x*ð Þ≥2*f x*ð Þ, (8)

2

2 *:*

<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>~</sup>*f*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>≥</sup>2*<sup>m</sup> <sup>f</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup> :* (10)

<sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*

<sup>2</sup> � *<sup>f</sup>*ð Þ <sup>0</sup> ð Þ *<sup>x</sup>* � *<sup>y</sup>*

2

<sup>2</sup>*<sup>m</sup> <sup>f</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* ,

2

*:* ð Þ *by* ð Þ 10

� (9)

, �

<sup>2</sup> <sup>≥</sup> *<sup>f</sup>*ð Þ <sup>0</sup> ð Þ *<sup>x</sup>* � *<sup>y</sup>*

*f x*ð Þ *=y*

<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>~</sup>*f*ð Þ *<sup>x</sup>*, *<sup>y</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>~</sup>*f*ð Þ *<sup>x</sup>*, *<sup>y</sup>* n o

n o<sup>≥</sup> *<sup>f</sup>*ð Þ <sup>0</sup> ð Þ *<sup>x</sup>* � *<sup>y</sup>*

*:*

*<sup>ρ</sup>*ð Þ *A*<sup>0</sup> .

*<sup>ρ</sup>*ð Þ *A*<sup>0</sup> .

� �<sup>2</sup>

Definition 1 F*op* is the class of functions *f* : ð Þ! 0, þ∞ ð Þ 0, þ∞ such that:

$$\mathbf{1}\_f f(\mathbf{1}) = \mathbf{1}.$$

2.*tf t*�<sup>1</sup> ð Þ¼ *f t*ð Þ.

3.*f* is operator monotone.

Example 1. Examples of elements of F*op* are given by the following:

$$f\_{RLD}(\mathbf{x}) = \frac{2\mathbf{x}}{\mathbf{x} + \mathbf{1}}, \quad f\_{W\mathbf{Y}}(\mathbf{x}) = \left(\frac{\sqrt{\mathbf{x}} + \mathbf{1}}{2}\right)^2, \quad f\_{RKM}(\mathbf{x}) = \frac{\mathbf{x} - \mathbf{1}}{\log \mathbf{x}},$$

$$f\_{SLD}(\mathbf{x}) = \frac{\mathbf{x} + \mathbf{1}}{2}, \quad f\_{W\mathbf{Y}D}(\mathbf{x}) = a(\mathbf{1} - a) \frac{(\mathbf{x} - \mathbf{1})^2}{(\mathbf{x}^a - \mathbf{1})(\mathbf{x}^{1-a} - \mathbf{1})}, \quad a \in (\mathbf{0}, \mathbf{1}).$$

Remark 1. Any *f* ∈ F*op* satisfies

$$\frac{2\varkappa}{\varkappa+1} \le f(\varkappa) \le \frac{\varkappa+1}{2}, \quad \varkappa > 0.$$

For *f* ∈ F*op*, we define *f*ð Þ¼ 0 lim *<sup>x</sup>*!<sup>0</sup>*f x*ð Þ. We introduce the sets of regular and non-regular functions

$$\mathcal{F}\_{op}^r = \left\{ f \in \mathcal{F}\_{op} | f(\mathbf{0}) \neq \mathbf{0} \right\}, \quad \mathcal{F}\_{op}^n \left\{ f \in \mathcal{F}\_{op} | f(\mathbf{0}) = \mathbf{0} \right\}.$$

and notice that trivially <sup>F</sup>*op* ¼ F*<sup>r</sup> op* <sup>∪</sup> <sup>F</sup>*<sup>n</sup> op*. Definition 2. For *<sup>f</sup>* <sup>∈</sup> <sup>F</sup>*<sup>r</sup> op*, we set

$$\tilde{f}(\mathbf{x}) = \frac{1}{2} \left[ (\mathbf{x} + \mathbf{1}) - (\mathbf{x} - \mathbf{1})^2 \frac{f(\mathbf{0})}{f(\mathbf{x})} \right], \quad \mathbf{x} > \mathbf{0}.$$

Theorem 1.5 ([10]). The correspondence *<sup>f</sup>* ! <sup>~</sup>*<sup>f</sup>* is a bijection between <sup>F</sup>*<sup>r</sup> op* and <sup>F</sup>*<sup>n</sup> op*.

#### **4.2 Metric adjusted skew information**

In the Kubo-Ando theory [11] of matrix means, a mean is associated with each operator monotone function *f* ∈ F*op* by the following formula:

$$\mathcal{m}\_f(A,B) = A^{1/2} f\left(A^{-1/2} B A^{-1/2}\right) A^{1/2},$$

where *A*, *B*∈ *Mn*,þð Þ . Using the notion of matrix means, the class of monotone metrics can be defined by the following formula:

$$\langle A, B \rangle\_{\rho\_f^f} = \operatorname{Tr} \left[ A \cdot m\_f \left( L\_\rho, R\_\rho \right)^{-1} (B) \right],$$

where *Lρ*ð Þ¼ *A ρA*, *Rρ*ð Þ¼ *A Aρ*.

Definition 3. For *A* ∈ *Mn*,*sa*ð Þ , we define as follows:

*Uncertainty Relations DOI: http://dx.doi.org/10.5772/intechopen.92137*

monotone function is said to be symmetric if *f x*ð Þ¼ *xf x*�<sup>1</sup> ð Þ and normalized if

Definition 1 F*op* is the class of functions *f* : ð Þ! 0, þ∞ ð Þ 0, þ∞ such that:

Example 1. Examples of elements of F*op* are given by the following:

<sup>2</sup> , *fWYD*ð Þ¼ *<sup>x</sup> <sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup>

ffiffiffi *<sup>x</sup>* <sup>p</sup> <sup>þ</sup> <sup>1</sup> 2 � �<sup>2</sup>

*x* þ 1

For *f* ∈ F*op*, we define *f*ð Þ¼ 0 lim *<sup>x</sup>*!<sup>0</sup>*f x*ð Þ. We introduce the sets of regular and

*op* <sup>∪</sup> <sup>F</sup>*<sup>n</sup> op*.

<sup>2</sup> ð Þ� *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup> *<sup>f</sup>*ð Þ <sup>0</sup>

Theorem 1.5 ([10]). The correspondence *<sup>f</sup>* ! <sup>~</sup>*<sup>f</sup>* is a bijection between <sup>F</sup>*<sup>r</sup>*

� �

In the Kubo-Ando theory [11] of matrix means, a mean is associated with each

*f A*�1*=*<sup>2</sup>

where *A*, *B*∈ *Mn*,þð Þ . Using the notion of matrix means, the class of monotone

h i *A*, *B <sup>ρ</sup>*,*<sup>f</sup>* ¼ *Tr A* � *m <sup>f</sup> Lρ*, *R<sup>ρ</sup>*

*BA*�1*=*<sup>2</sup> � �

� ��<sup>1</sup>

h i

<sup>2</sup> , *<sup>x</sup>*>0*:*

*f x*ð Þ

, *fBKM*ð Þ¼ *x*

*op <sup>f</sup>* <sup>∈</sup> <sup>F</sup>*op*j*f*ð Þ¼ <sup>0</sup> <sup>0</sup> � �

, *x*>0*:*

*A*<sup>1</sup>*=*<sup>2</sup> ,

ð Þ *B*

,

*op* and <sup>F</sup>*<sup>n</sup>*

*op*.

*<sup>x</sup>*ð Þ *<sup>α</sup>* � <sup>1</sup> *<sup>x</sup>*ð Þ <sup>1</sup>�*<sup>α</sup>* � <sup>1</sup> , *<sup>α</sup>*∈ð Þ 0, 1 *:*

*x* � 1 log *<sup>x</sup>* ,

, *fWY*ð Þ¼ *x*

2*x*

*<sup>x</sup>* <sup>þ</sup> <sup>1</sup> <sup>≤</sup>*f x*ð Þ<sup>≤</sup>

*op* <sup>¼</sup> *<sup>f</sup>* <sup>∈</sup> <sup>F</sup>*op*j*f*ð Þ <sup>0</sup> 6¼ <sup>0</sup> � �, <sup>F</sup>*<sup>n</sup>*

*op*, we set

operator monotone function *f* ∈ F*op* by the following formula:

*<sup>m</sup> <sup>f</sup>*ð Þ¼ *<sup>A</sup>*, *<sup>B</sup> <sup>A</sup>*<sup>1</sup>*=*<sup>2</sup>

Definition 3. For *A* ∈ *Mn*,*sa*ð Þ , we define as follows:

*f*ð Þ¼ 1 1.

1.*f*ð Þ¼ 1 1.

*Quantum Mechanics*

2.*tf t*�<sup>1</sup> ð Þ¼ *f t*ð Þ.

3.*f* is operator monotone.

*fRLD*ð Þ¼ *x*

*fSLD*ð Þ¼ *x*

non-regular functions

F*r*

Definition 2. For *<sup>f</sup>* <sup>∈</sup> <sup>F</sup>*<sup>r</sup>*

and notice that trivially <sup>F</sup>*op* ¼ F*<sup>r</sup>*

**4.2 Metric adjusted skew information**

<sup>~</sup>*f x*ð Þ¼ <sup>1</sup>

metrics can be defined by the following formula:

where *Lρ*ð Þ¼ *A ρA*, *Rρ*ð Þ¼ *A Aρ*.

**52**

2*x x* þ 1

*x* þ 1

Remark 1. Any *f* ∈ F*op* satisfies

$$I\_{\rho}^{f}(A) = \frac{f(\mathbf{0})}{2} \langle i[\rho, A], i[\rho, A] \rangle\_{\rho f},$$

$$\mathbf{C}\_{\rho}^{f}(A) = \operatorname{Tr} \left[ \mathfrak{m}\_{f} \left( \mathbf{L}\_{\rho}, \mathbf{R}\_{\rho} \right) (A) \cdot A \right],$$

$$U\_{\rho}^{f}(A) = \sqrt{\mathbf{V}\_{\rho}(A)^{2} - \left( V\_{\rho}(A) - I\_{\rho}^{f}(A) \right)^{2}}.$$

Quantity *I <sup>f</sup> <sup>ρ</sup>* ð Þ *A* is referred to as the metric adjusted skew information, and h i *A*, *B <sup>ρ</sup>*,*<sup>f</sup>* is referred to as the metric adjusted correlation measure.

Proposition 1. The following holds:

$$\begin{aligned} \mathbf{1}. \mathbf{1}\_{\rho}^{f}(A) &= \mathbf{1}\_{\rho}^{f}(A\_{0}) = \operatorname{Tr}(\rho A\_{0}^{2}) - \operatorname{Tr}\Big(\mathfrak{m}\_{\hat{f}}\big(\mathbf{L}\_{\rho}, \mathbf{R}\_{\rho}\big)(A\_{0}) \cdot A\_{0}\big) = \mathbf{V}\_{\rho}(A) - \operatorname{G}\_{\rho}^{\hat{f}}(A\_{0}). \\\\ \mathbf{2}. \mathbf{J}\_{\rho}^{f}(A) &= \operatorname{Tr}\big(\rho A\_{0}^{2}\big) + \operatorname{Tr}\Big(\mathfrak{m}\_{\hat{f}}(\mathbf{L}\_{\rho}, \mathbf{R}\_{\rho})(A\_{0}) \cdot A\_{0}\big) = \mathbf{V}\_{\rho}(A) + \mathbf{G}\_{\rho}^{\hat{f}}(A\_{0}). \\\\ \mathbf{3}. \mathbf{0} \le \mathbf{I}\_{\rho}^{f}(A) &\le \mathbf{U}\_{\rho}^{f}(A) \le \mathbf{V}\_{\rho}(A). \\\\ \mathbf{4}. \mathbf{U}\_{\rho}^{f}(A) &= \sqrt{\mathbf{I}\_{\rho}^{f}(A) \cdot \mathbf{J}\_{\rho}^{f}(A)}. \end{aligned}$$

Theorem 1.6 ([6]). For *f* ∈*F<sup>r</sup> op*, if

$$
\frac{\infty + 1}{2} + \tilde{f}(\infty) \ge 2f(\infty),
\tag{8}
$$

then it holds that

$$\left| U^{f}\_{\rho}(\mathbb{A}) \cdot U^{f}\_{\rho}(\mathbb{B}) \geq \! f(\mathbb{O}) \right| \left| Tr(\rho[\mathbb{A}, B]) \right|^{2}, \tag{9}$$

where *A*, *B*∈ *Mn*,*sa*ð Þ .

To prove Theorem 1.6, several lemmas are used. Lemma 1. If (8) holds, then the following inequality is satisfied:

$$\left(\frac{\varkappa+\mathcal{y}}{2}\right)^2 - m\_{\tilde{f}}(\varkappa,\mathcal{y})^2 \ge f(\mathbf{0})(\varkappa-\mathcal{y})^2.$$

Proof of Lemma 1. By (8), we have

$$\frac{\infty + \mathcal{y}}{2} + m\_{\tilde{f}}(\infty, \mathcal{y}) \ge 2m\_f(\infty, \mathcal{y}). \tag{10}$$

Since

$$m\_{\tilde{f}}(\mathbf{x}, \mathbf{y}) = y \tilde{\tilde{f}}\left(\frac{\mathbf{x}}{\mathbf{y}}\right) = \frac{y}{2} \left\{ \frac{\mathbf{x}}{\mathbf{y}} + \mathbf{1} - \left(\frac{\mathbf{x}}{\mathbf{y}} - \mathbf{1}\right)^2 \frac{f(\mathbf{0})}{f(\mathbf{x}/\mathbf{y})} \right\} = \frac{\mathbf{x} + \mathbf{y}}{2} - \frac{f(\mathbf{0})(\mathbf{x} - \mathbf{y})^2}{2m\_f(\mathbf{x}, \mathbf{y})}, \quad \mathbf{x} \in \mathbb{R}^2$$

we have

$$\begin{aligned} \left(\frac{\mathbf{x}+\mathbf{y}}{2}\right)^2 - m\_{\hat{f}}(\mathbf{x},\mathbf{y})^2 &= \left\{\frac{\mathbf{x}+\mathbf{y}}{2} - m\_{\hat{f}}(\mathbf{x},\mathbf{y})\right\} \left\{\frac{\mathbf{x}+\mathbf{y}}{2} + m\_{\hat{f}}(\mathbf{x},\mathbf{y})\right\} \\ &= \frac{f(\mathbf{0})(\mathbf{x}-\mathbf{y})^2}{2m\_{f}(\mathbf{x},\mathbf{y})} \left\{\frac{\mathbf{x}+\mathbf{y}}{2} + m\_{\hat{f}}(\mathbf{x},\mathbf{y})\right\} \ge f(\mathbf{0})(\mathbf{x}-\mathbf{y})^2. \end{aligned} \tag{9}$$

#### *Quantum Mechanics*

Lemma 2. Let j*ϕ*1i, j*ϕ*2i, ⋯, j*ϕ<sup>n</sup>* f gi be a basis of eigenvectors of *ρ*, corresponding to the eigenvalues f g *λ*1, *λ*2, ⋯, *λ<sup>n</sup>* . We set *ajk* ¼ *ϕ<sup>j</sup>* j*A*0j*ϕ<sup>k</sup>* D E, *bjk* <sup>¼</sup> *<sup>ϕ</sup><sup>j</sup>* j*B*0j*ϕ<sup>k</sup>* D E. Then, we have

$$I\_{\rho}^{f}(A) = \frac{1}{2} \sum\_{j,k} (\lambda\_{j} + \lambda\_{k}) a\_{jk} a\_{kj} - \sum\_{j,k} m\_{\bar{f}}(\lambda\_{j}, \lambda\_{k}) a\_{jk} a\_{kj},$$

$$I\_{\rho}^{f}(A) = \frac{1}{2} \sum\_{j,k} (\lambda\_{j} + \lambda\_{k}) a\_{jk} a\_{kj} + \sum\_{j,k} m\_{\bar{f}}(\lambda\_{j}, \lambda\_{k}) a\_{jk} a\_{kj},$$

$$\left( U\_{\rho}^{f}(A) \right)^{2} = \frac{1}{4} \left( \sum\_{j,k} (\lambda\_{j} + \lambda\_{k}) \left| a\_{jk} \right|^{2} \right)^{2} - \left( \sum\_{j,k} m\_{\bar{f}}(\lambda\_{j}, \lambda\_{k}) \left| a\_{jk} \right|^{2} \right)^{2}.$$

Proof of Theorem 1.6. Since

$$\operatorname{Tr}(\rho[A,B]) = \operatorname{Tr}(\rho[A\_0, B\_0]) = \sum\_{j,k} (\lambda\_j - \lambda\_k) a\_{jk} b\_{kj},$$

we have

$$\begin{split} \left| f(\mathbf{0}) | \mathrm{Tr}(\rho[A,B]) \right|^{2} &\leq \left( \sum\_{j,k} f(\mathbf{0})^{1/2} |\boldsymbol{\lambda}\_{j} - \boldsymbol{\lambda}\_{k}| |a\_{jk}| |b\_{kj}| \right)^{2} \\ &\leq \left( \sum\_{j,k} \left\{ \left( \frac{\boldsymbol{\lambda}\_{j} + \boldsymbol{\lambda}\_{k}}{2} \right)^{2} - m\_{\hat{f}}(\boldsymbol{\lambda}\_{j}, \boldsymbol{\lambda}\_{k})^{2} \right\}^{1/2} |a\_{jk}| |b\_{kj}| \right)^{2} \\ &\leq \left( \sum\_{j,k} \left\{ \frac{\boldsymbol{\lambda}\_{j} + \boldsymbol{\lambda}\_{k}}{2} - m\_{\hat{f}}(\boldsymbol{\lambda}\_{j}, \boldsymbol{\lambda}\_{k}) \right\} |a\_{jk}|^{2} \right) \\ &\times \left( \sum\_{j,k} \left\{ \frac{\boldsymbol{\lambda}\_{j} + \boldsymbol{\lambda}\_{k}}{2} + m\_{\hat{f}}(\boldsymbol{\lambda}\_{j}, \boldsymbol{\lambda}\_{k}) \right\} |b\_{kj}|^{2} \right) = I\_{\rho}^{f}(A) I\_{\rho}^{f}(B). \end{split}$$

We also have

$$I\_{\rho}^{f}(B)J\_{\rho}^{f}(A) \ge f(\mathbf{0})|Tr(\rho[A,B])|^{2}.$$

Thus, we have the final result (9). □

Definition 4. For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ we define the following:

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup> Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*g*

*I*

q

*Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup>*

*g*

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>* is referred to as the generalized metric adjusted skew information, and *Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>*, *<sup>B</sup>* is referred to as the generalized metric adjusted correlation measure.

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>J</sup>*

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>B</sup>* <sup>≥</sup> *Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* <sup>ð</sup>*A*, *<sup>B</sup>*Þj

*<sup>g</sup>* ð Þ *x* ≥ ℓ*f x*ð Þ for some ℓ>0*:*

� <sup>h</sup> � �

h i

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>X</sup>*, *<sup>X</sup>* , *<sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X* is referred to as the generalized quasi-metric adjusted skew information,

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* is referred to as the generalized quasi-metric adjusted correlation

� � �

2. (Heisenberg type) For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ , we assume the

*<sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>B</sup>* <sup>≥</sup>*k*<sup>ℓ</sup> *Tr <sup>ρ</sup>*½*A*, *<sup>B</sup>*��j

In this section, we present general uncertainty relations for non-Hermitian

Definition 5. For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B* ∈ *Mn*,þð Þ we define the following:

<sup>¼</sup> *kTr X*<sup>∗</sup> ð Þ *LA* � *RB <sup>m</sup> <sup>f</sup>*ð Þ *LA*, *RB* �<sup>1</sup>

<sup>¼</sup> *Tr X*<sup>∗</sup> *mg*ð Þ *LA*, *RB <sup>Y</sup>* � � � *Tr X*<sup>∗</sup> *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>*, *<sup>Y</sup> Tr X*<sup>∗</sup> *mg*ð Þ *LA*, *RB <sup>Y</sup>* � � <sup>þ</sup> *Tr X*<sup>∗</sup> *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>* <sup>Ψ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

� �*B*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup>*

*<sup>ρ</sup>* ð Þ *A*, *A*

*Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i <sup>þ</sup> *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>*

*:*

2 *:*

> 2 *:*

ð Þ *LA* � *RB Y*

ð Þ *LA*, *RB Y* h i,

ð Þ *LA*, *RB Y* h i,

> *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *X J*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*

*:*

*b*

*g*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *X*

*g*

*Lρ*, *R<sup>ρ</sup>* � �*B*<sup>0</sup> h i*:*

*g*

*Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i*:*

*g*

*Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i*:*

*Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>*, *<sup>B</sup> k i*h i ½ � *<sup>ρ</sup>*, *<sup>A</sup>*<sup>0</sup> , *<sup>i</sup>*½ � *<sup>ρ</sup>*, *<sup>B</sup>*<sup>0</sup> *<sup>f</sup>*

*I*

*<sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>*

Theorem 1.7 ([7]). Under condition (A), the following holds:

1. (Schrödinger type) For *A*, *B*∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>I</sup>*

*g x*ð Þþ <sup>Δ</sup> *<sup>f</sup>*

**6. Generalized quasi-metric adjusted skew information**

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *X*, *Y k L* h i ð Þ *<sup>A</sup>* � *RB X*,ð Þ *LA* � *RB Y <sup>f</sup>*

¼ *Tr A*0*mg Lρ*, *R<sup>ρ</sup>*

*Uncertainty Relations*

ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup> Tr A*0*mg <sup>L</sup>ρ*, *<sup>R</sup><sup>ρ</sup>*

*J*

*I*

Then,

� �*A*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

� �*A*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup>*

*I*

following condition (B):

observables *X*, *Y* ∈ *Mn*ð Þ .

Γð Þ *<sup>g</sup>*,*<sup>f</sup>*

Ψð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*,*X* , *J*

ð Þ *g*,*f*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*I* ð Þ *g*,*f*

> *I* ð Þ *g*,*f*

and Γð Þ *<sup>g</sup>*,*<sup>f</sup>*

measure.

**55**

¼ *Tr A*0*mg Lρ*, *R<sup>ρ</sup>*

#### **5. Generalized metric adjusted skew information**

We assume that *<sup>f</sup>* <sup>∈</sup> <sup>F</sup>*<sup>r</sup> op* satisfies the following condition (A):

$$\lg(\varkappa) \ge k \frac{\left(\varkappa - 1\right)^2}{f(\varkappa)},\\\text{for some } k > 0.$$

Let

$$\Delta\_{\mathcal{g}}^f(\mathbf{x}) = \mathbf{g}(\mathbf{x}) - k \frac{\left(\mathbf{x} - \mathbf{1}\right)^2}{f(\mathbf{x})} \in F\_{op}.$$

**54**

Definition 4. For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ we define the following:

*Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>*, *<sup>B</sup> k i*h i ½ � *<sup>ρ</sup>*, *<sup>A</sup>*<sup>0</sup> , *<sup>i</sup>*½ � *<sup>ρ</sup>*, *<sup>B</sup>*<sup>0</sup> *<sup>f</sup>* ¼ *Tr A*0*mg Lρ*, *R<sup>ρ</sup>* � �*B*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup> g Lρ*, *R<sup>ρ</sup>* � �*B*<sup>0</sup> h i*: I* ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup> Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *A*, *A* ¼ *Tr A*0*mg Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup> g Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup> g Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i*: J* ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup> Tr A*0*mg <sup>L</sup>ρ*, *<sup>R</sup><sup>ρ</sup>* � �*A*<sup>0</sup> � � � *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup> g Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i <sup>þ</sup> *Tr A*0*m*<sup>Δ</sup> *<sup>f</sup> g Lρ*, *R<sup>ρ</sup>* � �*A*<sup>0</sup> h i*: <sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ� *<sup>A</sup> <sup>J</sup>* ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>* q *:*

*I* ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>* is referred to as the generalized metric adjusted skew information, and *Corr*ð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>ρ</sup>* ð Þ *<sup>A</sup>*, *<sup>B</sup>* is referred to as the generalized metric adjusted correlation measure. Theorem 1.7 ([7]). Under condition (A), the following holds:

1. (Schrödinger type) For *A*, *B*∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ ,

$$\left| I\_{\rho}^{(\mathbf{g}f)}(A) \cdot I\_{\rho}^{(\mathbf{g}f)}(B) \right| \geq \left| Corr\_{\rho}^{(\mathbf{g}f)}(A,B) \right|^2.$$

2. (Heisenberg type) For *A*, *B* ∈ *Mn*,*sa*ð Þ , *ρ*∈ *Mn*,þ,1ð Þ , we assume the following condition (B):

$$\text{g}(\mathfrak{x}) + \Delta\_{\mathfrak{g}}^f(\mathfrak{x}) \ge \ell f(\mathfrak{x}) \text{ for some } \ell > \mathbf{0}.$$

Then,

Lemma 2. Let j*ϕ*1i, j*ϕ*2i, ⋯, j*ϕ<sup>n</sup>* f gi be a basis of eigenvectors of *ρ*, corresponding

� �*ajkakj* �<sup>X</sup>

� �*ajkakj* <sup>þ</sup><sup>X</sup>

*λ <sup>j</sup>* þ *λ<sup>k</sup>*

*λ <sup>j</sup>* þ *λ<sup>k</sup>*

*λ <sup>j</sup>* þ *λ<sup>k</sup>* � � *ajk* � � � � 2

*Tr*ð Þ¼ *ρ*½ � *A*, *B Tr*ð Þ¼ *ρ*½ � *A*0, *B*<sup>0</sup>

*j*, *k*

0 @

≤ X *j*, *k*

≤ X *j*, *k*

� <sup>X</sup> *j*, *k*

0 @

*I f <sup>ρ</sup>* ð Þ *<sup>B</sup> <sup>J</sup> <sup>f</sup>*

**5. Generalized metric adjusted skew information**

Δ *f*

*g x*ð Þ≥*<sup>k</sup>* ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup>

*<sup>g</sup>* ð Þ¼ *<sup>x</sup> g x*ð Þ� *<sup>k</sup>* ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup>

0 @

0 @ *<sup>f</sup>*ð Þ <sup>0</sup> <sup>1</sup>*=*<sup>2</sup>

*λ <sup>j</sup>* þ *λ<sup>k</sup>* 2 � �<sup>2</sup>

*λ <sup>j</sup>* þ *λ<sup>k</sup>*

*λ <sup>j</sup>* þ *λ<sup>k</sup>*

j*A*0j*ϕ<sup>k</sup>* D E

*m*~*<sup>f</sup> λ <sup>j</sup>*, *λ<sup>k</sup>*

*m*~*<sup>f</sup> λ <sup>j</sup>*, *λ<sup>k</sup>*

*λ <sup>j</sup>* � *λ<sup>k</sup>* � �*ajkbkj*,

> 1 A

2

*ajk* � � � � 2

*bkj* � � � � 2

*:*

1 A

1 <sup>A</sup> <sup>¼</sup> *<sup>I</sup><sup>f</sup>*

� �*ajkakj*,

� �*ajkakj*,

*m*~*<sup>f</sup> λ <sup>j</sup>*, *λ<sup>k</sup>* � � *ajk* � � � � 2

*j*, *k*

*j*, *k*

X *j*, *k*

j*λ <sup>j</sup>* � *λk*k*ajk*k*bkj*j

( )<sup>1</sup>*=*<sup>2</sup>

<sup>2</sup> � *<sup>m</sup>*<sup>~</sup>*<sup>f</sup> <sup>λ</sup> <sup>j</sup>*, *<sup>λ</sup><sup>k</sup>* � � � �

<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>~</sup>*<sup>f</sup> <sup>λ</sup> <sup>j</sup>*, *<sup>λ</sup><sup>k</sup>* � � � �

*<sup>ρ</sup>* ð Þ *<sup>A</sup>* <sup>≥</sup>*f*ð Þ <sup>0</sup> j j *Tr*ð*ρ*½*A*, *<sup>B</sup>*�Þ <sup>2</sup>

*op* satisfies the following condition (A):

*f x*ð Þ , for some *<sup>k</sup>*<sup>&</sup>gt; <sup>0</sup>*:*

*f x*ð Þ <sup>∈</sup>*Fop:*

Thus, we have the final result (9). □

� *m*<sup>~</sup>*<sup>f</sup> λ <sup>j</sup>*, *λ<sup>k</sup>* � �<sup>2</sup>

� <sup>X</sup> *j*, *k*

0 @

1 A

2

, *bjk* ¼ *ϕ<sup>j</sup>*

j*B*0j*ϕ<sup>k</sup>* D E

> 1 A

j*ajk*k*bkj*j

*<sup>ρ</sup>* ð Þ *<sup>A</sup> <sup>J</sup> <sup>f</sup>*

*<sup>ρ</sup>* ð Þ *B :*

1 A

2

2 *:* . Then,

to the eigenvalues f g *λ*1, *λ*2, ⋯, *λ<sup>n</sup>* . We set *ajk* ¼ *ϕ<sup>j</sup>*

*If <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>* <sup>1</sup> 2 X *j*, *k*

*J f <sup>ρ</sup>* ð Þ¼ *<sup>A</sup>* <sup>1</sup> 2 X *j*, *k*

Proof of Theorem 1.6. Since

*<sup>f</sup>*ð Þ <sup>0</sup> j j *Tr*ð*ρ*½*A*, *<sup>B</sup>*�Þ <sup>2</sup> <sup>≤</sup> <sup>X</sup>

¼ 1 4

X *j*, *k*

0 @

*U f <sup>ρ</sup>* ð Þ *A* � �<sup>2</sup>

we have

We also have

We assume that *<sup>f</sup>* <sup>∈</sup> <sup>F</sup>*<sup>r</sup>*

Let

**54**

we have

*Quantum Mechanics*

$$\left| U\_{\rho}^{(\mathfrak{g}f)}(A) \cdot U\_{\rho}^{(\mathfrak{g}f)}(B) \right| \geq k\ell \left| Tr\left[ \rho[A,B] \right] \right|^2.$$

#### **6. Generalized quasi-metric adjusted skew information**

In this section, we present general uncertainty relations for non-Hermitian observables *X*, *Y* ∈ *Mn*ð Þ .

Definition 5. For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B* ∈ *Mn*,þð Þ we define the following:

$$\Gamma\_{A,B}^{\{gf\}}(X,Y) = k \langle (L\_A - R\_B)X, (L\_A - R\_B)Y \rangle\_f$$

$$= k \operatorname{Tr} \left[ X^\* \left( L\_A - R\_B \right) m\_f(L\_A, R\_B)^{-1} (L\_A - R\_B) Y \right]$$

$$= \operatorname{Tr} \left[ X^\* \operatorname{m}\_{\mathcal{S}}(L\_A, R\_B)Y \right] - \operatorname{Tr} \left[ X^\* \operatorname{m}\_{\Delta\_{\tilde{\mathcal{S}}}^f}(L\_A, R\_B)Y \right],$$

$$\Psi\_{A,B}^{\{gf\}}(X,Y) = \operatorname{Tr} \left[ X^\* \operatorname{m}\_{\mathcal{S}}(L\_A, R\_B)Y \right] + \operatorname{Tr} \left[ X^\* \operatorname{m}\_{\Delta\_{\tilde{\mathcal{S}}}^f}(L\_A, R\_B)Y \right],$$

$$\Psi\_{A,B}^{\{gf\}}(X) = \Gamma\_{A,B}^{\{gf\}}(X,X), \ f\_{A,B}^{\{gf\}}(X) = \Psi\_{A,B}^{\{gf\}}(X,X), \ U\_{A,B}^{\{gf\}}(X) = \sqrt{\operatorname{I}\_{A,B}^{\{gf\}}}(X) \cdot \frac{\mathbf{1}\_{\mathcal{S},B}^{\{gf\}}(X)}{\mathbf{1}\_{\mathcal{S},B}(X)}.$$

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* is referred to as the generalized quasi-metric adjusted skew information, and Γð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* is referred to as the generalized quasi-metric adjusted correlation measure.

*I*

Theorem 1.8 ([12]). Under condition (A), the following holds:

1. (Schrödinger type) For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B*∈ *Mn*,þð Þ ,

$$\left| I\_{A,B}^{(\mathbf{g}f)}(\mathbf{X}) \cdot I\_{A,B}^{(\mathbf{g}f)}(\mathbf{Y}) \right| \geq \left| \Gamma\_{A,B}^{(\mathbf{g}f)}(\mathbf{X}, \mathbf{Y}) \right|^2 \geq \frac{1}{\mathbf{1}\mathbf{6}} \left( I\_{A,B}^{(\mathbf{g}f)}(\mathbf{X} + \mathbf{Y}) - I\_{A,B}^{(\mathbf{g}f)}(\mathbf{X} - \mathbf{Y}) \right)^2.$$

¼ 1 4 *I* ð Þ *g*,*f*

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

� � � 2 ¼ 1 <sup>16</sup> *<sup>I</sup>* ð Þ *g*,*f*

Thus,

� � �

Γð Þ *<sup>g</sup>*,*<sup>f</sup> <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y*

*Uncertainty Relations*

Lemma 3

We then have

*mg*ð Þ *x*, *y*

<sup>2</sup> � *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup> g* ð Þ *x*, *y*

Proof of 2 in Theorem 1.8. Let

*I* ð Þ *g*,*f*

*J* ð Þ *g*,*f*

Since

we have

**57**

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>* <sup>X</sup>

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>Y</sup>* <sup>X</sup>

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *X* þ *Y I*

≥ 1 <sup>16</sup> *<sup>I</sup>* ð Þ *g*,*f*

We use the following lemma to prove 2:

*mg* ð Þ *x*, *y*

*m*<sup>Δ</sup> *<sup>f</sup> g*

≥

*<sup>A</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

*i*, *j*

*i*, *j*

<sup>∣</sup>*LA* � *RB*<sup>∣</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*k x*ð Þ � *y* 2

*m <sup>f</sup>*ð Þ *x*, *y*

Proof of Lemma 3. By conditions (A) and (B), we have

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y* � � <sup>þ</sup> *iIm*fΓð Þ *<sup>g</sup>*,*<sup>f</sup>*

> ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

� �<sup>2</sup>

ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> mg* ð Þ� *<sup>x</sup>*, *<sup>y</sup> <sup>k</sup>* ð Þ *<sup>x</sup>* � *<sup>y</sup>*

*mg* ð Þþ *x*, *y m*Δ*<sup>g</sup> <sup>f</sup>*ð Þ *x*, *y* ≥ℓ*m <sup>f</sup>*ð Þ *x*, *y :*

*g* ð Þ *x*, *y* n o *mg*ð Þþ *<sup>x</sup>*, *<sup>y</sup> <sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

ℓ*m <sup>f</sup>*ð Þ¼ *x*, *y k*ℓð Þ *x* � *y*

\* +

*g*

*g*

∣*λ<sup>i</sup>* � *μ <sup>j</sup>*

*i*¼1

*λi*, *μ <sup>j</sup>* n o � � <sup>∣</sup>h*ϕi*∣*<sup>X</sup> <sup>ψ</sup> <sup>j</sup>*<sup>i</sup>

*λi*, *μ <sup>j</sup>* n o � � <sup>∣</sup>h*ϕi*∣*<sup>Y</sup> <sup>ψ</sup> <sup>j</sup>*<sup>i</sup>

∣*L*∣*ϕi*ih*ϕi*<sup>∣</sup>*R*∣*<sup>ψ</sup> <sup>j</sup>*ih*<sup>ψ</sup> <sup>j</sup>*<sup>∣</sup>,

*μi*j*ψ<sup>i</sup>*

<sup>2</sup> <sup>¼</sup> *mg*ð Þ� *<sup>x</sup>*, *<sup>y</sup> <sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

*<sup>λ</sup>i*∣*ϕi*<sup>i</sup> *<sup>ϕ</sup>i*j, *<sup>B</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

be the spectral decompositions of *A* and *B*, respectively. Then, we have

� � � *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

� � <sup>þ</sup> *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

X*n j*¼1

*mg λi*, *μ <sup>j</sup>*

*mg λi*, *μ <sup>j</sup>*

*i*¼1

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

<sup>2</sup> <sup>≥</sup>*k*ℓð Þ *<sup>x</sup>* � *<sup>y</sup>*

� �<sup>2</sup>

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *X* þ *Y I*

<sup>2</sup> � *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup> g* ð Þ *x*, *y*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *X* þ *Y I*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y :*

<sup>þ</sup> *Im* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*:*

2 *:*

2 *<sup>m</sup> <sup>f</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* ,

> 2 *:*

h*ψi*∣

� � �

� � � � � � 2 ,

� � � 2 ,

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* � � n o <sup>2</sup>

> *g* ð Þ *x*, *y*

n o

□

□

2. (Heisenberg type) For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B*∈ *Mn*,þð Þ , we assume condition (B). Then,

$$\|U\_{A,B}^{(\mathbf{g}f)}(X)\cdot U\_{A,B}^{(\mathbf{g}f)}(Y) \ge k\ell \|Tr\left[X^\* \,|L\_A - R\_B|Y\right]\|^2.$$

In particular,

$$\begin{split} \left| \mathcal{k}\ell \left| \operatorname{Tr} \left[ \mathbf{X}^\* \left| L\_A - R\_B \middle| \mathbf{X} \right| \right] \right|^2 \leq & \operatorname{Tr} \left[ \mathbf{X}^\* \left( m\_\mathfrak{g} (L\_A, R\_B) - m\_{\Delta\_\mathfrak{g}^f} (L\_A, R\_B) \right) \mathbf{X} \right] \\ \quad \times & \operatorname{Tr} \left[ \mathbf{X}^\* \left( m\_\mathfrak{g} (L\_A, R\_B) + m\_{\Delta\_\mathfrak{g}^f} (L\_A, R\_B) \right) \mathbf{X} \right], \end{split} \tag{11}$$

where *X* ∈ *Mn*ð Þ and *A*, *B* ∈ *Mn*,þð Þ . Proof of 1 in Theorem 1.8. By the Schwarz inequality, we have

$$I\_{A,B}^{(\mathfrak{g}f)}(\mathbf{X}) \cdot I\_{A,B}^{(\mathfrak{g}f)}(Y) = \Gamma\_{A,B}^{(\mathfrak{g}f)}(\mathbf{X}, \mathbf{X}) \cdot \Gamma\_{A,B}^{(\mathfrak{g}f)}(Y, \mathbf{Y}) \ge \left| \Gamma\_{A,B}^{(\mathfrak{g}f)}(\mathbf{X}, \mathbf{Y}) \right|^2.$$

Now, we prove the second inequality. Since

$$\begin{aligned} I\_{A,B}^{(gf)}(X+Y) &= \operatorname{Tr}\left[ (X^\* + Y^\*)m\_{\mathfrak{g}}(L\_A, R\_B)(X+Y) \right] \\ &- \operatorname{Tr}\left[ (X^\* + Y^\*)m\_{\mathfrak{\Delta}\_{\mathfrak{f}}^f}(L\_A, R\_B)(X+Y) \right], \\ I\_{A,B}^{(gf)}(X-Y) &= \operatorname{Tr}\left[ (X^\* - Y^\*)m\_{\mathfrak{g}}(L\_A, R\_B)(X-Y) \right] \\ &- \operatorname{Tr}\left[ (X^\* - Y^\*)m\_{\mathfrak{\Delta}\_{\mathfrak{f}}^f}(L\_A, R\_B)(X-Y) \right], \end{aligned}$$

we have

$$\begin{split} &I\_{A,\mathcal{B}}^{(\mathcal{g}f)}(\mathcal{X}+\mathcal{Y})-I\_{A,\mathcal{B}}^{(\mathcal{g}f)}(\mathcal{X}-\mathcal{Y}) \\ &=2\operatorname{Tr}\left[\mathcal{X}^{\*}\mathcal{m}\_{\mathcal{B}}(L\_{A},R\_{\mathcal{B}})\mathcal{Y}\right]+2\operatorname{Tr}\mathcal{Y}^{\*}\mathcal{m}\_{\mathcal{B}}(L\_{A},R\_{\mathcal{B}})\mathcal{X}\right]-2\operatorname{Tr}\left[\mathcal{X}^{\*}\mathcal{m}\_{\Delta\_{\mathcal{X}}^{\prime}}(L\_{A},R\_{\mathcal{B}})\mathcal{Y}\right] \\ &-2\operatorname{Tr}\left[\mathcal{Y}^{\*}\mathcal{m}\_{\Delta\_{\mathcal{X}}^{\prime}}(L\_{A},R\_{\mathcal{B}})\mathcal{X}\right]-2\Gamma\_{A,\mathcal{B}}^{(\mathcal{g}f)}(\mathcal{X},\mathcal{Y})+2\Gamma\_{A,\mathcal{B}}^{(\mathcal{g}f)}(\mathcal{Y},\mathcal{X})=4\operatorname{Re}\left\{\Gamma\_{A,\mathcal{B}}^{(\mathcal{g}f)}(\mathcal{X},\mathcal{Y})\right\}. \end{split}$$

Similarly, we have

$$I\_{A,B}^{(\mathbf{g}f)}(X+Y) + I\_{A,B}^{(\mathbf{g}f)}(X-Y) = 2\left(I\_{A,B}^{(\mathbf{g}f)}(X) + I\_{A,B}^{(\mathbf{g}f)}(Y)\right).$$

Then,

$$\Gamma\_{A,B}^{\mathfrak{g}f}(X,Y) = \operatorname{Re}\left\{ \Gamma\_{A,B}^{(\mathfrak{g}f)}(X,Y) \right\} + i\operatorname{Im}\left\{ \Gamma\_{A,B}^{(\mathfrak{g}f)}(X,Y) \right\},$$

*Uncertainty Relations DOI: http://dx.doi.org/10.5772/intechopen.92137*

$$\mathcal{I} = \frac{1}{4} \left( I\_{A,B}^{(\mathfrak{g}f)}(X+Y) - I\_{A,B}^{(\mathfrak{g}f)}(X-Y) \right) + i \operatorname{Im} \{ \Gamma\_{A,B}^{(\mathfrak{g}f)}(X,Y) \}.$$

Thus,

Theorem 1.8 ([12]). Under condition (A), the following holds:

1. (Schrödinger type) For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B*∈ *Mn*,þð Þ ,

*<sup>A</sup>*,*<sup>B</sup>* ð*X*, *Y*Þj

2 ≥ 1 <sup>16</sup> *<sup>I</sup>* ð Þ *g*,*f*

2. (Heisenberg type) For *X*, *Y* ∈ *Mn*ð Þ , *A*, *B*∈ *Mn*,þð Þ , we assume condition

�*Tr X*<sup>∗</sup> *mg* ð Þþ *LA*, *RB <sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *<sup>X</sup>*,*<sup>X</sup>* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>* <sup>þ</sup> *<sup>Y</sup> Tr X*<sup>∗</sup> <sup>þ</sup> *<sup>Y</sup>* <sup>∗</sup> ð Þ*mg* ð Þ *LA*, *RB* ð Þ *<sup>X</sup>* <sup>þ</sup> *<sup>Y</sup>* � � � *Tr X*<sup>∗</sup> <sup>þ</sup> *<sup>Y</sup>* <sup>∗</sup> ð Þ*m*<sup>Δ</sup> *<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>X</sup>* � *<sup>Y</sup> Tr X*<sup>∗</sup> � *<sup>Y</sup>* <sup>∗</sup> ð Þ*mg* ð Þ *LA*, *RB* ð Þ *<sup>X</sup>* � *<sup>Y</sup>* � � � *Tr X*<sup>∗</sup> � *<sup>Y</sup>* <sup>∗</sup> ð Þ*m*<sup>Δ</sup> *<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þþ *<sup>X</sup>*, *<sup>Y</sup>* <sup>2</sup>Γð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *X* � *Y* 2 *I*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* n o

<sup>¼</sup> <sup>2</sup>*Tr X*<sup>∗</sup> *mg* ð Þ *LA*, *RB <sup>Y</sup>* � � <sup>þ</sup> <sup>2</sup>*TrY* <sup>∗</sup> *mg* ð Þ *LA*, *RB <sup>X</sup>*� � <sup>2</sup>*Tr X*<sup>∗</sup> *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

<sup>¼</sup> <sup>2</sup>Γð Þ *<sup>g</sup>*,*<sup>f</sup>*

ð Þ *g*,*f*

*<sup>A</sup>*,*<sup>B</sup>*ð Þ¼ *<sup>X</sup>*, *<sup>Y</sup> Re* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>Y</sup>* <sup>≥</sup>*k*ℓ∣*Tr X*<sup>∗</sup> <sup>∣</sup>*LA* � *RB*j j *<sup>Y</sup>*� <sup>2</sup>

h

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *X* þ *Y I*

*g*

*g*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>Y</sup>*, *<sup>Y</sup>* <sup>≥</sup> <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*g*

*g*

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þþ *X I*

<sup>þ</sup> *iIm* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

h i

h i

� � �

ð Þ *LA*, *RB* ð Þ *X* þ *Y*

ð Þ *LA*, *RB* ð Þ *X* � *Y*

� �

h i

� �

h i

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

*:*

ð Þ *LA*, *RB*

ð Þ *LA*, *RB*

*X*

*X*

*<sup>A</sup>*,*<sup>B</sup>* ð*X*, *Y*Þj

,

2 *:*

,

,

ð Þ *LA*, *RB Y* h i

*:*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* n o

*:*

*g*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>Y</sup>*,*<sup>X</sup>* <sup>4</sup> *Re* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *Y*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *Y* n o

� �

*:*

(11)

� �<sup>2</sup>

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *X I*

*Quantum Mechanics*

(B). Then,

In particular,

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *X I*

> *I* ð Þ *g*,*f*

> *I* ð Þ *g*,*f*

> > ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

ð Þ *LA*, *RB X* h i

*<sup>A</sup>*,*<sup>B</sup>* ð Þþ *X* þ *Y I*

we have

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *X* þ *Y I*

�2*Tr Y* <sup>∗</sup> *<sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

Then,

**56**

Similarly, we have

*g*

*I* ð Þ *g*,*f*

Γ*<sup>g</sup>*,*f*<sup>Þ</sup>

*I* ð Þ *g*,*f* ð Þ *g*,*f*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>Y</sup>* <sup>≥</sup> <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*U*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

where *X* ∈ *Mn*ð Þ and *A*, *B* ∈ *Mn*,þð Þ .

ð Þ *g*,*f*

Now, we prove the second inequality. Since

*<sup>A</sup>*,*<sup>B</sup>* ð Þ¼ *<sup>Y</sup>* <sup>Γ</sup>ð Þ *<sup>g</sup>*,*<sup>f</sup>*

� � � �

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *<sup>X</sup> <sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*<sup>k</sup>*<sup>ℓ</sup> *Tr X*<sup>∗</sup> j j ½ � <sup>j</sup>*LA* � *RB*j*<sup>X</sup>* <sup>2</sup> <sup>≤</sup>*Tr X*<sup>∗</sup> *mg*ð Þ� *LA*, *RB <sup>m</sup>*<sup>Δ</sup> *<sup>f</sup>*

Proof of 1 in Theorem 1.8. By the Schwarz inequality, we have

$$\begin{aligned} \left| \Gamma\_{A,B}^{\langle gf \rangle} (\mathbf{X}, Y) \right|^2 &= \frac{1}{\mathbf{1} \mathsf{6}} \left( I\_{A,B}^{\langle gf \rangle} (\mathbf{X} + \mathbf{Y}) - I\_{A,B}^{\langle gf \rangle} (\mathbf{X} - \mathbf{Y}) \right)^2 + \left( \mathrm{Im} \left\{ \Gamma\_{A,B}^{\langle gf \rangle} (\mathbf{X}, Y) \right\} \right)^2 \\ &\geq \frac{1}{\mathbf{1} \mathsf{6}} \left( I\_{A,B}^{\langle gf \rangle} (\mathbf{X} + \mathbf{Y}) - I\_{A,B}^{\langle gf \rangle} (\mathbf{X} - \mathbf{Y}) \right)^2. \end{aligned}$$

We use the following lemma to prove 2: Lemma 3

$$\left(m\_{\mathfrak{g}}(\boldsymbol{\kappa},\boldsymbol{\jmath})^{2} - m\_{\boldsymbol{\Delta}\_{\mathfrak{k}}^{f}}(\boldsymbol{\kappa},\boldsymbol{\jmath})^{2} \geq k\ell\left(\boldsymbol{\kappa} - \boldsymbol{\jmath}\right)^{2}\right)$$

Proof of Lemma 3. By conditions (A) and (B), we have

$$\begin{aligned} m\_{\Delta\_{\mathbb{Z}}^f}(\mathfrak{x}, \mathfrak{y}) &= m\_{\mathfrak{g}}(\mathfrak{x}, \mathfrak{y}) - k \frac{\left(\mathfrak{x} - \mathfrak{y}\right)^2}{m\_f(\mathfrak{x}, \mathfrak{y})}, \\ m\_{\mathfrak{g}}(\mathfrak{x}, \mathfrak{y}) + m\_{\Delta\_{\mathfrak{z}}f}(\mathfrak{x}, \mathfrak{y}) &\geq \ell m\_f(\mathfrak{x}, \mathfrak{y}). \end{aligned}$$

We then have

$$\begin{aligned} \left(m\_{\mathfrak{x}}(\mathbf{x},\boldsymbol{\mathcal{y}})\right)^{2} - m\_{\Delta\_{\mathfrak{x}}^{f}}(\mathbf{x},\boldsymbol{\mathcal{y}})^{2} &= \left\{m\_{\mathfrak{x}}(\mathbf{x},\boldsymbol{\mathcal{y}}) - m\_{\Delta\_{\mathfrak{x}}^{f}}(\mathbf{x},\boldsymbol{\mathcal{y}})\right\} \left\{m\_{\mathfrak{x}}(\mathbf{x},\boldsymbol{\mathcal{y}}) + m\_{\Delta\_{\mathfrak{x}}^{f}}(\mathbf{x},\boldsymbol{\mathcal{y}})\right\} \\ &\geq \frac{k(\mathbf{x}-\boldsymbol{\mathcal{y}})^{2}}{m\_{f}(\mathbf{x},\boldsymbol{\mathcal{y}})} \ell m\_{f}(\mathbf{x},\boldsymbol{\mathcal{y}}) = k\ell(\mathbf{x}-\boldsymbol{\mathcal{y}})^{2} .\end{aligned}$$

□

Proof of 2 in Theorem 1.8. Let

$$A = \sum\_{i=1}^{n} \lambda\_i |\phi\_i\rangle \left\langle \phi\_i|, \quad B = \sum\_{i=1}^{n} \mu\_i |\psi\_i\rangle \langle \psi\_i| \right\rangle$$

be the spectral decompositions of *A* and *B*, respectively. Then, we have

$$I\_{A,B}^{(\mathbf{g},f)}(\mathbf{X}) = \sum\_{i,j} \left\{ m\_{\mathbf{g}}\left(\lambda\_i, \mu\_j\right) - m\_{\Delta\_{\mathbf{g}}^f}\left(\lambda\_i, \mu\_j\right) \right\} |\langle \phi\_i | \mathbf{X} \left| \boldsymbol{\mu}\_j \right\rangle|^2,$$

$$J\_{A,B}^{(\mathbf{g},f)}(\mathbf{Y}) = \sum\_{i,j} \left\{ m\_{\mathbf{g}}\left(\lambda\_i, \mu\_j\right) + m\_{\Delta\_{\mathbf{g}}^f}\left(\lambda\_i, \mu\_j\right) \right\} |\langle \phi\_i | \mathbf{Y} \left| \boldsymbol{\mu}\_j \right\rangle|^2,$$

Since

$$|L\_A - R\_B| = \sum\_{i=1}^n \sum\_{j=1}^n |\lambda\_i - \mu\_j| L\_{|\phi\_i\rangle\langle\phi\_i|} R\_{|\nu\_j\rangle\langle\psi\_j|},$$

we have

*Quantum Mechanics*

$$\operatorname{Tr}\left[\mathbf{X}^\* | L\_A - R\_B | Y\right] = \sum\_{i=1}^n \sum\_{j=1}^n |\lambda\_i - \mu\_j| \overline{\left\langle \phi\_i | \mathbf{X} | \boldsymbol{\mu}\_j \right\rangle} \left\langle \phi\_i | Y | \boldsymbol{\mu}\_j \right\rangle.$$

*f*

00ðÞ¼ *<sup>s</sup> Tr A*1�*<sup>s</sup>*

<sup>¼</sup> *Tr A*1�*<sup>s</sup>*

*Uncertainty Relations*

<sup>¼</sup> *Tr A*1�*<sup>s</sup>*

<sup>¼</sup> *Tr A*1�*<sup>s</sup>*

Let

<sup>¼</sup> *Tr A*ð Þ <sup>1</sup>�*<sup>s</sup> <sup>=</sup>*<sup>2</sup>

Then, we have

And since

we have

Then, we have

Therefore,

**59**

ð Þ log *<sup>A</sup>* <sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

ð Þ log *<sup>A</sup>* <sup>2</sup> *Bs*

<sup>¼</sup> *Tr A*1�*<sup>s</sup>* log *<sup>A</sup>*ð Þ log *<sup>A</sup>* � log *<sup>B</sup> Bs* � � � *Tr A*1�*<sup>s</sup>*

ð Þ log *<sup>A</sup>* � log *<sup>B</sup> Bs*

ð Þ log *<sup>A</sup>* � log *<sup>B</sup> Bs* log *<sup>A</sup>* � � � *Tr A*1�*<sup>s</sup>*

ð Þ log *<sup>A</sup>* � log *<sup>B</sup>* � �

ð Þ log *<sup>A</sup>* � log *<sup>B</sup> Bs*

*<sup>A</sup>* <sup>¼</sup> <sup>X</sup> *i*

*<sup>B</sup>* <sup>¼</sup> <sup>X</sup> *j μ j* ∣*ψ <sup>j</sup>* i *ψ <sup>j</sup>*

*Tr A*½ �¼ <sup>X</sup>

*i*, *j*

<sup>∣</sup>*LA* � *RB*<sup>∣</sup> <sup>¼</sup> <sup>X</sup>

<sup>∣</sup>*LA* � *RB*∣*<sup>I</sup>* <sup>¼</sup> <sup>X</sup>

*Tr A*<sup>½</sup> <sup>þ</sup> *<sup>B</sup>*�j*LA* � *RB*j*I*� ¼ <sup>X</sup>

*λi*∣h*ϕ<sup>i</sup> ψ <sup>j</sup>*i � � �

� � � 2

*i*, *j*

*i*, *j*

*Tr*½ �¼ <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* <sup>X</sup>

∣*λ<sup>i</sup>* � *μ <sup>j</sup>*

∣*λ<sup>i</sup>* � *μ <sup>j</sup>*

*i*, *j*

*i*, *j*

∣*λ<sup>i</sup>* � *μ <sup>j</sup>*

*λ<sup>i</sup>* þ *μ <sup>j</sup>*

*Bs* � *<sup>A</sup>*1�*<sup>s</sup>* log *AB<sup>s</sup>* log *<sup>B</sup>* h i � *Tr A*1*<sup>s</sup>*

ð Þ log *<sup>A</sup>* � log *<sup>B</sup> <sup>A</sup>*ð Þ <sup>1</sup>�*<sup>s</sup> <sup>=</sup>*<sup>2</sup> h i<sup>≥</sup> <sup>0</sup>*:*

and (12), respectively. Thus, we only prove the following inequality:

*<sup>λ</sup>i*∣*ϕi*<sup>i</sup> *<sup>ϕ</sup>i*j¼ <sup>X</sup>

h i � *Tr A*1�*<sup>s</sup>* log *<sup>A</sup>* log *BB<sup>s</sup>* � � � *Tr* log *<sup>B</sup>* log *AA*1�*<sup>s</sup>*

log *AB<sup>s</sup>* log *<sup>B</sup>* � *<sup>A</sup>*1�*<sup>s</sup>*

ð Þ log *<sup>A</sup>* � log *<sup>B</sup>* log *BBs* � �

ð Þ log *<sup>A</sup>* � log *<sup>B</sup>* log *BBs* � �

*ϕi*j*ψ <sup>j</sup>* D Eh*<sup>ψ</sup> <sup>j</sup>*

*ϕi*j*ψ <sup>j</sup>* D Eh*<sup>ψ</sup> <sup>j</sup>*∣*:*

∣,

*f s*ð Þ is convex in *<sup>s</sup>*. □ Proof of Theorem 1.9. The third and fourth inequalities follow from Lemma 4

*Tr A*½ � <sup>þ</sup> *<sup>B</sup>*�j*LA* � *RB*j*<sup>I</sup>* <sup>≤</sup>2*Tr A*<sup>1</sup>�*αB<sup>α</sup>* � � ð Þ <sup>0</sup> <sup>≤</sup>*<sup>α</sup>* <sup>≤</sup><sup>1</sup> *:*

*i*, *j λi*j*ϕ<sup>i</sup>*

\* +

, *Tr B*½ �¼ <sup>X</sup>

*i*, *j*

∣*L*∣*ϕi*ih*ϕi*<sup>∣</sup>*R*∣*<sup>ψ</sup> <sup>j</sup>*ih*<sup>ψ</sup> <sup>j</sup>*<sup>∣</sup>,

k*ϕi*i *ϕi*j*ψ <sup>j</sup>* D Eh*<sup>ψ</sup> <sup>j</sup>*

> kh*ϕ<sup>i</sup> ψ <sup>j</sup>* i � � �

�j*λ<sup>i</sup>* � *μ <sup>j</sup>*

*μ <sup>j</sup>*∣h*ϕ<sup>i</sup> ψ <sup>j</sup>*i � � �

∣*:*

� � � 2 *:*

j � �∣h*ϕ<sup>i</sup> <sup>ψ</sup> <sup>j</sup>*

i � � �

� � � 2 *:*

� � � 2 *:*

j¼ <sup>X</sup> *i*, *j μ j* j*ϕi*

\* +

ð Þ log *<sup>B</sup>* <sup>2</sup> h i

*Bs* � � <sup>þ</sup> *Tr A*1�*<sup>s</sup>*

*Bs*

ð Þ log *<sup>B</sup>* <sup>2</sup> *Bs*

h i

Then, by Lemma 3, we have

$$\begin{split} \lambda \ell \prime |Tr[X^\* | L\_A - R\_B | Y]|^2 &\le \left\{ \sum\_{i=1}^n \sum\_{j=1}^n \sqrt{k \ell} |\lambda\_i - \mu\_j| \big| \left\langle \phi\_i | X | \psi\_j \right\rangle |\langle \phi\_j | Y | \psi\_i \rangle| \right\}^2 \\ &\le \left\{ \sum\_{i=1}^n \sum\_{j=1}^n (m\_\mathfrak{g}\left(\dot{\mu}\_i, \mu\_j\right)^2 - m\_{\varDelta\_\mathfrak{g}'}(\lambda\_i, \mu\_j)^2) |\langle \phi\_i | X | \psi\_j \rangle| |\langle \phi\_j | Y | \psi\_i \rangle| \right\}^2 \\ &\le \left\{ \sum\_{i=1}^n \sum\_{j=1}^n (m\_\mathfrak{g}\left(\dot{\lambda}\_i, \mu\_j\right) - m\_{\varDelta\_\mathfrak{g}'}(\lambda\_i, \mu\_j)) |\langle \phi\_i | X \Big| \phi\_j \rangle| \right\}^2 \end{split}$$
 
$$\left\{ \sum\_{i=1}^n \sum\_{j=1}^n (m\_\mathfrak{g}\left(\dot{\lambda}\_i, \mu\_j\right) + m\_{\varDelta\_\mathfrak{g}'}(\lambda\_i, \mu\_j)) |\langle \phi\_j | Y | \psi\_i \rangle|^2 \right\}$$
 
$$= I\_{A,B}^{\mathfrak{g}(f)}(X) \cdot I\_{A,B}^{\mathfrak{g}(f)}(Y).$$

Similarly, we have *<sup>k</sup>*<sup>ℓ</sup> *Tr X*<sup>∗</sup> j j ½ � <sup>j</sup>*LA* � *RB*j*<sup>Y</sup>* <sup>2</sup> <sup>≤</sup> *<sup>I</sup>* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *Y J* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* . Therefore,

$$\mathbb{E}\left(U\_{A,B}^{(\mathbf{g}f)}(\mathbf{X})\cdot U\_{A,B}^{(\mathbf{g}f)}(\mathbf{Y}) \geq k\ell |\mathrm{Tr}\left[\mathbf{X}^\* | L\_A - R\_B |Y]|^2\right.\tag{7}$$

When *A* ¼ *B* ¼ *ρ*∈ *Mn*,þ,1ð Þ , *X* ¼ *A* ∈ *Mn*ð ÞÞ , and *Y* ¼ *B* ∈ *Mn*ð Þ , we obtain the result in Theorem 1.7.

We assume that

$$g(\mathbf{x}) = \frac{\mathbf{x} + \mathbf{1}}{2}, \ f(\mathbf{x}) = a(\mathbf{1} - a) \frac{\left(\mathbf{x} - \mathbf{1}\right)^2}{\left(\mathbf{x}^a - \mathbf{1}\right)\left(\mathbf{x}^{1-a} - \mathbf{1}\right)}, \ k = \frac{f(\mathbf{0})}{2}, \ \ell = 2.$$

We then obtain the following trace inequality by substituting *X* ¼ *I* in (11).

$$a(\mathbf{1} - a)(Tr[|L\_A - R\_B|I])^2 \le \left(\frac{1}{2}Tr[A + B]\right)^2 - \left(\frac{1}{2}Tr[A^a B^{1-a} + A^{1-a} B^a]\right)^2. \tag{12}$$

This is a generalization of the trace inequality provided in [13]. In addition, we produce the following new inequality by combining a Chernoff-type inequality with Theorem 1.8.

Theorem 1.9 ([14]). We have the following:

$$\frac{1}{2}Tr[A+B-|L\_A-R\_B|I] \le \inf\_{0 \le a \le 1} Tr\left[A^{1-a}B^a\right] \le Tr\left[A^{1/2}B^{1/2}\right]$$

$$\le \frac{1}{2}Tr\left[A^aB^{1-a}+A^{1-a}B^a\right] \le \sqrt{\left(\frac{1}{2}Tr[A+B]\right)^2-a(1-a)(Tr[|L\_A-R\_B|I])^2}.$$

The following lemma is necessary to prove Theorem 1.9.

Lemma 4. Let *f s*ðÞ¼ *Tr A*<sup>1</sup>�*<sup>s</sup> <sup>B</sup><sup>s</sup>* � � for *<sup>A</sup>*, *<sup>B</sup>*<sup>∈</sup> *Mn*ð Þ and 0<sup>≤</sup> *<sup>s</sup>*≤1. Then *f s*ð Þ is convex in *s*.

Proof of Lemma 4. *f* 0 ðÞ¼ *<sup>s</sup> Tr* �*A*<sup>1</sup>�*<sup>s</sup>* log *AB<sup>s</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>1</sup>�*<sup>s</sup> Bs* log *B* � �. And then *Uncertainty Relations DOI: http://dx.doi.org/10.5772/intechopen.92137*

$$f''(s) = \operatorname{Tr}\left[A^{1-s}(\log A)^2B' - A^{1-s}\log AB'\log B\right] - \operatorname{Tr}\left[A^{1-s}\log AB'\log B - A^{1-s}B'(\log B)^2\right]$$

$$= \operatorname{Tr}\left[A^{1-s}(\log A)^2B'\right] - \operatorname{Tr}\left[A^{1-s}\log A\log B B'\right] - \operatorname{Tr}\left[\log B\log A A^{1-s}B'\right] + \operatorname{Tr}\left[A^{1-s}(\log B)^2B'\right]$$

$$= \operatorname{Tr}\left[A^{1-s}\log A(\log A - \log B)B'\right] - \operatorname{Tr}\left[A^{1-s}(\log A - \log B)\log B'\right]$$

$$= \operatorname{Tr}\left[A^{1-s}(\log A - \log B)B'\log A\right] - \operatorname{Tr}\left[A^{1-s}(\log A - \log B)\log B'\right]$$

$$= \operatorname{Tr}\left[A^{1-s}(\log A - \log B)B'(\log A - \log B)\right]$$

$$= \operatorname{Tr}\left[A^{(1-s)/2}(\log A - \log B)B'(\log A - \log B)A^{(1-s)/2}\right] \ge 0.$$

*f s*ð Þ is convex in *<sup>s</sup>*. □ Proof of Theorem 1.9. The third and fourth inequalities follow from Lemma 4 and (12), respectively. Thus, we only prove the following inequality:

$$\operatorname{Tr}[A + B - |L\_A - R\_B|I] \le 2\operatorname{Tr}\left[A^{1-a}B^a\right] \quad (0 \le a \le 1).$$

Let

*Tr X*<sup>∗</sup> <sup>½</sup> <sup>j</sup>*LA* � *RB*j*Y*� ¼ <sup>X</sup>*<sup>n</sup>*

*i*¼1

<sup>≤</sup> <sup>X</sup>*<sup>n</sup> i*¼1

<sup>≤</sup> <sup>X</sup>*<sup>n</sup> i*¼1

> X*n i*¼1

¼ *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *X J*

Similarly, we have *<sup>k</sup>*<sup>ℓ</sup> *Tr X*<sup>∗</sup> j j ½ � <sup>j</sup>*LA* � *RB*j*<sup>Y</sup>* <sup>2</sup> <sup>≤</sup> *<sup>I</sup>*

X*n j*¼1

X*n j*¼1

X*n j*¼1

X*n j*¼1

Then, by Lemma 3, we have

*<sup>k</sup>*<sup>ℓ</sup> *Tr X*<sup>∗</sup> j j ½ � <sup>j</sup>*LA* � *RB*j*<sup>Y</sup>* <sup>2</sup> <sup>≤</sup> <sup>X</sup>*<sup>n</sup>*

*Quantum Mechanics*

*U*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

the result in Theorem 1.7. We assume that

*g x*ð Þ¼ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup>

*<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *Tr*½ � <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* <sup>2</sup> <sup>≤</sup>

1 2

with Theorem 1.8.

≤ 1 2

convex in *s*.

**58**

*<sup>A</sup>*,*<sup>B</sup>* ð Þ� *<sup>X</sup> <sup>U</sup>*ð Þ *<sup>g</sup>*,*<sup>f</sup>*

*i*¼1

ffiffiffiffiffi

ð*mg λi*, *μ <sup>j</sup>* � �<sup>2</sup>

ð*mg λi*, *μ <sup>j</sup>* � �

ð*mg λi*, *μ <sup>j</sup>* � �

> ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *Y :*

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>Y</sup>* <sup>≥</sup>*k*ℓ∣*Tr X*<sup>∗</sup> <sup>∣</sup>*LA* � *RB*j j *<sup>Y</sup>*� <sup>2</sup>

When *A* ¼ *B* ¼ *ρ*∈ *Mn*,þ,1ð Þ , *X* ¼ *A* ∈ *Mn*ð ÞÞ , and *Y* ¼ *B* ∈ *Mn*ð Þ , we obtain

We then obtain the following trace inequality by substituting *X* ¼ *I* in (11).

This is a generalization of the trace inequality provided in [13]. In addition, we produce the following new inequality by combining a Chernoff-type inequality

0 ≤*α* ≤1

*Tr A*½ � þ *B* � �<sup>2</sup>

1 2

ðÞ¼ *<sup>s</sup> Tr* �*A*<sup>1</sup>�*<sup>s</sup>* log *AB<sup>s</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>1</sup>�*<sup>s</sup>*

s

The following lemma is necessary to prove Theorem 1.9.

� <sup>1</sup> 2

*Tr A*½ � þ *B* � �<sup>2</sup>

h

<sup>2</sup> , *f x*ð Þ¼ *<sup>α</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>x</sup>* � <sup>1</sup> <sup>2</sup>

1 2

Theorem 1.9 ([14]). We have the following:

*Tr A<sup>α</sup>B*<sup>1</sup>�*<sup>α</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>1</sup>�*<sup>α</sup>B<sup>α</sup>* � �<sup>≤</sup>

0

Lemma 4. Let *f s*ðÞ¼ *Tr A*<sup>1</sup>�*<sup>s</sup>*

Proof of Lemma 4. *f*

*Tr A*½ � þ *B*�j*LA* � *RB*j*I* ≤ inf

X*n j*¼1

∣*λ<sup>i</sup>* � *μ <sup>j</sup>*

*<sup>k</sup>*<sup>ℓ</sup> <sup>p</sup> <sup>j</sup>*λ<sup>i</sup>* � *<sup>μ</sup> <sup>j</sup>*<sup>k</sup> *<sup>ϕ</sup>i*j*X*j*<sup>ψ</sup> <sup>j</sup>*

( )<sup>2</sup>

� *m*<sup>Δ</sup> *<sup>f</sup> g* ð*λi*, *μ <sup>j</sup>*Þ 2

� *m*<sup>Δ</sup> *<sup>f</sup> g*

> ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ� *Y J*

*<sup>x</sup>*ð Þ *<sup>α</sup>* � <sup>1</sup> *<sup>x</sup>*ð Þ <sup>1</sup>�*<sup>α</sup>* � <sup>1</sup> , *<sup>k</sup>* <sup>¼</sup> *<sup>f</sup>*ð Þ <sup>0</sup>

*Tr A*<sup>1</sup>�*<sup>α</sup>B<sup>α</sup>* � �≤ *Tr A*<sup>1</sup>*=*<sup>2</sup>

*<sup>B</sup><sup>s</sup>* � � for *<sup>A</sup>*, *<sup>B</sup>*<sup>∈</sup> *Mn*ð Þ and 0<sup>≤</sup> *<sup>s</sup>*≤1. Then *f s*ð Þ is

*Bs* log *B* � �. And then

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ *m*<sup>Δ</sup> *<sup>f</sup> g*

( )

( )

∣ *ϕi*j*X*j*ψ <sup>j</sup>* D E

D E

( )<sup>2</sup>

*ϕi*j*Y*j*ψ <sup>j</sup>* D E

kh*ϕj*j*Y*j*ψi*ij

ð*λi*, *μ <sup>j</sup>*ÞÞjh*ϕi*j*X ϕj*i

<sup>ð</sup>*λi*, *<sup>μ</sup> <sup>j</sup>*ÞÞjh*ϕj*j*<sup>Y</sup> <sup>ψ</sup><sup>i</sup>* j ji <sup>2</sup>

ð Þ *g*,*f*

*Tr A<sup>α</sup>B*<sup>1</sup>�*<sup>α</sup>* <sup>þ</sup> *<sup>A</sup>*<sup>1</sup>�*αB<sup>α</sup>* � � � �<sup>2</sup>

*<sup>B</sup>*<sup>1</sup>*=*<sup>2</sup> h i

� *α*ð Þ 1 � *α Tr*½ Þ ∣*LA* � *RB*∣*I* <sup>2</sup> ð *:*

� � � *:*

Þjh*ϕi*j*X*j*ψ <sup>j</sup>*ikh*ϕj*j*Y*j*ψi*ij

� � � 2

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X* . Therefore,

*:* □

<sup>2</sup> , <sup>ℓ</sup> <sup>¼</sup> <sup>2</sup>*:*

*:* (12)

$$A = \sum\_{i} \lambda\_i |\phi\_i\rangle \left\langle \phi\_i | = \sum\_{i,j} \lambda\_i |\phi\_i\rangle \left\langle \phi\_i | \psi\_j \right\rangle \langle \psi\_j |, \right.$$

$$B = \sum\_{j} \mu\_j |\psi\_j\rangle \left\langle \psi\_j | = \sum\_{i,j} \mu\_j |\phi\_i\rangle \left\langle \phi\_i | \psi\_j \right\rangle \langle \psi\_j |.$$

Then, we have

$$Tr[A] = \sum\_{i,j} \lambda\_i |\langle \phi\_i \Big| \boldsymbol{\nu}\_j \rangle|^2, \quad Tr[B] = \sum\_{i,j} \mu\_j |\langle \phi\_i \Big| \boldsymbol{\nu}\_j \rangle|^2.$$

And since

$$|L\_A - R\_B| = \sum\_{i,j} |\lambda\_i - \mu\_j| L\_{|\phi\_i\rangle\langle\phi\_i|} R\_{|\psi\_j\rangle\langle\psi\_j|} \star$$

we have

$$|L\_A - R\_B|I = \sum\_{i,j} |\lambda\_i - \mu\_j| |\phi\_i\rangle \left\langle \phi\_i | \boldsymbol{\nu}\_j \right\rangle \langle \boldsymbol{\nu}\_j |.$$

Then, we have

$$\operatorname{Tr}[|L\_A - R\_B|I] = \sum\_{i,j} |\lambda\_i - \mu\_j| |\langle \phi\_i | \boldsymbol{\nu}\_j \rangle|^2.$$

Therefore,

$$Tr[A + B - |L\_A - R\_B|I] = \sum\_{i,j} \left(\lambda\_i + \mu\_j - |\lambda\_i - \mu\_j|\right) |\langle \phi\_i | \boldsymbol{\psi}\_j \rangle|^2.$$

However, since we have

$$A^{a} = \sum\_{i} \lambda\_{i}^{a} |\phi\_{i}\rangle \left\langle \phi\_{i} | = \sum\_{i,j} \lambda\_{i}^{a} |\phi\_{i}\rangle \left\langle \phi\_{i} | \boldsymbol{\nu}\_{j} \right\rangle \langle \boldsymbol{\nu}\_{j} |,$$

$$B^{1-a} = \sum\_{j} \mu\_{j}^{1-a} |\boldsymbol{\nu}\_{j}\rangle \left\langle \boldsymbol{\nu}\_{j} | = \sum\_{i,j} \mu\_{j}^{1-a} |\phi\_{i}\rangle \left\langle \phi\_{i} | \boldsymbol{\nu}\_{j} \right\rangle \langle \boldsymbol{\nu}\_{j} |,$$

$$A^{a} B^{1-a} = \sum\_{i,j} \lambda\_{i}^{a} \mu\_{j}^{1-a} |\phi\_{i}\rangle \langle \phi\_{i} | \boldsymbol{\nu}\_{j} \rangle \langle \boldsymbol{\nu}\_{j} |.$$

For any quantum state ∣*ϕ*i, we define the two probability distributions

� �, *<sup>Q</sup>* <sup>¼</sup> *qi*

*pi* log *pi*

. Let

*H P*ð Þþ *H Q*ð Þ≥ � 2 log *c*,

, *H Q*ð Þ¼�X*<sup>n</sup>*

�2*πiωt dt:*

� �*:*

ð Þ , <sup>∥</sup>*ψ*∥<sup>2</sup> <sup>¼</sup> 1 satisfies *<sup>ψ</sup>*, *<sup>ψ</sup>*^ <sup>∈</sup> *<sup>Q</sup>*ð Þ , then

*dt*, *S*ð Þ¼� *ψ*^

Theorem 1.11 ([18]). For any *X*, *Y* ∈ *Mn*ð Þ , *A*, *B* ∈ *Mn*,þð Þ , the following holds:

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X* þ *Y* ,*I*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

ð Þ *g*,*f*

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* þ *Y*

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* þ *Y*

q

q

2 ,

> ð<sup>∞</sup> �∞

ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

,

,

max <sup>∥</sup>*<sup>X</sup>* <sup>þ</sup> *<sup>Y</sup>*∥<sup>2</sup>

� � q

� � q

*I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* � *Y*

*I* ð Þ *g*,*f <sup>A</sup>:<sup>B</sup>* ð Þ *X* � *Y*

n o*:*

j j *<sup>ψ</sup>*^ð Þ*<sup>t</sup>* <sup>2</sup> log j j *<sup>ψ</sup>*^ð Þ*<sup>t</sup>* <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*:*

.

, <sup>∥</sup>*<sup>X</sup>* � *<sup>Y</sup>*∥<sup>2</sup> � �*:* (13)

*dt:*

, *q*2, ⋯, *qn* � �,

*j*¼1

*q <sup>j</sup>* log *q <sup>j</sup>*

ð Þ is defined as

*dt*< ∞

*P* ¼ *p*1, *p*2, ⋯, *pn*

*H P*ð Þ¼�X*<sup>n</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

D E∣.

Definition 6. The Fourier transformation of *ψ* ∈*L*<sup>2</sup>

*<sup>Q</sup>*ð Þ¼ *<sup>f</sup>* <sup>∈</sup>*L*<sup>2</sup>

j j *<sup>ψ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup> log j j *<sup>ψ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup>

<sup>2</sup> max *I*

≥ max

≤2 max

<sup>2</sup> <sup>∥</sup>*<sup>X</sup>* <sup>þ</sup> *<sup>Y</sup>*∥<sup>2</sup> <sup>þ</sup> <sup>∥</sup>*<sup>X</sup>* � *<sup>Y</sup>*∥<sup>2</sup> � �<sup>≥</sup>

*ψ ω* ^ð Þ¼ <sup>ð</sup><sup>∞</sup>

�∞ *ψ*ð Þ*t e*

ð Þ ; ð<sup>∞</sup> �∞ *t* 2 j j *f t*ð Þ <sup>2</sup>

*<sup>S</sup>*ð Þþ *<sup>ψ</sup> <sup>S</sup>*ð Þ *<sup>ψ</sup>*^ <sup>≥</sup> log *<sup>e</sup>*

, *<sup>q</sup> <sup>j</sup>* <sup>¼</sup> <sup>∣</sup>h*<sup>ψ</sup> <sup>j</sup>*j j *<sup>ϕ</sup>*<sup>i</sup> <sup>2</sup>

*i*¼1

be the Shannon entropies of *P* and *Q*, respectively. Theorem 1.10. The following uncertainty relation holds:

where *pi* <sup>¼</sup> *<sup>ϕ</sup><sup>i</sup>* j j h i <sup>j</sup>*<sup>ϕ</sup>* <sup>2</sup>

*Uncertainty Relations*

where *c* ¼ max *<sup>i</sup>*,*<sup>j</sup>*∣ *ϕi*j*ψ <sup>j</sup>*

For details, see [15, 16].

Proposition 2. If *ψ* ∈*L*<sup>2</sup>

*S*ð Þ¼� *ψ*

For details, see [17].

*<sup>A</sup>*,*<sup>B</sup>* ð Þþ *X*, *Y I*

þ

þ

q

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*

<sup>∥</sup>*X*∥<sup>2</sup> <sup>þ</sup> <sup>∥</sup>*Y*∥<sup>2</sup> <sup>¼</sup> <sup>1</sup>

ð<sup>∞</sup> �∞

> ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *<sup>Y</sup>* <sup>≥</sup> <sup>1</sup>

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *Y*

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *Y*

Proof 1. The Hilbert-Schmidt norm ∥ � ∥ satisfies

We also define

where

1.*I* ð Þ *g*,*f*

2.

3.

**61**

q

q

Then,

$$\operatorname{Tr}\left[A^a B^{1-a}\right] = \sum\_{i,j} \lambda\_i^a \mu\_j^{1-a} |\langle \phi\_i | \boldsymbol{\mu}\_j \rangle|^2.$$

Thus,

$$2Tr\left[A^a B^{1-a}\right] - Tr[A + B - |L\_A - R\_B|I] = \sum\_{i,j} \left\{2\lambda\_i^a \mu\_j^{1-a} - \left(\lambda\_i + \mu\_j - |\lambda\_i - \mu\_j|\right)\right\} |\langle \phi\_i | \boldsymbol{\nu}\_j \rangle|^2 \boldsymbol{\nu}\_j$$

Since 2*xαy*<sup>1</sup>�*<sup>α</sup>* � ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*�j*<sup>x</sup>* � *<sup>y</sup>*<sup>j</sup> <sup>≥</sup> 0 for *<sup>x</sup>*, *<sup>y</sup>* <sup>&</sup>gt;0, 0<sup>≤</sup> *<sup>α</sup>*≤1 in general, we can obtain Theorem 1.9. □

$$\begin{aligned} \text{Remark 2. We note the following 1, 2:}\\ \text{1. } \frac{1}{2}Tr[A + B - |A - B|] \leq \inf\_{0 \leq a \leq 1} Tr\left[A^{1-a}B^{a}\right] \leq Tr\left[A^{1/2}B^{1/2}\right], \end{aligned}$$

$$\leq \sqrt{\left(\frac{1}{2}Tr[A+B]\right)^2 - \frac{1}{4}\left(Tr[|A-B|]\right)^2}.$$

2. There is no relationship between *Tr*½ � j*LA* � *RB*j*I* and *Tr*½ � j*A* � *B*j . When

$$A = \begin{pmatrix} \mathbf{3} & \mathbf{1} \\ \mathbf{2} & \mathbf{2} \\ \mathbf{1} & \mathbf{3} \\ \mathbf{2} & \mathbf{2} \end{pmatrix}, \quad B = \begin{pmatrix} \mathbf{4} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{pmatrix},$$

we have *Tr*½ �¼ <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* 3, *Tr*½ �¼ <sup>j</sup>*<sup>A</sup>* � *<sup>B</sup>*<sup>j</sup> ffiffiffiffiffi <sup>10</sup> <sup>p</sup> . When

$$A = \begin{pmatrix} \frac{13}{2} & \frac{7}{2} \\ \frac{7}{2} & \frac{13}{2} \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix},$$

we have *Tr*½ �¼ <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* 8, *Tr*½ �¼ <sup>j</sup>*<sup>A</sup>* � *<sup>B</sup>*<sup>j</sup> ffiffiffiffiffi <sup>58</sup> <sup>p</sup> .

#### **7. Sum type of uncertainty relations**

Let *A*, *B*∈ *Mn*,*sa*ð Þ have the following spectral decompositions:

$$A = \sum\_{i=1}^{n} \lambda\_i |\phi\_i\rangle \left\langle \phi\_i|, \quad B = \sum\_{i=1}^{n} \mu\_i |\psi\_i\rangle \langle \psi\_i|.\right\rangle$$

However, since we have

*Quantum Mechanics*

Then,

Thus,

1. <sup>1</sup>

**60**

*<sup>A</sup><sup>α</sup>* <sup>¼</sup> <sup>X</sup> *i λα*

*j*

*<sup>A</sup><sup>α</sup>B*1�*<sup>α</sup>* <sup>¼</sup> <sup>X</sup>

*<sup>B</sup>*1�*<sup>α</sup>* <sup>¼</sup> <sup>X</sup>

<sup>2</sup>*Tr A<sup>α</sup>B*<sup>1</sup>�*<sup>α</sup>* � � � *Tr A*<sup>½</sup> <sup>þ</sup> *<sup>B</sup>*�j*LA* � *RB*j*I*� ¼ <sup>X</sup>

Remark 2. We note the following 1, 2:

≤

s

<sup>2</sup> *Tr A*½ � þ *B*�j*A* � *B*j ≤ inf

*<sup>i</sup>* <sup>∣</sup>*ϕi*<sup>i</sup> *<sup>ϕ</sup>i*j¼ <sup>X</sup>

*<sup>μ</sup>*1�*<sup>α</sup> <sup>j</sup>* <sup>∣</sup>*<sup>ψ</sup> <sup>j</sup>*<sup>i</sup> *<sup>ψ</sup> <sup>j</sup>*j¼ <sup>X</sup>

*i*, *j λα*

*Tr A<sup>α</sup>B*1�*<sup>α</sup>* � � <sup>¼</sup> <sup>X</sup>

0 ≤*α*≤1

*Tr A*½ � þ *B* � �<sup>2</sup>

> 3 2

0

B@

13 2

0

B@

7 2

Let *A*, *B*∈ *Mn*,*sa*ð Þ have the following spectral decompositions:

*<sup>λ</sup>i*∣*ϕi*<sup>i</sup> *<sup>ϕ</sup>i*j, *<sup>B</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

1

3 2

> 7 2

1

13 2

1 2

*A* ¼

we have *Tr*½ �¼ <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* 3, *Tr*½ �¼ <sup>j</sup>*<sup>A</sup>* � *<sup>B</sup>*<sup>j</sup> ffiffiffiffiffi

*A* ¼

we have *Tr*½ �¼ <sup>j</sup>*LA* � *RB*j*<sup>I</sup>* 8, *Tr*½ �¼ <sup>j</sup>*<sup>A</sup>* � *<sup>B</sup>*<sup>j</sup> ffiffiffiffiffi

*<sup>A</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

**7. Sum type of uncertainty relations**

*i*, *j λα <sup>i</sup>* j*ϕ<sup>i</sup>*

*i*, *j*

*i*, *j λα*

*i*, *j*

Since 2*xαy*<sup>1</sup>�*<sup>α</sup>* � ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*�j*<sup>x</sup>* � *<sup>y</sup>*<sup>j</sup> <sup>≥</sup> 0 for *<sup>x</sup>*, *<sup>y</sup>* <sup>&</sup>gt;0, 0<sup>≤</sup> *<sup>α</sup>*≤1 in general, we can obtain Theorem 1.9. □

2. There is no relationship between *Tr*½ � j*LA* � *RB*j*I* and *Tr*½ � j*A* � *B*j . When

2*λ<sup>α</sup>*

*Tr A*<sup>1</sup>�*αB<sup>α</sup>* � �≤*Tr A*<sup>1</sup>*=*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� 1

CA, *<sup>B</sup>* <sup>¼</sup> 4 0

*<sup>i</sup> <sup>μ</sup>*1�*<sup>α</sup> <sup>j</sup>* <sup>∣</sup>*ϕi*<sup>i</sup> *<sup>ϕ</sup>i*j*<sup>ψ</sup> <sup>j</sup>*

\* +

*<sup>μ</sup>*1�*<sup>α</sup> <sup>j</sup>* <sup>j</sup>*ϕ<sup>i</sup>*

D E

*<sup>i</sup> <sup>μ</sup>*1�*<sup>α</sup> <sup>j</sup>* <sup>∣</sup>h*ϕ<sup>i</sup> <sup>ψ</sup> <sup>j</sup>*<sup>i</sup> � � �

*ϕi*j*ψ <sup>j</sup>* D E

h*ψ <sup>j</sup>*∣,

h*ψ <sup>j</sup>*∣,

*ϕi*j*ψ <sup>j</sup>* D E

h*ψ <sup>j</sup>*∣*:*

� � � 2 *:*

*<sup>i</sup> <sup>μ</sup>*<sup>1</sup>�*<sup>α</sup> <sup>j</sup>* � *<sup>λ</sup><sup>i</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>j</sup>*�j*λ<sup>i</sup>* � *<sup>μ</sup> <sup>j</sup>*<sup>j</sup> n o � �

> *B*<sup>1</sup>*=*<sup>2</sup> h i

> > *:*

<sup>4</sup> ð Þ *Tr*½ � <sup>j</sup>*<sup>A</sup>* � *<sup>B</sup>*<sup>j</sup> <sup>2</sup>

0 1 � �

> 0 5 � �

10 <sup>p</sup> . When

CA, *<sup>B</sup>* <sup>¼</sup> 2 0

<sup>58</sup> <sup>p</sup> .

*i*¼1

\* +

*μi*j*ψ<sup>i</sup>*

h*ψi*∣*:*

,

,

∣h*ϕ<sup>i</sup> ψ <sup>j</sup>*i � � �

� � � 2 *:*

\* +

For any quantum state ∣*ϕ*i, we define the two probability distributions

$$P = (p\_1, p\_2, \cdots, p\_n), \quad Q = (q\_i, q\_2, \cdots, q\_n), \dots$$

where *pi* <sup>¼</sup> *<sup>ϕ</sup><sup>i</sup>* j j h i <sup>j</sup>*<sup>ϕ</sup>* <sup>2</sup> , *<sup>q</sup> <sup>j</sup>* <sup>¼</sup> <sup>∣</sup>h*<sup>ψ</sup> <sup>j</sup>*j j *<sup>ϕ</sup>*<sup>i</sup> <sup>2</sup> . Let

$$H(P) = -\sum\_{i=1}^{n} p\_i \log p\_i, \quad H(Q) = -\sum\_{j=1}^{n} q\_j \log q\_j$$

be the Shannon entropies of *P* and *Q*, respectively. Theorem 1.10. The following uncertainty relation holds:

$$H(P) + H(Q) \ge -2\log c,$$

where *c* ¼ max *<sup>i</sup>*,*<sup>j</sup>*∣ *ϕi*j*ψ <sup>j</sup>* D E∣.

For details, see [15, 16].

Definition 6. The Fourier transformation of *ψ* ∈*L*<sup>2</sup> ð Þ is defined as

$$
\hat{\psi}(a) = \int\_{-\infty}^{\infty} \psi(t) e^{-2\pi iat} dt.
$$

We also define

$$Q(\mathbb{R}) = \left\{ f \in L^2(\mathbb{R}) ; \int\_{-\infty}^{\infty} t^2 |f(t)|^2 dt < \infty \right\}.$$

Proposition 2. If *ψ* ∈*L*<sup>2</sup> ð Þ , <sup>∥</sup>*ψ*∥<sup>2</sup> <sup>¼</sup> 1 satisfies *<sup>ψ</sup>*, *<sup>ψ</sup>*^ <sup>∈</sup> *<sup>Q</sup>*ð Þ , then

$$
\mathcal{S}(\tilde{\boldsymbol{\nu}}) + \mathcal{S}(\hat{\boldsymbol{\nu}}) \ge \log \frac{e}{2},
$$

where

$$S(\boldsymbol{\nu}) = -\int\_{-\infty}^{\infty} \left| \boldsymbol{\nu}(t) \right|^2 \log \left| \boldsymbol{\nu}(t) \right|^2 dt, \quad S(\boldsymbol{\hat{\nu}}) = -\int\_{-\infty}^{\infty} \left| \boldsymbol{\hat{\nu}}(t) \right|^2 \log \left| \boldsymbol{\hat{\nu}}(t) \right|^2 dt.$$

For details, see [17].

Theorem 1.11 ([18]). For any *X*, *Y* ∈ *Mn*ð Þ , *A*, *B* ∈ *Mn*,þð Þ , the following holds:

$$\begin{aligned} \mathbb{1}\_{A,B}^{(gf)}(X,Y) + I\_{A,B}^{(gf)}(Y) &\geq \frac{1}{2} \max\left\{ I\_{A,B}^{(gf)}(X+Y), I\_{A,B}^{(gf)}(X-Y) \right\}. \\\\ \mathbb{1}\_{A,B}\sqrt{I\_{A,B}^{(gf)}(X)} + \sqrt{I\_{A,B}^{(gf)}(Y)} &\geq \max\left\{ \sqrt{I\_{A,B}^{(gf)}(X+Y)}, \sqrt{I\_{A,B}^{(gf)}(X-Y)} \right\}. \\\\ \mathbb{1}\_{A,\sqrt{I\_{A,B}^{(gf)}(X)}} + \sqrt{I\_{A,B}^{(gf)}(Y)} &\leq 2\max\left\{ \sqrt{I\_{A,B}^{(gf)}(X+Y)}, \sqrt{I\_{A,B}^{(gf)}(X-Y)} \right\}. \end{aligned}$$

Proof 1. The Hilbert-Schmidt norm ∥ � ∥ satisfies

$$\|\|X\|^2 + \|Y\|^2 = \frac{1}{2} \left( \|X + Y\|^2 + \|X - Y\|^2 \right) \ge \frac{1}{2} \max\left\{ \|X + Y\|^2, \|X - Y\|^2 \right\}.\tag{13}$$

Since *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *X* is the second power of the Hilbert-Schmidt norm, ∥*X*∥ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* q . We then obtain the result by substituting (13),

2. By the triangle inequality of a general norm, we apply the triangle inequality for ∥*X*∥ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X* q .

3. We prove the following norm inequality:

$$\|\|X\|\| + \|Y\| \le \|X + Y\| + \|X - Y\|. \tag{14}$$

Since

$$\|\|X\|\| = \|\frac{1}{2}(X+Y) + \frac{1}{2}(X-Y)\| \le \frac{1}{2} \|X+Y\| + \frac{1}{2} \|X-Y\|.$$

and

$$\|Y\| = \|\frac{1}{2}(Y+X) + \frac{1}{2}(Y-X)\| \le \frac{1}{2} \|Y+X\| + \frac{1}{2} \|Y-X\|,$$

we add two inequalities and obtain (14). □

**References**

*Uncertainty Relations*

267-288

[1] Lieb EH. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Advances in Mathematics. 1973;**11**:

*DOI: http://dx.doi.org/10.5772/intechopen.92137*

[10] Gibilisco P, Hansen F, Isola T. On a correspondence between regular and non-regular operator monotone functions. Linear Algebra and its Applications. 2009;**430**:2225-2232

[11] Kubo F, Ando T. Means of positive linear operators. Mathematische Annalen. 1980;**246**:205-224

[12] Yanagi K. On the trace inequalities related to left-right multiplication operators and their applications. Linear and Nonlinear Analysis. 2018;**4**(3):

[13] Audenaert KMR, Calsamiglia J, Masancs LI, Munnoz-Tapia R, Acin A, Bagan E, et al. The quantum Chernoff bound. Review Letters. 2007;**98**:

[14] Yanagi K. Generalized trace inequalities related to fidelity and trace distance. Linear and Nonlinear Analysis.

[15] Maassen H, Uffink JBM. Generalized entropic uncertainty relations. Physical Review Letters. 1988;

Computation and Quantum

adjusted skew informations.

[16] Nielsen MA, Chuang IL. Quantum

[17] Hirschman II Jr. A note on entropy. American Journal of Mathematics. 1957;

[18] Yanagi K. Sum types of uncertainty relations for generalized quasi-metric

International Journal of Mathematical Analysis and Applications. 2018;**4**(4):

Information. Cambridge University

361-370

160501-1-160501-4

2016;**2**(2):263-270

**60**(12):1103-1106

Press; 2000

**79**:152-156

85-94

[2] Luo S, Zhang Q. On skew information. IEEE Transactions of Information Theory. 2004;**50**:1778-1782,

and Correction to "On skew information", IEEE Transactions of Information Theory. 2005;**51**:4432

2005;**16**:629-646

4401-4404

9909-9916

1-14

12-18

**63**

[3] Kosaki H. Matrix trace inequality related to uncertainty principle. International Journal of Mathematics.

[4] Yanagi K, Furuichi S, Kuriyama K. A generalized skew information and uncertainty relations. IEEE Transactions of Information Theory. 2005;**IT-51**(12):

[5] Hansen F. Metric adjusted skew information. Proceedings of the National Academy of Sciences of the United States of America. 2008;**105**:

[6] Yanagi K. Metric adjusted skew information and uncertainty relation. Journal of Mathematical Analysis and Applications. 2011;**380**(2):888-892

[7] Yanagi K, Furuichi S, Kuriyama K. Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. Journal of Uncertainty Analysis and Applications. 2013;**1**(12):

[8] Luo S. Heisenberg uncertainty relation for mixed states. Physical Review A. 2005;**72**:042110

[9] Yanagi K. Uncertainty relation on

information. Journal of Mathematical Analysis and Applications. 2010;**365**:

Wigner-Yanase-Dyson skew

### **Author details**

Kenjiro Yanagi1,2

1 Faculty of Science, Department of Mathematics, Josai University, Japan

2 Yamaguchi University, Japan

\*Address all correspondence to: yanagi@yamaguchi-u.ac.jp

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

Since *I*

*Quantum Mechanics*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*

for ∥*X*∥ ¼

Since

and

**Author details**

Kenjiro Yanagi1,2

**62**

2 Yamaguchi University, Japan

provided the original work is properly cited.

q

ð Þ *g*,*f*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *I* ð Þ *g*,*f <sup>A</sup>*,*<sup>B</sup>* ð Þ *X*

<sup>∥</sup>*X*<sup>∥</sup> <sup>¼</sup> <sup>∥</sup> <sup>1</sup>

<sup>∥</sup>*Y*<sup>∥</sup> <sup>¼</sup> <sup>∥</sup> <sup>1</sup>

2

2

ð Þþ *<sup>X</sup>* <sup>þ</sup> *<sup>Y</sup>* <sup>1</sup>

ð Þþ *<sup>Y</sup>* <sup>þ</sup> *<sup>X</sup>* <sup>1</sup>

2

2

1 Faculty of Science, Department of Mathematics, Josai University, Japan

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: yanagi@yamaguchi-u.ac.jp

.

3. We prove the following norm inequality:

*<sup>A</sup>*,*<sup>B</sup>* ð Þ *X*, *X* is the second power of the Hilbert-Schmidt norm, ∥*X*∥ ¼

2. By the triangle inequality of a general norm, we apply the triangle inequality

ð Þ *<sup>X</sup>* � *<sup>Y</sup>* ∥ ≤ <sup>1</sup>

ð Þ *<sup>Y</sup>* � *<sup>X</sup>* ∥ ≤ <sup>1</sup>

we add two inequalities and obtain (14). □

2

2

∥*X*∥ þ ∥*Y*∥≤∥*X* þ *Y*∥ þ ∥*X* � *Y*∥*:* (14)

∥*X* þ *Y*∥ þ

∥*Y* þ *X*∥ þ

1 2

1 2 ∥*X* � *Y*∥

∥*Y* � *X*∥,

. We then obtain the result by substituting (13),

[1] Lieb EH. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Advances in Mathematics. 1973;**11**: 267-288

[2] Luo S, Zhang Q. On skew information. IEEE Transactions of Information Theory. 2004;**50**:1778-1782, and Correction to "On skew information", IEEE Transactions of Information Theory. 2005;**51**:4432

[3] Kosaki H. Matrix trace inequality related to uncertainty principle. International Journal of Mathematics. 2005;**16**:629-646

[4] Yanagi K, Furuichi S, Kuriyama K. A generalized skew information and uncertainty relations. IEEE Transactions of Information Theory. 2005;**IT-51**(12): 4401-4404

[5] Hansen F. Metric adjusted skew information. Proceedings of the National Academy of Sciences of the United States of America. 2008;**105**: 9909-9916

[6] Yanagi K. Metric adjusted skew information and uncertainty relation. Journal of Mathematical Analysis and Applications. 2011;**380**(2):888-892

[7] Yanagi K, Furuichi S, Kuriyama K. Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. Journal of Uncertainty Analysis and Applications. 2013;**1**(12): 1-14

[8] Luo S. Heisenberg uncertainty relation for mixed states. Physical Review A. 2005;**72**:042110

[9] Yanagi K. Uncertainty relation on Wigner-Yanase-Dyson skew information. Journal of Mathematical Analysis and Applications. 2010;**365**: 12-18

[10] Gibilisco P, Hansen F, Isola T. On a correspondence between regular and non-regular operator monotone functions. Linear Algebra and its Applications. 2009;**430**:2225-2232

[11] Kubo F, Ando T. Means of positive linear operators. Mathematische Annalen. 1980;**246**:205-224

[12] Yanagi K. On the trace inequalities related to left-right multiplication operators and their applications. Linear and Nonlinear Analysis. 2018;**4**(3): 361-370

[13] Audenaert KMR, Calsamiglia J, Masancs LI, Munnoz-Tapia R, Acin A, Bagan E, et al. The quantum Chernoff bound. Review Letters. 2007;**98**: 160501-1-160501-4

[14] Yanagi K. Generalized trace inequalities related to fidelity and trace distance. Linear and Nonlinear Analysis. 2016;**2**(2):263-270

[15] Maassen H, Uffink JBM. Generalized entropic uncertainty relations. Physical Review Letters. 1988; **60**(12):1103-1106

[16] Nielsen MA, Chuang IL. Quantum Computation and Quantum Information. Cambridge University Press; 2000

[17] Hirschman II Jr. A note on entropy. American Journal of Mathematics. 1957; **79**:152-156

[18] Yanagi K. Sum types of uncertainty relations for generalized quasi-metric adjusted skew informations. International Journal of Mathematical Analysis and Applications. 2018;**4**(4): 85-94

**Chapter 5**

**Abstract**

**1. Introduction**

**65**

Complex Space Nature of the

to Quantum Mechanics

*Ciann-Dong Yang and Shiang-Yi Han*

Quantum World: Return Causality

As one chapter, we about to begin a journey with exploring the limitation of the causality that rules the whole universe. Quantum mechanics is established on the basis of the phenomenology and the lack of ontology builds the wall which blocks the causality. It is very difficult to reconcile the probability and the causality in such a platform. A higher dimension consideration may leverage this dilemma by expanding the vision. Information may seem to be discontinuous or even so weird if only be viewed from a part of the degree of freedoms. Based on this premise, we reexamined the microscopic world within a complex space. Significantly, some knowledge beyond the empirical findings is revealed and paves the way for a more detailed exploration of the quantum world. The random quantum motion is essential for atomic particle and exhibits a wave-related property with a bulk of trajectories. It seems we can break down the wall which forbids the causality entering the quantum kingdom and connect quantum mechanics with classical mechanics. The causality returns to the quantum world without any assumption in terms of the quantum random motion under the optimal guidance law in complex space. Thereby hangs a tale, we briefly introduce this new formulation from the fundamental theoretical description to the practical technology applications.

**Keywords:** random quantum trajectory, optimal guidance law, complex space

most elegant beauty of nature. As precise as physics.

It took scientists nearly two centuries from first observation of flower powder's Brownian motion to propose a mathematical qualitative description [1]. Time is an arrow launched from the past to the future, every event happens for a reason. "The world is woven from billions of lives, every strand crossing every other. What we call premonition is just movement of the web. If you could attenuate to every strand of quivering data, the future would be entirely calculable. As inevitable as mathematics [2]." All physical phenomena are connected to the same web. As long as we can see through the quivering data and cut into the very core, we can glimpse the

It took nearly 30 years for physicists to establish quantum mechanics but nearly 100 years to seek for its essence. Quantum mechanics is the most precise theory to describe the microscopic world but also is the most obscure one among all theories. It collects lots data but not all. Just like what we can observed is the shadow on the

#### **Chapter 5**

## Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics

*Ciann-Dong Yang and Shiang-Yi Han*

#### **Abstract**

As one chapter, we about to begin a journey with exploring the limitation of the causality that rules the whole universe. Quantum mechanics is established on the basis of the phenomenology and the lack of ontology builds the wall which blocks the causality. It is very difficult to reconcile the probability and the causality in such a platform. A higher dimension consideration may leverage this dilemma by expanding the vision. Information may seem to be discontinuous or even so weird if only be viewed from a part of the degree of freedoms. Based on this premise, we reexamined the microscopic world within a complex space. Significantly, some knowledge beyond the empirical findings is revealed and paves the way for a more detailed exploration of the quantum world. The random quantum motion is essential for atomic particle and exhibits a wave-related property with a bulk of trajectories. It seems we can break down the wall which forbids the causality entering the quantum kingdom and connect quantum mechanics with classical mechanics. The causality returns to the quantum world without any assumption in terms of the quantum random motion under the optimal guidance law in complex space. Thereby hangs a tale, we briefly introduce this new formulation from the fundamental theoretical description to the practical technology applications.

**Keywords:** random quantum trajectory, optimal guidance law, complex space

#### **1. Introduction**

It took scientists nearly two centuries from first observation of flower powder's Brownian motion to propose a mathematical qualitative description [1]. Time is an arrow launched from the past to the future, every event happens for a reason. "The world is woven from billions of lives, every strand crossing every other. What we call premonition is just movement of the web. If you could attenuate to every strand of quivering data, the future would be entirely calculable. As inevitable as mathematics [2]." All physical phenomena are connected to the same web. As long as we can see through the quivering data and cut into the very core, we can glimpse the most elegant beauty of nature. As precise as physics.

It took nearly 30 years for physicists to establish quantum mechanics but nearly 100 years to seek for its essence. Quantum mechanics is the most precise theory to describe the microscopic world but also is the most obscure one among all theories. It collects lots data but not all. Just like what we can observed is the shadow on the

ground not the actual object in the air. It is impossible to see the whole appearance of the object by observing its shadow. The development of the quantum era seems started in such circumstances and missed something we call the essence of nature. In this chapter, we hope to recover the missing part by considering a higher dimension to capture the actual appearance of nature. At the end, we will find out that nature dominates the web where we live as well as the theories we develop. Everything should follow the law of the nature, and there is no exception.

the Nano-scale is demonstrated in Section 4. We consider the quantum potential relation to the electronic channel in a 2D Nano-structure. In addition, the conductance quantization is realized in terms of the quantum potential which shows that the lower potential region is where the most electrons pass through the channel.

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

In the macroscopic world, it is natural to see an object moving along with a specific path which is determined by the resultant optimal action function. However, in the microscopic world, we cannot repetitively carry out this observation since there is no definition of the trajectory for a quantum particle. With the limit on the observation, only a part of trajectory, more precisely, the trajectory in the real part of complex space can be detected. As particle passing or staying in the imaginary part of complex space it disappears from our visible world and becomes untraceable. The particle randomly transits in and out of the real part and imaginary part of complex space, causes a discontinuous trajectory viewed from the observable space. Therefore, it can only be empirically described by the probability in

In this section, we briefly introduce how particle's motion can be fully described by the optimal guidance law in the complex plane [28]. Then we will discuss under what condition the statistical distribution of an ensemble of trajectories in the complex plane will be compatible with the quantum mechanical and classical results. In the following, we consider a complex plane for the purpose of simplicity; however, there should be no problem to implement the optimal guidance law in complex space. Let us consider a particle with random motion in the complex plane

where *x* represents a vector, *u* is the guidance law needed to be determined, *w* is

where *Et*,*<sup>x</sup>* represents the expectation of the cost function over all infinite trajec-

*u t*, *<sup>t</sup>* ½ �*<sup>f</sup>*

Instead of using the variational method, we apply the dynamic programming method to Eq. (3) for the random motion. We then have the following expression

> 1 2

*tr g<sup>T</sup>*ð Þ *<sup>x</sup>*, *<sup>u</sup>*

*∂*2 *V t*ð Þ , *x <sup>∂</sup>x*<sup>2</sup> *g x*ð Þ , *<sup>u</sup>*

� � � � , (4)

Wiener process with properties h i *dw* <sup>¼</sup> 0 and *dw*<sup>2</sup> � � <sup>¼</sup> *dt*, *f t*ð Þ , *<sup>x</sup>*, *<sup>u</sup>* is the drift velocity, and *g x*ð Þ , *u* is the diffusion velocity. The cost function for *x t*ð Þ with

> ð*tf t*

tories launched from the single initial condition, *x t*ðÞ¼ *x* in time interval *t*, *tf*

*V t*ð Þ¼ , *x* min

*J t*ð Þ¼ , *x*, *u Et*,*<sup>x</sup>*

find the minimum cost function, we define the value function,

*<sup>∂</sup>V t*ð Þ , *<sup>x</sup> <sup>∂</sup><sup>x</sup> <sup>f</sup>* <sup>þ</sup>

*L* þ

*dx* ¼ *f t*ð Þ , *x*, *u dt* þ *g x*ð Þ , *u dw*, *x* ¼ *xR* þ *ixI* ∈ , (1)

*L*ð Þ *τ*, *x*ð Þ*τ* , *u*ð Þ*τ dτ*

� �, (2)

*J t*ð Þ , *x*, *u :* (3)

� �. To

And then, concluding remarks are presented in Section 5.

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

**2. Random quantum motion in the complex plane**

quantum mechanics.

whose dynamic evolution reads

randomness property reads

after having the Taylor expansion:

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> min *u t*, *<sup>t</sup>* ½ �*<sup>f</sup>*

� *<sup>∂</sup>V t*ð Þ , *<sup>x</sup>*

**67**

Trajectory is a typical classical feature of the macroscopic object solved by the equation of motion. The trajectory of the microscopic particle is supposed to be observed if the law of nature remains consistent all the way down to the atomic scale. However, such an observation cannot be made till 2011. Kocsis and his coworkers propose an observation of the average trajectories of single photons in a two-slit interferometer on the basis of weak measurement [3]. Since then quantum trajectories are observed for many quantum systems, such as superconducting quantum bit, mechanical resonator, and so on [4–6]. Weak measurement provides the weak value which is a measurable quantity definable to any quantum observable under the weak coupling between the system and the measurement apparatus [7]. The significant characteristic of the weak value does not lie within the range of eigenvalues and is complex. It is pointed out that the real part of the complex weak value represents the average quantum value [8], and the imaginary part is related to the rate of variation in the interference observation [9].

The trajectory interpretation of quantum mechanics is developed on the basis of de Broglie's matter wave and Bohm's guidance law. In recent years, the importance of the quantum trajectory in theoretical treatment and experimental test has been discussed in complex space [10–21]. All these research indirectly or directly show that the complex space extension is more than a mathematical tool, it implies a causal essence of the quantum world.

On the other hand, it is found out that the real part of momentum's weak value is the Bohmian momentum representing the average momentum conditioned on a position detection; while its imaginary part is proportional to the osmotic velocity that describes the logarithmic derivative of the probability density for measuring the particular position directed along the flow generated by the momentum [22]. This not only implies the existence of randomness in a quantum system, but also discloses that the random motion occurs in complex space. Numerous studies with the complex initial condition and the random property have been discussed [23–25]. A stochastic interpretation of quantum mechanics is proposed which regards the random motion as a nature property of the quantum world not the interference made by the measurement devices [26, 27]. These investigations suggest that a complex space and the random motion are two important features of the quantum world.

Based on the complex space structure, we propose a new perspective of quantum mechanics that allows one to reexamine quantum phenomena in a classical way. We will see in this chapter how the quantum motion can provide the classical description for the quantum kingdom and is in line with the probability distribution. One thing particular needed to be emphasized is that the stochastic Hamilton Jacobi Bellman equation can reduce to the Schrödinger equation under some specific conditions. In other words, the Schrödinger equation is one special case of all kinds of random motions in complex space. A further discussion of the relationship between the trajectory interpretation and probability interpretation is presented in Section 2. In particular, the solvable nodal issue is put into discussion, and the continuity equation for the complex probability density function is proposed. In Section 3, we demonstrate how the quantum force could play the crucial role in the force balanced condition within the hydrogen atom and how the quantum potential forms the shell structure where the orbits are quantized. A practical application to

the Nano-scale is demonstrated in Section 4. We consider the quantum potential relation to the electronic channel in a 2D Nano-structure. In addition, the conductance quantization is realized in terms of the quantum potential which shows that the lower potential region is where the most electrons pass through the channel. And then, concluding remarks are presented in Section 5.

#### **2. Random quantum motion in the complex plane**

In the macroscopic world, it is natural to see an object moving along with a specific path which is determined by the resultant optimal action function. However, in the microscopic world, we cannot repetitively carry out this observation since there is no definition of the trajectory for a quantum particle. With the limit on the observation, only a part of trajectory, more precisely, the trajectory in the real part of complex space can be detected. As particle passing or staying in the imaginary part of complex space it disappears from our visible world and becomes untraceable. The particle randomly transits in and out of the real part and imaginary part of complex space, causes a discontinuous trajectory viewed from the observable space. Therefore, it can only be empirically described by the probability in quantum mechanics.

In this section, we briefly introduce how particle's motion can be fully described by the optimal guidance law in the complex plane [28]. Then we will discuss under what condition the statistical distribution of an ensemble of trajectories in the complex plane will be compatible with the quantum mechanical and classical results. In the following, we consider a complex plane for the purpose of simplicity; however, there should be no problem to implement the optimal guidance law in complex space. Let us consider a particle with random motion in the complex plane whose dynamic evolution reads

$$d\boldsymbol{x} = \boldsymbol{f}(t, \boldsymbol{x}, \boldsymbol{u})dt + \mathbf{g}(\boldsymbol{x}, \boldsymbol{u})dw, \quad \boldsymbol{\kappa} = \boldsymbol{\kappa}\_{\mathbb{R}} + i\boldsymbol{\kappa}\_{I} \in \mathbb{C},\tag{1}$$

where *x* represents a vector, *u* is the guidance law needed to be determined, *w* is Wiener process with properties h i *dw* <sup>¼</sup> 0 and *dw*<sup>2</sup> � � <sup>¼</sup> *dt*, *f t*ð Þ , *<sup>x</sup>*, *<sup>u</sup>* is the drift velocity, and *g x*ð Þ , *u* is the diffusion velocity. The cost function for *x t*ð Þ with randomness property reads

$$J(t, \varkappa, u) = E\_{t, \varkappa} \left[ \int\_t^{t\_f} L(\tau, \varkappa(\tau), u(\tau)) d\tau \right],\tag{2}$$

where *Et*,*<sup>x</sup>* represents the expectation of the cost function over all infinite trajectories launched from the single initial condition, *x t*ðÞ¼ *x* in time interval *t*, *tf* � �. To find the minimum cost function, we define the value function,

$$V(t, \mathfrak{x}) = \min\_{u \mid [t, t\_f]} J(t, \mathfrak{x}, u). \tag{3}$$

Instead of using the variational method, we apply the dynamic programming method to Eq. (3) for the random motion. We then have the following expression after having the Taylor expansion:

$$-\frac{\partial V(t,\mathbf{x})}{\partial t} = \min\_{u\left[t,t\_f\right]} \left\{ L + \frac{\partial V(t,\mathbf{x})}{\partial \mathbf{x}} f + \frac{1}{2} tr \left[ \mathbf{g}^T(\mathbf{x},u) \frac{\partial^2 V(t,\mathbf{x})}{\partial \mathbf{x}^2} \mathbf{g}(\mathbf{x},u) \right] \right\},\tag{4}$$

ground not the actual object in the air. It is impossible to see the whole appearance of the object by observing its shadow. The development of the quantum era seems started in such circumstances and missed something we call the essence of nature. In this chapter, we hope to recover the missing part by considering a higher dimension to capture the actual appearance of nature. At the end, we will find out that nature dominates the web where we live as well as the theories we develop. Every-

Trajectory is a typical classical feature of the macroscopic object solved by the equation of motion. The trajectory of the microscopic particle is supposed to be observed if the law of nature remains consistent all the way down to the atomic scale. However, such an observation cannot be made till 2011. Kocsis and his coworkers propose an observation of the average trajectories of single photons in a two-slit interferometer on the basis of weak measurement [3]. Since then quantum trajectories are observed for many quantum systems, such as superconducting quantum bit, mechanical resonator, and so on [4–6]. Weak measurement provides the weak value which is a measurable quantity definable to any quantum observable under the weak coupling between the system and the measurement apparatus [7]. The significant characteristic of the weak value does not lie within the range of eigenvalues and is complex. It is pointed out that the real part of the complex weak value represents the average quantum value [8], and the imaginary part is related to

The trajectory interpretation of quantum mechanics is developed on the basis of de Broglie's matter wave and Bohm's guidance law. In recent years, the importance of the quantum trajectory in theoretical treatment and experimental test has been discussed in complex space [10–21]. All these research indirectly or directly show that the complex space extension is more than a mathematical tool, it implies a

On the other hand, it is found out that the real part of momentum's weak value is the Bohmian momentum representing the average momentum conditioned on a position detection; while its imaginary part is proportional to the osmotic velocity that describes the logarithmic derivative of the probability density for measuring the particular position directed along the flow generated by the momentum [22]. This not only implies the existence of randomness in a quantum system, but also discloses that the random motion occurs in complex space. Numerous studies with the complex initial condition and the random property have been discussed [23–25]. A stochastic interpretation of quantum mechanics is proposed which regards the random motion as a nature property of the quantum world not the interference made by the measurement devices [26, 27]. These investigations suggest that a complex space

and the random motion are two important features of the quantum world.

Based on the complex space structure, we propose a new perspective of quantum mechanics that allows one to reexamine quantum phenomena in a classical way. We will see in this chapter how the quantum motion can provide the classical description for the quantum kingdom and is in line with the probability distribution. One thing particular needed to be emphasized is that the stochastic Hamilton Jacobi Bellman equation can reduce to the Schrödinger equation under some specific conditions. In other words, the Schrödinger equation is one special case of all kinds of random motions in complex space. A further discussion of the relationship between the trajectory interpretation and probability interpretation is presented in Section 2. In particular, the solvable nodal issue is put into discussion, and the continuity equation for the complex probability density function is proposed. In Section 3, we demonstrate how the quantum force could play the crucial role in the force balanced condition within the hydrogen atom and how the quantum potential forms the shell structure where the orbits are quantized. A practical application to

thing should follow the law of the nature, and there is no exception.

the rate of variation in the interference observation [9].

causal essence of the quantum world.

*Quantum Mechanics*

**66**

which is recognized as the Hamilton-Jacobi-Bellman (HJB) equation and *∂*2 *V t*ð Þ , *<sup>x</sup> <sup>=</sup>∂x*<sup>2</sup> is Jacobi matrix. Finding the minimum of the cost function leads to the momentum for the optimal path,

$$p = \frac{\partial L(t, \mathbf{x}, u)}{\partial u} = \frac{\partial L(t, \mathbf{x}, \dot{\mathbf{x}})}{\partial \dot{\mathbf{x}}} = -\nabla V(t, \mathbf{x}), \tag{5}$$

and determines the optimal guidance law,

$$\left.u\right| = u(t, \mathfrak{x}, p)|\_{p=-\operatorname{TV}}.\tag{6}$$

This becomes a solid evidence to support the deduction that the matter wave is

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

guidance law, we consider a particle experiencing a randomness,

guidance law can be expressed in terms of the wave function,

*dx* <sup>¼</sup> �*i*<sup>ℏ</sup> *m*

*dx* ¼ *u t*ð Þ , *x*, *p dt* þ

*u t*ð Þ¼ , *<sup>x</sup>*, *<sup>p</sup>* �*i*<sup>ℏ</sup>

∇*Ψ*ð Þ *t*, *x Ψ*ð Þ *t*, *x*

*<sup>x</sup>*\_ <sup>¼</sup> �*i*<sup>ℏ</sup> *m*

quantum harmonic oscillator in *n* ¼ 1 state. The trajectory interpretation is

collecting all crossovers on the real axis of an ensemble of CRTs as the dots

bution and the quantum mechanical probability distribution [36].

supported by the excellent agreement of the statistical spatial distribution made by

displayed in **Figure 2(a)**. It shows a good agreement of the statistical spatial distri-

In most text book of quantum mechanics, the nodes of the probability of harmonic oscillator either be ignored or be regarded as the quantum characteristic. Only the classical-like curve of the averaged probability has been mentioned. The other significant finding brought out by the CRT interpretation is the nodal vanished condition given by the statistical distribution of the collection of all pointes be projected onto the real axis as **Figure 2(b)** shows. It starts to approach the classical probability distribution for high quantum number as **Figure 2(c)** presents. The leverage of complex space structure deals with the probability nodes, and even further to reach the classical region dominated by Newtonian mechanics (more detail refers to [36]). After the matter wave can be interpreted by an ensemble of trajectories in both theoretical and experimental results [3, 18, 34, 35], the CRT interpretation shows both quantum mechanical and classical compatible

Therefore, Eq. (11) can be recast into the following expression:

To fully understand the property of these trajectories under the influence of the

where we have replaced *f t*ð Þ , *x*, *u* by the optimal guidance law *u t*ð Þ , *x*, *p* , and

*m*

*dt* þ

Eq. (13) will reduce to the equation of motion given by the quantum HJ equation

since the random motion in Eq. (13) has zero mean. This result shows that the quantum HJ equation represents the mean motion of the particle. The trajectory in the complex plane solved from Eq. (13) is random and will become the mean trajectory solved from Eq. (14) after being averaged out. **Figure 1** illustrates this property by demonstrating the quantum motion of the Gaussian wave packet [28]. The first question we would like to answer by the complex random trajectory (CRT) interpretation is its connection to the probability interpretation. In quantum mechanics, the amplitude square of the wave function gives the probability density of physical quantities as shown in **Figure 2(a)**, in which the solid line stands for the

∇*Ψ*ð Þ *t*, *x*

ffiffiffiffiffiffiffiffi �*i*ℏ *m*

�*iℏ=<sup>m</sup>* <sup>p</sup> into Eq. (1). Combining Eqs. (6) and (10), the optimal

ffiffiffiffiffiffiffiffi �*i*ℏ *m*

r

∇*Ψ*ð Þ *t*, *x*

*dw*, (11)

*<sup>Ψ</sup>*ð Þ *<sup>t</sup>*, *<sup>x</sup> :* (12)

*<sup>Ψ</sup>*ð Þ *<sup>t</sup>*, *<sup>x</sup>* , (14)

*dw:* (13)

r

formed by a huge number of trajectories.

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

(7) if we take the average of both sides,

**69**

assigned *g x*ð Þ¼ , *<sup>u</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

If one replaces Lagrange *<sup>L</sup>* by Hamiltonian *H t*ð Þ¼ , *<sup>x</sup>*, *<sup>p</sup> pTu* � *L t*ð Þ , *<sup>x</sup>*, *<sup>u</sup>* , defines the action function as *S t*ð Þ¼� , *<sup>x</sup> V t*ð Þ , *<sup>x</sup>* and let *g x*ð Þ¼ , *<sup>u</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi �*iℏ=<sup>m</sup>* <sup>p</sup> , Eq. (4) can be transferred to the quantum Hamilton-Jacobi (HJ) equation,

$$\frac{\partial \mathbf{S}}{\partial t} + H(t, \mathbf{x}, p)|\_{p=\nabla \mathbf{S}} + \frac{i\hbar}{2m} \nabla^2 \mathbf{S} = \mathbf{0}.\tag{7}$$

Please notice that the last term in Eq. (7) is what makes the quantum HJ equation differs from its classical counterpart. It is called the quantum potential,

$$Q = \frac{i\hbar}{2m}\nabla^2 \mathbf{S} \tag{8}$$

in dBB theory, Bohmian mechanics, and quantum Hamilton mechanics [29–33]. Even the quantum potential we derive here has the same expression appeared in Bohmian mechanics, its relation to the random motion should be noticed. However, it is not yet suitable to claim that the random motion attributes to the quantum potential or vice versa. It is worthwhile to bring into discussion. Before inspecting this question more deeply, we still can take advantage of the quantum potential to describe or even explain some quantum phenomena.

We can transfer the quantum HJ equation (7) to the Schrödinger equation,

$$i\hbar\frac{\partial\Psi(t,\mathbf{x})}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi(t,\mathbf{x}) + U\Psi(t,\mathbf{x})\tag{9}$$

via the relation between the action function and wave function,

$$S(t,x) = -i\hbar \ln \Psi(t,x),\tag{10}$$

where *U* represents the external potential. This simple relation reveals a connection between the trajectory and the wave description. In classical mechanics, a particle follows the principle of least action; while the wave picture took place in quantum mechanics. Eq. (10) implies that if we collect all action functions determined by different initial conditions which satisfy the initial probability distribution, a collection of corresponding wave patterns arise and eventually forms the solution wave function of the Schrödinger equation. This process is the same as what Schrödinger attempted to cope with the observable wave and tried to deduce the suitable wave equation based on the classical wave theory. The only difference is that Schrödinger started his deduction from the wave perspective; however, we start from the particle perspective. Even the wave-particle duality troubles physicists to inspect advanced about the essence of nature, the recent experiment confirms relation (10) by observing an ensemble of quantum trajectories [3].

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

This becomes a solid evidence to support the deduction that the matter wave is formed by a huge number of trajectories.

To fully understand the property of these trajectories under the influence of the guidance law, we consider a particle experiencing a randomness,

$$d\mathbf{x} = u(t, \mathbf{x}, p)dt + \sqrt{\frac{-i\hbar}{m}}dw,\tag{11}$$

where we have replaced *f t*ð Þ , *x*, *u* by the optimal guidance law *u t*ð Þ , *x*, *p* , and assigned *g x*ð Þ¼ , *<sup>u</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi �*iℏ=<sup>m</sup>* <sup>p</sup> into Eq. (1). Combining Eqs. (6) and (10), the optimal guidance law can be expressed in terms of the wave function,

$$u(t, \boldsymbol{x}, p) = \frac{-i\hbar}{m} \frac{\nabla \Psi(t, \boldsymbol{x})}{\Psi(t, \boldsymbol{x})}.\tag{12}$$

Therefore, Eq. (11) can be recast into the following expression:

$$d\infty = \frac{-i\hbar}{m} \frac{\nabla \Psi^{\prime}(t, \varkappa)}{\Psi^{\prime}(t, \varkappa)} dt + \sqrt{\frac{-i\hbar}{m}} dw. \tag{13}$$

Eq. (13) will reduce to the equation of motion given by the quantum HJ equation (7) if we take the average of both sides,

$$\dot{\boldsymbol{x}} = \frac{-i\hbar}{m} \frac{\nabla \Psi^{\nu}(t, \boldsymbol{x})}{\Psi(t, \boldsymbol{x})},\tag{14}$$

since the random motion in Eq. (13) has zero mean. This result shows that the quantum HJ equation represents the mean motion of the particle. The trajectory in the complex plane solved from Eq. (13) is random and will become the mean trajectory solved from Eq. (14) after being averaged out. **Figure 1** illustrates this property by demonstrating the quantum motion of the Gaussian wave packet [28].

The first question we would like to answer by the complex random trajectory (CRT) interpretation is its connection to the probability interpretation. In quantum mechanics, the amplitude square of the wave function gives the probability density of physical quantities as shown in **Figure 2(a)**, in which the solid line stands for the quantum harmonic oscillator in *n* ¼ 1 state. The trajectory interpretation is supported by the excellent agreement of the statistical spatial distribution made by collecting all crossovers on the real axis of an ensemble of CRTs as the dots displayed in **Figure 2(a)**. It shows a good agreement of the statistical spatial distribution and the quantum mechanical probability distribution [36].

In most text book of quantum mechanics, the nodes of the probability of harmonic oscillator either be ignored or be regarded as the quantum characteristic. Only the classical-like curve of the averaged probability has been mentioned. The other significant finding brought out by the CRT interpretation is the nodal vanished condition given by the statistical distribution of the collection of all pointes be projected onto the real axis as **Figure 2(b)** shows. It starts to approach the classical probability distribution for high quantum number as **Figure 2(c)** presents. The leverage of complex space structure deals with the probability nodes, and even further to reach the classical region dominated by Newtonian mechanics (more detail refers to [36]). After the matter wave can be interpreted by an ensemble of trajectories in both theoretical and experimental results [3, 18, 34, 35], the CRT interpretation shows both quantum mechanical and classical compatible

which is recognized as the Hamilton-Jacobi-Bellman (HJB) equation and

*<sup>∂</sup><sup>u</sup>* <sup>¼</sup> *<sup>∂</sup>L t*ð Þ , *<sup>x</sup>*, *<sup>x</sup>*\_

*V t*ð Þ , *<sup>x</sup> <sup>=</sup>∂x*<sup>2</sup> is Jacobi matrix. Finding the minimum of the cost function leads to the

If one replaces Lagrange *<sup>L</sup>* by Hamiltonian *H t*ð Þ¼ , *<sup>x</sup>*, *<sup>p</sup> pTu* � *L t*ð Þ , *<sup>x</sup>*, *<sup>u</sup>* , defines

Please notice that the last term in Eq. (7) is what makes the quantum HJ equation

in dBB theory, Bohmian mechanics, and quantum Hamilton mechanics [29–33]. Even the quantum potential we derive here has the same expression appeared in Bohmian mechanics, its relation to the random motion should be noticed. However, it is not yet suitable to claim that the random motion attributes to the quantum potential or vice versa. It is worthwhile to bring into discussion. Before inspecting this question more deeply, we still can take advantage of the quantum potential to

We can transfer the quantum HJ equation (7) to the Schrödinger equation,

where *U* represents the external potential. This simple relation reveals a connection between the trajectory and the wave description. In classical mechanics, a particle follows the principle of least action; while the wave picture took place in quantum mechanics. Eq. (10) implies that if we collect all action functions determined by different initial conditions which satisfy the initial probability distribution, a collection of corresponding wave patterns arise and eventually forms the solution wave function of the Schrödinger equation. This process is the same as what Schrödinger attempted to cope with the observable wave and tried to deduce the suitable wave equation based on the classical wave theory. The only difference is that Schrödinger started his deduction from the wave perspective; however, we start from the particle perspective. Even the wave-particle duality troubles physicists to inspect advanced about the essence of nature, the recent experiment confirms relation (10) by observing an ensemble of quantum trajectories [3].

2*m* ∇2

*i*ℏ 2*m* ∇2

*<sup>∂</sup>x*\_ ¼ �∇*V t*ð Þ , *<sup>x</sup>* , (5)

�*iℏ=<sup>m</sup>* <sup>p</sup> , Eq. (4) can be

*S* ¼ 0*:* (7)

*S* (8)

*Ψ*ð Þþ *t*, *x UΨ*ð Þ *t*, *x* (9)

*S t*ð Þ¼� , *x i*ℏln*Ψ*ð Þ *t*, *x* , (10)

*<sup>u</sup>* <sup>¼</sup> *u t*ð Þj , *<sup>x</sup>*, *<sup>p</sup> <sup>p</sup>*¼�∇*V:* (6)

*∂*2

*Quantum Mechanics*

**68**

momentum for the optimal path,

*<sup>p</sup>* <sup>¼</sup> *<sup>∂</sup>L t*ð Þ , *<sup>x</sup>*, *<sup>u</sup>*

the action function as *S t*ð Þ¼� , *<sup>x</sup> V t*ð Þ , *<sup>x</sup>* and let *g x*ð Þ¼ , *<sup>u</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> *H t*ð Þj , *<sup>x</sup>*, *<sup>p</sup> <sup>p</sup>*¼∇*<sup>S</sup>* <sup>þ</sup>

differs from its classical counterpart. It is called the quantum potential,

*<sup>Q</sup>* <sup>¼</sup> *<sup>i</sup>*<sup>ℏ</sup> 2*m* ∇2

transferred to the quantum Hamilton-Jacobi (HJ) equation,

*∂S ∂t*

describe or even explain some quantum phenomena.

*<sup>∂</sup>Ψ*ð Þ *<sup>t</sup>*, *<sup>x</sup>*

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>ℏ</sup><sup>2</sup>

via the relation between the action function and wave function,

iℏ

and determines the optimal guidance law,

**Figure 1.**

*100,000 trajectories solved from Eq. (13) with the same initial condition of the Gaussian wave packet in the complex plane: (a) the time evolution on the real axis for which the mean is denoted by the blue line; (b) the time evolution on the imaginary axis with zero mean represented by the blue line. The complex trajectory solved from Eq. (14) with one initial condition: (c) the time evolution on the real axis; (d) the time imaginary part of the motion. This figure reveals that the mean of the CRT is the trajectory solved from the quantum Hamilton equations of motion [28].*

results under two kinds of point collections. In other words, Bohr's correspondence principle can be interpreted by the CRT interpretation without loss of generality [36].

The second question we try to cope with by means of the CRT interpretation is the conservation of the complex probability. In quantum mechanics, the continuity equation for the probability density function is given by Bohr's law *<sup>ρ</sup>QM* <sup>¼</sup> j j <sup>Ψ</sup> <sup>2</sup> , and the current density *J*,

$$\frac{\partial \rho\_{\rm QM}}{\partial t} = -\nabla \cdot f. \tag{15}$$

where *x* denotes the mean of valuable *x*. From Eq. (17) we can see that the complex probability density is conserved in the complex plane, neither on the real axis nor imaginary axis. **Figure 2(d)** illustrates the good agreement between the solution solved from Eq. (17) (blue dotted line) and the statistical spatial distribution (black solid line) contributed by all points collected by the projections onto the real axis. This result verifies that the analytical solution coheres with the statistical distribution made by CRT. It shows that the same results obtained from two differ-

*(a) The quantum mechanical compatible outcome proposed by point collections of an ensemble of CRTs crossing the real axis for quantum harmonic oscillator in* n ¼ 1 *state with coefficient correlation,* Γ ¼ 0*:*995*. (b) The dismissed nodal condition is given by the same trajectory ensemble but is composed of all projected points onto the real axis. (c) The classical-like probability distribution is presented by collecting all projection points on the real axis for* n ¼ 70 *state with coefficient correlation,* Γ ¼ 0*:*9412*. (d) The analytical solution of the complex probability density function solved from the Fokker-Planck equation shows good agreement with the spatial distribution composed of all projection points on the real axis with coefficient correlation,* Γ ¼ 0*:*9975 *[36].*

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

In quantum mechanics, the quantized orbits of the electron in the hydrogen atom is determined by solving the Schrödinger equation for different eigen states. There is no further description of these orbits, especially no explanation about the force balanced condition under the influence of the Coulomb force. Less study reports the role that the quantum potential plays in atomic analysis. In this section, a quest for describing the hydrogen atom is stretching underlying the quantum potential in complex space. We show our most equations in dimensionless form for

Let us consider the quantum Hamiltonian with Coulomb potential in complex

ent ways stand from the equal footing of the classical concept.

**3. Shell structure in hydrogen atom**

the purposes of simplifying the question.

space [37],

**71**

**Figure 2.**

The probability density function of the CRT interpretation satisfies the Fokker-Planck equation,

$$\frac{\partial \rho(t,\boldsymbol{x})}{\partial t} = -\nabla \cdot \left(\dot{\overline{\boldsymbol{x}}}(t,\boldsymbol{x})\rho(t,\boldsymbol{x})\right) - \frac{i\hbar}{2m}\nabla^2 \rho(t,\boldsymbol{x}),\tag{16}$$

and has the complex value. Multiplying Eq. (16) and its complex conjugate then dividing by 2, we obtain the continuity equation for complex probability density,

$$\frac{\partial \rho(t,\overline{\mathfrak{x}})}{\partial t} = -\nabla \cdot \left(\dot{\overline{\mathfrak{x}}} \rho(t,\overline{\mathfrak{x}})\right),\tag{17}$$

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

#### **Figure 2.**

results under two kinds of point collections. In other words, Bohr's correspondence principle can be interpreted by the CRT interpretation without loss of generality [36]. The second question we try to cope with by means of the CRT interpretation is the conservation of the complex probability. In quantum mechanics, the continuity equation for the probability density function is given by Bohr's law *<sup>ρ</sup>QM* <sup>¼</sup> j j <sup>Ψ</sup> <sup>2</sup>

*100,000 trajectories solved from Eq. (13) with the same initial condition of the Gaussian wave packet in the complex plane: (a) the time evolution on the real axis for which the mean is denoted by the blue line; (b) the time evolution on the imaginary axis with zero mean represented by the blue line. The complex trajectory solved from Eq. (14) with one initial condition: (c) the time evolution on the real axis; (d) the time imaginary part of the motion. This figure reveals that the mean of the CRT is the trajectory solved from the quantum Hamilton*

*<sup>∂</sup><sup>t</sup>* ¼ �<sup>∇</sup> � *<sup>J</sup>:* (15)

*ρ*ð Þ *t*, *x* , (16)

2*m* ∇2

*<sup>∂</sup><sup>t</sup>* ¼ �<sup>∇</sup> � *<sup>x</sup>*\_ *<sup>ρ</sup>*ð Þ *<sup>t</sup>*, *<sup>x</sup>* , (17)

*∂ρQM*

The probability density function of the CRT interpretation satisfies the

\_ð Þ , *<sup>x</sup> <sup>ρ</sup>*ð Þ *<sup>t</sup>*, *<sup>x</sup>* � *<sup>i</sup>*<sup>ℏ</sup>

and has the complex value. Multiplying Eq. (16) and its complex conjugate then dividing by 2, we obtain the continuity equation for complex probability density,

the current density *J*,

*equations of motion [28].*

*Quantum Mechanics*

**Figure 1.**

Fokker-Planck equation,

**70**

*<sup>∂</sup>ρ*ð Þ *<sup>t</sup>*, *<sup>x</sup>*

*<sup>∂</sup><sup>t</sup>* ¼ �<sup>∇</sup> � *x t*

*<sup>∂</sup>ρ*ð Þ *<sup>t</sup>*, *<sup>x</sup>*

, and

*(a) The quantum mechanical compatible outcome proposed by point collections of an ensemble of CRTs crossing the real axis for quantum harmonic oscillator in* n ¼ 1 *state with coefficient correlation,* Γ ¼ 0*:*995*. (b) The dismissed nodal condition is given by the same trajectory ensemble but is composed of all projected points onto the real axis. (c) The classical-like probability distribution is presented by collecting all projection points on the real axis for* n ¼ 70 *state with coefficient correlation,* Γ ¼ 0*:*9412*. (d) The analytical solution of the complex probability density function solved from the Fokker-Planck equation shows good agreement with the spatial distribution composed of all projection points on the real axis with coefficient correlation,* Γ ¼ 0*:*9975 *[36].*

where *x* denotes the mean of valuable *x*. From Eq. (17) we can see that the complex probability density is conserved in the complex plane, neither on the real axis nor imaginary axis. **Figure 2(d)** illustrates the good agreement between the solution solved from Eq. (17) (blue dotted line) and the statistical spatial distribution (black solid line) contributed by all points collected by the projections onto the real axis. This result verifies that the analytical solution coheres with the statistical distribution made by CRT. It shows that the same results obtained from two different ways stand from the equal footing of the classical concept.

#### **3. Shell structure in hydrogen atom**

In quantum mechanics, the quantized orbits of the electron in the hydrogen atom is determined by solving the Schrödinger equation for different eigen states. There is no further description of these orbits, especially no explanation about the force balanced condition under the influence of the Coulomb force. Less study reports the role that the quantum potential plays in atomic analysis. In this section, a quest for describing the hydrogen atom is stretching underlying the quantum potential in complex space. We show our most equations in dimensionless form for the purposes of simplifying the question.

Let us consider the quantum Hamiltonian with Coulomb potential in complex space [37],

$$H = \frac{1}{2m} \left[ \left( \frac{\partial \mathcal{S}}{\partial r} \right)^2 + \frac{\hbar}{i} \left( \frac{2}{r} \frac{\partial \mathcal{S}}{\partial r} + \frac{\partial^2 \mathcal{S}}{\partial r^2} \right) \right] + \frac{1}{2mr^2} \left[ \left( \frac{\partial \mathcal{S}}{\partial \theta} \right)^2 + \frac{\hbar}{i} \left( \cot \theta \frac{\partial \mathcal{S}}{\partial \theta} + \frac{\partial^2 \mathcal{S}}{\partial \theta^2} \right) \right. \tag{18}$$
 
$$+ \frac{1}{\sin^2 \theta} \left( \left( \frac{\partial \mathcal{S}}{\partial \phi} \right)^2 + \frac{\hbar}{i} \frac{\partial^2 \mathcal{S}}{\partial \phi^2} \right) \right] + \frac{-Ze^2}{4\pi \varepsilon\_0 r}, \tag{19}$$

where *S* is the action function. Hamiltonian (18) is state dependent if we apply the simple relation (9) to it. We can therefore have the dimensionless total potential in terms of the wave function,

$$V\_{nlml} = -\frac{2}{r} + \left[\frac{1}{4r^2} \left(4 + \cot^2 \theta\right) - \frac{d^2 \ln R\_{nl}(r)}{dr^2} - \frac{1}{r^2} \frac{d^2 \ln \Theta\_{lm}(\theta)}{d\theta^2}\right],\tag{19}$$

meaning between the classical shell layers and the quantum probability. Furthermore, it may help us to realize the probabilistic electron cloud in a classical

*(a) The variations of three potentials in radial direction for the ground state. (b) The total radial force in the ground state which is composed of the coulomb force and quantum force with zero value at the Bohr radius [37].*

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

Let us consider ð Þ¼ *n*, *l*, *ml* ð Þ 2, 0, 0 state, which has the total potential as

*(a) The shell structure of* ð Þ¼ n, l, ml ð Þ 2, 0, 0 *state in radial direction. (b) The dynamic equilibrium points locate where the total force equals to zero. (c) Electron's motion in* r � *θ plane, and (d) illustrated in the shell*

1 ð Þ 2 � *r*

<sup>2</sup> þ 1

<sup>4</sup>*r*<sup>2</sup> <sup>4</sup> <sup>þ</sup> cot <sup>2</sup> *<sup>θ</sup>* � � " #, (23)

*r* þ

*<sup>V</sup>*<sup>200</sup> <sup>¼</sup> *<sup>V</sup>* <sup>þ</sup> *<sup>Q</sup>* ¼ � <sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

standpoint.

**Figure 4.**

*plane [37].*

**73**

**Figure 3.**

where *n*, *l*, and *ml* denote the principle quantum number, azimuthal quantum number, and magnetic quantum number, respectively. The first term in Eq. (19) is recognized as the Coulomb potential; while the remaining terms are the components of the quantum potential. **Figure 3(a)** illustrates the three potentials varying in radial direction of ð Þ¼ *n*, *l*, *ml* ð Þ 1, 0, 0 state; they are the total potential, Coulomb potential, and quantum potential. The quantum potential yields the opposite spatial distribution to the Coulomb potential, therefore, the total potential performs a neutral situation. When the electron is too close (less than the Bohr radius) to the nucleus, the total potential forms a solid wall that forbids the electron getting closer. The total potential holds an appropriate distribution such that the electron is subject to an attractive force when it is too far away from the nucleus. From the perspective of the electron, it is quantum potential maintains the orbit stable and stop the disaster of crashing on the nucleus.

From Eq. (19) we can obtain the total forces for ð Þ¼ *n*, *l*, *ml* ð Þ 1, 0, 0 state:

$$f\_{100}^r = -\frac{2}{r^2} + \frac{1}{2r^3} \left( 4 + \cot^2 \theta \right), \quad f\_{100}^\theta = \frac{1}{2r^2} \frac{\cos \theta}{\sin^3 \theta}, \quad f\_{100}^\phi = 0. \tag{20}$$

Under a specific condition *f r* <sup>100</sup> ¼ *f θ* <sup>100</sup> ¼ 0, the electron stays stationary at the equilibrium position ð Þ¼ *r*, *θ* ð Þ 1, *π=*2 for which *r* ¼ 1 corresponds to the Bohr radius. The motion of electron at the equilibrium point is determined by

$$f\_{100}^r(r, \pi/2) = f\_Q^r + f\_V^r = \frac{2}{r^3} - \frac{2}{r^2},\tag{21}$$

where the first and the second term represent the repulsive quantum force and the attractive Coulomb force with lower label *Q* and *V*, respectively. As the distance between the electron and the nucleus changes, the two forces take the lead in turn as **Figure 3(b)** illustrates. It is clear to see that the zero force location happens at *r* ¼ 1 (Bohr radius) owing to the force balancing formed by the Coulomb force and quantum force.

In quantum mechanics, the maximum probability of finding the electron is at the Bohr radius according to

$$\frac{d}{dr}P\_{10}(r) = \frac{d}{dr}\left(4\pi r^2 e^{-2r}\right) = 0.\tag{22}$$

The balanced force and the probability are totally different concepts; however, present the same description of the hydrogen atom. This may reflect the equivalent *Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

*<sup>H</sup>* <sup>¼</sup> <sup>1</sup> 2*m*

*Quantum Mechanics*

þ

*∂S ∂r* � �<sup>2</sup>

1 sin <sup>2</sup> *θ*

in terms of the wave function,

*r* þ

disaster of crashing on the nucleus.

*r*<sup>2</sup> þ

Under a specific condition *f*

1

<sup>2</sup>*r*<sup>3</sup> <sup>4</sup> <sup>þ</sup> cot <sup>2</sup>

*f r*

*d*

*dr <sup>P</sup>*10ð Þ¼ *<sup>r</sup>*

*θ* � �, *f*

*r* <sup>100</sup> ¼ *f θ*

<sup>100</sup>ð Þ¼ *<sup>r</sup>*, *<sup>π</sup>=*<sup>2</sup> *<sup>f</sup> <sup>r</sup>*

*f r*

quantum force.

**72**

the Bohr radius according to

<sup>100</sup> ¼ � <sup>2</sup>

*Vnlml* ¼ � <sup>2</sup>

þ ℏ *i* 2 *r ∂S ∂r* þ *∂*2 *S ∂r*2

*∂S ∂ϕ* � �<sup>2</sup>

1

" # � �

þ ℏ *i ∂*2 *S ∂ϕ*<sup>2</sup>

<sup>4</sup>*r*<sup>2</sup> <sup>4</sup> <sup>þ</sup> cot <sup>2</sup>

!#

þ 1 2*mr*<sup>2</sup>

<sup>þ</sup> �*Ze*<sup>2</sup> 4*πϵ*0*r*

where *S* is the action function. Hamiltonian (18) is state dependent if we apply the simple relation (9) to it. We can therefore have the dimensionless total potential

" #

where *n*, *l*, and *ml* denote the principle quantum number, azimuthal quantum number, and magnetic quantum number, respectively. The first term in Eq. (19) is recognized as the Coulomb potential; while the remaining terms are the components of the quantum potential. **Figure 3(a)** illustrates the three potentials varying in radial direction of ð Þ¼ *n*, *l*, *ml* ð Þ 1, 0, 0 state; they are the total potential, Coulomb potential, and quantum potential. The quantum potential yields the opposite spatial distribution to the Coulomb potential, therefore, the total potential performs a neutral situation. When the electron is too close (less than the Bohr radius) to the nucleus, the total potential forms a solid wall that forbids the electron getting closer. The total potential holds an appropriate distribution such that the electron is subject to an attractive force when it is too far away from the nucleus. From the perspective of the electron, it is quantum potential maintains the orbit stable and stop the

From Eq. (19) we can obtain the total forces for ð Þ¼ *n*, *l*, *ml* ð Þ 1, 0, 0 state:

equilibrium position ð Þ¼ *r*, *θ* ð Þ 1, *π=*2 for which *r* ¼ 1 corresponds to the Bohr radius. The motion of electron at the equilibrium point is determined by

*θ* <sup>100</sup> <sup>¼</sup> <sup>1</sup> 2*r*<sup>2</sup>

*<sup>Q</sup>* <sup>þ</sup> *<sup>f</sup> <sup>r</sup>*

where the first and the second term represent the repulsive quantum force and the attractive Coulomb force with lower label *Q* and *V*, respectively. As the distance between the electron and the nucleus changes, the two forces take the lead in turn as **Figure 3(b)** illustrates. It is clear to see that the zero force location happens at *r* ¼ 1 (Bohr radius) owing to the force balancing formed by the Coulomb force and

In quantum mechanics, the maximum probability of finding the electron is at

The balanced force and the probability are totally different concepts; however, present the same description of the hydrogen atom. This may reflect the equivalent

*d dr* <sup>4</sup>*π<sup>r</sup>* 2 *e*

*<sup>V</sup>* <sup>¼</sup> <sup>2</sup> *<sup>r</sup>*<sup>3</sup> � <sup>2</sup>

*<sup>θ</sup>* � � � *<sup>d</sup>*<sup>2</sup> ln *Rnl*ð Þ*<sup>r</sup>*

*∂S ∂θ* � �<sup>2</sup> "

*dr*<sup>2</sup> � <sup>1</sup>

*r*2

cos *θ* sin <sup>3</sup> *θ* , *f ϕ*

<sup>100</sup> ¼ 0, the electron stays stationary at the

�2*<sup>r</sup>* � � <sup>¼</sup> <sup>0</sup>*:* (22)

*d*<sup>2</sup> ln Θ*lml*

*dθ*<sup>2</sup>

ð Þ*θ*

þ ℏ *<sup>i</sup>* cot *<sup>θ</sup>* *∂S ∂θ* þ

, (18)

� �

*∂*2 *S ∂θ*<sup>2</sup>

, (19)

<sup>100</sup> ¼ 0*:* (20)

*<sup>r</sup>*<sup>2</sup> , (21)

*(a) The variations of three potentials in radial direction for the ground state. (b) The total radial force in the ground state which is composed of the coulomb force and quantum force with zero value at the Bohr radius [37].*

meaning between the classical shell layers and the quantum probability. Furthermore, it may help us to realize the probabilistic electron cloud in a classical standpoint.

Let us consider ð Þ¼ *n*, *l*, *ml* ð Þ 2, 0, 0 state, which has the total potential as

$$V\_{200} = V + Q = -\frac{2}{r} + \left[\frac{1}{(2-r)^2} + \frac{1}{4r^2} \left(4 + \cot^2 \theta\right)\right],\tag{23}$$

#### **Figure 4.**

*(a) The shell structure of* ð Þ¼ n, l, ml ð Þ 2, 0, 0 *state in radial direction. (b) The dynamic equilibrium points locate where the total force equals to zero. (c) Electron's motion in* r � *θ plane, and (d) illustrated in the shell plane [37].*

and the force distributions in three directions:

$$f\_{200}^{\prime} = -\frac{2}{r^2} + \left[ -\frac{1}{\left(2 - r\right)^3} + \frac{1}{2r^3} \left(4 + \cot^2 \theta\right) \right] ,\\ f\_{200}^{\theta} = \frac{1}{2r^2} \frac{\cos \theta}{\sin^3 \theta} ,\\ f\_{200}^{\phi} = 0, \quad \text{(24)}$$

which indicates the same equilibrium point location *req*, *<sup>θ</sup>eq* � � <sup>¼</sup> <sup>3</sup> � ffiffi <sup>5</sup> <sup>p</sup> , *<sup>π</sup>=*<sup>2</sup> � � given by the equations of motion from Eq. (14):

$$\frac{dr}{dt} = 4i \frac{r^2 - \mathfrak{G}r + 4}{r(r - 2)},\\\frac{d(\cos \theta)}{dt} = i \frac{\cos \theta}{r^2},\\\frac{d\phi}{dt} = \mathbf{0},\tag{25}$$

where *<sup>m</sup>*<sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*067*me* is the effective mass of the electron, and *<sup>E</sup>* is the total energy of the incident electron. The general solution of Eq. (26) has the form as

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

�*iknx* � �*ϕn*ð Þ*<sup>y</sup>* , *<sup>ϕ</sup>n*ð Þ¼ *<sup>y</sup>* sin *<sup>n</sup><sup>π</sup>*

where *N* is the number of mode, *w* is the width of the channel, and *kn* is the

direction due to the presence of the infinite square well. From Eq. (28), we have the

The function *Bneiknx* <sup>þ</sup> *Cne*�*iknx* in Eq. (27) is the free-particle wave function in the *x* direction, and *ϕn*ð Þ*y* is an eigen function for the infinite well in the *y* direction satisfying the boundary condition *ϕn*ð Þ*y* ð Þ¼ *w=*2 *ϕn*ðÞ� *y* ð Þ¼ *w=*2 0. The coefficients *Bn* and *Cn* are uniquely determined by the incident energy *E* and incident angle *ϕ* . (More detail refers to [38].) The quantum potential in the channel can be obtained by combing Eqs. (8), (10) and the wave function (27) (in dimensionless form),

> *∂*2 *∂x*<sup>2</sup> þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*m*<sup>∗</sup> ð Þ *<sup>E</sup>* � *En <sup>=</sup>*ℏ<sup>2</sup>

*∂*2 *∂y*<sup>2</sup> � � ln *<sup>ψ</sup><sup>C</sup>*

The quantum potential provides fully information of electron's motion, its characteristic of inverse proportional to the probability density displays more knowl-

> *∂ψ<sup>C</sup> k ∂x* � �<sup>2</sup>

which represents that the high quantum potential region corresponds to the low probability of electrons passing through as **Figure 6** displays; and **Figure 7** illustrates how the quantum potential gradually form the quantized channels as the incident angle increases, which shows the state dependent characteristic of the

The other quantum feature originating from the quantum potential is the quantization of conductance in the channel as **Figure 8** presents. We will show that the high conductance region is where the most electrons gather. To simplify the system, we firstly replace the motion in 2D channel by a motion in 1D square barriers [39].

þ

*∂ψ<sup>C</sup> k ∂y*

� �<sup>2</sup> " #, (31)

<sup>ℏ</sup><sup>2</sup> ð Þ *<sup>E</sup>* � *Vn <sup>ψ</sup>n*ð Þ¼ *<sup>x</sup>* 0, (32)

*Ex* <sup>þ</sup> *Ey* <sup>¼</sup> ð Þ *kn*<sup>ℏ</sup> <sup>2</sup>

*<sup>w</sup> <sup>y</sup>* <sup>þ</sup>

<sup>2</sup>*m*<sup>∗</sup> <sup>þ</sup> *En* <sup>¼</sup> *<sup>E</sup>*, (28)

*<sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> ð Þ is the free particle energy in the *<sup>x</sup>*

*<sup>π</sup>*2*<sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> *<sup>w</sup>*<sup>2</sup> ð Þ, *<sup>n</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, is quantized energy in the *<sup>y</sup>*

*w* 2 h i � � , (27)

*:* (29)

*<sup>k</sup>* ð Þ *x*, *y :* (30)

*ψ<sup>C</sup>*

*<sup>k</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* <sup>X</sup>

in which *Ex* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup>

wave number read

quantum potential.

Schrödinger equation,

**75**

direction, and *Ey* <sup>¼</sup> *En* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*ℏ*<sup>2</sup>

*N*

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

*Bne*

*iknx* <sup>þ</sup> *Cne*

wave number which satisfies the energy conservation law:

*<sup>x</sup><sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> ð Þ¼ ð Þ *kn<sup>ℏ</sup>* <sup>2</sup>

*kn* ¼

*Q x*ð Þ¼� , *y*

edge in the channel. The inverse proportional relation reads

j j *Q x*ð Þ , *<sup>y</sup>* <sup>¼</sup> <sup>1</sup>

*d*2 *ψn*ð Þ *x dx*<sup>2</sup> þ

*P x*ð Þ , *y*

Therefore, we consider the wave function *ψn*ð Þ *x* satisfying the following

2*m*<sup>∗</sup>

q

*n*¼1

under the zero resultant force condition and the electron dynamic equilibrium condition. **Figure 4(a)** presents the shell structures in radial direction according to Eq. (24). The range of the layers are constrained by the total potential and divided into two different parts. The two equilibrium points individually correspond to the zero force locations in the two shells as **Figure 4(b)** indicates. Eq. (25) offers how electron move in this state. **Figure 4(c)** illustrates electron's trajectory in the *r* � *θ* plane; while **Figure 4(d)** embodies trajectory in the shell structure.

#### **4. Channelized quantum potential and conductance quantization in 2D Nano-channels**

The practical technology usage of the proposed formalism is applied to 2D Nanochannels in this section. Instead of the probability density function offered by the conventional quantum mechanics, we stay in line with causalism to perceive what role played by the quantum potential. Consider a 2D straight channel made by GaAs-GaAlAs and is surrounded by infinite potential barrier except the two reservoirs and the channel. The schematic plot of the channel refers to **Figure 5**. The dynamic evolution of the wave function *ψ*ð Þ *x*, *y* in the channel is described by the Schrödinger equation,

$$-\frac{\hbar^2}{2m^\*}\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2}\right)\psi(\mathbf{x}, \mathbf{y}) = E\psi(\mathbf{x}, \mathbf{y}),\tag{26}$$

#### **Figure 5.**

*(a) A single quantum wire and an expanded view showing schematically the single degree of freedom in the x direction. (b) 2D straight channel made up of quantum wire with length* 2d *and width* w *connects the left reservoir to the right reservoir.*

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

where *<sup>m</sup>*<sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*067*me* is the effective mass of the electron, and *<sup>E</sup>* is the total energy of the incident electron. The general solution of Eq. (26) has the form as

$$\left(\varphi\_k^C(x,y) = \sum\_{n=1}^N \left(B\_n e^{ik\_n x} + C\_n e^{-ik\_n x}\right) \phi\_n(y), \phi\_n(y) = \sin\left[\frac{n\pi}{\omega}\left(y + \frac{\omega}{2}\right)\right],\tag{27}$$

where *N* is the number of mode, *w* is the width of the channel, and *kn* is the wave number which satisfies the energy conservation law:

$$E\_{\chi} + E\_{\chi} = \frac{\left(k\_n \hbar\right)^2}{2m^\*} + E\_n = E,\tag{28}$$

in which *Ex* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> *<sup>x</sup><sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> ð Þ¼ ð Þ *kn<sup>ℏ</sup>* <sup>2</sup> *<sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> ð Þ is the free particle energy in the *<sup>x</sup>* direction, and *Ey* <sup>¼</sup> *En* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*ℏ*<sup>2</sup> *<sup>π</sup>*2*<sup>=</sup>* <sup>2</sup>*m*<sup>∗</sup> *<sup>w</sup>*<sup>2</sup> ð Þ, *<sup>n</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, is quantized energy in the *<sup>y</sup>* direction due to the presence of the infinite square well. From Eq. (28), we have the wave number read

$$k\_n = \sqrt{2m^\* \left(E - E\_n\right)/\hbar^2}.\tag{29}$$

The function *Bneiknx* <sup>þ</sup> *Cne*�*iknx* in Eq. (27) is the free-particle wave function in the *x* direction, and *ϕn*ð Þ*y* is an eigen function for the infinite well in the *y* direction satisfying the boundary condition *ϕn*ð Þ*y* ð Þ¼ *w=*2 *ϕn*ðÞ� *y* ð Þ¼ *w=*2 0. The coefficients *Bn* and *Cn* are uniquely determined by the incident energy *E* and incident angle *ϕ* . (More detail refers to [38].) The quantum potential in the channel can be obtained by combing Eqs. (8), (10) and the wave function (27) (in dimensionless form),

$$Q(\mathbf{x}, \mathbf{y}) = -\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2}\right) \ln \boldsymbol{\upmu}\_k^C(\mathbf{x}, \mathbf{y}). \tag{30}$$

The quantum potential provides fully information of electron's motion, its characteristic of inverse proportional to the probability density displays more knowledge in the channel. The inverse proportional relation reads

$$|Q(\mathbf{x}, \boldsymbol{\upchi})| = \frac{\mathbf{1}}{P(\mathbf{x}, \boldsymbol{\upchi})} \left[ \left( \frac{\partial \boldsymbol{\upmu}\_k^C}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \boldsymbol{\upmu}\_k^C}{\partial \boldsymbol{\upmu}} \right)^2 \right],\tag{31}$$

which represents that the high quantum potential region corresponds to the low probability of electrons passing through as **Figure 6** displays; and **Figure 7** illustrates how the quantum potential gradually form the quantized channels as the incident angle increases, which shows the state dependent characteristic of the quantum potential.

The other quantum feature originating from the quantum potential is the quantization of conductance in the channel as **Figure 8** presents. We will show that the high conductance region is where the most electrons gather. To simplify the system, we firstly replace the motion in 2D channel by a motion in 1D square barriers [39]. Therefore, we consider the wave function *ψn*ð Þ *x* satisfying the following Schrödinger equation,

$$\frac{d^2\psi\_n(\mathbf{x})}{d\mathbf{x}^2} + \frac{2m^\*}{\hbar^2}(E - V\_n)\psi\_n(\mathbf{x}) = \mathbf{0},\tag{32}$$

and the force distributions in three directions:

1

*<sup>θ</sup>* � � " #

which indicates the same equilibrium point location *req*, *θeq*

plane; while **Figure 4(d)** embodies trajectory in the shell structure.

*<sup>r</sup>*<sup>2</sup> � <sup>6</sup>*<sup>r</sup>* <sup>þ</sup> <sup>4</sup> *r r*ð Þ � <sup>2</sup> ,

<sup>2</sup>*r*<sup>3</sup> <sup>4</sup> <sup>þ</sup> cot <sup>2</sup>

, *f θ* <sup>200</sup> <sup>¼</sup> <sup>1</sup> 2*r*<sup>2</sup>

> cos *θ <sup>r</sup>*<sup>2</sup> ,

*dϕ*

*ψ*ð Þ¼ *x*, *y Eψ*ð Þ *x*, *y* , (26)

*d*ð Þ cos *θ dt* <sup>¼</sup> *<sup>i</sup>*

under the zero resultant force condition and the electron dynamic equilibrium condition. **Figure 4(a)** presents the shell structures in radial direction according to Eq. (24). The range of the layers are constrained by the total potential and divided into two different parts. The two equilibrium points individually correspond to the zero force locations in the two shells as **Figure 4(b)** indicates. Eq. (25) offers how electron move in this state. **Figure 4(c)** illustrates electron's trajectory in the *r* � *θ*

**4. Channelized quantum potential and conductance quantization in 2D**

The practical technology usage of the proposed formalism is applied to 2D Nanochannels in this section. Instead of the probability density function offered by the conventional quantum mechanics, we stay in line with causalism to perceive what role played by the quantum potential. Consider a 2D straight channel made by GaAs-GaAlAs and is surrounded by infinite potential barrier except the two reservoirs and the channel. The schematic plot of the channel refers to **Figure 5**. The dynamic evolution of the wave function *ψ*ð Þ *x*, *y* in the channel is described by the

cos *θ* sin <sup>3</sup> *θ* , *f ϕ*

� � <sup>¼</sup> <sup>3</sup> � ffiffi

*dt* <sup>¼</sup> 0, (25)

<sup>200</sup> ¼ 0, (24)

<sup>5</sup> <sup>p</sup> , *<sup>π</sup>=*<sup>2</sup> � �

1 ð Þ <sup>2</sup> � *<sup>r</sup>* <sup>3</sup> <sup>þ</sup>

given by the equations of motion from Eq. (14):

*dr dt* <sup>¼</sup> <sup>4</sup>*<sup>i</sup>*

*r*<sup>2</sup> þ �

**Nano-channels**

Schrödinger equation,

**Figure 5.**

**74**

*reservoir to the right reservoir.*

� <sup>ℏ</sup><sup>2</sup> 2*m*<sup>∗</sup>

*∂*2 *∂x*<sup>2</sup> þ

*∂*2 *∂y*<sup>2</sup> � �

*(a) A single quantum wire and an expanded view showing schematically the single degree of freedom in the x direction. (b) 2D straight channel made up of quantum wire with length* 2d *and width* w *connects the left*

*f r* <sup>200</sup> ¼ � <sup>2</sup>

*Quantum Mechanics*

#### **Figure 6.**

*The incident energy* <sup>E</sup> <sup>¼</sup> <sup>11</sup> *and the incident angle <sup>ϕ</sup>* <sup>¼</sup> <sup>40</sup>° *for: (a) the probability density function; (b) the corresponding quantum potential of the cross-section in the channel. The bright regions of the quantum potential in (b) represent the lower potential barriers which are in accord with the bright regions in (a) where are the locations with higher probability of finding electrons [38].*

where *Vn* is the equivalent square barrier,

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

*Vn* ¼

*Tn*ð Þ¼ *ξ* 1 þ

where *<sup>ξ</sup>* <sup>¼</sup> ffiffiffi

**Figure 8.**

<sup>p</sup> *increases [39].*

*<sup>ξ</sup>* <sup>¼</sup> ffiffiffi E

*T*ð Þ *<sup>N</sup>*

*<sup>N</sup>* <sup>¼</sup> 2 as an example, *<sup>T</sup>*ð Þ *<sup>N</sup>*

**77**

*E*

*Total*ð Þ¼ <sup>ξ</sup> <sup>X</sup>

*N*

*n*¼1

2 4

*Tn*ð Þ¼ *<sup>ξ</sup>* <sup>X</sup>

*T*ð Þ<sup>2</sup> *Total*ð Þξ ≈

*N*

2 4

*n*¼1

*n*<sup>2</sup>ℏ<sup>2</sup> *π*2 <sup>2</sup>*m*<sup>∗</sup> *<sup>w</sup>*<sup>2</sup> , j j *<sup>x</sup>* <sup>≤</sup> *<sup>d</sup>* 0, j j *x* > *d*

*The conductance G of a narrow channel shows plateaus at integer multiples of* 2e<sup>2</sup>*=*h *as the electron's energy*

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

Please notice that potential *Vn* depends on the eigen state, hence, the electron will encounter different heights of the potential barrier in different eigen states. Furthermore, it makes electron with different energy either transmitting or going through the barrier by tunneling. When electrons transmit the channel, the con-

*n*<sup>4</sup> sin <sup>2</sup> *πd* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>4</sup>*ξ*<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> � *<sup>n</sup>*<sup>2</sup> � �

<sup>p</sup> , *<sup>d</sup>* <sup>¼</sup> <sup>2</sup>*d=<sup>w</sup>* is the aspect ratio of the channel. To display the

quantization of the conductance, we conduct a combination consisting of all transmission coefficients which represents all electrons transmitting through all potential barriers. This combination is expressed in terms of the total transmission coefficients,

1 þ

8 ><

**Figure 9** illustrates the quantization of the total transmission coefficient. Take

*Total*ð Þξ is composed of *T*1ð Þ*ξ* and*T*2ð Þ*ξ* :

0, *ξ*<1 1, 1≤*ξ*<2 2, *ξ*≥2

*<sup>ξ</sup>*<sup>2</sup> � *<sup>n</sup>*<sup>2</sup> � � p

*n*<sup>4</sup> sin <sup>2</sup> *πd* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>4</sup>*ξ*<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> � *<sup>n</sup>*<sup>2</sup> � �

: *:* (33)

3 5

*<sup>ξ</sup>*<sup>2</sup> � *<sup>n</sup>*<sup>2</sup> � � p

>: , (36)

3 5

�1

�1

, (34)

*:* (35)

8 <

ductance will be changed and is expected to have the quantized value. Let us express the transmission coefficient in dimensionless form as

#### **Figure 7.**

*The variation of the quantum potential with respect to the incident angle ϕ for a fixed incident energy* E ¼ 11*. It is seen that the channelized structure becomes more and more apparent with the increasing incident angle ϕ [38].*

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

#### **Figure 8.**

**Figure 6.**

*Quantum Mechanics*

**Figure 7.**

**76**

*angle ϕ [38].*

*locations with higher probability of finding electrons [38].*

*The incident energy* <sup>E</sup> <sup>¼</sup> <sup>11</sup> *and the incident angle <sup>ϕ</sup>* <sup>¼</sup> <sup>40</sup>° *for: (a) the probability density function; (b) the corresponding quantum potential of the cross-section in the channel. The bright regions of the quantum potential in (b) represent the lower potential barriers which are in accord with the bright regions in (a) where are the*

*The variation of the quantum potential with respect to the incident angle ϕ for a fixed incident energy* E ¼ 11*. It is seen that the channelized structure becomes more and more apparent with the increasing incident*

*The conductance G of a narrow channel shows plateaus at integer multiples of* 2e<sup>2</sup>*=*h *as the electron's energy <sup>ξ</sup>* <sup>¼</sup> ffiffiffi E <sup>p</sup> *increases [39].*

where *Vn* is the equivalent square barrier,

$$V\_n = \begin{cases} \frac{n^2 \hbar^2 \pi^2}{2m^\* w^2}, & |x| \le d \\ 0, & |x| > d \end{cases} \tag{33}$$

Please notice that potential *Vn* depends on the eigen state, hence, the electron will encounter different heights of the potential barrier in different eigen states. Furthermore, it makes electron with different energy either transmitting or going through the barrier by tunneling. When electrons transmit the channel, the conductance will be changed and is expected to have the quantized value.

Let us express the transmission coefficient in dimensionless form as

$$T\_n(\xi) = \left[ 1 + \frac{n^4 \sin^2 \left( \pi \overline{d} \sqrt{\xi^2 - n^2} \right)}{4 \xi^2 \left( \xi^2 - n^2 \right)} \right]^{-1},\tag{34}$$

where *<sup>ξ</sup>* <sup>¼</sup> ffiffiffi *E* <sup>p</sup> , *<sup>d</sup>* <sup>¼</sup> <sup>2</sup>*d=<sup>w</sup>* is the aspect ratio of the channel. To display the quantization of the conductance, we conduct a combination consisting of all transmission coefficients which represents all electrons transmitting through all potential barriers. This combination is expressed in terms of the total transmission coefficients,

$$T\_{Total}^{(N)}(\xi) = \sum\_{n=1}^{N} T\_n(\xi) = \sum\_{n=1}^{N} \left[ 1 + \frac{n^4 \sin^2 \left( \pi \overline{d} \sqrt{\xi^2 - n^2} \right)}{4 \xi^2 \left( \xi^2 - n^2 \right)} \right]^{-1}. \tag{35}$$

**Figure 9** illustrates the quantization of the total transmission coefficient. Take *<sup>N</sup>* <sup>¼</sup> 2 as an example, *<sup>T</sup>*ð Þ *<sup>N</sup> Total*ð Þξ is composed of *T*1ð Þ*ξ* and*T*2ð Þ*ξ* :

$$T\_{Total}^{(2)}(\xi) \approx \begin{cases} 0, & \xi < 1 \\ 1, & 1 \le \xi < 2 \\ 2, & \xi \ge 2 \end{cases},\tag{36}$$

naturally arises in the process of finding the minimum cost function. From the view point of the space geometry, the quantum potential exposits the geometric variation for the particle to lead its motion. This is what makes the quantum world quite different to the classical world as many quantum phenomena reveal. The quantum potential is so charming and plays the most important part that bridges the gap

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

Probability is a prescription to deal with the empirical data not to represent the essence of nature in such a small scale. We have demonstrated how to emerge the trajectory from the probability by expanding the dimensions to complex space. As meanwhile, we have pointed out how to reach the classical limit with increasing quantum numbers from the same ensemble of trajectories by adopting different statistical collection method. Take the advantage of the quantum potential, we are allowed to explain the force balanced condition in the hydrogen atom, moreover, we illustrate the formation of the shell structures which cohere with the shape of the electron clouds. The channels in 2D Nano-structure are shown to be related to the quantum potential and so does the conductance. We confirm that the quantized conductance is originated from the electron's transmission behavior. The ontology renders the reality of the identity to the quantum object. It cannot be done without the complex space structure. Complex space is essential for the quantum world and becomes the most crucial part of solving the quantum puzzle. It may proper to say that the causality returns to the quantum world and throughout the whole universe.

between the quantum and classical world.

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

**Author details**

**79**

Ciann-Dong Yang<sup>1</sup> and Shiang-Yi Han2

Taiwan (R.O.C.), Republic of China

provided the original work is properly cited.

Tainan, Taiwan (R.O.C.), Republic of China

\*Address all correspondence to: syhan.taiwan@gmail.com

\*

1 Department of Aeronautics and Astronautics, National Cheng Kung University,

2 Department of Applied Physics, National University of Kaohsiung, Kaohsiung,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Figure 9.**

*The total transmission coefficients* Tð Þ <sup>N</sup> Totalð Þξ *display the step shape with the increasing of incident energy ξ for* N ¼ 1, 2, 3, 4 *with* d ¼ 10*. [39].*

where we have ignored the rapid oscillations parts in the transmission coefficient (more detail refers to [39]). Eq. (36) shows the step structure illustrated in **Figure 9**, which has the same steps shape of the conductance shown in **Figure 8**. We have demonstrated that the total transmission coefficient is proportional to the total number of electrons passing the channel and it is relevant to the conductance in the channel.

#### **5. Concluding remarks**

Looking for the unifying theory of quantum and classical mechanics lasts for decades. Several approaches have been proposed, they share some viewpoints and contributions. We have learned that the quantum potential plays a switch role between the quantum and classical world. When the mass is getting larger and larger, the quantum potential will become smaller and smaller, and eventually becomes ignorable. Causality exists everywhere in the universe but hides itself in the microscopic world. What makes physicists miss the link that connects the two scale worlds is the statistical expression of the quantum world. It is impossible to extract the fundamental law from the probability interpretation. As the higher dimension is demanded, there are more evidences of causality emerging from the backbone of quantum mechanics. The complex weak measurement proposes the solid evidence of the complex space structure nature of the quantum world, and evokes the ontology return to the quantum kingdom. All quantum motions happen in complex space. All we can observe is a part of the whole appearance.

In Bohmian mechanics, the quantum potential is a product given by the transformation process which starts from the Schrödinger equation to the quantum HJ equation. In optimal guidance quantum motion formulation, the quantum potential

#### *Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

naturally arises in the process of finding the minimum cost function. From the view point of the space geometry, the quantum potential exposits the geometric variation for the particle to lead its motion. This is what makes the quantum world quite different to the classical world as many quantum phenomena reveal. The quantum potential is so charming and plays the most important part that bridges the gap between the quantum and classical world.

Probability is a prescription to deal with the empirical data not to represent the essence of nature in such a small scale. We have demonstrated how to emerge the trajectory from the probability by expanding the dimensions to complex space. As meanwhile, we have pointed out how to reach the classical limit with increasing quantum numbers from the same ensemble of trajectories by adopting different statistical collection method. Take the advantage of the quantum potential, we are allowed to explain the force balanced condition in the hydrogen atom, moreover, we illustrate the formation of the shell structures which cohere with the shape of the electron clouds. The channels in 2D Nano-structure are shown to be related to the quantum potential and so does the conductance. We confirm that the quantized conductance is originated from the electron's transmission behavior. The ontology renders the reality of the identity to the quantum object. It cannot be done without the complex space structure. Complex space is essential for the quantum world and becomes the most crucial part of solving the quantum puzzle. It may proper to say that the causality returns to the quantum world and throughout the whole universe.

#### **Author details**

where we have ignored the rapid oscillations parts in the transmission coefficient (more detail refers to [39]). Eq. (36) shows the step structure illustrated in **Figure 9**, which has the same steps shape of the conductance shown in **Figure 8**. We have demonstrated that the total transmission coefficient is proportional to the total number of electrons passing the channel and it is relevant to the conductance

Totalð Þξ *display the step shape with the increasing of incident energy ξ for*

Looking for the unifying theory of quantum and classical mechanics lasts for decades. Several approaches have been proposed, they share some viewpoints and contributions. We have learned that the quantum potential plays a switch role between the quantum and classical world. When the mass is getting larger and larger, the quantum potential will become smaller and smaller, and eventually becomes ignorable. Causality exists everywhere in the universe but hides itself in the microscopic world. What makes physicists miss the link that connects the two scale worlds is the statistical expression of the quantum world. It is impossible to extract the fundamental law from the probability interpretation. As the higher dimension is demanded, there are more evidences of causality emerging from the backbone of quantum mechanics. The complex weak measurement proposes the solid evidence of the complex space structure nature of the quantum world, and evokes the ontology return to the quantum kingdom. All quantum motions happen

in complex space. All we can observe is a part of the whole appearance.

In Bohmian mechanics, the quantum potential is a product given by the transformation process which starts from the Schrödinger equation to the quantum HJ equation. In optimal guidance quantum motion formulation, the quantum potential

in the channel.

**78**

**Figure 9.**

**5. Concluding remarks**

*The total transmission coefficients* Tð Þ <sup>N</sup>

N ¼ 1, 2, 3, 4 *with* d ¼ 10*. [39].*

*Quantum Mechanics*

Ciann-Dong Yang<sup>1</sup> and Shiang-Yi Han2 \*

1 Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan (R.O.C.), Republic of China

2 Department of Applied Physics, National University of Kaohsiung, Kaohsiung, Taiwan (R.O.C.), Republic of China

\*Address all correspondence to: syhan.taiwan@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Einstein A. The motion of elements suspended in static liquids as claimed in the molecular kinetic theory of heat. Annals of Physics. 1905;**17**:549-560

[2] Holmes S. The Six Thatchers. Season 4 Episode 1. Available from: https:// scatteredquotes.com/call-premonitionjust-movement-web/

[3] Kocsis S et al. Observing the average trajectories of single photons in a twoslit interferometer. Science. 2011;**332**: 1170-1173. DOI: 10.1126/science. 1202218

[4] Murch KW, Weber SJ, Macklin C, Siddiqi I. Observing single quantum trajectories of a superconducting quantum bit. Nature. 2013;**502**:211-214

[5] Rossi M, Mason D, Chen J, Schliesser A. Observing and verifying the quantum trajectory of a mechanical resonator. Physical Review Letters. 2019;**123**:163601

[6] Zhou ZQ, Liu X, et al. Experimental observation of anomalous trajectories of single photons. Physical Review A. 2017; **95**:042121

[7] Aharonov Y, Albert DZ, Vaidman L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Physical Review Letters. 1988;**60**:1351-1354

[8] Aharonov Y, Botero A. Quantum averages of weak values. Physical Review A. 2005;**72**:052111

[9] Mori T, Tsutsui I. Quantum trajectories based on the weak value. Progress of Theoretical and Experimental Physics. 2015;**2015**:043A01

[10] Shudo A, Ikeda KS. Complex classical trajectories and chaotic tunneling. Physical Review Letters. 1995;**74**:682-685

[11] Yang CD. Wave-particle duality in complex space. Annals of Physics. 2005; **319**:444-470

quantum theories using a metamaterial. Nature Communications. 2017;**8**:15044

*DOI: http://dx.doi.org/10.5772/intechopen.91669*

[31] de Castro AS, de Dutra AS. On the quantum Hamilton-Jacobi formalism. Foundations of Physics. 1991;**21**:649-663

[32] John MV. Modified de Broglie-Bohm approach to quantum mechanics. Foundation of Physics Letters. 2002;**15**:

[33] Yang CD. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. Annals of Physics. 2006;**321**:

[34] Gondran M. Numerical simulation of the double slit interference with ultracold atom. American Journal of

[35] Floyd ER. Trajectory representation of a quantum Young's diffraction experiment. Foundations of Physics.

[36] Yang CD, Han SY. Trajectory interpretation of correspondence principle: Solution of nodal issue. 2019. Available from: https://arxiv.org/abs/

[37] Yang CD. Quantum dynamics of hydrogen atom in complex space. Annals of Physics. 2005;**319**:399-443

[38] Yang CD, Lee CB. Nonlinear quantum motions in 2D nano-channels part I: Complex potential and quantum trajectories. International Journal of Nonlinear Sciences and Numerical Simulation. 2010;**11**:297-318

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2010;**11**:319-336

Physics. 2005;**73**:507-515

2007;**37**:1403-1420

1911.04747

329-343

*Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics*

2876-2926

[21] Davidson M. Bohmian trajectories for Kerr-Newman particles in complex space-time. Foundations of Physics.

[22] Dressel J, Jordan AN. Significance of the imaginary part of the weak value. Physical Review A. 2012;**85**:012107

[23] de Aguiar MAM, Vitiello SA, Grigolo A. An initial value

representation for the coherent state propagator with complex trajectories. Chemical Physics. 2010;**370**:42-50

[24] Kedem Y. Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values. Physical Review A. 2012;**85**:

[25] Petersen J, Kay KG. Wave packet propagation across barriers by

semicalssical initial value methods. The Journal of Chemical Physics. 2015;**143**:

[27] Wang MS. Stochastic interpretation of quantum mechanics in complex space. Physical Review Letters. 1997;**79**:

[29] Bohm D. A suggested interpretation of quantum theory in terms of 'hidden' variables, I and II. Physical Review.

[30] Leacock RA, Padgett MJ. Hamilton-Jacobi theory and the quantum action variable. Physical Review Letters. 1983;

[26] Rosenbrock HH. A stochastic variational treatment of quantum mechanics. Proceedings: Mathematical and Physical Sciences. 1995;**450**:417-437

[28] Yang CD, Cheng LL. Optimal guidance law in quantum mechanics. Annals of Physics. 2013;**338**:167-185

2018;**11**:1590-1616

060102

014107

3319-3322

1952;**85**:166-193

**50**:3-6

**81**

[12] Goldfarb Y, Degani I, Tannor DJ. Bohmian mechanics with complex action: A new trajectory-based formulation of quantum mechanics. The Journal of Chemical Physics. 2006;**125**: 231103

[13] Chou CC, Wyatt RE. Quantum trajectories in complex space: Onedimensional stationary scattering problems. The Journal of Chemical Physics. 2008;**128**:154106

[14] John MV. Probability and complex quantum trajectories. Annals of Physics. 2008;**324**:220-231

[15] Sanz AS, Miret-Artes. Interplay of causticity and verticality within the complex quantum Hamilton-Jacobi formalism. Chemical Physics Letters. 2008;**458**:239-243

[16] Poirier B, Tannor D. An action principle for complex quantum trajectories. International Journal at the Interface Between Chemistry and Physics. 2012;**110**:897-908. DOI: 10.1080/00268976.2012.681811

[17] Dey S, Fring A. Bohm quantum trajectories from coherent states. Physical Review A. 2013;**88**:022116

[18] Yang CD, Su KC. Reconstructing interference fringes in slit experiments by complex quantum trajectories. International Journal of Quantum Chemistry. 2013;**113**:1253-1263

[19] Mahler DH, Rozema L, et al. Experimental nonlocal and surreal Bohmian trajectories. Science Advances. 2016;**2**:1501466

[20] Procopio LM, Rozema LA, et al. Single-photon test of hyper-complex *Complex Space Nature of the Quantum World: Return Causality to Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.91669*

quantum theories using a metamaterial. Nature Communications. 2017;**8**:15044

**References**

*Quantum Mechanics*

just-movement-web/

1202218

[1] Einstein A. The motion of elements suspended in static liquids as claimed in the molecular kinetic theory of heat. Annals of Physics. 1905;**17**:549-560

[11] Yang CD. Wave-particle duality in complex space. Annals of Physics. 2005;

[12] Goldfarb Y, Degani I, Tannor DJ. Bohmian mechanics with complex action: A new trajectory-based

formulation of quantum mechanics. The Journal of Chemical Physics. 2006;**125**:

[13] Chou CC, Wyatt RE. Quantum trajectories in complex space: Onedimensional stationary scattering problems. The Journal of Chemical

[14] John MV. Probability and complex quantum trajectories. Annals of Physics.

[15] Sanz AS, Miret-Artes. Interplay of causticity and verticality within the complex quantum Hamilton-Jacobi formalism. Chemical Physics Letters.

[16] Poirier B, Tannor D. An action principle for complex quantum

[17] Dey S, Fring A. Bohm quantum trajectories from coherent states. Physical Review A. 2013;**88**:022116

[18] Yang CD, Su KC. Reconstructing interference fringes in slit experiments by complex quantum trajectories. International Journal of Quantum Chemistry. 2013;**113**:1253-1263

[19] Mahler DH, Rozema L, et al. Experimental nonlocal and surreal Bohmian trajectories. Science Advances.

[20] Procopio LM, Rozema LA, et al. Single-photon test of hyper-complex

2016;**2**:1501466

trajectories. International Journal at the Interface Between Chemistry and Physics. 2012;**110**:897-908. DOI: 10.1080/00268976.2012.681811

Physics. 2008;**128**:154106

2008;**324**:220-231

2008;**458**:239-243

**319**:444-470

231103

[2] Holmes S. The Six Thatchers. Season 4 Episode 1. Available from: https:// scatteredquotes.com/call-premonition-

[3] Kocsis S et al. Observing the average trajectories of single photons in a twoslit interferometer. Science. 2011;**332**: 1170-1173. DOI: 10.1126/science.

[4] Murch KW, Weber SJ, Macklin C, Siddiqi I. Observing single quantum trajectories of a superconducting quantum bit. Nature. 2013;**502**:211-214

Schliesser A. Observing and verifying the quantum trajectory of a mechanical resonator. Physical Review Letters.

[6] Zhou ZQ, Liu X, et al. Experimental observation of anomalous trajectories of single photons. Physical Review A. 2017;

[7] Aharonov Y, Albert DZ, Vaidman L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Physical Review Letters. 1988;**60**:1351-1354

[8] Aharonov Y, Botero A. Quantum averages of weak values. Physical Review A. 2005;**72**:052111

[9] Mori T, Tsutsui I. Quantum trajectories based on the weak value. Progress of Theoretical and Experimental

[10] Shudo A, Ikeda KS. Complex classical trajectories and chaotic tunneling. Physical Review Letters.

Physics. 2015;**2015**:043A01

1995;**74**:682-685

**80**

[5] Rossi M, Mason D, Chen J,

2019;**123**:163601

**95**:042121

[21] Davidson M. Bohmian trajectories for Kerr-Newman particles in complex space-time. Foundations of Physics. 2018;**11**:1590-1616

[22] Dressel J, Jordan AN. Significance of the imaginary part of the weak value. Physical Review A. 2012;**85**:012107

[23] de Aguiar MAM, Vitiello SA, Grigolo A. An initial value representation for the coherent state propagator with complex trajectories. Chemical Physics. 2010;**370**:42-50

[24] Kedem Y. Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values. Physical Review A. 2012;**85**: 060102

[25] Petersen J, Kay KG. Wave packet propagation across barriers by semicalssical initial value methods. The Journal of Chemical Physics. 2015;**143**: 014107

[26] Rosenbrock HH. A stochastic variational treatment of quantum mechanics. Proceedings: Mathematical and Physical Sciences. 1995;**450**:417-437

[27] Wang MS. Stochastic interpretation of quantum mechanics in complex space. Physical Review Letters. 1997;**79**: 3319-3322

[28] Yang CD, Cheng LL. Optimal guidance law in quantum mechanics. Annals of Physics. 2013;**338**:167-185

[29] Bohm D. A suggested interpretation of quantum theory in terms of 'hidden' variables, I and II. Physical Review. 1952;**85**:166-193

[30] Leacock RA, Padgett MJ. Hamilton-Jacobi theory and the quantum action variable. Physical Review Letters. 1983; **50**:3-6

[31] de Castro AS, de Dutra AS. On the quantum Hamilton-Jacobi formalism. Foundations of Physics. 1991;**21**:649-663

[32] John MV. Modified de Broglie-Bohm approach to quantum mechanics. Foundation of Physics Letters. 2002;**15**: 329-343

[33] Yang CD. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. Annals of Physics. 2006;**321**: 2876-2926

[34] Gondran M. Numerical simulation of the double slit interference with ultracold atom. American Journal of Physics. 2005;**73**:507-515

[35] Floyd ER. Trajectory representation of a quantum Young's diffraction experiment. Foundations of Physics. 2007;**37**:1403-1420

[36] Yang CD, Han SY. Trajectory interpretation of correspondence principle: Solution of nodal issue. 2019. Available from: https://arxiv.org/abs/ 1911.04747

[37] Yang CD. Quantum dynamics of hydrogen atom in complex space. Annals of Physics. 2005;**319**:399-443

[38] Yang CD, Lee CB. Nonlinear quantum motions in 2D nano-channels part I: Complex potential and quantum trajectories. International Journal of Nonlinear Sciences and Numerical Simulation. 2010;**11**:297-318

[39] Yang CD, Lee CB. Nonlinear quantum motions in 2D nano-channels part II: Quantization and wave motion. International Journal of Nonlinear Sciences and Numerical Simulation. 2010;**11**:319-336

**Chapter 6**

*Paul Bracken*

**Abstract**

appeared.

irreversible

**83**

**1. Introduction**

Entropy in Quantum Mechanics

Nonequilibrium Thermodynamics

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have

**Keywords:** classical, quantum, partition function, temperatures, entropy,

nature [1, 2]. It is not surprising then to find they have a very wide range of applications beyond their original scope, such as to gravitation. The analogy between properties of black holes and thermodynamics could be extended to a complete correspondence, since a black hole in free space had been shown to radiate thermally with a temperature *T* ¼ *κ=*2*π*, where *κ* is the surface gravity. One should be able to assign an entropy to a black hole given by *SH* ¼ *AH=*4 where *AH* is the surface area of the black hole [3]. In the nineteenth century, the problem of reconciling time asymmetric behavior with time symmetric microscopic dynamics became a central issue in this area of physics [4]. Lord Kelvin wrote about the subjection of physical phenomenon to microscopic dynamical law. If then the motion of every particle of matter in the universe were precisely reversed at any instant, the course of nature would be simply reversed for ever after [5]. Physical processes, on the other hand, are irreversible, such as conduction of heat and diffusion processes [6, 7]. It subsequently became apparent that not only is there no conflict between reversible microscopic laws and irreversible microscopic behavior, but there are extremely strong reasons to expect the latter from the former. There are many reasons; for example, there exists a great disparity between microscopic

The laws of thermodynamics are fundamental to the present understanding of

and Applications to

#### **Chapter 6**

## Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics

*Paul Bracken*

#### **Abstract**

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have appeared.

**Keywords:** classical, quantum, partition function, temperatures, entropy, irreversible

#### **1. Introduction**

The laws of thermodynamics are fundamental to the present understanding of nature [1, 2]. It is not surprising then to find they have a very wide range of applications beyond their original scope, such as to gravitation. The analogy between properties of black holes and thermodynamics could be extended to a complete correspondence, since a black hole in free space had been shown to radiate thermally with a temperature *T* ¼ *κ=*2*π*, where *κ* is the surface gravity. One should be able to assign an entropy to a black hole given by *SH* ¼ *AH=*4 where *AH* is the surface area of the black hole [3]. In the nineteenth century, the problem of reconciling time asymmetric behavior with time symmetric microscopic dynamics became a central issue in this area of physics [4]. Lord Kelvin wrote about the subjection of physical phenomenon to microscopic dynamical law. If then the motion of every particle of matter in the universe were precisely reversed at any instant, the course of nature would be simply reversed for ever after [5]. Physical processes, on the other hand, are irreversible, such as conduction of heat and diffusion processes [6, 7]. It subsequently became apparent that not only is there no conflict between reversible microscopic laws and irreversible microscopic behavior, but there are extremely strong reasons to expect the latter from the former. There are many reasons; for example, there exists a great disparity between microscopic

and macroscopic scales and the fact that the events we observe in the macroworld are determined not only by the microscopic dynamics but also by the initial conditions or state of the system.

*SG*ð Þ¼� *ρ kB*

*<sup>ρ</sup>M*ð Þ¼ **<sup>x</sup>** j j <sup>Γ</sup>*<sup>M</sup>* �<sup>1</sup>

same as that for the microcanonical and equivalent to the canonical or

*SBSG* ¼

*<sup>d</sup><sup>μ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>3</sup>*Npd*3*Nq*

*<sup>S</sup>* ¼ �<sup>X</sup> *i*

In attempting to translate these considerations to the quantum domain, it is

Although the situation is in many ways similar in quantum mechanics, it is not identical. The irreversible incompressible flow in phase space is replaced by the unitary evolution of wave functions in Hilbert space and velocity reversal of *x* by

when the integrand is integrable.

In this case, (5) gives the entropy to be

**85**

immediately clear that a perfect analogy does not exist.

ð *dν*

(

microcanonical ensemble associated with a macrostate *M*

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

Then clearly

ð Γ

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

In (2), *ρ*ð Þ **x** is the probability for the microscopic state of the system to be found in the phase space volume element *d***x**. Suppose *ρ*ð Þ **x** is taken to be the generalized

The probability density for the system in the equilibrium macrostate *ρMeq* is the

grandcanonical ensemble when the system is of macroscopic size. The time development of *SB* and *SG* subsequent to some initial time when *ρ* ¼ *ρ<sup>M</sup>* is very different unless *M* ¼ *Meq* when there is no further systematic change in *M* or *ρ*. In fact, *SG*ð Þ*ρ* never changes in time as long as **x** evolves according to Hamiltonian evolution, so *ρ* evolves according to the Liouville equation. Then *SG* does not give any indication that the system is evolving towards equilibrium. Thus the relevant entropy for understanding the time evolution of macroscopic systems is *SB* and not *SG*.

From the standpoint of mathematics, these expressions for classical entropies can be unified under the heading of the Boltzmann-Shannon-Gibbs entropy [16]. A very general form of entropy which includes those mentioned can be defined in a mathematically rigorous way. To do so, let ð Þ Ω, *A*, *μ* be a finite measure space, *ν* a probability measure that is absolutely continuous with respect to *μ*, and its Radon-Nikodym derivative *dν=dμ* exists. The generalized *BSG* entropy is defined to be

*<sup>d</sup><sup>μ</sup>* � log *<sup>d</sup><sup>ν</sup>*

This includes the classical Boltzmann-Gibbs entropy when *dμ* and *dν* are given by

It also includes the Shannon entropy appearing in information theory in which

Ω ¼ f g 1, 2, … , *μ*ð Þ¼ f g1 *μ*ð Þ¼ f g2 … ¼ 1, *ν*ð Þ¼ f g*i* 1*:* (7)

*dμ*

� �*dν*, (5)

<sup>ℏ</sup><sup>3</sup>*<sup>N</sup>* , *<sup>d</sup><sup>ν</sup>* <sup>¼</sup> *<sup>ρ</sup>cl <sup>d</sup>μ:* (6)

*ρ<sup>i</sup>* log *ρ<sup>i</sup>* ð Þ*:* (8)

, **x**∈ Γ; 0, otherwise*:*

*SG*ð Þ¼ *ρ<sup>M</sup> kB* log ∣Γ*M*∣ ¼ *SB*ð Þ *M :* (4)

*ρ*ð Þ **x** log ð Þ *ρ*ð Þ **x** *d***x***:* (2)

(3)

In the twentieth century, it became clear that the microworld was described by a different kind of physics along with mathematical ideas that need not be taken into account in describing the macroworld. This is the subject of quantum mechanics. Even though the new quantum equations have similar symmetry properties as their classical counterparts, it also reveals numerous phenomena that can contribute at this level to the problems mentioned above. These physical phenomena which play various roles include the phenomenon of quantum entanglement, the effect of decoherence in general, and the theory of measurements as well.

The purpose of this is to study the subject of entropy as it applies to quantum mechanics [8, 9]. Its definition is to be relevant to very small systems at the atomic and molecular level. Its relationship to entropies known at other scales can be examined. It is also important to relate this information from this new area of physics to the older and more established theories of thermodynamics and statistical physics [10–15]. To summarize, many good reasons dictate that the arrow of time is specified by the direction of increase of the Boltzmann entropy, the von Neumann macroscopic entropy. To relate the quantum Boltzmann approach to irreversibility to measurement theory, the measuring apparatus must be included as a part of the closed quantum mechanical system.

#### **2. Entropy and quantum mechanics**

Boltzmann's great insight was to connect the second law of thermodynamics with phase space volume. This he did by making the observation that for a dilute gas, log ∣Γ*M*∣ is proportional up to terms negligible compared to the system size, to the thermodynamic entropy of Clausius. He then extended his insight about the relation between thermodynamic entropy and log ∣Γ*M*∣ to all macroscopic systems, no matter what their composition. This gave a macroscopic definition of the observationally measureable entropy of equilibrium macroscopic systems. With this connection established, he generalized it to define an entropy for systems not in equilibrium.

Clearly, the macrostate *M*ð Þ **x** is determined by **x**, a point in phase space, and there are many such points, in fact a continuum, which correspond to the same *M*. Let Γ*<sup>M</sup>* then be the region in Γ consisting of all microstates **x** corresponding to a given macrostate *M*. Boltzmann associated with each microstate **x** of a macroscopic system *M* a number *SB*, which depends only on *M*ð Þ **x** , such that up to multiplicative and additive constants is given by

$$\mathcal{S}(\mathbf{x}) = \mathcal{S}\_B(\mathcal{M}(\mathbf{x})) = k\_B \log |\Gamma\_M|. \tag{1}$$

This *S* is called the Boltzmann entropy of a classical system. The constant *kB* ¼ <sup>1</sup>*:*<sup>38</sup> � <sup>10</sup>�<sup>16</sup> erg/K is called Boltzmann's constant, and if temperature is measured in ergs instead of Kelvin, it may be set to one. Boltzmann argued that due to large differences in the sizes of Γ*M*, *SB*ð Þ **x***<sup>t</sup>* will typically increase in a way which explains and describes the evolution of physical systems towards equilibrium.

The approach of Gibbs, which concentrates primarily on probability distributions or ensembles, is conceptually different from Boltzmann's. The entropy of Gibbs for a microstate **x** of a macroscopic system is defined for an ensemble density *ρ*ð Þ **x** to be

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

$$S\_G(\rho) = -k\_B \int\_{\Gamma} \rho(\mathbf{x}) \log \left( \rho(\mathbf{x}) \right) d\mathbf{x}.\tag{2}$$

In (2), *ρ*ð Þ **x** is the probability for the microscopic state of the system to be found in the phase space volume element *d***x**. Suppose *ρ*ð Þ **x** is taken to be the generalized microcanonical ensemble associated with a macrostate *M*

$$\rho\_{\mathcal{M}}(\mathbf{x}) = \begin{cases} \left| \Gamma\_{\mathcal{M}} \right|^{-1}, & \mathbf{x} \in \Gamma; \\ \mathbf{0}, & \text{otherwise.} \end{cases} \tag{3}$$

Then clearly

and macroscopic scales and the fact that the events we observe in the macroworld are determined not only by the microscopic dynamics but also by the initial

decoherence in general, and the theory of measurements as well.

In the twentieth century, it became clear that the microworld was described by a different kind of physics along with mathematical ideas that need not be taken into account in describing the macroworld. This is the subject of quantum mechanics. Even though the new quantum equations have similar symmetry properties as their classical counterparts, it also reveals numerous phenomena that can contribute at this level to the problems mentioned above. These physical phenomena which play various roles include the phenomenon of quantum entanglement, the effect of

The purpose of this is to study the subject of entropy as it applies to quantum mechanics [8, 9]. Its definition is to be relevant to very small systems at the atomic and molecular level. Its relationship to entropies known at other scales can be examined. It is also important to relate this information from this new area of physics to the older and more established theories of thermodynamics and statistical physics [10–15]. To summarize, many good reasons dictate that the arrow of time is specified by the direction of increase of the Boltzmann entropy, the von Neumann macroscopic entropy. To relate the quantum Boltzmann approach to irreversibility to measurement theory, the measuring apparatus must be included as a part of the

Boltzmann's great insight was to connect the second law of thermodynamics with phase space volume. This he did by making the observation that for a dilute gas, log ∣Γ*M*∣ is proportional up to terms negligible compared to the system size, to the thermodynamic entropy of Clausius. He then extended his insight about the relation between thermodynamic entropy and log ∣Γ*M*∣ to all macroscopic systems, no matter what their composition. This gave a macroscopic definition of the observationally measureable entropy of equilibrium macroscopic systems. With this connection established, he generalized it to define an entropy for systems not in

Clearly, the macrostate *M*ð Þ **x** is determined by **x**, a point in phase space, and there are many such points, in fact a continuum, which correspond to the same *M*. Let Γ*<sup>M</sup>* then be the region in Γ consisting of all microstates **x** corresponding to a given macrostate *M*. Boltzmann associated with each microstate **x** of a macroscopic system *M* a number *SB*, which depends only on *M*ð Þ **x** , such that up to multiplicative

This *S* is called the Boltzmann entropy of a classical system. The constant *kB* ¼ <sup>1</sup>*:*<sup>38</sup> � <sup>10</sup>�<sup>16</sup> erg/K is called Boltzmann's constant, and if temperature is measured in ergs instead of Kelvin, it may be set to one. Boltzmann argued that due to large differences in the sizes of Γ*M*, *SB*ð Þ **x***<sup>t</sup>* will typically increase in a way which explains

The approach of Gibbs, which concentrates primarily on probability distributions or ensembles, is conceptually different from Boltzmann's. The entropy of Gibbs for a microstate **x** of a macroscopic system is defined for an ensemble density

and describes the evolution of physical systems towards equilibrium.

*S*ð Þ¼ **x** *SB*ð Þ¼ *M*ð Þ **x** *kB* log ∣Γ*M*∣*:* (1)

conditions or state of the system.

*Quantum Mechanics*

closed quantum mechanical system.

and additive constants is given by

equilibrium.

*ρ*ð Þ **x** to be

**84**

**2. Entropy and quantum mechanics**

$$\mathcal{S}\_G(\rho\_M) = k\_B \log |\Gamma\_M| = \mathcal{S}\_B(\mathcal{M}).\tag{4}$$

The probability density for the system in the equilibrium macrostate *ρMeq* is the same as that for the microcanonical and equivalent to the canonical or grandcanonical ensemble when the system is of macroscopic size. The time development of *SB* and *SG* subsequent to some initial time when *ρ* ¼ *ρ<sup>M</sup>* is very different unless *M* ¼ *Meq* when there is no further systematic change in *M* or *ρ*. In fact, *SG*ð Þ*ρ* never changes in time as long as **x** evolves according to Hamiltonian evolution, so *ρ* evolves according to the Liouville equation. Then *SG* does not give any indication that the system is evolving towards equilibrium. Thus the relevant entropy for understanding the time evolution of macroscopic systems is *SB* and not *SG*.

From the standpoint of mathematics, these expressions for classical entropies can be unified under the heading of the Boltzmann-Shannon-Gibbs entropy [16]. A very general form of entropy which includes those mentioned can be defined in a mathematically rigorous way. To do so, let ð Þ Ω, *A*, *μ* be a finite measure space, *ν* a probability measure that is absolutely continuous with respect to *μ*, and its Radon-Nikodym derivative *dν=dμ* exists. The generalized *BSG* entropy is defined to be

$$S\_{\rm BSG} = \int \frac{d\nu}{d\mu} \cdot \log \left(\frac{d\nu}{d\mu}\right) d\nu,\tag{5}$$

when the integrand is integrable.

This includes the classical Boltzmann-Gibbs entropy when *dμ* and *dν* are given by

$$d\mu = \frac{d^{3N}p \, d^{3N}q}{\hbar^{3N}}, \qquad d\nu = \rho^{cl} d\mu. \tag{6}$$

It also includes the Shannon entropy appearing in information theory in which

$$\mathfrak{Q} = \{1, 2, \dots\}, \quad \mu(\{1\}) = \mu(\{2\}) = \dots = \mathbb{1}, \quad \nu(\{i\}) = \mathbb{1}.\tag{7}$$

In this case, (5) gives the entropy to be

$$\mathcal{S} = -\sum\_{i} \rho\_i \log \left( \rho\_i \right). \tag{8}$$

In attempting to translate these considerations to the quantum domain, it is immediately clear that a perfect analogy does not exist.

Although the situation is in many ways similar in quantum mechanics, it is not identical. The irreversible incompressible flow in phase space is replaced by the unitary evolution of wave functions in Hilbert space and velocity reversal of *x* by

complex conjugation of the wave function. The analogue of the Gibbs entropy (2) of an ensemble is the von Neumann entropy of a density matrix *ρ*:

$$\mathcal{S}\_{\upsilon N}(\rho) = -k\_B \operatorname{Tr}(\rho \log \rho). \tag{9}$$

classical system of whose macrostate we were unsure. The second part of (10) will be negligible compared to the first term for a macroscopic system, classical or

Entropy functions have a number of characteristic properties which should be briefly described in the quantum case. The set of observables will be the bounded, self-adjoint operators with discrete spectra in a Hilbert space. The set of normal states can be taken to be the density operators or positive operators of trace one.

> ≥ X *i*

*<sup>λ</sup>iS <sup>ρ</sup><sup>i</sup>* ð Þ�<sup>X</sup>

<sup>≤</sup>*S T*ð Þ *<sup>B</sup><sup>ρ</sup>* <sup>≤</sup>*S*ð Þ� *<sup>ρ</sup>* <sup>X</sup>

where the first equality holds iff *TBρ* ¼ *ρ* and the second iff *S ρ<sup>k</sup>* ð Þ¼ *S*ð Þ*ρ* for all *k*.

The formal expression will be interpreted as follows. If *A*, *B* are positive traceless

operators with complete orthonormal sets of eigenstates ∣*ai*i and ∣*bi*i, using a

<sup>h</sup>*ai*∣*<sup>A</sup>* log *<sup>A</sup>* <sup>∣</sup>*ai*i ¼ <sup>P</sup>

*i*

*k*

*S ρ*1j*ρ*<sup>2</sup> ð Þ¼ *Tr ρ*<sup>1</sup> log *ρ*<sup>1</sup> � *ρ*<sup>1</sup> log *ρ*<sup>2</sup> ð Þ*:* (18)

� � *<sup>b</sup> <sup>j</sup>*<sup>j</sup> log *<sup>A</sup>*j*ai*

� � <sup>¼</sup>

*<sup>i</sup>*,*<sup>j</sup> ai*j*A*j*b <sup>j</sup>*

*λiS ρ<sup>i</sup>* ð Þ, (15)

*λ<sup>i</sup>* log *λi:* (16)

*pk* log *pk* (17)

The entropy functional satisfies the following inequalities. Let *λ<sup>i</sup>* > 0 and

*S* X *i λiρ<sup>i</sup>* !

Subadditivity holds with equality if and only if *ρiρ <sup>j</sup>* ¼ 0, *i* 6¼ *j*

≤ X *i*

Note the difference that in the classical case, the state of the system is described by **x**∈ Γ*<sup>α</sup>* for some *α*, so the system is always in one of the macrostates *Mα*. For a quantum system described by *ρ* or Ψ, this is not the case. There is no analogue of (1) for general *ρ* or Ψ. Even when the system is in a macrostate corresponding to a definite microstate at *t*0, only the classical system will be in a unique macrostate at time *t*. The quantum system will in general evolve into a superposition of different macrostates, as is the case in the Schrödinger Cat paradox. In this wave function, Ψ corresponding to a particular macrostate evolves into a linear combination of wave functions associated with very different macrostates. The classical limit is obtained by a prescription in which the density matrix is identified with a probability distribution in phase space and the trace is replaced by integration over phase space. The superposition principle excludes partitions of the Hilbert space: an orthogonal decomposition is all that is

quantum, and going to zero when divided by the number of particles.

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

relevant.

P

and

P *i*,*j ai ai*j*b <sup>j</sup>*

**87**

**2.1 Properties of entropy functions**

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

with equality if all *λ<sup>i</sup>* are equal.

*<sup>i</sup> λ<sup>i</sup>* ¼ 1. Then *S* has the concavity property:

*S* X *i λiρ<sup>i</sup>* !

*S* X *i λiρ<sup>i</sup>* !

*i*

� � so that

The conditional entropy is defined to be

resolution of identity, P

� � log *ai <sup>b</sup> <sup>j</sup>*j*ai*

This formula was given by von Neumann. It generalizes the classical expression of Boltzmann and Gibbs to the realm of quantum mechanics. The density matrix with maximal entropy is the Gibbs state. The range of *SvN* is the whole of the extended real line 0, ½ � ∞ , so to every number *ζ* with 0<*ζ* ≤ ∞, there is a density matrix *ρ* such that *SvN*ð Þ¼ *ρ ζ*. Like the classical *SG*ð Þ*ρ* , this does not alter in time for an isolated system evolving under Schrödinger evolution. It has value zero whenever *ρ* represents a pure stare. Similar to *SG*ð Þ*ρ* , it is not most appropriate for describing the time symmetric behavior of isolated macroscopic systems. The Szilard engine composed of an atom is an example in which the entropy of a quantum object is made use of. von Neumann discusses the macroscopic entropy of a system, so a macrostate is described by specifying values of a set of commuting macroscopic observable operators *A*^, such as particle number, energy, and so forth, to each of the cells that make up the system corresponding to the eigenvalues *aα*, an orthogonal decomposition of the system's Hilbert space H into linear subspaces Γ^*<sup>α</sup>* in which the observables *A*^ take the values *aα*. Let Π*<sup>α</sup>* the projection into Γ^*α*. von Neumann then defines the macroscopic entropy of a system with density matrix ~*ρ* as

$$\tilde{\mathcal{S}}\_{mac}(\tilde{\rho}) = k\_{\mathcal{B}} \sum\_{a=1}^{N} p\_a(\tilde{\rho}) \log |\hat{\Gamma}\_a| - k\_{\mathcal{B}} \sum\_{a=1}^{N} p\_a(\tilde{\rho}) \log p\_a(\tilde{\rho}). \tag{10}$$

Here, *pα*ð Þ~*ρ* is the probability of finding the system with density matrix ~*ρ* in the microstate *M<sup>α</sup>*

$$p\_a(\tilde{\rho}) = \operatorname{Tr}(\Pi\_a \tilde{\rho}),\tag{11}$$

and ∣Γ^*α*∣ is the dimension of Γ^*α*. An analogous definition is made for a system which is represented by a wave function Ψ; simply replace *pα*ð Þ*ρ* by *pα*ð Þ¼ Ψ h i Ψ, Π*α*Ψ . In fact, ∣ΨihΨ∣ just corresponds to a particular pure density matrix.

von Neumann justifies (10) by noting that

$$\tilde{\mathcal{S}}\_{\text{mac}}(\rho) = -k\_B \text{Tr} [\tilde{\rho} \, \log \tilde{\rho}] = \mathcal{S}\_{vN}(\tilde{\rho}),\tag{12}$$

for

$$\tilde{\rho} = \sum\_{a} \frac{p\_a}{|\hat{\Gamma}\_a|} \Pi\_{a\flat} \tag{13}$$

and ~*ρ* is macroscopically indistinguishable from *ρ*.

A correspondence can be made between the partitioning of classical phase space Γ and the decomposition of Hilbert space H and to define the natural quantum analogues to Boltzmann's definition of *SB*ð Þ *M* in (1) as

$$
\hat{S}\_B(M\_a) = k\_B \log |\hat{\Gamma}\_{M\_a}| \tag{14}
$$

where ∣Γ^*<sup>M</sup><sup>α</sup>* ∣ is the dimension of Γ^*<sup>M</sup><sup>α</sup>* . With definition (14) the first term on the right of (10) is just what would be stated for the expected value of the entropy of a *Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

classical system of whose macrostate we were unsure. The second part of (10) will be negligible compared to the first term for a macroscopic system, classical or quantum, and going to zero when divided by the number of particles.

Note the difference that in the classical case, the state of the system is described by **x**∈ Γ*<sup>α</sup>* for some *α*, so the system is always in one of the macrostates *Mα*. For a quantum system described by *ρ* or Ψ, this is not the case. There is no analogue of (1) for general *ρ* or Ψ. Even when the system is in a macrostate corresponding to a definite microstate at *t*0, only the classical system will be in a unique macrostate at time *t*. The quantum system will in general evolve into a superposition of different macrostates, as is the case in the Schrödinger Cat paradox. In this wave function, Ψ corresponding to a particular macrostate evolves into a linear combination of wave functions associated with very different macrostates. The classical limit is obtained by a prescription in which the density matrix is identified with a probability distribution in phase space and the trace is replaced by integration over phase space. The superposition principle excludes partitions of the Hilbert space: an orthogonal decomposition is all that is relevant.

#### **2.1 Properties of entropy functions**

Entropy functions have a number of characteristic properties which should be briefly described in the quantum case. The set of observables will be the bounded, self-adjoint operators with discrete spectra in a Hilbert space. The set of normal states can be taken to be the density operators or positive operators of trace one.

P The entropy functional satisfies the following inequalities. Let *λ<sup>i</sup>* > 0 and *<sup>i</sup> λ<sup>i</sup>* ¼ 1. Then *S* has the concavity property:

$$S\left(\sum\_{i} \lambda\_i \rho\_i\right) \ge \sum\_{i} \lambda\_i S(\rho\_i),\tag{15}$$

with equality if all *λ<sup>i</sup>* are equal. Subadditivity holds with equality if and only if *ρiρ <sup>j</sup>* ¼ 0, *i* 6¼ *j*

$$S\left(\sum\_{i}\lambda\_{i}\rho\_{i}\right)\leq\sum\_{i}\lambda\_{i}S(\rho\_{i})-\sum\_{i}\lambda\_{i}\log\lambda\_{i}.\tag{16}$$

and

complex conjugation of the wave function. The analogue of the Gibbs entropy (2) of

This formula was given by von Neumann. It generalizes the classical expression of Boltzmann and Gibbs to the realm of quantum mechanics. The density matrix with maximal entropy is the Gibbs state. The range of *SvN* is the whole of the extended real line 0, ½ � ∞ , so to every number *ζ* with 0<*ζ* ≤ ∞, there is a density matrix *ρ* such that *SvN*ð Þ¼ *ρ ζ*. Like the classical *SG*ð Þ*ρ* , this does not alter in time for an isolated system evolving under Schrödinger evolution. It has value zero whenever *ρ* represents a pure stare. Similar to *SG*ð Þ*ρ* , it is not most appropriate for describing the time symmetric behavior of isolated macroscopic systems. The Szilard engine composed of an atom is an example in which the entropy of a quantum object is made use of. von Neumann discusses the macroscopic entropy of a system, so a macrostate is described by specifying values of a set of commuting macroscopic observable operators *A*^, such as particle number, energy, and so forth, to each of the cells that make up the system corresponding to the eigenvalues *aα*, an orthogonal decomposition of the system's Hilbert space H into linear subspaces Γ^*<sup>α</sup>* in which the observables *A*^ take the values *aα*. Let Π*<sup>α</sup>* the projection into Γ^*α*. von Neumann then defines the macroscopic entropy of a system with

*SvN*ð Þ¼� *ρ kB*Trð Þ *ρ* log *ρ :* (9)

an ensemble is the von Neumann entropy of a density matrix *ρ*:

density matrix ~*ρ* as

*Quantum Mechanics*

microstate *M<sup>α</sup>*

for

**86**

~

*Smac*ð Þ¼ ~*ρ kB*

von Neumann justifies (10) by noting that

~

and ~*ρ* is macroscopically indistinguishable from *ρ*.

analogues to Boltzmann's definition of *SB*ð Þ *M* in (1) as

^

X *N*

*<sup>p</sup>α*ð Þ~*<sup>ρ</sup>* log <sup>∣</sup>Γ^*α*<sup>∣</sup> � *kB*

Here, *pα*ð Þ~*ρ* is the probability of finding the system with density matrix ~*ρ* in the

and ∣Γ^*α*∣ is the dimension of Γ^*α*. An analogous definition is made for a system

which is represented by a wave function Ψ; simply replace *pα*ð Þ*ρ* by *pα*ð Þ¼ Ψ h i Ψ, Π*α*Ψ . In fact, ∣ΨihΨ∣ just corresponds to a particular pure density matrix.

> <sup>~</sup>*<sup>ρ</sup>* <sup>¼</sup> <sup>X</sup> *α*

*pα* ∣Γ^*α*∣

A correspondence can be made between the partitioning of classical phase space

where ∣Γ^*<sup>M</sup><sup>α</sup>* ∣ is the dimension of Γ^*<sup>M</sup><sup>α</sup>* . With definition (14) the first term on the right of (10) is just what would be stated for the expected value of the entropy of a

Γ and the decomposition of Hilbert space H and to define the natural quantum

X *N*

*pα*ð Þ~*ρ* log *pα*ð Þ~*ρ :* (10)

*α*¼1

*Smac*ð Þ¼� *ρ kB*Tr½ �¼ ~*ρ* log ~*ρ SvN*ð Þ~*ρ* , (12)

*SB*ð Þ¼ *<sup>M</sup><sup>α</sup> kB* log <sup>∣</sup>Γ^*<sup>M</sup><sup>α</sup>* <sup>∣</sup> (14)

*pα*ð Þ¼ ~*ρ Tr*ð Þ Π*α*~*ρ* , (11)

Π*α*, (13)

*α*¼1

$$\mathcal{S}\left(\sum\_{i}\lambda\_{i}\rho\_{i}\right)\leq\mathcal{S}(T\_{B}\rho)\leq\mathcal{S}(\rho)-\sum\_{k}p\_{k}\log p\_{k}\tag{17}$$

where the first equality holds iff *TBρ* ¼ *ρ* and the second iff *S ρ<sup>k</sup>* ð Þ¼ *S*ð Þ*ρ* for all *k*. The conditional entropy is defined to be

$$S(\rho\_1|\rho\_2) = Tr(\rho\_1 \log \rho\_1 - \rho\_1 \log \rho\_2). \tag{18}$$

The formal expression will be interpreted as follows. If *A*, *B* are positive traceless operators with complete orthonormal sets of eigenstates ∣*ai*i and ∣*bi*i, using a resolution of identity, P *i* <sup>h</sup>*ai*∣*<sup>A</sup>* log *<sup>A</sup>* <sup>∣</sup>*ai*i ¼ <sup>P</sup> *<sup>i</sup>*,*<sup>j</sup> ai*j*A*j*b <sup>j</sup>* � � *<sup>b</sup> <sup>j</sup>*<sup>j</sup> log *<sup>A</sup>*j*ai* � � <sup>¼</sup> P *i*,*j ai ai*j*b <sup>j</sup>* � � log *ai <sup>b</sup> <sup>j</sup>*j*ai* � � so that

$$\begin{split} \sum\_{j} \langle b\_{j} | A \log A - A \log B + B - A | b\_{j} \rangle &= \sum\_{j} \langle a\_{i} | A \log A - A \log B + B - A | a\_{i} \rangle \\ &= \sum\_{i,j} | \langle a\_{i} | b\_{i} \rangle |^{2} \langle a\_{i} \log a\_{i} - a\_{i} \log b\_{j} + b\_{j} - a\_{i} \rangle = S(A | B). \end{split} \tag{19}$$

Concavity of the function *x* log *x* ensures the terms of the final sum are nonnegative. In order that *S ρ*1j*ρ*<sup>2</sup> ð Þ< ∞, it is necessary that *πρ*<sup>1</sup> ≤*πρ*<sup>2</sup> where *π<sup>W</sup>* ¼ supp*W* is the support projection of *W*, so *ρ*<sup>1</sup> <*ρ*2. From the definition, *S ρ*1j*ρ*<sup>2</sup> ð Þ≥0 with equality if *ρ*<sup>1</sup> ¼ *ρ*2. If *λρ*<sup>1</sup> ≤*ρ*2, for some *λ*∈ ð Þ 0, 1 , *S ρ*1j*ρ*<sup>2</sup> ð Þ≤ � log *λ* from operator monotony of log *<sup>z</sup>*. If *<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup> *<sup>i</sup> λiρi*, then

$$\mathcal{S}(\rho) = \sum\_{i} \lambda\_i \mathcal{S}(\rho\_i) + \sum\_{i} \lambda\_i \mathcal{S}(\rho\_i|\rho),\tag{20}$$

*πα* ¼ 1 ⊗ Π*α*, *α* ¼ 1, … , *n:* (24)

*παρπα:* (25)

*αα*<sup>0</sup> *παρπα*<sup>0</sup> . When a sequence

*<sup>α</sup> παρπα* may be viewed as a loss of information

ð Þ *t*<sup>1</sup> ⋯*πα<sup>n</sup>* ð Þ *tn* � �, (26)

� �*:* (27)

*D α*, *α*<sup>0</sup> ð Þ¼ *δα*,*α*0*ρα:* (28)

*παρ*ð Þ 0 *πβ* ¼ 0, *α* 6¼ *β:* (29)

*πα*0*ρmπα* þ *παρmπα* ð Þ0 6¼ 0*:* (30)

*S ρ*1j*ρ*<sup>2</sup> ð Þ≥ 0*:* (31)

*S ρ*1j*ρ*<sup>2</sup> ð Þ¼ 0, *ρ*<sup>1</sup> ¼ *ρ*2*:* (32)

*<sup>α</sup> λαπα*Φ*<sup>α</sup>* such that P

, which is equivalent to ½ �¼ *πα*, *ρ*

*<sup>α</sup>*j j *λα* <sup>2</sup> <sup>¼</sup> 1 and <sup>Φ</sup>*<sup>α</sup>* <sup>∈</sup> <sup>H</sup>

Suppose *A* is measured on system *S*, initially in a state of the composite system described by a density matrix *ρ*. The value *p<sup>α</sup>* is obtained with probability *τα* ¼ Tr *ρπα* ð Þ. After the measurement, the state of the composite system is accounted for

> *ρα* <sup>¼</sup> <sup>1</sup> *τα*

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

of measurements is carried out and a time evolution is permitted to occur between measurements leads one to assign to a sequence of events *πα*<sup>1</sup> ð Þ *t*<sup>1</sup> *πα*<sup>2</sup> ð Þ *t*<sup>2</sup> ⋯*πα<sup>n</sup>* ð Þ *tn* the

where *ρ* ¼ *ρ*ð Þ 0 , over the set of histories, where the *π<sup>k</sup>* satisfy (22) with Π

*D α*<sup>0</sup> ð Þ¼ , *α Tr πα*<sup>1</sup> ð Þ *t*<sup>1</sup> ⋯*πα<sup>n</sup>* ð Þ *tn ρπα<sup>n</sup>* ð Þ *tn* ⋯*πα*<sup>1</sup> ð Þ *t*<sup>1</sup>

A state is called decoherent with respect to the set of *πα* if and only if

The following definition can now be stated. A history is said to *decohere* if and

0 for all *α*. In contrast to infinite systems where there is no need to refer to a choice of projections, decoherent mixed states over the macroscopic observables can be described by relations between the density matrix and the projectors. They would

The relative or conditional entropy between two states *S ρ*1j*ρ*<sup>2</sup> ð Þ was defined in (18), and it plays a crucial role. It is worth stating a few of its properties, as some are

*S λρ*<sup>1</sup> þ ð Þ 1 � *λ ρ*<sup>2</sup> ð Þ j*λσ*<sup>1</sup> þ ð Þ 1 � *λ σ*<sup>2</sup> ≤*λS ρ*<sup>1</sup> ð Þþ j*σ*<sup>1</sup> ð Þ 1 � *λ S ρ*<sup>2</sup> ð Þ j*σ*<sup>2</sup> *:* (33)

This is a mixture of states in each of which *A* has a definite value.

Pð Þ¼ *α* Tr *πα*<sup>1</sup> ð Þ *tn* ⋯*πα*<sup>1</sup> ð Þ *t*<sup>1</sup> *ρπα*<sup>1</sup>

contained in non-diagonal terms *ψαρπα*<sup>0</sup> with *<sup>α</sup>* 6¼ *<sup>β</sup>* in <sup>P</sup>

This implies that *Tr πα* ð Þ¼ <sup>0</sup>*ρπαA* 0 for all *α* 6¼ *α*<sup>0</sup>

X *α*6¼*α*<sup>0</sup>

When *γ* is a completely positive map, or embedding

be of the form *<sup>ρ</sup><sup>m</sup>* <sup>¼</sup> <sup>∣</sup>ΨihΨ<sup>∣</sup> with <sup>∣</sup>Ψi ¼ <sup>P</sup>

by the density matrix:

probability distribution:

only if

and satisfy

**89**

necessary for the theorem:

replaced by the *π*. Let us define

The transformation *<sup>ρ</sup>* ! <sup>~</sup>*<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

which gives (15) and (16). If *T* is a trace-preserving operator, then *ρ* <*Tρ*, and

$$\mathcal{S}(T\rho) = \mathcal{S}(\rho) + \mathcal{S}(\rho|T\rho). \tag{21}$$

This is to say that *T* is entropy-increasing.

The concept of irreversibility is clearly going to be relevant to the subject at hand, so some thoughts related to it will be given periodically in what follows. A possible way to account for irreversibility in a closed system in nature is by the various types of course-graining. There are also strong reasons to suggest the arrow of time is provided by the direction of increase of the quantum form of the Boltzmann entropy. The measuring apparatus should be included as part of the closed quantum mechanical system in order to relate the quantum Boltzmann approach to irreversibility to the concept of a measurement. Let *Sc* be a composite system consisting of a macroscopic system *S* coupled to a measuring instrument I, so *Sc* ¼ *S* þ I, where I is a large but finite *N*-particle system. A set of coursegrained mutually commuting extrinsic variables are provided whose eigenspaces correspond to the pointer positions of I. von Neumann's picture of the measurement process is basic to the approach, but according to which, the coupling of *S* to I leads to the following effects. A pure state of P *S* described by a linear combination *<sup>α</sup>cαψα* of its orthonormal energy eigenstates is converted into a statistical mixture of these states for which j j *<sup>c</sup><sup>α</sup>* <sup>2</sup> is the probability of finding the system in state *ψα*. It also sends a certain set of classical or intercommuting, macroscopic variables M of I to values indicated by pointer readings that indicate which of the states is realized.

There is an amplification process of the *S* � I coupling where different microstates of *S* give rise to macroscopically different states of I. If I is designed to have readings which are in one-to-one correspondence with the eigenstates of *S*, it may be assumed index *α* of its microstates goes from 1 to *n*. Denote the projection operator for subspace K by Π*α*, then

$$
\Pi\_a \Pi\_\beta = \Pi\_a \delta\_{a\beta}, \qquad \sum\_a \Pi\_a = \mathbf{1}\_{\tilde{\kappa}\_a}, \tag{22}
$$

and each element of the abelian subalgebra of ℬ takes the form with *M<sup>α</sup>* scalars

$$M = \sum\_{a} M\_{a} \Pi\_{a}.\tag{23}$$

Define the projection operators:

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

$$
\pi\_a = \mathbf{1} \otimes \Pi\_a, \qquad a = \mathbf{1}, \ldots, n. \tag{24}
$$

Suppose *A* is measured on system *S*, initially in a state of the composite system described by a density matrix *ρ*. The value *p<sup>α</sup>* is obtained with probability *τα* ¼ Tr *ρπα* ð Þ. After the measurement, the state of the composite system is accounted for by the density matrix:

$$
\rho\_a = \frac{1}{\tau\_a} \pi\_a \rho \pi\_a. \tag{25}
$$

This is a mixture of states in each of which *A* has a definite value.

The transformation *<sup>ρ</sup>* ! <sup>~</sup>*<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup> *<sup>α</sup> παρπα* may be viewed as a loss of information contained in non-diagonal terms *ψαρπα*<sup>0</sup> with *<sup>α</sup>* 6¼ *<sup>β</sup>* in <sup>P</sup> *αα*<sup>0</sup> *παρπα*<sup>0</sup> . When a sequence of measurements is carried out and a time evolution is permitted to occur between measurements leads one to assign to a sequence of events *πα*<sup>1</sup> ð Þ *t*<sup>1</sup> *πα*<sup>2</sup> ð Þ *t*<sup>2</sup> ⋯*πα<sup>n</sup>* ð Þ *tn* the probability distribution:

$$\mathcal{P}(a) = \operatorname{Tr}\left(\pi\_{a\_1}(t\_n)\cdots\pi\_{a\_1}(t\_1)\rho\pi\_{a\_1}(t\_1)\cdots\pi\_{a\_n}(t\_n)\right),\tag{26}$$

where *ρ* ¼ *ρ*ð Þ 0 , over the set of histories, where the *π<sup>k</sup>* satisfy (22) with Π replaced by the *π*. Let us define

$$D(a',a) = \operatorname{Tr}\left(\pi\_{a\_1}(t\_1)\cdots\pi\_{a\_n}(t\_n)\rho\pi\_{a\_n}(t\_n)\cdots\pi\_{a\_1}(t\_1)\right). \tag{27}$$

The following definition can now be stated. A history is said to *decohere* if and only if

$$D(a, a') = \delta\_{a, a'} \rho\_a. \tag{28}$$

A state is called decoherent with respect to the set of *πα* if and only if

$$
\pi\_a \rho(\mathbf{0}) \pi\_\beta = \mathbf{0}, \qquad a \neq \beta. \tag{29}
$$

This implies that *Tr πα* ð Þ¼ <sup>0</sup>*ρπαA* 0 for all *α* 6¼ *α*<sup>0</sup> , which is equivalent to ½ �¼ *πα*, *ρ* 0 for all *α*. In contrast to infinite systems where there is no need to refer to a choice of projections, decoherent mixed states over the macroscopic observables can be described by relations between the density matrix and the projectors. They would be of the form *<sup>ρ</sup><sup>m</sup>* <sup>¼</sup> <sup>∣</sup>ΨihΨ<sup>∣</sup> with <sup>∣</sup>Ψi ¼ <sup>P</sup> *<sup>α</sup> λαπα*Φ*<sup>α</sup>* such that P *<sup>α</sup>*j j *λα* <sup>2</sup> <sup>¼</sup> 1 and <sup>Φ</sup>*<sup>α</sup>* <sup>∈</sup> <sup>H</sup> and satisfy

$$\sum\_{a \neq a'} (\pi\_{a'} \rho\_m \pi\_a + \pi\_a \rho\_m \pi\_{a'}) \neq \mathbf{0}.\tag{30}$$

The relative or conditional entropy between two states *S ρ*1j*ρ*<sup>2</sup> ð Þ was defined in (18), and it plays a crucial role. It is worth stating a few of its properties, as some are necessary for the theorem:

$$\mathbf{S}(\rho\_1|\rho\_2) \ge \mathbf{0}.\tag{31}$$

$$\mathcal{S}(\rho\_1|\rho\_2) = \mathbf{0}, \qquad \rho\_1 = \rho\_2. \tag{32}$$

$$\mathcal{S}(\lambda \rho\_1 + (\mathbf{1} - \lambda)\rho\_2 | \lambda \sigma\_1 + (\mathbf{1} - \lambda)\sigma\_2) \le \lambda \mathcal{S}(\rho\_1 | \sigma\_1) + (\mathbf{1} - \lambda)\mathcal{S}(\rho\_2 | \sigma\_2). \tag{33}$$

When *γ* is a completely positive map, or embedding

X *j*

*Quantum Mechanics*

<sup>¼</sup> <sup>X</sup> *i*, *j*

monotony of log *<sup>z</sup>*. If *<sup>ρ</sup>* <sup>¼</sup> <sup>P</sup>

*b <sup>j</sup>*j*A* log *A* � *A* log *B* þ *B* � *A*j*b <sup>j</sup>* � � <sup>¼</sup> <sup>X</sup>

*ai* h i <sup>j</sup>*bi* j j<sup>2</sup>

*j*

� � <sup>¼</sup> *S A*ð Þ <sup>j</sup>*<sup>B</sup> :*

Concavity of the function *x* log *x* ensures the terms of the final sum are nonnegative. In order that *S ρ*1j*ρ*<sup>2</sup> ð Þ< ∞, it is necessary that *πρ*<sup>1</sup> ≤*πρ*<sup>2</sup> where *π<sup>W</sup>* ¼ supp*W* is the support projection of *W*, so *ρ*<sup>1</sup> <*ρ*2. From the definition, *S ρ*1j*ρ*<sup>2</sup> ð Þ≥0 with equality if *ρ*<sup>1</sup> ¼ *ρ*2. If *λρ*<sup>1</sup> ≤*ρ*2, for some *λ*∈ ð Þ 0, 1 , *S ρ*1j*ρ*<sup>2</sup> ð Þ≤ � log *λ* from operator

*<sup>λ</sup>iS <sup>ρ</sup><sup>i</sup>* ð Þþ<sup>X</sup>

which gives (15) and (16). If *T* is a trace-preserving operator, then *ρ* <*Tρ*, and

The concept of irreversibility is clearly going to be relevant to the subject at hand, so some thoughts related to it will be given periodically in what follows. A possible way to account for irreversibility in a closed system in nature is by the various types of course-graining. There are also strong reasons to suggest the arrow

*<sup>α</sup>cαψα* of its orthonormal energy eigenstates is converted into a statistical mixture of these states for which j j *<sup>c</sup><sup>α</sup>* <sup>2</sup> is the probability of finding the system in state *ψα*. It also sends a certain set of classical or intercommuting, macroscopic variables M of I to values indicated by pointer readings that indicate which of the states is realized. There is an amplification process of the *S* � I coupling where different microstates of *S* give rise to macroscopically different states of I. If I is designed to have readings which are in one-to-one correspondence with the eigenstates of *S*, it may be assumed index *α* of its microstates goes from 1 to *n*. Denote the projection

*α*

and each element of the abelian subalgebra of ℬ takes the form with *M<sup>α</sup>* scalars

<sup>Π</sup>*α*Π*<sup>β</sup>* <sup>¼</sup> <sup>Π</sup>*αδαβ*, <sup>X</sup>

*<sup>M</sup>* <sup>¼</sup> <sup>X</sup> *α*

of time is provided by the direction of increase of the quantum form of the Boltzmann entropy. The measuring apparatus should be included as part of the closed quantum mechanical system in order to relate the quantum Boltzmann approach to irreversibility to the concept of a measurement. Let *Sc* be a composite system consisting of a macroscopic system *S* coupled to a measuring instrument I, so *Sc* ¼ *S* þ I, where I is a large but finite *N*-particle system. A set of coursegrained mutually commuting extrinsic variables are provided whose eigenspaces correspond to the pointer positions of I. von Neumann's picture of the measurement process is basic to the approach, but according to which, the coupling of *S* to I

*i*

*S T*ð Þ¼ *ρ S*ð Þþ *ρ S*ð Þ *ρ*j*Tρ :* (21)

*ai* log *ai* � *ai* log *b <sup>j</sup>* þ *b <sup>j</sup>* � *ai*

*<sup>i</sup> λiρi*, then

*<sup>S</sup>*ð Þ¼ *<sup>ρ</sup>* <sup>X</sup> *i*

This is to say that *T* is entropy-increasing.

leads to the following effects. A pure state of

operator for subspace K by Π*α*, then

Define the projection operators:

P

**88**

*ai* h i j*A* log *A* � *A* log *B* þ *B* � *A*j*ai*

*λiS ρ<sup>i</sup>* ð Þ j*ρ* , (20)

*S* described by a linear combination

Π*<sup>α</sup>* ¼ 1<sup>K</sup>*<sup>α</sup>* , (22)

*Mα*Π*α:* (23)

(19)

*Quantum Mechanics*

$$\mathbb{S}(\rho\_1 \cdot \gamma | \rho\_2 \cdot \gamma) \le \mathbb{S}(\rho\_1 | \rho\_2). \tag{34}$$

**3. Quantum mechanics and nonequilibrium thermodynamics**

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

probability of finding the system in state ∣*n*i is

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

equilibrium. This breakup motivates us to define

Inverting Eq. (38) for *pn*, we can solve for

P

P

**91**

*<sup>k</sup> pk* ¼ 1 is

*<sup>n</sup>* ð Þ *dε<sup>n</sup> pn* which means

log ð Þ *Z* and the other to log *pn*

It remains to study

�*T* X *n*

*<sup>δ</sup><sup>W</sup>* <sup>¼</sup> <sup>X</sup> *n*

The average external energy *U* of the system is given as

*dU* <sup>¼</sup> <sup>X</sup> *n*

By integrating over the infinitesimal segments, we find *W* is

Some aspects of equilibrium thermodynamics are examined by considering an isothermal process. Since it is a quasistatic process, it may be decomposed into a sequence of infinitesimal processes. Assume initially the system has a Hamiltonian *H*ð Þ*γ* in thermal equilibrium at a temperature *T*. Boltzmann's constant is set to one. The state is given by the Gibbs density operator *ρ*. This expression can also be written in terms of the energy eigenvalues *ε<sup>n</sup>* and eigenvectors ∣*n*i of *H*. The

*pn* <sup>¼</sup> h i *<sup>n</sup>*j*ρ*j*<sup>n</sup>* <sup>¼</sup> *<sup>e</sup>*�*βε<sup>n</sup>*

*<sup>U</sup>* <sup>¼</sup> h i *<sup>U</sup>* <sup>¼</sup> Tr ð Þ¼ *<sup>H</sup><sup>ρ</sup>* <sup>X</sup>

When the parameter *γ* is changed to *γ* þ *dγ*, both *ε<sup>n</sup>* and *pn* as well as *U* change to

Each instantaneous infinitesimal process can be broken down into a part which is the work performed; the second is the heat transformed as the system relaxes to

ð Þ *<sup>d</sup>ε<sup>n</sup> pn*, *<sup>δ</sup><sup>Q</sup>* <sup>¼</sup> <sup>X</sup>

so *dU* ¼ *δQ* þ *δW*, and *δ* is used to indicate that heat and work are not exact

differentials. The free energy of the system is defined to be *F* ¼ �*T* log *Z*, so *dF* ¼

*ε<sup>n</sup>* ¼ �*T* log *Zpn*

Substituting into the relation for *δQ*, we get two terms, one proportional to

log ð Þ *<sup>Z</sup> dpn* ¼ �*<sup>T</sup>* log ð Þ *<sup>Z</sup> <sup>d</sup>* <sup>X</sup>

X *n*

*dpn* log *pn*

*δQ* ¼ �*T*

*dε<sup>n</sup> pn* þ *ε<sup>n</sup> dpn*

*n*

*n*

*<sup>Z</sup> :* (38)

*εnpn:* (39)

*ε<sup>n</sup> dpn*, (41)

� �*:* (40)

*δW* ¼ *dF:* (42)

� �*:* (44)

¼ 0, (45)

� �*:* (46)

*W* ¼ Δ*F* ¼ Δ*U* � *Q:* (43)

� �. The term with log ð Þ *<sup>Z</sup>* when the *pk* satisfy

*n pn* !

The last two inequalities are known as joint concavity and monotonicity of the relative entropy. The following result may be thought of as a quantum version of the second law.

**Theorem**: Suppose the initial density matrix is decoherent at zero time (29) with respect to *πα* and have finite entropy

$$\begin{aligned} \rho(\mathbf{0}) &= \sum\_{a} \pi\_{a} \rho(\mathbf{0}) \pi\_{a}, \\ S(\rho(\mathbf{0})) &= -k\_{\mathbb{B}} \operatorname{Tr} \left( \rho(\mathbf{0}) \log \left( \rho(\mathbf{0}) \right) \right) < \infty, \end{aligned} \tag{35}$$

and it is not an equilibrium state of the system. Let *ρ tf* � �, for *tf* >0, be any subsequent state of the system, possibly an equilibrium state. Then for an automorphic, unitary time evolution of the system between 0 ≤*t*≤*tf*

$$\mathcal{S}(\mathbf{0}) \le \mathcal{S}(t\_f),\tag{36}$$

where *S*ð Þ¼ 0 *S tf* � � if and only if **(e)** P *<sup>α</sup>*<*<sup>β</sup> παρ tf* � �*πβ* <sup>þ</sup> *πβρ tf* � �*πα* <sup>¼</sup> 0. Proof: Set *ρ*<sup>0</sup> *tf* � � <sup>¼</sup> <sup>P</sup> *<sup>α</sup> παρ tf* � �*πα* <sup>¼</sup> *<sup>ρ</sup> tf* � � � *<sup>γ</sup>*, so *<sup>ρ</sup>*<sup>0</sup> is obtained from *<sup>ρ</sup>* by means of a completely positive map. It follows that

$$\mathcal{S}\left(\boldsymbol{\rho}'\left(\mathbf{t}\_f\right)|\boldsymbol{\rho}'\left(\mathbf{0}\right)\right) = -\mathcal{S}\left(\boldsymbol{\rho}'\left(\mathbf{t}\_f\right)\right) - \mathbb{k}\_\mathbb{B} \sum\_a \text{Tr}\left(\boldsymbol{\rho}\left(\mathbf{t}\_f\right)\boldsymbol{\pi}\_a \log\left(\boldsymbol{\rho}(\mathbf{0})\right)\mathbf{z}\_a\right)$$

$$\mathbf{z} = -\mathcal{S}\left(\mathbf{t}\_f\right) - \mathbb{k}\_\mathbb{B} \text{Tr}\left(\boldsymbol{\rho}\left(\mathbf{t}\_f\right)\log\left(\boldsymbol{\rho}(\mathbf{0})\right)\right) \leq \mathcal{S}\left(\boldsymbol{\rho}\left(\mathbf{t}\_f\right)|\boldsymbol{\rho}(\mathbf{0})\right) = -\mathcal{S}\left(\boldsymbol{\rho}(\mathbf{0})\right) - \text{Tr}\left(\boldsymbol{\rho}\left(\mathbf{t}\_f\right)\log\left(\boldsymbol{\rho}(\mathbf{0})\right)\right). \tag{37}$$

The first equality uses the cyclic property of the trace and the definition of *ρ*<sup>0</sup> . The second equality uses decoherence of *ρ*ð Þ 0 , and the next inequality is a consequence of (34). The evolution is unitary and hence preserves entropy which is the last equality. This implies that *S t*ð Þ≥*S*ð Þ 0 and the equality condition (**e**) follows from (32). □

Of course, entropy growth as in the theorem is not necessarily monotonic in the time variable. For this reason, it is usual to refer to fixed initial and final states. For thermal systems, a natural choice of the final state is the equilibrium state of the system. It is the case in thermodynamics that irreversibility is manifested as a monotonic increase in the entropy. Thermodynamic entropy, it is thought, is related to the entropy of the states defined in both classical and quantum theory. Under an automorphic time evolution, the entropy is conserved. One application of an environment is to account for an increase. A type of coursegraining becomes necessary together with the right conditions on the initial state to account for the arrow of time. In quantum mechanics, the course-graining seems to be necessary and may be thought of as a restriction of the algebra and can also be interpreted as leaving out unobservable quantum correlations. This may, for example, correspond to decoherence effects important in quantum measurements. Competing effects arise such as the fact that correlations becoming unobservable may lead to entropy increase. There is also the effect that a decrease in entropy might be due to nonautomorphic processes. Although both effects lead to irreversibility, they are not cooperative but rather contrary to one another. The observation that the second law does hold implies these nonautomorphic events must be rare in comparison with time scales relevant to thermodynamics.

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

#### **3. Quantum mechanics and nonequilibrium thermodynamics**

Some aspects of equilibrium thermodynamics are examined by considering an isothermal process. Since it is a quasistatic process, it may be decomposed into a sequence of infinitesimal processes. Assume initially the system has a Hamiltonian *H*ð Þ*γ* in thermal equilibrium at a temperature *T*. Boltzmann's constant is set to one. The state is given by the Gibbs density operator *ρ*. This expression can also be written in terms of the energy eigenvalues *ε<sup>n</sup>* and eigenvectors ∣*n*i of *H*. The probability of finding the system in state ∣*n*i is

$$p\_n = \langle n|\rho|n\rangle = \frac{e^{-\beta\varepsilon\_n}}{Z}.\tag{38}$$

The average external energy *U* of the system is given as

$$U = \langle U \rangle = \operatorname{Tr} \left( H\rho \right) = \sum\_{n} \varepsilon\_{n} p\_{n}. \tag{39}$$

When the parameter *γ* is changed to *γ* þ *dγ*, both *ε<sup>n</sup>* and *pn* as well as *U* change to

$$dU = \sum\_{n} \left[ d\varepsilon\_{n} p\_{n} + \varepsilon\_{n} dp\_{n} \right]. \tag{40}$$

Each instantaneous infinitesimal process can be broken down into a part which is the work performed; the second is the heat transformed as the system relaxes to equilibrium. This breakup motivates us to define

$$
\delta \mathcal{W} = \sum\_{n} (d\varepsilon\_{n}) p\_{n}, \qquad \delta Q = \sum\_{n} \varepsilon\_{n} dp\_{n}, \tag{41}
$$

so *dU* ¼ *δQ* þ *δW*, and *δ* is used to indicate that heat and work are not exact P differentials. The free energy of the system is defined to be *F* ¼ �*T* log *Z*, so *dF* ¼ *<sup>n</sup>* ð Þ *dε<sup>n</sup> pn* which means

$$
\delta W = dF.\tag{42}
$$

By integrating over the infinitesimal segments, we find *W* is

$$W = \Delta F = \Delta U - Q.\tag{43}$$

Inverting Eq. (38) for *pn*, we can solve for

$$
\varepsilon\_n = -T \log \left( Z p\_n \right). \tag{44}
$$

Substituting into the relation for *δQ*, we get two terms, one proportional to log ð Þ *Z* and the other to log *pn* � �. The term with log ð Þ *<sup>Z</sup>* when the *pk* satisfy P *<sup>k</sup> pk* ¼ 1 is

$$-T\sum\_{n} \log \left( Z \right) dp\_n = -T \log \left( Z \right) d \left( \sum\_{n} p\_n \right) = 0,\tag{45}$$

It remains to study

$$
\delta Q = -T \sum\_{n} dp\_n \log \left( p\_n \right). \tag{46}
$$

*S ρ*<sup>1</sup> � *γ*j*ρ*<sup>2</sup> ð Þ � *γ* ≤ *S ρ*1j*ρ*<sup>2</sup> ð Þ*:* (34)

(35)

(37)

.

� �, for *tf* >0, be any

� �*πα* <sup>¼</sup> 0.

� �, (36)

� � � *<sup>γ</sup>*, so *<sup>ρ</sup>*<sup>0</sup> is obtained from *<sup>ρ</sup>* by means of

� �*πα* log ð Þ *<sup>ρ</sup>*ð Þ <sup>0</sup> *πα* � �

� �*πβ* <sup>þ</sup> *πβρ tf*

The last two inequalities are known as joint concavity and monotonicity of the relative entropy. The following result may be thought of as a quantum version of

**Theorem**: Suppose the initial density matrix is decoherent at zero time (29) with

*S*ð Þ¼� *ρ*ð Þ 0 *kB*Trð Þ *ρ*ð Þ 0 log ð Þ *ρ*ð Þ 0 < ∞,

*S*ð Þ 0 ≤*S tf*

*παρ*ð Þ 0 *πα*,

*<sup>α</sup>*<*<sup>β</sup> παρ tf*

X *α*

Tr *ρ tf*

� �j*ρ*ð Þ <sup>0</sup> � � ¼ �*S*ð Þ� *<sup>ρ</sup>*ð Þ <sup>0</sup> Tr *<sup>ρ</sup> tf* � � log ð Þ *<sup>ρ</sup>*ð Þ <sup>0</sup> � �*:* �

*<sup>ρ</sup>*ð Þ¼ <sup>0</sup> <sup>X</sup> *α*

subsequent state of the system, possibly an equilibrium state. Then for an automorphic, unitary time evolution of the system between 0 ≤*t*≤*tf*

� �*πα* <sup>¼</sup> *<sup>ρ</sup> tf*

� � � � � *kB*

The first equality uses the cyclic property of the trace and the definition of *ρ*<sup>0</sup>

manifested as a monotonic increase in the entropy. Thermodynamic entropy, it is thought, is related to the entropy of the states defined in both classical and quantum theory. Under an automorphic time evolution, the entropy is conserved. One application of an environment is to account for an increase. A type of coursegraining becomes necessary together with the right conditions on the initial state to account for the arrow of time. In quantum mechanics, the course-graining seems to be necessary and may be thought of as a restriction of the algebra and can also be interpreted as leaving out unobservable quantum correlations. This may, for example, correspond to decoherence effects important in quantum measurements. Competing effects arise such as the fact that correlations becoming unobservable may lead to entropy increase. There is also the effect that a decrease in entropy might be due to nonautomorphic processes. Although both effects lead to irreversibility, they are not cooperative but rather contrary to one

another. The observation that the second law does hold implies these

nonautomorphic events must be rare in comparison with time scales relevant to

The second equality uses decoherence of *ρ*ð Þ 0 , and the next inequality is a consequence of (34). The evolution is unitary and hence preserves entropy which is the last equality. This implies that *S t*ð Þ≥*S*ð Þ 0 and the equality condition (**e**) follows from (32). □ Of course, entropy growth as in the theorem is not necessarily monotonic in the time variable. For this reason, it is usual to refer to fixed initial and final states. For thermal systems, a natural choice of the final state is the equilibrium state of the system. It is the case in thermodynamics that irreversibility is

and it is not an equilibrium state of the system. Let *ρ tf*

� � if and only if **(e)** P

� � log ð Þ *<sup>ρ</sup>*ð Þ <sup>0</sup> <sup>≤</sup>*<sup>S</sup> <sup>ρ</sup> tf*

*<sup>α</sup> παρ tf*

ð Þ <sup>0</sup> � � ¼ �*<sup>S</sup> <sup>ρ</sup>*<sup>0</sup> *tf*

the second law.

*Quantum Mechanics*

where *S*ð Þ¼ 0 *S tf*

*S ρ*<sup>0</sup> *tf* � �j*ρ*<sup>0</sup>

� � � *kB*Tr *<sup>ρ</sup> tf*

� � <sup>¼</sup> <sup>P</sup>

a completely positive map. It follows that

Proof: Set *ρ*<sup>0</sup> *tf*

¼ �*S tf*

thermodynamics.

**90**

respect to *πα* and have finite entropy

By the chain rule

$$d\left(\sum\_{n} p\_n \log \left(p\_n\right)\right) = \sum\_{n} dp\_n \log \left(p\_n\right) + \sum\_{n} dp\_n = \sum\_{n} dp\_n \log \left(p\_n\right). \tag{47}$$

So *δQ* is not a function of the state but is related to the variation of something that is. Define the entropy *<sup>S</sup>* as usual from (9), *<sup>S</sup>* ¼ �<sup>P</sup> *<sup>n</sup> pn* log *pn* � �, and arrive at

$$
\delta Q = TdS.\tag{48}
$$

Consider nonunitary quantum dynamics. Initially, the system has Hamiltonian *Hi* ¼ *H γ<sup>i</sup>* ð Þ. The system was in thermal equilibrium with a bath at temperature *T*. The initial state of the system is the Gibbs thermal density matrix (38). Let *ε<sup>i</sup>*

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

Immediately after this measurement, *γ* changes from *γ*ð Þ¼ 0 *γ<sup>i</sup>* to *γ τ*ð Þ¼ *γ <sup>f</sup>* according to the rule *γ*ð Þ*t* . If it is assumed the contact with the bath is very weak

where *U* is the unitary evolution operator which satisfies Schrödinger's equation,

*<sup>n</sup>* measured is <sup>∣</sup>h*m*j j *ψ τ*ð Þi <sup>2</sup> <sup>¼</sup> <sup>∣</sup>h*m*∣*U*ð Þ*<sup>τ</sup>* j j *<sup>n</sup>*<sup>i</sup> <sup>2</sup>

*<sup>W</sup>* <sup>¼</sup> *<sup>ε</sup> <sup>f</sup>*

interpreted as the conditional probability a system in ∣*n*i will be in ∣*m*i after time *τ*. No heat has been exchanged with the environment, so any change in the environment has to be attributed to the work performed by the external agent and is

� � at the end and has energy levels *<sup>ε</sup> <sup>f</sup>*

*<sup>m</sup>* � *<sup>ε</sup><sup>i</sup>*

*<sup>n</sup>* is random due to thermal fluctuations and *<sup>ε</sup> <sup>f</sup>*

To get an expression for Pð Þ *W* obtained by repeating the process several times, this is a two-step measurement process. From probability theory, if *A*, *B* are two

where *p B*ð Þ is the probability *B* which occurs and *p A*ð Þ j*B* is the conditional probability *B* that has occurred. The probability of both events that have occurred is

<sup>∣</sup>h*m*∣*U*ð Þ*<sup>τ</sup>* j j *<sup>n</sup>*<sup>i</sup> <sup>2</sup>

*<sup>m</sup>* � *<sup>ε</sup><sup>i</sup>*

rather large number of allowed levels, and even more allowed differences *ε <sup>f</sup>*

*iyW* � � <sup>¼</sup>

2*π* ð<sup>∞</sup> �∞

Pð Þ¼ *<sup>W</sup>* <sup>1</sup>

*pn*. Since we are interested in the work performed, we write

And some over all allowed events, weighted by their probabilities, and arrange

ð<sup>∞</sup> �∞

Pð Þ *W e*

*dy*Gð Þ*y e*

*<sup>m</sup>* are fluctuating and change during each realization of the

*p A*ð Þ¼ , *B p A*ð Þ j*B p B*ð Þ, (54)

*<sup>m</sup>* � *<sup>ε</sup><sup>i</sup> n* � � � � *:* (55)

*<sup>n</sup>*. In most systems, there are present a

*iyWdW:* (56)

�*iyW:* (57)

*pn <sup>δ</sup> <sup>W</sup>* � *<sup>ε</sup> <sup>f</sup>*

<sup>∣</sup>*n*<sup>i</sup> denote the initial eigenvalues and eigenvectors of *Hi* as *<sup>ε</sup><sup>i</sup>*

during this process, the state of the system evolves according to

*<sup>n</sup> =Z*.

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

probability *pn* <sup>¼</sup> *<sup>e</sup>*�*βε<sup>i</sup>*

*<sup>i</sup>∂tU* <sup>¼</sup> *H t*ð Þ*U*, *<sup>U</sup>*ð Þ¼ <sup>0</sup> **<sup>1</sup>**.

so the probability *ε <sup>f</sup>*

where both *ε<sup>i</sup>*

<sup>∣</sup>h*m*∣*U*ð Þ*<sup>τ</sup>* j j *<sup>n</sup>*<sup>i</sup> <sup>2</sup>

experiment. The first *ε<sup>i</sup>*

The Hamiltonian is *H γ <sup>f</sup>*

*<sup>n</sup>* and *<sup>ε</sup> <sup>f</sup>*

to quantum fluctuations in *W* as a random variable by (53).

Pð Þ¼ *<sup>W</sup>* <sup>X</sup>

is more efficient to use the Fourier transform

This has the inverse Fourier transform

the terms according to the values *ε <sup>f</sup>*

Using (55), we obtain that

**93**

*n*, *m*

Gð Þ¼ *y e*

events, the total probability *p A*ð Þ j*B* that both events have occurred is

*<sup>n</sup>* and

*<sup>n</sup>* is obtained with

*<sup>m</sup>*, eigenvectors ∣*m*i,

*<sup>m</sup>* is random due

*<sup>m</sup>* � *<sup>ε</sup><sup>i</sup> <sup>n</sup>*. It

. This may be

*<sup>n</sup>*, (53)

∣*ψ*ð Þi ¼ *t U t*ð Þ∣*n*i, (52)

This relation only holds for infinitesimal processes. For finite and irreversible processes, there may be additional terms to the entropy change. This has been quite successful at describing many different types of physical system [17–19].

A deep insight has come recently into the properties of nonequilibrium thermodynamics which could be achieved by regarding work as a random variable. For example, consider a process in which a piston is used to compress a gas in a cylinder. Due to the nature of the gas and its chaotic motion, each time the piston is pressed, the gas molecules exert a back reaction with a different force. This means the work needed to achieve a given compression changes each time something is carried out.

Usually a knowledge of nonequilibrium processes is restricted to inequalities such as the Jarzynski inequality. He was able to show by interpreting work *W* as a random variable that an inequality can be obtained, even for a process performed arbitrarily far from equilibrium.

Suppose the system is always prepared in the same state initially. A process is carried out and the total work *W* performed is measured. Repeating this many times, a probability distribution for the work Pð Þ *W* can be constructed. An average for *W* can be computed using Pð Þ *W* as

$$
\langle \mathcal{W} \rangle = \int \mathcal{P}(\mathcal{W}) d\mathcal{W}.\tag{49}
$$

Jarzynski showed that the statistical average of *e*�*β<sup>W</sup>* satisfies

$$
\langle e^{-\beta W} \rangle = e^{-\beta \Delta F},
\tag{50}
$$

where Δ*F* ¼ *F T*, *γ <sup>f</sup>* � � � *F T*, *<sup>γ</sup><sup>i</sup>* ð Þ. It holds for a process performed arbitrarily far from equilibrium. Now the inequality *W* ≥ Δ*F* is contained in (50) and can be realized by applying Jensen's inequality, which states that *e*�*β<sup>W</sup>* � �≥*e*�*β*h i *<sup>W</sup>* .

In macroscopic systems, individual measurements are usually very close to the average by the law of large numbers. For mictoscopic systems, this is usually not true. In fact, the individual realizations of *W* may be smaller than Δ*F*. These cases would be local violations of the second law but for large systems become extremely rare. If the function Pð Þ *W* is known, the probability of a local violation of the second law is

$$\mathcal{P}(W < \Delta F) = \int\_{-\infty}^{\Delta F} \mathcal{P}(W) \, dW. \tag{51}$$

To get (50) requires detailed knowledge of the system's dynamics, be it classical, quantum, unitary, or whatever.

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

Consider nonunitary quantum dynamics. Initially, the system has Hamiltonian *Hi* ¼ *H γ<sup>i</sup>* ð Þ. The system was in thermal equilibrium with a bath at temperature *T*. The initial state of the system is the Gibbs thermal density matrix (38). Let *ε<sup>i</sup> <sup>n</sup>* and <sup>∣</sup>*n*<sup>i</sup> denote the initial eigenvalues and eigenvectors of *Hi* as *<sup>ε</sup><sup>i</sup> <sup>n</sup>* is obtained with probability *pn* <sup>¼</sup> *<sup>e</sup>*�*βε<sup>i</sup> <sup>n</sup> =Z*.

Immediately after this measurement, *γ* changes from *γ*ð Þ¼ 0 *γ<sup>i</sup>* to *γ τ*ð Þ¼ *γ <sup>f</sup>* according to the rule *γ*ð Þ*t* . If it is assumed the contact with the bath is very weak during this process, the state of the system evolves according to

$$|\psi(t)\rangle = U(t)|n\rangle,\tag{52}$$

where *U* is the unitary evolution operator which satisfies Schrödinger's equation, *<sup>i</sup>∂tU* <sup>¼</sup> *H t*ð Þ*U*, *<sup>U</sup>*ð Þ¼ <sup>0</sup> **<sup>1</sup>**.

The Hamiltonian is *H γ <sup>f</sup>* � � at the end and has energy levels *<sup>ε</sup> <sup>f</sup> <sup>m</sup>*, eigenvectors ∣*m*i, so the probability *ε <sup>f</sup> <sup>n</sup>* measured is <sup>∣</sup>h*m*j j *ψ τ*ð Þi <sup>2</sup> <sup>¼</sup> <sup>∣</sup>h*m*∣*U*ð Þ*<sup>τ</sup>* j j *<sup>n</sup>*<sup>i</sup> <sup>2</sup> . This may be

interpreted as the conditional probability a system in ∣*n*i will be in ∣*m*i after time *τ*. No heat has been exchanged with the environment, so any change in the envi-

ronment has to be attributed to the work performed by the external agent and is

$$\mathcal{W} = \varepsilon\_m^f - \varepsilon\_n^i,\tag{53}$$

where both *ε<sup>i</sup> <sup>n</sup>* and *<sup>ε</sup> <sup>f</sup> <sup>m</sup>* are fluctuating and change during each realization of the experiment. The first *ε<sup>i</sup> <sup>n</sup>* is random due to thermal fluctuations and *<sup>ε</sup> <sup>f</sup> <sup>m</sup>* is random due to quantum fluctuations in *W* as a random variable by (53).

To get an expression for Pð Þ *W* obtained by repeating the process several times, this is a two-step measurement process. From probability theory, if *A*, *B* are two events, the total probability *p A*ð Þ j*B* that both events have occurred is

$$p(A,B) = p(A|B)p(B),\tag{54}$$

where *p B*ð Þ is the probability *B* which occurs and *p A*ð Þ j*B* is the conditional probability *B* that has occurred. The probability of both events that have occurred is <sup>∣</sup>h*m*∣*U*ð Þ*<sup>τ</sup>* j j *<sup>n</sup>*<sup>i</sup> <sup>2</sup> *pn*. Since we are interested in the work performed, we write

$$\mathcal{P}(\mathcal{W}) = \sum\_{n,m} |\langle m|U(\mathfrak{r})|n\rangle|^2 p\_n \delta(\mathcal{W} - \left(\varepsilon\_m^f - \varepsilon\_n^i\right)). \tag{55}$$

And some over all allowed events, weighted by their probabilities, and arrange the terms according to the values *ε <sup>f</sup> <sup>m</sup>* � *<sup>ε</sup><sup>i</sup> <sup>n</sup>*. In most systems, there are present a rather large number of allowed levels, and even more allowed differences *ε <sup>f</sup> <sup>m</sup>* � *<sup>ε</sup><sup>i</sup> <sup>n</sup>*. It is more efficient to use the Fourier transform

$$\mathcal{G}(\mathbf{y}) = \left\langle e^{\mathbf{j}\cdot\mathbf{W}} \right\rangle = \int\_{-\infty}^{\infty} \mathcal{P}(\mathbf{W}) e^{\mathbf{j}\cdot\mathbf{W}} d\mathbf{W}.\tag{56}$$

This has the inverse Fourier transform

$$\mathcal{P}(W) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} dy \, \mathcal{G}(y) e^{-i\mathcal{y}W}.\tag{57}$$

Using (55), we obtain that

By the chain rule

*pn* log *pn* � � !

arbitrarily far from equilibrium.

where Δ*F* ¼ *F T*, *γ <sup>f</sup>*

quantum, unitary, or whatever.

second law is

**92**

for *W* can be computed using Pð Þ *W* as

� �

<sup>¼</sup> <sup>X</sup> *n*

*dpn* log *pn*

successful at describing many different types of physical system [17–19].

thing that is. Define the entropy *<sup>S</sup>* as usual from (9), *<sup>S</sup>* ¼ �<sup>P</sup>

� � þ<sup>X</sup> *n*

So *δQ* is not a function of the state but is related to the variation of some-

This relation only holds for infinitesimal processes. For finite and irreversible processes, there may be additional terms to the entropy change. This has been quite

A deep insight has come recently into the properties of nonequilibrium thermodynamics which could be achieved by regarding work as a random variable. For example, consider a process in which a piston is used to compress a gas in a cylinder. Due to the nature of the gas and its chaotic motion, each time the piston is pressed, the gas molecules exert a back reaction with a different force. This means the work needed to achieve a given compression changes each time something is carried out. Usually a knowledge of nonequilibrium processes is restricted to inequalities such as the Jarzynski inequality. He was able to show by interpreting work *W* as a random variable that an inequality can be obtained, even for a process performed

Suppose the system is always prepared in the same state initially. A process is carried out and the total work *W* performed is measured. Repeating this many times, a probability distribution for the work Pð Þ *W* can be constructed. An average

ð

from equilibrium. Now the inequality *W* ≥ Δ*F* is contained in (50) and can be realized by applying Jensen's inequality, which states that *e*�*β<sup>W</sup>* � �≥*e*�*β*h i *<sup>W</sup>* .

In macroscopic systems, individual measurements are usually very close to the average by the law of large numbers. For mictoscopic systems, this is usually not true. In fact, the individual realizations of *W* may be smaller than Δ*F*. These cases would be local violations of the second law but for large systems become extremely rare. If the function Pð Þ *W* is known, the probability of a local violation of the

> ð<sup>Δ</sup>*<sup>F</sup>* �∞

To get (50) requires detailed knowledge of the system's dynamics, be it classical,

h i *W* ¼

Jarzynski showed that the statistical average of *e*�*β<sup>W</sup>* satisfies

*e* �*β<sup>W</sup>* � � <sup>¼</sup> *<sup>e</sup>*

Pð Þ¼ *W* < Δ*F*

*dpn* <sup>¼</sup> <sup>X</sup> *n*

*δQ* ¼ *T dS:* (48)

Pð Þ *W dW:* (49)

�*β*Δ*<sup>F</sup>*, (50)

Pð Þ *W dW:* (51)

� *F T*, *γ<sup>i</sup>* ð Þ. It holds for a process performed arbitrarily far

*dpn* log *pn*

� �*:* (47)

� �, and

*<sup>n</sup> pn* log *pn*

*d* X *n*

*Quantum Mechanics*

arrive at

$$\begin{split} \mathcal{G}(\boldsymbol{\jmath}) &= \sum\_{n,m} |\langle m|U|n\rangle|^{2} p\_{n} e^{j\chi\left(\boldsymbol{\varepsilon}\_{n}^{\boldsymbol{f}} - \boldsymbol{\varepsilon}\_{n}^{\boldsymbol{i}}\right)} = \sum\_{n,m} \left\langle n|U^{\dagger} e^{j\boldsymbol{\varepsilon}\_{m}^{\boldsymbol{f}}}|m\rangle \left\langle m|U e^{-j\boldsymbol{\varepsilon}\_{n}^{\boldsymbol{i}}} p\_{n}|n\right\rangle \\ &= \sum\_{n,m} \left\langle n|U^{\dagger} e^{j\boldsymbol{\varepsilon}\_{\boldsymbol{f}}^{\boldsymbol{i}H}}|m\rangle \langle m|U e^{-j\boldsymbol{\varepsilon}\_{\boldsymbol{f}}^{\boldsymbol{i}}} \rho|n\rangle = \operatorname{Tr}\left(U^{\dagger}(\boldsymbol{\varepsilon}) e^{j\boldsymbol{\varepsilon}\_{\boldsymbol{f}}^{\boldsymbol{i}H}} U(\boldsymbol{\varepsilon}) e^{-j\boldsymbol{\varepsilon}\_{\boldsymbol{f}}^{\boldsymbol{i}}{}} \rho\right). \end{split} \tag{58}$$

Hence, it may be concluded that

$$\mathcal{G} = \operatorname{Tr}\left( U^{\dagger}(\tau) e^{i y H\_f} U(\tau) e^{-i y H\_i} \rho \right). \tag{59}$$

where *β* ¼ 1*=KBT*. The dynamics of an open quantum system is described by a quantum operator ~*ρ* ¼ *Sρ*, a linear trace-preserving, complete positive map of operators. Any such complete positive superoperator has an operator-sum

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

*<sup>S</sup><sup>ρ</sup>* <sup>¼</sup> <sup>X</sup> *α*

trace-preserving and conserves probability if P

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

such that the initial combined state is *ρ<sup>S</sup>* ⊗ *ρ<sup>B</sup>*

of the system dynamics can also be obtained:

*<sup>H</sup>* <sup>¼</sup> *<sup>H</sup><sup>S</sup>*

*eq* � �*U*† <sup>¼</sup> <sup>X</sup>

(66) implies the Krauss operators for this dynamics are

ð Þ*t* ⊗ *I*

*i*, *f*

¼ 1 *ZB* X *i*, *f e* �*βε<sup>B</sup>*

Here *U* is the unitary evolution operator of the total system

*U* ¼ exp

*Ai*,*<sup>f</sup>* <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffi *ZB* p *e*

and *TrB* is the partial trace over the bath degrees of freedom, *ε<sup>B</sup>*

*i* ℏ ð*t s H*ð Þ*τ dτ*

eigenvalues, f g <sup>j</sup>*b*<sup>i</sup> is the orthonormal energy eigenvectors of the bath, and *<sup>Z</sup><sup>B</sup>* is the bath partition function. Assume the bath energy states are nondegenerate. Then

�*βε<sup>B</sup>*

Suppose the environment is large, with a characteristic relaxation time short compared with the bath-system interactions, and the system-bath coupling *ε* is small. The environment remains near thermal equilibrium, unentangled and uncorrelated with the system. The system dynamics of each consecutive time

*<sup>i</sup> <sup>=</sup>*<sup>2</sup> *<sup>b</sup> <sup>f</sup>* <sup>j</sup>*U*j*bi*

Conversely, any operator-sum represents a complete positive superoperator. The set of operators f g *A<sup>α</sup>* is often called Krauss operators. The superoperator is

dynamics of an isolated quantum system is described by a single unitary operator

The interest here is in the dynamics of a quantum system governed by a timedependent Hamiltonian weakly coupled to an extended, thermal environment. Let

*<sup>B</sup>* <sup>þ</sup> **<sup>I</sup>**

Hamiltonian, *H<sup>B</sup>* the bath Hamiltonian, and *Hint* the bath-system interaction with *ε* a small parameter. Assume initially the system and environment are uncorrelated

By following the unitary dynamics of the combined total system for a finite time and measuring the final state of the environment, a quantum operator description

<sup>h</sup>*<sup>b</sup> <sup>f</sup>* <sup>∣</sup>*U*ð*ρ<sup>S</sup>* <sup>⊗</sup> <sup>ð</sup>

*<sup>B</sup>* are system and bath identity operators, *<sup>H</sup><sup>S</sup>*ð Þ*<sup>t</sup>* the system

*eq*, where *ρ<sup>B</sup>*

X *i*

*<sup>i</sup> b <sup>f</sup>* j*U*j*bi*

*e*�*βε<sup>B</sup> i*

� �*ρ<sup>S</sup> bi*j*U*†

*<sup>Z</sup><sup>B</sup>* <sup>∣</sup>*bi*<sup>i</sup> *bi*jÞÞ*U*†

� �, (67)

j*b f* � �*:*

*i*

� �*:* (68)

� � are the energy

j*b f* � �

(66)

*AαρA*†

*<sup>α</sup> A*†

*<sup>α</sup>:* (64)

*<sup>α</sup>A<sup>α</sup>* ¼ **I**. In the simplest case, the

*<sup>S</sup>* <sup>⊗</sup> *<sup>H</sup><sup>B</sup>* <sup>þ</sup> *<sup>ε</sup>Hint*, (65)

*eq* is the thermal density

representation

*<sup>U</sup>*† <sup>¼</sup> *<sup>U</sup>*�<sup>1</sup>

where **I**

**95**

.

the total Hamiltonian be

*<sup>S</sup>* and **I**

equilibrium matrix of the bath.

*S s*ð Þ , *<sup>t</sup> <sup>ρ</sup><sup>S</sup>* <sup>¼</sup> Tr*<sup>B</sup> <sup>U</sup> <sup>ρ</sup><sup>S</sup>* <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup>*

This turns out to be somewhat easier to work with than Pð Þ *W* , and (59) plays a similar role as *Z* in equilibrium statistical mechanics. From *G y* ð Þ, the statistical moments of *W* can be found by expanding

$$\mathcal{G}(\mathbf{y}) = \left< \epsilon^{\dot{\mathbf{y}} \cdot \mathbf{W}} \right> = \mathbf{1} + i \mathbf{y} \langle \mathbf{W} \rangle - \frac{\mathbf{y}^2}{2} \left< \mathbf{W}^2 \right> - \frac{\mathbf{y}^3}{6} \langle \mathbf{W}^3 \rangle + \cdots. \tag{60}$$

A formula for the quantum mechanical formula for the moments can be found as well. The average work is h i *W* ¼ *H <sup>f</sup>* � � � h i *Hi* , where for any operator *<sup>A</sup>*, we have h i *<sup>A</sup> <sup>t</sup>* <sup>¼</sup> *Tr U*†ð Þ*<sup>t</sup> AU t*ð Þ*<sup>ρ</sup>* � � as the expectation value of *<sup>A</sup>* at time *<sup>t</sup>*. This follows from the fact that the state of the system at *<sup>t</sup>* is *<sup>ρ</sup>*ðÞ¼ *<sup>t</sup> U t*ð Þ*ρU t*ð Þ† . From the definition of G, it ought to be the case that *G y*ð Þ¼ <sup>¼</sup> *<sup>i</sup><sup>β</sup> <sup>e</sup>*�*β<sup>W</sup>* � �. However, *<sup>ρ</sup>* in (38) and (59) yields

$$\mathcal{G}(i\boldsymbol{\beta}) = \frac{\mathbf{1}}{Z\_i} \operatorname{Tr} \left( U^\dagger e^{-\beta H\_f} U \right) = \frac{\mathbf{1}}{Z\_i} \operatorname{Tr} \left( e^{\beta H\_f} \right) = \frac{Z\_f}{Z\_i}.\tag{61}$$

Using *<sup>Z</sup>* <sup>¼</sup> *<sup>e</sup>*�*β<sup>F</sup>*, (61) yields (50)

$$\mathcal{G}(i\mathcal{y}) = \left\langle e^{-\beta \mathcal{W}} \right\rangle = e^{-\beta \Delta F}. \tag{62}$$

Nothing has been assumed about the speed of this process. Thus inequality (50) must hold for a process arbitrarily far from equilibrium.

#### **4. Heat flow from environment approach**

There is another somewhat different way in which the Jarzynski inequality can be generalized to quantum dynamics. In a classical system, the energy of the system can be continuously measured as well as the flow of heat and work. Continuous measurement is not possible in quantum mechanics without disrupting the dynamics of the system [20].

A more satisfactory approach is to realize that although work cannot be continuously measured, the heat flow from the environment can be measured. To this end, the system of interest is divided into a system of interest and a thermal bath. The ambient environment is large, and it rapidly decoheres and remains at thermal equilibrium, uncorrelated and unentangled with the system. Consequently, we can measure the change in energy of the bath ð Þ �*Q* without disturbing the dynamics of the system. The open-system Jarzynski identity is expressed as

$$
\langle \mathbf{e}^{-\beta \mathcal{W}} \rangle = \langle \mathbf{e}^{-\beta \mathcal{E}\_f} \mathbf{e}^{\beta \mathcal{Q}} \mathbf{e}^{\beta \mathcal{E}\_i} \rangle = \mathbf{e}^{-\beta \Delta F}. \tag{63}
$$

For a system that has equilibrated with Hamiltonian *H* interacting with a thermal bath at temperature *<sup>T</sup>*, the equilibrium density matrix is *<sup>ρ</sup>* <sup>¼</sup> *<sup>e</sup>β<sup>H</sup>=<sup>Z</sup>* <sup>¼</sup> *<sup>e</sup>*�*βF*�*β<sup>H</sup>*, *Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

where *β* ¼ 1*=KBT*. The dynamics of an open quantum system is described by a quantum operator ~*ρ* ¼ *Sρ*, a linear trace-preserving, complete positive map of operators. Any such complete positive superoperator has an operator-sum representation

$$\mathcal{S}\rho = \sum\_{a} A\_{a}\rho A\_{a}^{\dagger}.\tag{64}$$

Conversely, any operator-sum represents a complete positive superoperator. The set of operators f g *A<sup>α</sup>* is often called Krauss operators. The superoperator is trace-preserving and conserves probability if P *<sup>α</sup> A*† *<sup>α</sup>A<sup>α</sup>* ¼ **I**. In the simplest case, the dynamics of an isolated quantum system is described by a single unitary operator *<sup>U</sup>*† <sup>¼</sup> *<sup>U</sup>*�<sup>1</sup> .

The interest here is in the dynamics of a quantum system governed by a timedependent Hamiltonian weakly coupled to an extended, thermal environment. Let the total Hamiltonian be

$$H = H^S(t) \otimes I^B + \mathbf{I}^S \otimes H^B + eH^{int},\tag{65}$$

where **I** *<sup>S</sup>* and **I** *<sup>B</sup>* are system and bath identity operators, *<sup>H</sup><sup>S</sup>*ð Þ*<sup>t</sup>* the system Hamiltonian, *H<sup>B</sup>* the bath Hamiltonian, and *Hint* the bath-system interaction with *ε* a small parameter. Assume initially the system and environment are uncorrelated such that the initial combined state is *ρ<sup>S</sup>* ⊗ *ρ<sup>B</sup> eq*, where *ρ<sup>B</sup> eq* is the thermal density equilibrium matrix of the bath.

By following the unitary dynamics of the combined total system for a finite time and measuring the final state of the environment, a quantum operator description of the system dynamics can also be obtained:

$$\begin{split} S(\mathbf{s},t)\rho^{\mathcal{S}} = \operatorname{Tr}\_{\mathcal{B}} U\Big(\rho^{\mathcal{S}} \otimes \rho\_{\neq \mathbf{q}}^{\mathcal{B}}\Big)U^{\dagger} &= \sum\_{i,f} \langle b\_{f}|U(\rho^{\mathcal{S}} \otimes (\sum\_{i} \frac{e^{-\beta \boldsymbol{\kappa}\_{i}^{\mathcal{S}}}}{Z^{\mathcal{B}}}|b\_{i}\rangle\langle b\_{i}|)|U^{\dagger}|b\_{f}\rangle\\ &= \frac{1}{Z\_{\mathcal{B}}}\sum\_{i,f} e^{-\beta \boldsymbol{\kappa}\_{i}^{\mathcal{S}}} \langle b\_{f}|U|b\_{i}\rangle \rho^{\mathcal{S}} \langle b\_{i}|U^{\dagger}|b\_{f}\rangle. \end{split} \tag{66}$$

Here *U* is the unitary evolution operator of the total system

$$U = \exp\left(\frac{i}{\hbar} \int\_{s}^{t} H(\tau) d\tau\right),\tag{67}$$

and *TrB* is the partial trace over the bath degrees of freedom, *ε<sup>B</sup> i* � � are the energy eigenvalues, f g <sup>j</sup>*b*<sup>i</sup> is the orthonormal energy eigenvectors of the bath, and *<sup>Z</sup><sup>B</sup>* is the bath partition function. Assume the bath energy states are nondegenerate. Then (66) implies the Krauss operators for this dynamics are

$$A\_{if} = \frac{1}{\sqrt{Z\_B}} e^{-\beta c\_i^B/2} \langle b\_f | U | b\_i \rangle. \tag{68}$$

Suppose the environment is large, with a characteristic relaxation time short compared with the bath-system interactions, and the system-bath coupling *ε* is small. The environment remains near thermal equilibrium, unentangled and uncorrelated with the system. The system dynamics of each consecutive time

Gð Þ¼ *<sup>y</sup>* <sup>X</sup> *n*, *m*

*Quantum Mechanics*

<sup>¼</sup> <sup>X</sup> *n*, *m*

<sup>∣</sup>h*m*∣*U n*j ji <sup>2</sup>

*<sup>n</sup>*j*U*†*<sup>e</sup>*

Hence, it may be concluded that

moments of *W* can be found by expanding

Gð Þ¼ *y e*

well. The average work is h i *W* ¼ *H <sup>f</sup>*

Gð Þ¼ *iβ*

Using *<sup>Z</sup>* <sup>¼</sup> *<sup>e</sup>*�*β<sup>F</sup>*, (61) yields (50)

ics of the system [20].

**94**

*pn e iy ε <sup>f</sup>*

*<sup>m</sup>*�*ε<sup>i</sup>* ð Þ*<sup>n</sup>* <sup>¼</sup> <sup>X</sup>

*iyH <sup>f</sup>* <sup>j</sup>*<sup>m</sup>* � � *<sup>m</sup>*j*Ue*�*iyHi <sup>ρ</sup>*j*<sup>n</sup>* � � <sup>¼</sup> Tr *<sup>U</sup>*†ð Þ*<sup>τ</sup> <sup>e</sup>*

G ¼ *Tr U*†

*iyW* � � <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *iy W*h i � *<sup>y</sup>*<sup>2</sup>

the fact that the state of the system at *<sup>t</sup>* is *<sup>ρ</sup>*ðÞ¼ *<sup>t</sup> U t*ð Þ*ρU t*ð Þ†

1 *Zi*

must hold for a process arbitrarily far from equilibrium.

the system. The open-system Jarzynski identity is expressed as

*e* �*β<sup>W</sup>* � � <sup>¼</sup> *<sup>e</sup>*

**4. Heat flow from environment approach**

Tr *U*† *e* �*β<sup>H</sup> <sup>f</sup> <sup>U</sup>* � � <sup>¼</sup> <sup>1</sup>

Gð Þ¼ *iy e*

*n*, *m*

ð Þ*τ e*

similar role as *Z* in equilibrium statistical mechanics. From *G y* ð Þ, the statistical

*<sup>n</sup>*j*U*†*<sup>e</sup> iyε <sup>f</sup> <sup>m</sup>* j*m* D E

*iyH <sup>f</sup> <sup>U</sup>*ð Þ*<sup>τ</sup> <sup>e</sup>*

<sup>2</sup> *<sup>W</sup>*<sup>2</sup> � � � *<sup>y</sup>*<sup>3</sup>

*Zi* Tr *e βH <sup>f</sup>* � � <sup>¼</sup> *<sup>Z</sup> <sup>f</sup>*

This turns out to be somewhat easier to work with than Pð Þ *W* , and (59) plays a

A formula for the quantum mechanical formula for the moments can be found as

h i *<sup>A</sup> <sup>t</sup>* <sup>¼</sup> *Tr U*†ð Þ*<sup>t</sup> AU t*ð Þ*<sup>ρ</sup>* � � as the expectation value of *<sup>A</sup>* at time *<sup>t</sup>*. This follows from

it ought to be the case that *G y*ð Þ¼ <sup>¼</sup> *<sup>i</sup><sup>β</sup> <sup>e</sup>*�*β<sup>W</sup>* � �. However, *<sup>ρ</sup>* in (38) and (59) yields

�*β<sup>W</sup>* � � <sup>¼</sup> *<sup>e</sup>*

Nothing has been assumed about the speed of this process. Thus inequality (50)

There is another somewhat different way in which the Jarzynski inequality can be generalized to quantum dynamics. In a classical system, the energy of the system can be continuously measured as well as the flow of heat and work. Continuous measurement is not possible in quantum mechanics without disrupting the dynam-

A more satisfactory approach is to realize that although work cannot be continuously measured, the heat flow from the environment can be measured. To this end, the system of interest is divided into a system of interest and a thermal bath. The ambient environment is large, and it rapidly decoheres and remains at thermal equilibrium, uncorrelated and unentangled with the system. Consequently, we can measure the change in energy of the bath ð Þ �*Q* without disturbing the dynamics of

> �*β<sup>E</sup> fe <sup>β</sup><sup>Q</sup> e βEi* � � <sup>¼</sup> *<sup>e</sup>*

For a system that has equilibrated with Hamiltonian *H* interacting with a thermal bath at temperature *<sup>T</sup>*, the equilibrium density matrix is *<sup>ρ</sup>* <sup>¼</sup> *<sup>e</sup>β<sup>H</sup>=<sup>Z</sup>* <sup>¼</sup> *<sup>e</sup>*�*βF*�*β<sup>H</sup>*,

*<sup>m</sup>*j*Ue*�*iyε<sup>i</sup>*

*iyH <sup>f</sup> <sup>U</sup>*ð Þ*<sup>τ</sup> <sup>e</sup>* �*iyHiρ* � �*:*

�*iyHi ρ* � �*:* (59)

� � � h i *Hi* , where for any operator *<sup>A</sup>*, we have

*Zi*

�*β*Δ*<sup>F</sup>:* (62)

�*β*<sup>Δ</sup> *<sup>F</sup>:* (63)

*<sup>n</sup> pn*j*n* D E

<sup>6</sup> *<sup>W</sup>*<sup>3</sup> � � <sup>þ</sup> <sup>⋯</sup>*:* (60)

. From the definition of G,

*:* (61)

(58)

interval can be described by a superoperator derived as in (66) which can then be chained together to form a quantum Markov chain:

$$\rho(t) = \mathbb{S}(t-1,t)\cdots\mathbb{S}(s+1,s+2)\mathbb{S}(s,s+1)\rho. \tag{69}$$

The Hermitian operator of a von Neumann-type measurement can be broken up into a set of eigenvalues P *λσ* and orthonormal projection operators *πσ* such that *H* ¼ *<sup>σ</sup> λσπσ*. In a more general sense, the measured operator of a positive operatorvalued measurement need not be projectors or orthonormal. The probability of observing the *a*-th outcome is

$$p\_a = \text{Tr}\left(A\_a \rho A\_a^\dagger\right). \tag{70}$$

Begin with a composite system which consists of the bath, initially in thermal

*ρ<sup>S</sup>* ⊗ *ρ<sup>B</sup>*

*eq* � � *<sup>I</sup>*

*eq* � � *<sup>I</sup>*

Taking the trace over the bath degrees of freedom produces the final normalized system density matrix where trace over *S* gives the probability of observing the given initial and final bath eigenstates. Multiply by the Boltzmann weighted heat,

Measure the initial energy eigenstate of the bath so based on (76):

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

Now allow the system to evolve together with the bath for some time:

*eq* � � *<sup>I</sup>*

and sum over the initial and final bath states to obtain the desired average

*TrSTrB*ð*I*

*ρ<sup>S</sup>* ⊗ *ρ<sup>B</sup> eq* � �ð*<sup>I</sup>*

tion Hamiltonian can be omitted in the small coupling limit giving

*<sup>β</sup>HS=*<sup>2</sup> *TrB Ue*�*βHS=*<sup>2</sup>

X *α e βHS=*2 *Aαe*

*e*

¼ Tr*<sup>S</sup>*

*<sup>S</sup>* ⊗ ∣*bi*ih*bi*<sup>∣</sup> � � *<sup>ρ</sup><sup>S</sup>* <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup>*

*U I<sup>S</sup>* ⊗ ∣*bi*ih*bi*<sup>∣</sup> � � *<sup>ρ</sup><sup>S</sup>* <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup>*

Finally, measure the final energy eigenstate of the bath:

*eq:* (76)

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>j</sup>*ih*<sup>b</sup> <sup>j</sup>*<sup>∣</sup> � �*:* (77)

*<sup>S</sup>* ⊗ ∣*bi*ih*bi*∣<sup>Þ</sup>

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>j</sup>*ih*<sup>b</sup> <sup>j</sup>*∣Þ*:*

�*βHS=*<sup>2</sup> ⊗ *I <sup>B</sup>* � �*U*† *<sup>e</sup>*

*βHS=*2

*<sup>S</sup>*R*ρ<sup>S</sup>* � �*:* (83)

ð Þ*t* ⊗ *I*

*:* (78)

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>f</sup>*ih*<sup>b</sup> <sup>f</sup>* <sup>∣</sup> � �*:* (79)

(80)

*<sup>B</sup>* � *<sup>ε</sup>Hint*. The

*<sup>β</sup>HS=*<sup>2</sup> ⊗ *I <sup>B</sup>* � �

(81)

(82)

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>j</sup>*ih*<sup>b</sup> <sup>j</sup>*<sup>∣</sup> � �*U*†

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>j</sup>*ih*<sup>b</sup> <sup>j</sup>*<sup>∣</sup> � �*U*† *<sup>I</sup>*

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>f</sup>*ih*<sup>b</sup> <sup>f</sup>* <sup>∣</sup>Þ*U*ð*<sup>I</sup>*

ð*I*

*<sup>S</sup>* <sup>⊗</sup> *<sup>H</sup><sup>B</sup>* <sup>¼</sup> *<sup>H</sup>* � *<sup>H</sup><sup>S</sup>*

*eq* � � *<sup>e</sup>*

*<sup>S</sup>* ⊗ ∣*<sup>b</sup> <sup>j</sup>*ih*<sup>b</sup> <sup>j</sup>*∣Þ*U*†

total Hamiltonian commutes with the unitary dynamics and cancels. The interac-

⊗ *I <sup>B</sup>* � � *<sup>ρ</sup><sup>S</sup>* <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup>*

Krauss operators describing the reduced dynamics of the system, the result is

*<sup>β</sup><sup>Q</sup>* � � <sup>¼</sup> Tr R�<sup>1</sup>

Collecting the terms acting on the bath and system separately and replacing the

*ρS e* �*βHS=*<sup>2</sup> � � <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup> eqU*† � �*<sup>e</sup>*

To summarize, it has been found that the average Boltzmann weighted heat flow

where *S* represents the reduced dynamics of the system. The Hermitian map

�*βHS=*<sup>2</sup> *ρS e βHS=*2 *A*† *αe* �*βHS=*<sup>2</sup> *:*

equilibrium weakly coupled to the system:

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

*I*

*<sup>S</sup>* ⊗ ∣*bi*ih*<sup>b</sup> <sup>f</sup>* <sup>∣</sup> � �*U I<sup>S</sup>* ⊗ ∣*bi*ih*bi*<sup>∣</sup> � � *<sup>ρ</sup><sup>S</sup>* <sup>⊗</sup> *<sup>ρ</sup><sup>B</sup>*

Boltzmann weighted heat flow:

*i*, *f e* �*β ε<sup>B</sup> f* �*ε<sup>B</sup> i* � �

Replace the heat bath Hamiltonian by *I*

*<sup>β</sup>HS=*<sup>2</sup> ⊗ *I*

*<sup>S</sup>* � �*U e*�*β=*2*H<sup>S</sup>*

*e <sup>β</sup> <sup>Q</sup>* � � <sup>¼</sup> <sup>X</sup>

*<sup>β</sup><sup>Q</sup>* � � <sup>¼</sup> Tr*S*Tr*<sup>B</sup> <sup>e</sup>*

*e*

is represented by

**97**

*<sup>β</sup><sup>Q</sup>* � � <sup>¼</sup> Tr*<sup>S</sup> <sup>e</sup>*

superoperator R*<sup>t</sup>* is given by

*I*

*e*

The state of the system after this interaction is

$$\tilde{\rho}\_a = \frac{A\_a \rho A\_a^\dagger}{\text{Tr}\left(A\_a \rho A\_a^\dagger\right)}.\tag{71}$$

The result of the measurement can be represented by using a Hermitian map superoperator A:

$$\mathcal{A} = \sum\_{a} a\_{a} \mathcal{A}\_{a} \rho A\_{a}^{\dagger}. \tag{72}$$

An operator-value sum maps Hermitian operators into Hermitian operators:

$$\left[\mathcal{A}H\right]^\dagger = \left[a\_a A\_a H A\_a^\dagger\right]^\dagger = \sum\_a a\_a \left(A^\dagger\right)^\dagger H^\dagger A\_a^\dagger = \mathcal{A}H.\tag{73}$$

In the other direction, any Hermitian map has an operator-value-mean representation. Hermitian maps provide a particularly concise and convenient representation of sequential measurements and correlation functions. For example, suppose Hermitian map A represents a measurement at time 0, C is a different measurement at time *t*, and the quantum operation *St* represents the system evolution between the measurements. The expectation value of a single measurement is

$$\langle a \rangle = \text{Tr} \left( \mathcal{A} \rho \right) = \sum\_{a} a\_{a} \text{Tr} A\_{a} \rho A\_{a}^{\dagger} = \sum\_{a} p\_{a} a\_{a}. \tag{74}$$

The correlation function h i *b t*ð Þ*a*ð Þ 0 can be expressed as

$$\langle b(t)a(\mathbf{0})\rangle = \operatorname{Tr}\left(B\mathcal{S}\_t\mathcal{A}\rho(\mathbf{0})\right) = \sum\_{a,\beta} a\_a b\_\beta \operatorname{Tr}B\_a\left(\mathcal{S}\_t\left(A\_a\rho(\mathbf{0})A\_a^\dagger\right)\right) \mathbf{B}\_\beta^\dagger. \tag{75}$$

It may be shown that just as every Hermitian operator represents some measurement on the Hilbert space of pure states, every Hermitian map can be associated with some measurement on the Liouville space of mixed states.

A Hermitian map representation of heat flow can now be constructed under assumptions that the bath and system Hamiltonian are constant during the measurement and the bath-system coupling is very small. A measurement on the total system is constructed, and thus the bath degrees of freedom are projected out. This leaves a Hermitian map superoperator that acts on the system density matrix alone. Let us describe the measurement process and mathematical formulation together.

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

Begin with a composite system which consists of the bath, initially in thermal equilibrium weakly coupled to the system:

$$
\rho^{\mathbb{S}} \otimes \rho\_{eq}^{\mathbb{B}}.\tag{76}
$$

Measure the initial energy eigenstate of the bath so based on (76):

$$(I^S \otimes |b\_i\rangle\langle b\_i|) \Big(\rho^S \otimes \rho\_{eq}^B\Big) (I^S \otimes |b\_j\rangle\langle b\_j|). \tag{77}$$

Now allow the system to evolve together with the bath for some time:

$$U\left(I^{S}\otimes|b\_{i}\rangle\langle b\_{i}|\right)\left(\rho^{S}\otimes\rho\_{eq}^{B}\right)\left(I^{S}\otimes|b\_{j}\rangle\langle b\_{j}|\right)U^{\dagger}.\tag{78}$$

Finally, measure the final energy eigenstate of the bath:

$$\left(\left(\Gamma^{\mathbb{S}}\otimes|b\_{i}\rangle\langle b\_{f}|\right)U\left(\Gamma^{\mathbb{S}}\otimes|b\_{i}\rangle\langle b\_{i}|\right)\left(\rho^{\mathbb{S}}\otimes\rho\_{eq}^{\mathbb{B}}\right)\left(\Gamma^{\mathbb{S}}\otimes|b\_{j}\rangle\langle b\_{j}|\right)U^{\dagger}\left(\Gamma^{\mathbb{S}}\otimes|b\_{f}\rangle\langle b\_{f}|\right). \tag{79}$$

Taking the trace over the bath degrees of freedom produces the final normalized system density matrix where trace over *S* gives the probability of observing the given initial and final bath eigenstates. Multiply by the Boltzmann weighted heat, and sum over the initial and final bath states to obtain the desired average Boltzmann weighted heat flow:

$$
\langle e^{\rho Q} \rangle = \sum\_{i,f} e^{-\beta \left( \epsilon\_f^{\mathcal{S}} - \epsilon\_i^{\mathcal{S}} \right)} Tr\_{\mathcal{S}} Tr\_{\mathcal{B}}(I^{\mathcal{S}} \otimes |b\_f\rangle\langle b\_f|) U(I^{\mathcal{S}} \otimes |b\_i\rangle\langle b\_i|) \tag{80}
$$

$$
\left(\rho^{\mathcal{S}} \otimes \rho\_{eq}^{\mathcal{B}}\right)(I^{\mathcal{S}} \otimes |b\_j\rangle\langle b\_j|) U^{\dagger}(I^{\mathcal{S}} \otimes |b\_j\rangle\langle b\_j|).
$$

Replace the heat bath Hamiltonian by *I <sup>S</sup>* <sup>⊗</sup> *<sup>H</sup><sup>B</sup>* <sup>¼</sup> *<sup>H</sup>* � *<sup>H</sup><sup>S</sup>* ð Þ*t* ⊗ *I <sup>B</sup>* � *<sup>ε</sup>Hint*. The total Hamiltonian commutes with the unitary dynamics and cancels. The interaction Hamiltonian can be omitted in the small coupling limit giving

$$\left< \boldsymbol{e}^{\boldsymbol{\theta} \mathbf{Q}} \right> = \operatorname{Tr}\_{\mathbf{S}} \operatorname{Tr}\_{\mathbf{B}} \left( \boldsymbol{e}^{\boldsymbol{\theta} \mathbf{H}^{\S}/2} \otimes \boldsymbol{I}^{\mathbf{S}} \right) U \Big( \boldsymbol{e}^{-\boldsymbol{\beta}/2 \mathbf{H}^{\S}} \otimes \boldsymbol{I}^{\mathbf{B}} \Big) \left( \boldsymbol{\rho}^{\mathbf{S}} \otimes \boldsymbol{\rho}\_{\mathbf{q}}^{\mathbf{B}} \right) \left( \boldsymbol{e}^{-\boldsymbol{\beta} \mathbf{H}^{\S}/2} \otimes \boldsymbol{I}^{\mathbf{B}} \right) U^{\dagger} \left( \boldsymbol{e}^{\boldsymbol{\theta} \mathbf{H}^{\S}/2} \otimes \boldsymbol{I}^{\mathbf{B}} \right) \tag{81}$$

Collecting the terms acting on the bath and system separately and replacing the Krauss operators describing the reduced dynamics of the system, the result is

$$\begin{split} \langle \boldsymbol{\varrho}^{\theta \mathbf{Q}} \rangle &= \operatorname{Tr}\_{\mathbf{S}} \mathbf{e}^{\theta \mathbf{H}^{\mathbf{S}}/2} \Big( \operatorname{Tr}\_{\mathbf{B}} \left( \mathbf{U} \mathbf{e}^{-\beta \mathbf{H}^{\mathbf{S}}/2} \boldsymbol{\rho}^{\mathbf{S}} \mathbf{e}^{-\beta \mathbf{H}^{\mathbf{S}}/2} \right) \otimes \boldsymbol{\rho}^{\mathbf{B}}\_{\mathbf{eq}} \mathbf{U}^{\dagger} \Big) \mathbf{e}^{\theta \mathbf{H}^{\mathbf{S}}/2} \\ &= \operatorname{Tr}\_{\mathbf{S}} \sum\_{a} \mathbf{e}^{\beta \mathbf{H}^{\mathbf{S}}/2} A\_{a} \mathbf{e}^{-\beta \mathbf{H}^{\mathbf{S}}/2} \boldsymbol{\rho}^{\mathbf{S}} \mathbf{e}^{\theta \mathbf{H}^{\mathbf{S}}/2} A\_{a}^{\dagger} \mathbf{e}^{-\beta \mathbf{H}^{\mathbf{S}}/2} . \end{split} \tag{82}$$

To summarize, it has been found that the average Boltzmann weighted heat flow is represented by

$$
\langle e^{\beta Q} \rangle = \text{Tr} \left( \mathcal{R}^{-1} \mathcal{S} \mathcal{R} \rho^{\mathcal{S}} \right). \tag{83}
$$

where *S* represents the reduced dynamics of the system. The Hermitian map superoperator R*<sup>t</sup>* is given by

interval can be described by a superoperator derived as in (66) which can then be

The Hermitian operator of a von Neumann-type measurement can be broken up

*<sup>σ</sup> λσπσ*. In a more general sense, the measured operator of a positive operatorvalued measurement need not be projectors or orthonormal. The probability of

*pa* <sup>¼</sup> Tr *AaρA*†

<sup>~</sup>*ρ<sup>a</sup>* <sup>¼</sup> *AaρA*†

A ¼ <sup>X</sup> *α*

*α* � �† <sup>¼</sup> <sup>X</sup>

*α*

*α*, *β*

It may be shown that just as every Hermitian operator represents some measurement on the Hilbert space of pure states, every Hermitian map can be associ-

A Hermitian map representation of heat flow can now be constructed under assumptions that the bath and system Hamiltonian are constant during the measurement and the bath-system coupling is very small. A measurement on the total system is constructed, and thus the bath degrees of freedom are projected out. This leaves a Hermitian map superoperator that acts on the system density matrix alone. Let us describe the measurement process and mathematical formulation together.

The result of the measurement can be represented by using a Hermitian map

An operator-value sum maps Hermitian operators into Hermitian operators:

*α*

In the other direction, any Hermitian map has an operator-value-mean representation. Hermitian maps provide a particularly concise and convenient representation of sequential measurements and correlation functions. For example, suppose Hermitian map A represents a measurement at time 0, C is a different measurement at time *t*, and the quantum operation *St* represents the system evolution between the measurements. The expectation value of a single measure-

*aα*Tr*AαρA*†

*ρ*ðÞ¼ *t S t*ð Þ � 1, *t* ⋯*S s*ð Þ þ 1, *s* þ 2 *S s*ð Þ , *s* þ 1 *ρ:* (69)

*a*

*a* Tr *AaρA*†

*aαAαρA*†

*<sup>a</sup><sup>α</sup> <sup>A</sup>*† � �†

*H*† *A*†

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*

*<sup>a</sup>αb<sup>β</sup>* Tr *<sup>B</sup><sup>α</sup> St <sup>A</sup>αρ*ð Þ <sup>0</sup> *<sup>A</sup>*†

*a*

*λσ* and orthonormal projection operators *πσ* such that *H* ¼

� �*:* (70)

� � *:* (71)

*<sup>α</sup>:* (72)

*<sup>α</sup>* ¼ A*H:* (73)

*pαaα:* (74)

*<sup>β</sup>:* (75)

*α* � � � � *B*†

chained together to form a quantum Markov chain:

The state of the system after this interaction is

½ � <sup>A</sup>*<sup>H</sup>* † <sup>¼</sup> *<sup>a</sup>αAαHA*†

h i *<sup>a</sup>* <sup>¼</sup> Trð Þ¼ <sup>A</sup>*<sup>ρ</sup>* <sup>X</sup>

The correlation function h i *b t*ð Þ*a*ð Þ 0 can be expressed as

ated with some measurement on the Liouville space of mixed states.

h i *b t*ð Þ*a*ð Þ <sup>0</sup> <sup>¼</sup> Trð Þ¼ *BSt*A*ρ*ð Þ <sup>0</sup> <sup>X</sup>

into a set of eigenvalues

*Quantum Mechanics*

superoperator A:

ment is

**96**

observing the *a*-th outcome is

P

$$\mathcal{R}\_t \rho = \mathbf{e}^{-\beta \mathcal{H}\_t/2} \rho \mathbf{e}^{\beta \mathcal{H}\_t/2}. \tag{84}$$

In units where ℏ is one, *B*<sup>0</sup> represents the characteristic precession frequency of

�*H=<sup>T</sup>* � � <sup>¼</sup> 2 cosh *<sup>B</sup>*<sup>0</sup>

**B** ¼ *B*ð Þ sin ð Þ *ωt* , cosð Þ *ωt* , 0 *:* (91)

<sup>2</sup> *<sup>σ</sup><sup>z</sup>* sin ð Þþ *<sup>ω</sup><sup>t</sup> <sup>σ</sup><sup>y</sup>* cosð Þ *<sup>ω</sup><sup>t</sup>* � �*:* (92)

*<sup>i</sup>ωtσz=*<sup>2</sup>*V t*ð Þ*:* (93)

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *H t* <sup>~</sup> ð Þ*V*, *<sup>V</sup>*ð Þ¼ <sup>0</sup> 1, (94)

2

*<sup>α</sup>* <sup>¼</sup> *<sup>σ</sup><sup>x</sup>* <sup>þ</sup> 2 sin *<sup>α</sup>* cos *ασ<sup>y</sup>* sin <sup>2</sup>

� �, (96)

*<sup>α</sup>*, (97)

2*T*

*T*

� �, (89)

� �*:* (90)

the spin. Since *Ho* is diagonal in the ∣�i basis that diagonalizes *σz*, the matrix

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

, *Z* ¼ Tr *e*

If we set *σ*~ to be the equilibrium magnetization of the system, *σ*~ ¼ h i *σ<sup>x</sup> th*, the

The work segment is implemented by introducing a very small field of amplitude *B* rotating in the *xy* plane with frequency *ω*. The work parameter is governed

Typically, *B*0≈*ωT* and *B*≈0*:*01*T*, so we may approximate *B* < < *B*0. The total

The oscillating field plays the role of a perturbation which although weak may initiate transitions between the up and down spin states and will be most frequent at the resonance condition *ω* ¼ *B*0, so the driving frequency matches the natural

The time evolution operator *U t*ð Þ is calculated now. To do this, define a new

Substituting (43) into the evolution equation for *U t*ð Þ, *<sup>i</sup>∂tU* <sup>¼</sup> *H t*ð Þ*U*, *<sup>U</sup>*ð Þ¼ <sup>0</sup> 1.

*<sup>i</sup>ωσz=*<sup>2</sup> � �*V t*ð Þ*:* (95)

� � � *<sup>i</sup>σ<sup>z</sup>* sin *<sup>ω</sup><sup>t</sup>*

*U t*ðÞ¼ *e*

<sup>2</sup> *ωσ<sup>z</sup>* � *<sup>B</sup>*0*σ<sup>z</sup>* � *Be*�*iωtσz=*<sup>2</sup> *<sup>σ</sup><sup>x</sup>* sin ð Þþ *<sup>ω</sup><sup>t</sup> <sup>σ</sup><sup>y</sup>* cosð Þ *<sup>ω</sup><sup>t</sup>* � �*<sup>e</sup>*

�*iωσ<sup>z</sup>* <sup>¼</sup> **<sup>1</sup>** cos *<sup>ω</sup><sup>t</sup>*

Using the commutation relations of the Pauli matrices and the fact that

it is found that the terms in the evolution equation can be simplified

2

*<sup>i</sup>ασ<sup>z</sup>* <sup>¼</sup> ð Þ **<sup>1</sup>** cos *<sup>α</sup>* � *<sup>i</sup>σ<sup>x</sup>* sin *<sup>α</sup> <sup>σ</sup>x*ð Þ **<sup>1</sup>** cos *<sup>α</sup>* <sup>þ</sup> *<sup>i</sup>σ<sup>z</sup>* sin *<sup>α</sup>*

� �, *<sup>σ</sup>*<sup>~</sup> <sup>¼</sup> tanh *<sup>B</sup>*<sup>0</sup>

1 þ *σ*~ 0 0 1 � *σ*~

and *σ*~ corresponds to the parametric response of a spin-1*=*2 particle.

exponential and partition function are given by

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

0 *e*�*B*0*=*2*<sup>T</sup>* !

*<sup>ρ</sup>* <sup>¼</sup> *<sup>ρ</sup>th* <sup>¼</sup> <sup>1</sup>

2

*H t*ðÞ¼� *<sup>B</sup>*<sup>0</sup>

It is found that *V t*ð Þ obeys the Schrödinger equation:

*i ∂V*

*e*

<sup>¼</sup> *<sup>σ</sup><sup>x</sup>* <sup>þ</sup> 2 sin *<sup>α</sup>* cos *ασ<sup>y</sup>* � <sup>2</sup>*iσzσ<sup>y</sup>* sin <sup>2</sup>

<sup>2</sup> *<sup>σ</sup><sup>z</sup>* � *<sup>B</sup>*

�*H=<sup>T</sup>* <sup>¼</sup> *<sup>e</sup>B*0*=*2*<sup>T</sup>* <sup>0</sup>

thermal density matrix is

Hamiltonian is the combination

operator *V t*ð Þ by means of the equation

It is found that *V t*ð Þ satisfies

*e* �*iασzσxe*

oscillation frequency.

*i ∂V <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup>

**99**

*e*

by the field

The paired Hermitian map superoperators act at the start and end of a time interval. They give a measure of the change in the energy of the system over that interval. This procedure does not disturb the system beyond that already incurred by coupling the system to the environment. The Jarzynski inequality now follows by applying this Hermitian map and quantum formalism. Discretize the experimental time into a series of discrete intervals labeled by an integer *t*.

The system Hamiltonian is fixed within each interval. It changes only in discrete jumps at the boundaries. The heat flow can be measured by wrapping the superoperator time evolution of each time interval *St* along with the corresponding Hermitian map measurements R�<sup>1</sup> *<sup>t</sup> S*R*t*. In a similar fashion, the measurement of the Boltzmann weighted energy change of the system can be measured with *<sup>e</sup>*�*<sup>β</sup>* <sup>Δ</sup>*<sup>E</sup>* � � <sup>¼</sup> *Tr* <sup>R</sup>*τS*R�<sup>1</sup> *<sup>τ</sup>* . The average Boltzmann weighted work of a driven, dissipative quantum system can be expressed as

$$\langle e^{-\beta \mathcal{W}} \rangle = \operatorname{Tr} \left( \mathcal{R}\_{\tau} \prod\_{t} \left( \mathcal{R}\_{t}^{-1} \mathbb{S}\_{t} \mathcal{R}\_{t} \right) \mathcal{R}\_{\tau}^{-1} \rho\_{0}^{eq} \right), \tag{85}$$

In (85), *ρ<sup>t</sup> eq* is the system equilibrium density matrix when the system Hamiltonian is *H<sup>S</sup> t* .

This product actually telescopes due to the structure of the energy change Hermitian map (84) and the equilibrium density matrix (65). This leaves only the free energy difference between the initial and final equilibrium ensembles, as can be seen by writing out the first few terms

$$\begin{split} \left< e^{-\beta W} \right> &= \operatorname{Tr} \left[ \mathcal{R}\_{\tau} \left( \mathcal{R}\_{\tau}^{-1} \mathcal{S}\_{\tau} \mathcal{R}\_{\tau} \right) \cdots \left( \mathcal{R}\_{2}^{-1} \mathcal{S}\_{2} \mathcal{R}\_{2} \right) \left( \mathcal{R}\_{1}^{-1} \mathcal{S}\_{1} \mathcal{R}\_{1} \right) \mathcal{R}\_{0}^{-1} \rho\_{eq}^{0} \right] \\ &= \operatorname{Tr} \left[ \tau \left( \mathcal{R}\_{\tau}^{-1} \mathcal{S}\_{\tau} \mathcal{R}\_{\tau} \right) \cdots \left( \mathcal{R}\_{2}^{-1} \mathcal{S}\_{2} \mathcal{R}\_{2} \right) \left( \mathcal{R}\_{1}^{-1} \mathcal{S}\_{1} \mathcal{R}\_{1} \right) \frac{I}{Z(0)} \right] \\ &= \operatorname{Tr} \left[ \mathcal{R}\_{\tau} \left( \mathcal{R}\_{\tau}^{-1} \mathcal{S}\_{\tau} \mathcal{R}\_{\tau} \right) \cdots \left( \mathcal{R}\_{2}^{-1} \mathcal{S}\_{2} \mathcal{R}\_{2} \right) \left( \mathcal{R}\_{1}^{-1} \mathcal{S}\_{1} \rho\_{eq}^{1} \frac{Z(1)}{Z(0)} \right) \right] \\ & \cdots = \frac{Z(\tau)}{Z(\mathcal{Q})} = e^{-\beta \Delta F} = e^{-\beta \Delta F}. \end{split} \tag{86}$$

In the limit in which the time intervals are reduced to zero, the inequality can be expressed in the continuous Lindblad form:

$$\langle e^{-\beta W} \rangle = \text{Tr} \, \mathcal{R}(t) \exp \left[ \int\_0^t \mathcal{R}(\xi)^{-1} \mathcal{S}(\xi) \mathcal{R}(\xi) d\xi \right] \mathcal{R}(0)^{-1} \rho\_0^{eq} = e^{-\beta \Delta F}. \tag{87}$$

#### **5. A model quantum spin system**

A magnetic resonance experiment can be used to illustrate how these ideas can be applied in practice. A sample of noninteracting spin-1*=*2 particles are placed in a strong magnetic field *B*<sup>0</sup> which is directed along the *z* direction. Denote by *σ <sup>j</sup>*, *j* ¼ *x*, *y*, *z* the usual Pauli matrices and **1** the 2 � 2 identity matrix. It is assumed the motion of the system is unitary. Then the spin is governed by the Hamiltonian:

$$H\_0 = -\frac{1}{2}B\_0\sigma\_\text{z.}\tag{88}$$

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

In units where ℏ is one, *B*<sup>0</sup> represents the characteristic precession frequency of the spin. Since *Ho* is diagonal in the ∣�i basis that diagonalizes *σz*, the matrix exponential and partition function are given by

$$e^{-H/T} = \begin{pmatrix} e^{B\_0/2T} & 0\\ 0 & e^{-B\_0/2T} \end{pmatrix}, \qquad Z = \text{Tr}\left(e^{-H/T}\right) = 2\cosh\left(\frac{B\_0}{2T}\right),\tag{89}$$

If we set *σ*~ to be the equilibrium magnetization of the system, *σ*~ ¼ h i *σ<sup>x</sup> th*, the thermal density matrix is

$$\rho = \rho\_{th} = \frac{1}{2} \begin{pmatrix} 1 + \tilde{\sigma} & 0 \\ 0 & 1 - \tilde{\sigma} \end{pmatrix}, \qquad \tilde{\sigma} = \tanh\left(\frac{B\_0}{T}\right). \tag{90}$$

and *σ*~ corresponds to the parametric response of a spin-1*=*2 particle.

The work segment is implemented by introducing a very small field of amplitude *B* rotating in the *xy* plane with frequency *ω*. The work parameter is governed by the field

$$\mathbf{B} = B\left(\sin\left(at\right), \cos\left(at\right), \mathbf{0}\right). \tag{91}$$

Typically, *B*0≈*ωT* and *B*≈0*:*01*T*, so we may approximate *B* < < *B*0. The total Hamiltonian is the combination

$$H(t) = -\frac{B\_0}{2}\sigma\_x - \frac{B}{2}\left(\sigma\_x \sin\left(\alpha t\right) + \sigma\_\circ \cos\left(\alpha t\right)\right). \tag{92}$$

The oscillating field plays the role of a perturbation which although weak may initiate transitions between the up and down spin states and will be most frequent at the resonance condition *ω* ¼ *B*0, so the driving frequency matches the natural oscillation frequency.

The time evolution operator *U t*ð Þ is calculated now. To do this, define a new operator *V t*ð Þ by means of the equation

$$U(t) = \mathfrak{e}^{iat\sigma\_{\mathfrak{z}}/2} V(t). \tag{93}$$

Substituting (43) into the evolution equation for *U t*ð Þ, *<sup>i</sup>∂tU* <sup>¼</sup> *H t*ð Þ*U*, *<sup>U</sup>*ð Þ¼ <sup>0</sup> 1. It is found that *V t*ð Þ obeys the Schrödinger equation:

$$i\frac{\partial V}{\partial t} = \tilde{H}(t)\,\mathrm{V}, \qquad V(0) = 1,\tag{94}$$

It is found that *V t*ð Þ satisfies

$$i\frac{\partial V}{\partial t} = \frac{1}{2} \left( \rho \sigma\_{\overline{x}} - B\_0 \sigma\_{\overline{x}} - B e^{-i\alpha t \sigma\_{\overline{x}}/2} \left( \sigma\_{\overline{x}} \sin \left( \alpha t \right) + \sigma\_{\overline{\gamma}} \cos \left( \alpha t \right) \right) e^{i\alpha \sigma\_{\overline{x}}/2} \right) V(t). \tag{95}$$

Using the commutation relations of the Pauli matrices and the fact that

$$e^{-i\alpha\sigma\_{\overline{x}}} = \mathbf{1}\cos\left(\frac{\alpha t}{2}\right) - i\sigma\_{\overline{x}}\sin\left(\frac{\alpha t}{2}\right),\tag{96}$$

it is found that the terms in the evolution equation can be simplified

$$\begin{aligned} e^{-ia\sigma\_{\mathfrak{x}}}\sigma\_{\mathfrak{x}}e^{ia\sigma\_{\mathfrak{x}}} &= (\mathbf{1}\cos a - i\sigma\_{\mathfrak{x}}\sin a)\sigma\_{\mathfrak{x}}(\mathbf{1}\cos a + i\sigma\_{\mathfrak{x}}\sin a) \\ = \sigma\_{\mathfrak{x}} + 2\sin a \cos a\sigma\_{\mathfrak{y}} - 2i\sigma\_{\mathfrak{x}}\sigma\_{\mathfrak{y}}\sin^{2}a &= \sigma\_{\mathfrak{x}} + 2\sin a \cos a\sigma\_{\mathfrak{y}}\sin^{2}a,\end{aligned} \tag{97}$$

R*<sup>t</sup> ρ* ¼ *e*

mental time into a series of discrete intervals labeled by an integer *t*.

Hermitian map measurements R�<sup>1</sup>

tum system can be expressed as

*e*

seen by writing out the first few terms

�*β<sup>W</sup>* � � <sup>¼</sup> *Tr* <sup>R</sup>*<sup>τ</sup>* <sup>R</sup>�<sup>1</sup>

<sup>¼</sup> Tr *<sup>τ</sup>* <sup>R</sup>�<sup>1</sup>

<sup>¼</sup> Tr½R*<sup>τ</sup>* R�<sup>1</sup>

expressed in the continuous Lindblad form:

�*β<sup>W</sup>* � � <sup>¼</sup> Tr Rð Þ*<sup>t</sup>* exp

**5. A model quantum spin system**

<sup>⋯</sup> <sup>¼</sup> *<sup>Z</sup>*ð Þ*<sup>τ</sup>*

�*β<sup>W</sup>* � � <sup>¼</sup> Tr <sup>R</sup>*<sup>τ</sup>*

*<sup>τ</sup> Sτ*R*<sup>τ</sup>* � �<sup>⋯</sup> R�<sup>1</sup>

*<sup>τ</sup> Sτ*R*<sup>τ</sup>* � �<sup>⋯</sup> R�<sup>1</sup>

*<sup>τ</sup> Sτ*R*<sup>τ</sup>* � �<sup>⋯</sup> R�<sup>1</sup>

*<sup>Z</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>e</sup>*

ð*t* 0

Rð Þ*<sup>ξ</sup>* �<sup>1</sup>

*Tr* <sup>R</sup>*τS*R�<sup>1</sup>

*Quantum Mechanics*

In (85), *ρ<sup>t</sup>*

*t* .

*e*

*e*

**98**

nian is *H<sup>S</sup>*

jumps at the boundaries. The heat flow can be measured by wrapping the

�*βHt=*2 *ρe βHt=*2

The paired Hermitian map superoperators act at the start and end of a time interval. They give a measure of the change in the energy of the system over that interval. This procedure does not disturb the system beyond that already incurred by coupling the system to the environment. The Jarzynski inequality now follows by applying this Hermitian map and quantum formalism. Discretize the experi-

The system Hamiltonian is fixed within each interval. It changes only in discrete

superoperator time evolution of each time interval *St* along with the corresponding

Boltzmann weighted energy change of the system can be measured with *<sup>e</sup>*�*<sup>β</sup>* <sup>Δ</sup>*<sup>E</sup>* � � <sup>¼</sup>

Y *t*

This product actually telescopes due to the structure of the energy change Hermitian map (84) and the equilibrium density matrix (65). This leaves only the free energy difference between the initial and final equilibrium ensembles, as can be

*<sup>τ</sup>* . The average Boltzmann weighted work of a driven, dissipative quan-

R�<sup>1</sup> *<sup>t</sup> St*R*<sup>t</sup>* � �R�<sup>1</sup>

<sup>2</sup> *S*2R<sup>2</sup> � � R�<sup>1</sup>

<sup>2</sup> *S*2R<sup>2</sup> � � R�<sup>1</sup>

�*β*Δ*<sup>F</sup>:*

In the limit in which the time intervals are reduced to zero, the inequality can be

� �

*S*ð ÞR*ξ* ð Þ*ξ dξ*

A magnetic resonance experiment can be used to illustrate how these ideas can be applied in practice. A sample of noninteracting spin-1*=*2 particles are placed in a strong magnetic field *B*<sup>0</sup> which is directed along the *z* direction. Denote by *σ <sup>j</sup>*, *j* ¼ *x*, *y*, *z* the usual Pauli matrices and **1** the 2 � 2 identity matrix. It is assumed the motion of the system is unitary. Then the spin is governed by the Hamiltonian:

> *<sup>H</sup>*<sup>0</sup> ¼ � <sup>1</sup> 2

<sup>2</sup> *S*2R<sup>2</sup> � � R�<sup>1</sup>

�*β*Δ*<sup>F</sup>* <sup>¼</sup> *<sup>e</sup>*

� �

h i

!

*eq* is the system equilibrium density matrix when the system Hamilto-

*:* (84)

, (85)

(86)

�*β*Δ*<sup>F</sup>:* (87)

*<sup>t</sup> S*R*t*. In a similar fashion, the measurement of the

*<sup>τ</sup> ρ eq* 0

<sup>1</sup> *S*1R<sup>1</sup> � �R�<sup>1</sup>

> <sup>1</sup> *S*1*ρ*<sup>1</sup> *eq Z*ð Þ1 *Z*ð Þ 0

Rð Þ <sup>0</sup> �<sup>1</sup> *ρ eq* <sup>0</sup> ¼ *e*

� �

<sup>1</sup> *S*1R<sup>1</sup> � � *I* <sup>0</sup> *ρ*<sup>0</sup> *eq*

*Z*ð Þ 0

*B*0*σz:* (88)

*e* �*iασzσye <sup>i</sup>ασ<sup>z</sup>* <sup>¼</sup> *<sup>σ</sup><sup>y</sup>* cos *<sup>α</sup>* � *<sup>i</sup>σxσ<sup>y</sup>* sin *<sup>α</sup>*ð**<sup>1</sup>** cos *<sup>α</sup>* <sup>þ</sup> *<sup>i</sup>σ<sup>z</sup>* sin *<sup>α</sup>*Þ ¼ *<sup>σ</sup><sup>y</sup>* � 2 sin *<sup>α</sup>* cos *ασ<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*iσz<sup>σ</sup>* sin <sup>2</sup> *α:* � (98)

By means of these results, it remains to simplify

*e* �*iωtσz=*<sup>2</sup> *<sup>σ</sup><sup>z</sup>* sin ð Þþ *<sup>ω</sup><sup>t</sup> <sup>σ</sup><sup>y</sup>* cosð Þ *<sup>ω</sup><sup>t</sup>* � �*<sup>e</sup> iωtσz=*2 <sup>¼</sup> *<sup>σ</sup>z*ðsin*ω<sup>t</sup>* � sin*ω<sup>t</sup>* <sup>þ</sup> cos *<sup>ω</sup>t*sin*ω<sup>t</sup>* � cos*ωt*sin*ωt*Þ þ *<sup>σ</sup><sup>y</sup>* sin <sup>2</sup> *<sup>ω</sup><sup>t</sup>* <sup>þ</sup> cos *<sup>ω</sup><sup>t</sup>* � cos*ω<sup>t</sup>* <sup>þ</sup> cos <sup>2</sup> *<sup>ω</sup><sup>t</sup>* � � <sup>¼</sup> *<sup>σ</sup>y:* (98a)

Taking these results to (95), we arrive at

$$i\frac{\partial V}{\partial t} = H\_1 V, \quad H\_1 = -\frac{1}{2}(B\_0 - \alpha)\sigma\_x - \frac{1}{2}B\sigma\_y. \tag{99}$$

This means *V t*ð Þ evolves according to a time-dependent Hamiltonian, so the solution can be written as

$$V(t) = e^{-iH\_1t},\tag{100}$$

*e*

The functions *u t*ð Þ and *v t*ð Þ in (107) are given as

<sup>þ</sup> *<sup>i</sup>*sin *<sup>ϑ</sup>* sin <sup>Ω</sup>

� � � � , *v t*ðÞ¼ *<sup>e</sup>*

the evolution operator is then given by

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

Ω 2 *t* � �

will occur. Since the unitarity condition *U*†

*u t*ðÞ¼ *e*

If *A* is replaced by *σ<sup>z</sup>* in (111), we obtain

*<sup>u</sup>*<sup>∗</sup> ðÞ � *<sup>t</sup> v t*ð Þ *<sup>v</sup>* <sup>∗</sup> ð Þ*<sup>t</sup> u t*ð Þ !*σ<sup>z</sup>*

*<sup>ϑ</sup>* <sup>þ</sup> sin <sup>2</sup>

2

¼ *σ*~ j j *u*

<sup>2</sup> � j j *<sup>v</sup>*

*<sup>ϑ</sup>* cosð Þ <sup>Ω</sup>*<sup>t</sup>* � � <sup>¼</sup> tanh *<sup>B</sup>*<sup>0</sup>

, this takes the form

<sup>2</sup> � � <sup>¼</sup> *<sup>σ</sup>*<sup>~</sup> <sup>1</sup> � <sup>2</sup>j j *<sup>v</sup>*

*<sup>i</sup>ωt=*<sup>2</sup> cos

*B* 2 *t*

h i *<sup>A</sup> <sup>t</sup>* <sup>¼</sup> *Tr U*†

*u t*ðÞ¼ *e*

conclude j j *u*

simplify to

h i *σ<sup>z</sup> <sup>t</sup>* ¼ Tr

Substituting j j *v*

h i *<sup>σ</sup><sup>z</sup> <sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>~</sup> cos <sup>2</sup>

**101**

*<sup>i</sup>ωt=*<sup>2</sup> cos

found in state ∣�i after time *t* is

*<sup>i</sup>ωσzt=*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>iωt=*<sup>2</sup> <sup>0</sup>

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

*U t*ðÞ¼ *u t*ð Þ *v t*ð Þ

2 *t*

Apart from a phase factor, the final result depends only on Ω and *ϑ*, and these in turn depend on *B*0, *B*, and *ω* through (108). To understand the physics of *U t*ð Þ a bit better, suppose the system is initially in the pure state ∣þi. The probability will be

<sup>∣</sup>h�∣*U t*ð Þ þi j j<sup>2</sup> <sup>¼</sup> j j *<sup>v</sup>*

sin *ϑ*, which gives a physical meaning to *ϑ*. It represents the transition probability and reaches a maximum when *ω* ¼ *B*<sup>0</sup> at resonance where Ω ¼ *B*, so *u* and *v*

� �, *v t*ðÞ¼ *<sup>e</sup>*

Now that *U t*ð Þ is known, the evolution of any observable *A* can be calculated

*u t*ð Þ *v t*ð Þ

! <sup>1</sup>

�*<sup>v</sup>* <sup>∗</sup> ð Þ*<sup>t</sup> <sup>u</sup>*<sup>∗</sup> ð Þ*<sup>t</sup>*

Consider the average work. Suppose *B*< <*B*0, so the unperturbed Hamiltonian *H*<sup>0</sup> can be used instead of the full Hamiltonian *H t*ð Þ when expectation values of

<sup>2</sup> � �*:*

2*T* � � cos <sup>2</sup>

This expression represents the transition probability per unit time a transition

2

<sup>2</sup> is the probability when no transition occurs. Note *v* is proportional to

*U* ¼ **1** implies that j j *u*

2

*<sup>i</sup>ωt=*<sup>2</sup> sin *<sup>B</sup>*

2 *t*

ð Þ*<sup>t</sup> AU t*ð Þ*<sup>ρ</sup>* � �*:* (111)

1 þ *σ*~ 0

!

0 1 � *σ*~

*<sup>ϑ</sup>* <sup>þ</sup> sin <sup>2</sup> *<sup>ϑ</sup>* cosð Þ <sup>Ω</sup>*<sup>t</sup>* � �*:*

0 *e*�*iωt=*<sup>2</sup> !

� �*:* (107)

*<sup>i</sup>ωt=*<sup>2</sup> � sin *<sup>ϑ</sup>* � sin <sup>Ω</sup>

*:* (109)

<sup>2</sup> <sup>þ</sup> j j *<sup>v</sup>*

� �*:* (110)

<sup>2</sup> <sup>¼</sup> 1, we

(112)

(113)

�*<sup>v</sup>* <sup>∗</sup> ð Þ*<sup>t</sup> <sup>u</sup>*<sup>∗</sup> ð Þ*<sup>t</sup>*

(106)

2 *t* � �*:*

(108)

and the full-time evolution operator is given by

$$U(t) = e^{i\alpha t \sigma\_z/2} e^{-iH\_1 t}.\tag{101}$$

Since the operators *σ<sup>y</sup>* and *σ<sup>z</sup>* do not commute, the exponentials in (101) cannot be using the usual addition rule.

To express (100) otherwise, suppose **<sup>M</sup>** is an arbitrary matrix such that **<sup>M</sup>**<sup>2</sup> <sup>¼</sup> **<sup>1</sup>**. When *α* is an arbitrary parameter, power series expansion of *e*�*iα***<sup>M</sup>** yields

$$e^{-ia\mathbf{M}} = \mathbf{1}\cos\left(a\right) - i\mathbf{M}\sin\left(a\right). \tag{102}$$

Now *H*<sup>1</sup> can be put in equivalent form

$$H\_1 = \frac{\Omega}{2} \left(\sigma\_x \cos \theta + \sigma\_\mathcal{V} \sin \theta\right),$$

$$\Omega = \sqrt{\left(B\_0 - \alpha\right)^2 + B^2}, \qquad \tan \theta = \frac{B}{B\_0 - \alpha},\tag{103}$$

Since *σ*<sup>2</sup> *<sup>i</sup>* ¼ **1**, it follows that

$$\left(\sigma\_x \cos \theta + \sigma\_y \sin \theta\right)^2 = \mathbf{1}.\tag{104}$$

Consequently, (100) can be used to prove that *V t*ð Þ is given by

$$e^{-iHt} = \mathbf{1}\cos\left(\frac{\Omega}{2}t\right) + i\left(\sigma\_x\cos\theta + \sigma\_y\sin\theta\right)\sin\left(\frac{\Omega}{2}t\right)$$

$$= \begin{pmatrix} \cos\left(\frac{\Omega}{2}t\right) + i\cos\theta\sin\left(\frac{\Omega}{2}t\right) & \sin\theta\sin\left(\frac{\Omega}{2}t\right) \\\\ -\sin\theta\sin\left(\frac{\Omega}{2}t\right) & \cos\left(\frac{\Omega}{2}t\right) - i\cos\theta\sin\left(\frac{\Omega}{2}t\right) \end{pmatrix} \tag{105}$$

Since

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

$$e^{i\alpha\sigma\_{\rm t}t/2} = \begin{pmatrix} e^{i\alpha t/2} & \mathbf{0} \\ \mathbf{0} & e^{-i\alpha t/2} \end{pmatrix} \tag{106}$$

the evolution operator is then given by

*e* �*iασzσye*

*Quantum Mechanics*

*<sup>i</sup>ασ<sup>z</sup>* <sup>¼</sup> *<sup>σ</sup><sup>y</sup>* cos *<sup>α</sup>* � *<sup>i</sup>σxσ<sup>y</sup>* sin *<sup>α</sup>*ð**<sup>1</sup>** cos *<sup>α</sup>* <sup>þ</sup> *<sup>i</sup>σ<sup>z</sup>* sin *<sup>α</sup>*Þ ¼ *<sup>σ</sup><sup>y</sup>* � 2 sin *<sup>α</sup>* cos *ασ<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*iσz<sup>σ</sup>* sin <sup>2</sup> *α:* �

*iωtσz=*2

2

ð Þ *<sup>B</sup>*<sup>0</sup> � *<sup>ω</sup> <sup>σ</sup><sup>z</sup>* � <sup>1</sup>

�*iα***<sup>M</sup>** <sup>¼</sup> **<sup>1</sup>** cosð Þ� *<sup>α</sup> <sup>i</sup>***<sup>M</sup>** sin ð Þ *<sup>α</sup> :* (102)

*<sup>B</sup>*<sup>0</sup> � *<sup>ω</sup>*,

2 *t* � �

> 2 *t* � �

1

CCCA

2 *t* � �

� *<sup>i</sup>* cos *<sup>ϑ</sup>* sin <sup>Ω</sup>

*<sup>σ</sup><sup>z</sup>* cos *<sup>ϑ</sup>* <sup>þ</sup> *<sup>σ</sup><sup>y</sup>* sin *<sup>ϑ</sup>* � �<sup>2</sup> <sup>¼</sup> **<sup>1</sup>***:* (104)

sin *<sup>ϑ</sup>* sin <sup>Ω</sup>

*<sup>ω</sup><sup>t</sup>* <sup>þ</sup> cos *<sup>ω</sup><sup>t</sup>* � cos*ω<sup>t</sup>* <sup>þ</sup> cos <sup>2</sup> *<sup>ω</sup><sup>t</sup>* � � <sup>¼</sup> *<sup>σ</sup>y:*

, (100)

*:* (101)

*Bσy:* (99)

�*iωtσz=*<sup>2</sup> *<sup>σ</sup><sup>z</sup>* sin ð Þþ *<sup>ω</sup><sup>t</sup> <sup>σ</sup><sup>y</sup>* cosð Þ *<sup>ω</sup><sup>t</sup>* � �*<sup>e</sup>*

2

�*iH*1*t*

*iωtσz=*2 *e* �*iH*1*t*

Since the operators *σ<sup>y</sup>* and *σ<sup>z</sup>* do not commute, the exponentials in (101) cannot

To express (100) otherwise, suppose **<sup>M</sup>** is an arbitrary matrix such that **<sup>M</sup>**<sup>2</sup> <sup>¼</sup> **<sup>1</sup>**.

<sup>2</sup> *<sup>σ</sup><sup>z</sup>* cos *<sup>ϑ</sup>* <sup>þ</sup> *<sup>σ</sup><sup>y</sup>* sin *<sup>ϑ</sup>* � �,

<sup>þ</sup> *<sup>i</sup> <sup>σ</sup><sup>z</sup>* cos *<sup>ϑ</sup>* <sup>þ</sup> *<sup>σ</sup><sup>y</sup>* sin *<sup>ϑ</sup>* � � sin <sup>Ω</sup>

Ω 2 *t* � �

cos

, tan *<sup>ϑ</sup>* <sup>¼</sup> *<sup>B</sup>*

This means *V t*ð Þ evolves according to a time-dependent Hamiltonian, so the

*V t*ðÞ¼ *e*

*U t*ðÞ¼ *e*

When *α* is an arbitrary parameter, power series expansion of *e*�*iα***<sup>M</sup>** yields

By means of these results, it remains to simplify

*e*

<sup>¼</sup> *<sup>σ</sup>z*ðsin*ω<sup>t</sup>* � sin*ω<sup>t</sup>* <sup>þ</sup> cos *<sup>ω</sup>t*sin*ω<sup>t</sup>* � cos*ωt*sin*ωt*Þ þ *<sup>σ</sup><sup>y</sup>* sin <sup>2</sup>

and the full-time evolution operator is given by

*e*

*<sup>H</sup>*<sup>1</sup> <sup>¼</sup> <sup>Ω</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>B</sup>*<sup>0</sup> � *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>B</sup>*<sup>2</sup>

Consequently, (100) can be used to prove that *V t*ð Þ is given by

2 *t* � �

Now *H*<sup>1</sup> can be put in equivalent form

Ω ¼

*<sup>i</sup>* ¼ **1**, it follows that

�*iH*1*<sup>t</sup>* <sup>¼</sup> **<sup>1</sup>** cos

� sin *<sup>ϑ</sup>* sin <sup>Ω</sup>

Ω 2 *t* � �

Since *σ*<sup>2</sup>

¼

Since

**100**

*e*

cos

0

BBB@

q

Ω 2 *t* � �

<sup>þ</sup> *<sup>i</sup>* cos *<sup>ϑ</sup>* sin <sup>Ω</sup>

2 *t* � �

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>H</sup>*1*V*, *<sup>H</sup>*<sup>1</sup> ¼ � <sup>1</sup>

Taking these results to (95), we arrive at

*i ∂V*

solution can be written as

be using the usual addition rule.

(98)

(98a)

(103)

(105)

$$U(t) = \begin{pmatrix} u(t) & v(t) \\ -v^\*(t) & u^\*(t) \end{pmatrix}. \tag{107}$$

The functions *u t*ð Þ and *v t*ð Þ in (107) are given as

$$u(t) = e^{iat/2} \left( \cos\left(\frac{\Omega}{2}t\right) + i\sin\theta\sin\left(\frac{\Omega}{2}t\right) \right), \qquad v(t) = e^{iat/2} \cdot \sin\theta \cdot \sin\left(\frac{\Omega}{2}t\right). \tag{108}$$

Apart from a phase factor, the final result depends only on Ω and *ϑ*, and these in turn depend on *B*0, *B*, and *ω* through (108). To understand the physics of *U t*ð Þ a bit better, suppose the system is initially in the pure state ∣þi. The probability will be found in state ∣�i after time *t* is

$$\left| \langle -|U(t)|+\rangle \right|^2 = \left| \boldsymbol{v} \right|^2. \tag{109}$$

This expression represents the transition probability per unit time a transition will occur. Since the unitarity condition *U*† *U* ¼ **1** implies that j j *u* <sup>2</sup> <sup>þ</sup> j j *<sup>v</sup>* <sup>2</sup> <sup>¼</sup> 1, we conclude j j *u* <sup>2</sup> is the probability when no transition occurs. Note *v* is proportional to sin *ϑ*, which gives a physical meaning to *ϑ*. It represents the transition probability and reaches a maximum when *ω* ¼ *B*<sup>0</sup> at resonance where Ω ¼ *B*, so *u* and *v* simplify to

$$u(t) = e^{i\alpha t/2} \cos\left(\frac{B}{2}t\right), \qquad v(t) = e^{i\alpha t/2} \sin\left(\frac{B}{2}t\right). \tag{110}$$

Now that *U t*ð Þ is known, the evolution of any observable *A* can be calculated

$$
\langle A \rangle\_t = \operatorname{Tr} \left( U^\dagger(t) A \, U(t) \, \rho \right). \tag{111}
$$

If *A* is replaced by *σ<sup>z</sup>* in (111), we obtain

$$
\langle \sigma\_{\mathbf{z}} \rangle\_t = \operatorname{Tr} \begin{pmatrix} u^\*(t) & -v(t) \\ & v^\*(t) & u(t) \end{pmatrix} \sigma\_{\mathbf{z}} \begin{pmatrix} u(t) & v(t) \\ & -v^\*(t) & u^\*(t) \end{pmatrix} \frac{1}{2} \begin{pmatrix} 1+\tilde{\sigma} & 0 \\ & 0 & 1-\tilde{\sigma} \end{pmatrix} \tag{112}
$$

$$
= \tilde{\sigma} \left( |u|^2 - |v|^2 \right) = \tilde{\sigma} \left( 1 - 2|v|^2 \right).
$$

Substituting j j *v* 2 , this takes the form

$$
\langle \sigma\_{\mathbf{z}} \rangle\_{\mathbf{t}} = \bar{\sigma} (\cos^2 \theta + \sin^2 \theta \cos(\Omega \mathbf{t})) = \tanh \left( \frac{B\_0}{2T} \right) (\cos^2 \theta + \sin^2 \theta \cos(\Omega \mathbf{t})).\tag{113}
$$

Consider the average work. Suppose *B*< <*B*0, so the unperturbed Hamiltonian *H*<sup>0</sup> can be used instead of the full Hamiltonian *H t*ð Þ when expectation values of

quantities are calculated which are related to the energy. Let us determine the energy of the system at any *t* by taking operator *A* to be *H*0:

$$\begin{split} \langle H\_{0} \rangle\_{t} &= -\frac{B\_{0}}{2} \langle \sigma\_{\bar{s}} \rangle\_{t} = -\frac{1}{2} B\_{0} \operatorname{Tr} \, U^{\dagger}(t) \sigma\_{\bar{s}} U(t) \rho \\ &= -\frac{1}{2} B\_{0} \operatorname{Tr} \left( \begin{pmatrix} u^{\*} & -v \\ v^{\*} & u \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} u & v \\ -v^{\*} & u^{\*} \end{pmatrix} \frac{1}{2} \begin{pmatrix} 1+\bar{\sigma} & 0 \\ 0 & 1-\bar{\sigma} \end{pmatrix} \right) \\ &= -\frac{B\_{0}}{4} \operatorname{Tr} \left( \begin{pmatrix} u^{\*} & -v \\ v^{\*} & u \end{pmatrix} \begin{pmatrix} u & v \\ v^{\*} & -u^{\*} \end{pmatrix} \begin{pmatrix} 1+\bar{\sigma} & 0 \\ 0 & 1-\bar{\sigma} \end{pmatrix} \right) = -\frac{1}{4} B\_{0} \bar{\sigma} \Big( \mathbf{1} - 2|v|^{2} \Big). \end{split} \tag{114}$$

The average work at time *t* is simply the difference between the energy st time *t*<sup>1</sup> and *t* ¼ 0. Since *v*ð Þ¼ 0 0, this difference is

$$\begin{split} \langle \mathcal{W} \rangle\_t &= -\frac{B\_0}{2} \tilde{\sigma} \Big( \mathbf{1} - 2|\boldsymbol{v}|^2 \Big) + \frac{B\_0}{2} \tilde{\sigma} = \tilde{\sigma} B\_0 |\boldsymbol{v}|^2 \\ &= \tilde{\sigma} B\_0 \sin^2 \theta \sin^2 \left( \frac{\Omega}{2} t \right) = \tilde{\sigma} B\_0 \frac{B}{\Omega^2} \sin^2 \left( \frac{\Omega}{2} t \right), \end{split} \tag{115}$$

since sin <sup>2</sup> *<sup>ϑ</sup>* <sup>¼</sup> <sup>1</sup> � cos <sup>2</sup> *<sup>ϑ</sup>* <sup>¼</sup> *<sup>B</sup>*<sup>2</sup> *=*Ω<sup>2</sup> . The average work oscillates indefinitely with frequency Ω*=*2. This is a consequence of the fact the time evolution is unitary. The amplitude multiplying the average work is proportional to the initial magnetization *σ*~ and *B*<sup>2</sup> *=*Ω<sup>2</sup> , so the ratio is a Lorentzian function.

The equilibrium free energy is *F* ¼ �*T* log *Z* where *Z* ¼ 2 cosh ð Þ *B*0*=*2*T* . The free energy of the initial state at *t* ¼ 0 and final state at any arbitrary time is the same yielding

$$
\Delta F = \mathbf{0}.\tag{116}
$$

Substituting (119) into (118), we can conclude

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

*e*

Gð Þ¼ *y* j j *u*

varð Þ¼ *<sup>W</sup> <sup>W</sup>*<sup>2</sup> � � � h i *<sup>W</sup>* <sup>2</sup> <sup>¼</sup> *<sup>B</sup>*<sup>2</sup>

Now Pð Þ *W* is the inverse Fourier transform of Gð Þ*y* :

Pð Þ¼ *W*

*dy u t* j j ð Þ <sup>2</sup> <sup>þ</sup> j j *v t*ð Þ <sup>2</sup> <sup>1</sup>

<sup>2</sup> j j *v t*ð Þ <sup>2</sup>

*<sup>δ</sup>*ð Þþ *<sup>W</sup>* <sup>1</sup>

a power series

¼ 1 2*π* ð<sup>∞</sup> �∞

> ¼ j j *u* 2

with no spin flip.

from (115).

**103**

�*β<sup>W</sup>* � � ¼ Gð Þ¼ *<sup>i</sup><sup>β</sup>* j j *<sup>u</sup>*

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

<sup>2</sup> <sup>þ</sup> j j *<sup>v</sup>*

h i *W* ¼ *σ*~*B*0j j *v*

As a consequence, the variance of the work can be determined

Pð Þ¼ *<sup>W</sup>* <sup>1</sup>

1 2*π* ð<sup>∞</sup> �∞

2

second law are always exceptions to the rule and never dominate.

*δ*ð Þþ *W* � *B*<sup>0</sup>

ð Þ 1 þ *σ*~ *e*

� � � � *<sup>e</sup>*

Work taken as a random variable can take three values *W* ¼ 0, þ *B*0, � *B*<sup>0</sup> where *B*<sup>0</sup> is the energy spacing between the up and down states. The event *W* ¼ *B*<sup>0</sup> corresponds to the case where the spin was originally up and then reversed, so an up-down transition. The energy change is ð Þ� � *B*0*=*2 ð Þ¼ *B*0*=*2 *B*0. Similarly, *W* ¼ �*B*<sup>0</sup> is the opposite flip from this one, and *W* ¼ 0 is the case

<sup>0</sup>j j *v* <sup>2</sup> � *<sup>σ</sup>*~<sup>2</sup> *B*2 <sup>0</sup>j j *v*

> 2*π* ð<sup>∞</sup> �∞

This is the Jarzynski inequality, since it is the case that Δ*F* ¼ 0 here. The statistical moments of the work can be obtained by writing an expression for G into

From (121), the first and second moments can be obtained; for example

2

<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>i</sup>σ*~*B*0*<sup>y</sup>* � <sup>1</sup>

A final calculation that may be considered is the full distribution of work Pð Þ *W* .

Using the Fourier integral form of the delta function, (124) can be written as

*dy*Gð Þ*y e*

1 2 j j *v t*ð Þ <sup>2</sup>

The second law would have us think that *W* >0, but a down-up flip should have *W* ¼ �*B*0, so Pð Þ *W* ¼ �*B*<sup>0</sup> is the probability of observing a local violation of the second law. However, since Pð Þ *W* ¼ �*B*<sup>0</sup> is proportional to 1 � *σ*~, up-down flips are always more likely than down-up. This ensures that h i *W* ≥ 0, so violations of the

The work performed by an external magnetic field on a single spin-1*=*2 particle has been studied so far. The energy differences mentioned correspond to the work. For noninteracting particles, energy is additive. Hence the total work h i W which is performed during a certain process is the sum of works performed on each individual particle W ¼ *W*<sup>1</sup> þ ⋯ þ *WN*. Since the spins are all independent and energy is an extensive variable, it follows that h i W ¼ *N W*h i. where h i *W* is the average work

*iB*0*<sup>y</sup>* <sup>þ</sup> 1 2

*dy*Gð Þ*y e*

�*iyW*

ð Þ 1 � *σ*~ *e*

�*iB*0*y*

ð Þ 1 þ *σ*~ *δ*ð Þ *W* þ *B*<sup>0</sup> *:*

, *<sup>W</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>B</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> j j *<sup>v</sup>*

2 *B*2 <sup>0</sup>*y*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> � �*:* (121)

> <sup>4</sup> � *<sup>B</sup>*<sup>2</sup> <sup>0</sup>j j *v*

<sup>0</sup>j j *v* 2

<sup>2</sup> <sup>1</sup> � *<sup>σ</sup>*~<sup>2</sup>

j j *v* <sup>2</sup> � �*:* (123)

�*iyW:* (124)

�*iyW:*

(125)

<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:* (120)

*:* (122)

This is a consequence of the fact that *B*< <*B*0. According to h i *W* ≥*F*, it should be expected that

$$
\langle W \rangle\_t \ge \Delta F = \mathbf{0}.\tag{117}
$$

Given the matrices for *U t*ð Þ and *ρ* that have been determined so far, the function G can be computed:

$$\mathcal{G}(\boldsymbol{y}) = \operatorname{Tr} \left( U^{\dagger}(\boldsymbol{y}) \epsilon^{j k H\_{j}} U(\boldsymbol{y}) \epsilon^{-j p\_{i}} \boldsymbol{\rho} \right)$$

$$\boldsymbol{\rho} = \operatorname{Tr} \begin{pmatrix} \boldsymbol{u}^{\*} & -\boldsymbol{v} \\ \boldsymbol{v}^{\*} & \boldsymbol{u} \end{pmatrix} \begin{pmatrix} \epsilon^{-j p\_{0}/2} & \mathbf{0} \\ \mathbf{0} & \epsilon^{j p\_{0}/2} \end{pmatrix} \begin{pmatrix} \boldsymbol{u} & \boldsymbol{v} \\ -\boldsymbol{v}^{\*} & \boldsymbol{u}^{\*} \end{pmatrix} \begin{pmatrix} \epsilon^{j p\_{0}/2} & \mathbf{0} \\ \mathbf{0} & \epsilon^{-j p\_{0}/2} \end{pmatrix} \begin{pmatrix} \frac{1}{2} (1 + \bar{\sigma}) & \mathbf{0} \\ \mathbf{0} & \frac{1}{2} (1 - \bar{\sigma}) \end{pmatrix}. \tag{10.8}$$

$$= \left| \boldsymbol{u} \right|^{2} + \frac{1}{2} \left( (\mathbf{1} + \bar{\sigma}) \boldsymbol{e}^{j \eta \boldsymbol{B}\_{0}} + \left( \mathbf{1} - \bar{\sigma} \boldsymbol{e}^{-j \boldsymbol{B}\_{0}} \right) \left| \boldsymbol{v} \right|^{2} \right. \tag{10.9}$$

Set *x* ¼ *iβ* and recall use definition (42) for *σ*~ in the second term of (118) to give

$$\begin{aligned} \left(\mathbf{1} + \tanh\left(\frac{\beta}{2}B\_0\right)\right)e^{-\beta B\_0} + \left(\mathbf{1} - \tanh\left(\frac{\beta}{2}B\_0\right)\right)e^{\beta B\_0} \\ = \mathbf{2}\cosh\left(\beta B\_0\right) - \mathbf{2}\tanh\left(\frac{\beta}{2}B\_0\right)\sinh\left(\beta B\_0\right) = \mathbf{1}. \end{aligned} \tag{119}$$

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

Substituting (119) into (118), we can conclude

quantities are calculated which are related to the energy. Let us determine the

ð Þ*t σzU t*ð Þ*ρ*Þ

! <sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>~</sup> <sup>0</sup>

! *u v*

The average work at time *t* is simply the difference between the energy st time *t*<sup>1</sup>

þ *B*0

frequency Ω*=*2. This is a consequence of the fact the time evolution is unitary. The amplitude multiplying the average work is proportional to the initial magnetization

The equilibrium free energy is *F* ¼ �*T* log *Z* where *Z* ¼ 2 cosh ð Þ *B*0*=*2*T* . The free energy of the initial state at *t* ¼ 0 and final state at any arbitrary time is the

This is a consequence of the fact that *B*< <*B*0. According to h i *W* ≥*F*, it should

Given the matrices for *U t*ð Þ and *ρ* that have been determined so far, the function

*ixH <sup>f</sup> U y* ð Þ*<sup>e</sup>* �*iyHi ρ* � �

! *eiyB*0*=*<sup>2</sup> 0

*iyB*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � *<sup>σ</sup>*~*<sup>e</sup>*

�*βB*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � tanh *<sup>β</sup>*

Set *x* ¼ *iβ* and recall use definition (42) for *σ*~ in the second term of (118) to give

2 *B*0 � �

ð Þ*y e*

�*<sup>v</sup>* <sup>∗</sup> *<sup>u</sup>*<sup>∗</sup>

<sup>2</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>~</sup> *<sup>e</sup>*

*e*

�*<sup>v</sup>* <sup>∗</sup> *<sup>u</sup>*<sup>∗</sup>

0 1 � *σ*~

<sup>2</sup> *<sup>σ</sup>*<sup>~</sup> <sup>¼</sup> *<sup>σ</sup>*~*B*0j j *<sup>v</sup>*

*B* <sup>Ω</sup><sup>2</sup> sin <sup>2</sup> <sup>Ω</sup>

¼ *σ*~*B*<sup>0</sup>

!

! !

1 2

2

2 *t* � � ,

. The average work oscillates indefinitely with

Δ*F* ¼ 0*:* (116)

h i *W <sup>t</sup>* ≥ Δ*F* ¼ 0*:* (117)

1 2

*e βB*<sup>0</sup>

sinh ð Þ¼ *βB*<sup>0</sup> 1*:* (119)

0

BB@

*:*

2 *B*0 ð Þ 1 þ *σ*~ 0

1

CCA

(118)

<sup>0</sup> <sup>1</sup> 2 ð Þ 1 � *σ*~

0 *e*�*iyB*0*=*<sup>2</sup> !

�*iyB*<sup>0</sup> � �j j *<sup>v</sup>* <sup>2</sup>

� � � �

1 þ *σ*~ 0 0 1 � *σ*~

> *B*0*σ*~ 1 � 2j j *v* <sup>2</sup> � �

*:*

(114)

(115)

¼ � <sup>1</sup> 4

0 �1

! !

*<sup>v</sup>* <sup>∗</sup> �*u*<sup>∗</sup>

<sup>2</sup> *<sup>σ</sup>*<sup>~</sup> <sup>1</sup> � <sup>2</sup>j j *<sup>v</sup>* <sup>2</sup> � �

*<sup>ϑ</sup>* sin <sup>2</sup> <sup>Ω</sup> 2 *t* � �

*=*Ω<sup>2</sup>

, so the ratio is a Lorentzian function.

Gð Þ¼ *<sup>y</sup> Tr U*†

1

�

0 *eiyB*0*=*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> tanh *<sup>β</sup>*

� � � �

<sup>¼</sup> j j *<sup>u</sup>* <sup>2</sup> <sup>þ</sup>

! *u v*

2 *B*0

<sup>¼</sup> 2 cosh ð Þ� *<sup>β</sup>B*<sup>0</sup> 2 tanh *<sup>β</sup>*

energy of the system at any *t* by taking operator *A* to be *H*0:

*B*0*Tr U*†

! 1 0

2

*<sup>u</sup>*<sup>∗</sup> �*<sup>v</sup> v* <sup>∗</sup> *u*

! *u v*

h i *<sup>H</sup>*<sup>0</sup> *<sup>t</sup>* ¼ � *<sup>B</sup>*<sup>0</sup>

*Quantum Mechanics*

¼ � <sup>1</sup> 2 *B*0Tr

since sin <sup>2</sup>

*=*Ω<sup>2</sup>

*σ*~ and *B*<sup>2</sup>

same yielding

be expected that

G can be computed:

! *e*�*iyB*0*=*<sup>2</sup> 0

<sup>¼</sup> Tr *<sup>u</sup>*<sup>∗</sup> �*<sup>v</sup> v* <sup>∗</sup> *u*

**102**

¼ � *<sup>B</sup>*<sup>0</sup> 4 Tr <sup>2</sup> h i *<sup>σ</sup><sup>z</sup> <sup>t</sup>* ¼ � <sup>1</sup>

*<sup>u</sup>*<sup>∗</sup> �*<sup>v</sup> v* <sup>∗</sup> *u*

and *t* ¼ 0. Since *v*ð Þ¼ 0 0, this difference is

h i *<sup>W</sup> <sup>t</sup>* ¼ � *<sup>B</sup>*<sup>0</sup>

*<sup>ϑ</sup>* <sup>¼</sup> <sup>1</sup> � cos <sup>2</sup> *<sup>ϑ</sup>* <sup>¼</sup> *<sup>B</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>σ</sup>*~*B*<sup>0</sup> sin <sup>2</sup>

$$\left\langle e^{-\beta W} \right\rangle = \mathcal{G}(i\beta) = \left| u \right|^2 + \left| v \right|^2 = \mathbf{1}.\tag{120}$$

This is the Jarzynski inequality, since it is the case that Δ*F* ¼ 0 here. The statistical moments of the work can be obtained by writing an expression for G into a power series

$$\mathcal{G}(y) = |u|^2 + |v|^2 \left( \mathbf{1} + i\tilde{\sigma}B\_0 y - \frac{\mathbf{1}}{2}B\_0^2 y^2 + \cdots \right). \tag{121}$$

From (121), the first and second moments can be obtained; for example

$$
\langle \mathcal{W} \rangle = \tilde{\sigma} B\_0 |v|^2, \qquad \left\langle \mathcal{W}^2 \right\rangle = B\_0^2 |v|^2. \tag{122}
$$

As a consequence, the variance of the work can be determined

$$\text{var}\left(\mathcal{W}\right) = \left\langle \mathcal{W}^2 \right\rangle - \left\langle \mathcal{W} \right\rangle^2 = B\_0^2 |v|^2 - \tilde{\sigma}^2 B\_0^2 |v|^4 - B\_0^2 |v|^2 \left(\mathbb{1} - \tilde{\sigma}^2 |v|^2\right). \tag{123}$$

A final calculation that may be considered is the full distribution of work Pð Þ *W* . Now Pð Þ *W* is the inverse Fourier transform of Gð Þ*y* :

$$\mathcal{P}(W) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} dy \, \mathcal{G}(y) \, e^{-i\mathcal{y}W}. \tag{124}$$

Using the Fourier integral form of the delta function, (124) can be written as

$$\mathcal{P}(W) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} dy \, \mathcal{G}(y) e^{-i\eta W}$$

$$= \frac{1}{2\pi} \int\_{-\infty}^{\infty} dy \left( |u(t)|^2 + |v(t)|^2 \left[ \frac{1}{2} (1 + \bar{\sigma}) e^{iB\_0 \eta} + \frac{1}{2} (1 - \bar{\sigma}) e^{-iB\_0 \eta} \right] \right) e^{-i\eta W}. \tag{125}$$

$$= |u|^2 \delta(\mathcal{W}) + \frac{1}{2} |v(t)|^2 \delta(\mathcal{W} - B\_0) + \frac{1}{2} |v(t)|^2 (1 + \bar{\sigma}) \delta(\mathcal{W} + B\_0).$$

Work taken as a random variable can take three values *W* ¼ 0, þ *B*0, � *B*<sup>0</sup> where *B*<sup>0</sup> is the energy spacing between the up and down states. The event *W* ¼ *B*<sup>0</sup> corresponds to the case where the spin was originally up and then reversed, so an up-down transition. The energy change is ð Þ� � *B*0*=*2 ð Þ¼ *B*0*=*2 *B*0. Similarly, *W* ¼ �*B*<sup>0</sup> is the opposite flip from this one, and *W* ¼ 0 is the case with no spin flip.

The second law would have us think that *W* >0, but a down-up flip should have *W* ¼ �*B*0, so Pð Þ *W* ¼ �*B*<sup>0</sup> is the probability of observing a local violation of the second law. However, since Pð Þ *W* ¼ �*B*<sup>0</sup> is proportional to 1 � *σ*~, up-down flips are always more likely than down-up. This ensures that h i *W* ≥ 0, so violations of the second law are always exceptions to the rule and never dominate.

The work performed by an external magnetic field on a single spin-1*=*2 particle has been studied so far. The energy differences mentioned correspond to the work. For noninteracting particles, energy is additive. Hence the total work h i W which is performed during a certain process is the sum of works performed on each individual particle W ¼ *W*<sup>1</sup> þ ⋯ þ *WN*. Since the spins are all independent and energy is an extensive variable, it follows that h i W ¼ *N W*h i. where h i *W* is the average work from (115).

## **6. Conclusions**

We have tried to give an introduction to this frontier area that lies in between that of thermodynamics and quantum mechanics in such a way as to be comprehensible. There are many other areas of investigation presently which have had interesting repercussions for this area as well. There is a growing awareness that entanglement facilitates reaching equilibrium [21–23]. It is then worth mentioning that the ideas of einselection and entanglement with the environment can lead to a time-independent equilibrium in an individual quantum system and statistical mechanics can be done without ensembles. However, there is really a lot of work yet to be done in these blossoming areas and will be left for possible future expositions.

**References**

Scientific; 1993

Society; 2008

pp. 241-263

NY: Dover; 1956

[1] Boltzmann L. Vorlesungen über Gastheorie. Leipzig: Barth; 1872

*DOI: http://dx.doi.org/10.5772/intechopen.91831*

systems. Communications in

Physics. 1973;**40**:147-151

553-569

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics*

1957;**29**:74-93

1978;**50**:221-260

627-633

**9**:316-398

Mathematical Physics. 1974;**39**:111-119

[12] Lindblad G. Completely positive maps and entropy inequalities. Communications in Mathematical

[13] Lieb E. The stability of matter. Reviews of Modern Physics. 1976;**48**:

[14] Fano U. Description of states in quantum mechanics and density matrix techniques. Reviews of Modern Physics.

[15] Ribeiro W, Landi GT, Semião F. Quantum thermodynamics and work fluctuations with applications to magnetic resonance. American Journal

[16] Wehrl A. General properties of entropy. Reviews of Modern Physics.

[17] Bracken P. A quantum Carnot engine in three dimensions. Advanced Studies in Theoretical Physics. 2014;**8**:

[18] Lieb E. The classical limit of

Communications in Mathematical

[19] Lieb E, Liebowitz J. The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Advances in Mathematics. 1972;

[20] Crooks J. On the Jarzynski relation for dissipative systems. Journal of Statistical Mechanics: Theory and Experiment. 2008;**10023**:1-9

[21] Vedral V. The role of relative entropy in quantum information theory.

quantum spin systems.

Physics. 1973;**31**:327-340

of Physics. 2016;**84**:948-957

[2] Landau L, Lifschitz E. Statistical Mechanics. Oxford Press, Oxford; 1978

[3] Gibbons E, Hawking S. Euclidean Quantum Gravity. Singapore: World

[4] Lebowitz J. From time-symmetric microscopic dynamics to timeasymmetric macroscopic behavior an overview. In: Gallavotti G, Reuter W, Yngyason J, editors. Boltzmann's Legacy. Zürich: European Mathematical

[5] Bracken P. A quantum version of the classical Szilard engine. Central European Journal of Physics. 2014;**12**:1-8

[6] Bracken P. Quantum dynamics, entropy and quantum versions of Maxwell's demon. In: Bracken P, editor.

Dynamics. Croatia: IntechOpen; 2016.

[7] Fermi E. Thermodynamics. Mineola,

[9] Narnhofer H, Wreszinski WF. On reduction of the wave packet, decoherence, irreversibility and the second law of thermodynamics. Physics

[10] Linblad G. Entropy, information

Communications in Mathematical

[11] Lindblad G. Expectations and entropy inequalities for finite quantum

Recent Advances in Quantum

[8] Allahverdyan A, Balian R, Nieuwenhuisen T. Understanding quantum measurement from the solution of dynamical models. Physics

Reports. 2013;**525**:1-166

Reports. 2014;**541**:249-273

and quantum mechanics.

Physics. 1973;**33**:305-322

**105**

### **Author details**

Paul Bracken Department of Mathematics, University of Texas, Edinburg, TX, USA

\*Address all correspondence to: paul.bracken@utrgv.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Entropy in Quantum Mechanics and Applications to Nonequilibrium Thermodynamics DOI: http://dx.doi.org/10.5772/intechopen.91831*

#### **References**

**6. Conclusions**

*Quantum Mechanics*

expositions.

**Author details**

Department of Mathematics, University of Texas, Edinburg, TX, USA

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: paul.bracken@utrgv.edu

provided the original work is properly cited.

Paul Bracken

**104**

We have tried to give an introduction to this frontier area that lies in between that of thermodynamics and quantum mechanics in such a way as to be comprehensible. There are many other areas of investigation presently which have had interesting repercussions for this area as well. There is a growing awareness that entanglement facilitates reaching equilibrium [21–23]. It is then worth mentioning that the ideas of einselection and entanglement with the environment can lead to a time-independent equilibrium in an individual quantum system and statistical mechanics can be done without ensembles. However, there is really a lot of work yet to be done in these blossoming areas and will be left for possible future

[1] Boltzmann L. Vorlesungen über Gastheorie. Leipzig: Barth; 1872

[2] Landau L, Lifschitz E. Statistical Mechanics. Oxford Press, Oxford; 1978

[3] Gibbons E, Hawking S. Euclidean Quantum Gravity. Singapore: World Scientific; 1993

[4] Lebowitz J. From time-symmetric microscopic dynamics to timeasymmetric macroscopic behavior an overview. In: Gallavotti G, Reuter W, Yngyason J, editors. Boltzmann's Legacy. Zürich: European Mathematical Society; 2008

[5] Bracken P. A quantum version of the classical Szilard engine. Central European Journal of Physics. 2014;**12**:1-8

[6] Bracken P. Quantum dynamics, entropy and quantum versions of Maxwell's demon. In: Bracken P, editor. Recent Advances in Quantum Dynamics. Croatia: IntechOpen; 2016. pp. 241-263

[7] Fermi E. Thermodynamics. Mineola, NY: Dover; 1956

[8] Allahverdyan A, Balian R, Nieuwenhuisen T. Understanding quantum measurement from the solution of dynamical models. Physics Reports. 2013;**525**:1-166

[9] Narnhofer H, Wreszinski WF. On reduction of the wave packet, decoherence, irreversibility and the second law of thermodynamics. Physics Reports. 2014;**541**:249-273

[10] Linblad G. Entropy, information and quantum mechanics. Communications in Mathematical Physics. 1973;**33**:305-322

[11] Lindblad G. Expectations and entropy inequalities for finite quantum systems. Communications in Mathematical Physics. 1974;**39**:111-119

[12] Lindblad G. Completely positive maps and entropy inequalities. Communications in Mathematical Physics. 1973;**40**:147-151

[13] Lieb E. The stability of matter. Reviews of Modern Physics. 1976;**48**: 553-569

[14] Fano U. Description of states in quantum mechanics and density matrix techniques. Reviews of Modern Physics. 1957;**29**:74-93

[15] Ribeiro W, Landi GT, Semião F. Quantum thermodynamics and work fluctuations with applications to magnetic resonance. American Journal of Physics. 2016;**84**:948-957

[16] Wehrl A. General properties of entropy. Reviews of Modern Physics. 1978;**50**:221-260

[17] Bracken P. A quantum Carnot engine in three dimensions. Advanced Studies in Theoretical Physics. 2014;**8**: 627-633

[18] Lieb E. The classical limit of quantum spin systems. Communications in Mathematical Physics. 1973;**31**:327-340

[19] Lieb E, Liebowitz J. The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Advances in Mathematics. 1972; **9**:316-398

[20] Crooks J. On the Jarzynski relation for dissipative systems. Journal of Statistical Mechanics: Theory and Experiment. 2008;**10023**:1-9

[21] Vedral V. The role of relative entropy in quantum information theory. Reviews of Modern Physics. 2002;**74**: 197-234

**Chapter 7**

Spin)

**Abstract**

**1. Introduction**

**107**

*Vahram Mekhitarian*

Equations of Relativistic and

**Keywords:** quantum mechanics, relativistic invariant equations

*I must begin all over with new eyes, I must rethink everything!*

*both dispense with the necessity of reflection.*

*from seeing anything beyond it!*

*–To doubt everything or to believe everything are two equally convenient solutions;*

*–I know, I know, but suppose – just suppose! – the purity of the circle has blinded us*

**Henri Poincaré (1854-1912)**

**Hypathia (360-415 AD)**

Quantum Mechanics (without

A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.

[22] Zurek WH. Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics. 2000;**9**:715-775

[23] Zurek WH. Eliminating ensembles from equilibrium statistical physics, Maxwell's demon, Szilard's engine and thermodynamics via entanglement. Physics Reports. 2018;**755**:1-21

#### **Chapter 7**

Reviews of Modern Physics. 2002;**74**:

einselection, and the quantum origins of

[23] Zurek WH. Eliminating ensembles from equilibrium statistical physics, Maxwell's demon, Szilard's engine and thermodynamics via entanglement. Physics Reports. 2018;**755**:1-21

[22] Zurek WH. Decoherence,

Physics. 2000;**9**:715-775

the classical. Reviews of Modern

197-234

*Quantum Mechanics*

**106**

## Equations of Relativistic and Quantum Mechanics (without Spin)

*Vahram Mekhitarian*

#### **Abstract**

A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.

**Keywords:** quantum mechanics, relativistic invariant equations

#### **1. Introduction**

*–To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.*

**Henri Poincaré (1854-1912)**

*–I know, I know, but suppose – just suppose! – the purity of the circle has blinded us from seeing anything beyond it! I must begin all over with new eyes, I must rethink everything!*

**Hypathia (360-415 AD)**

In 1913, Bohr, based on the Balmer empirical formulas, constructed a model of atom based on the quantization of the orbital momentum [1], which was subsequently supplemented by the more general Sommerfeld quantization rules. In those years, naturally, the presence of a spin or an intrinsic magnetic moment of the particle or, especially, spin-orbit interaction, or interaction with the nuclear spin, was not supposed.

In 1916, Sommerfeld, within the framework of relativistic approaches, derived a formula for the energy levels of a hydrogen-like atom, without taking into account the spin [2]. Sommerfeld proceeded from the model of the Bohr atom and used the relativistic relation between the momentum **p** and the energy *E* of a free particle with the mass *m***.**

$$E^2 \text{--} (\mathbf{p}c)^2 = \left(mc^2\right)^2,\tag{1}$$

*En*,*<sup>l</sup>* <sup>¼</sup> *mc*<sup>2</sup>

vuuut

*Equations of Relativistic and Quantum Mechanics (without Spin)*

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*En*,*<sup>j</sup>* <sup>¼</sup> *mc*<sup>2</sup>

vuut

was used in the representation

the form *U r*ð Þ¼ *Ze*<sup>2</sup>

function

**109**

an external field

*=r*.

*En*,*<sup>j</sup>* <sup>¼</sup> *mc*<sup>2</sup> � ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup>

*i*ℏ *∂ ∂t* � �<sup>2</sup>

*i*ℏ *∂ ∂t* � *qφ* � �<sup>2</sup> Ψ � *c*

Ψ � *c*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup> *<sup>n</sup>*� ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup> *l*þ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p � �<sup>2</sup>

where *n* ¼ *n*<sup>r</sup> þ *j* þ 1*=*2 ¼ 1, 2, 3, … , *j* ¼ 1*=*2, 3*=*2, 5*=*2, … , *n* þ 1*=*2, and l possess the values *l* ¼ 0 at *j* ¼ 1*=*2 and *l* ¼ *j* � 1*=*2 for others. This formula coincides with the result of an exact solution of the relativistic Dirac equations in 1928 [5] for a particle with the spin 1*=*2 with the classical expression for the potential energy of an immobile charge in the Coulomb field of a nucleus with an atomic number *Z* in

Formula (7) also indicated a strange limitation of value the charge of a nucleus with the atomic number *Z* <137, above which the formula is losing its meaning. It was also evident that within the framework of the approaches outlined, the strong and gravitational interactions, the motions of the planets are not described. The problem *Z* < 137 or *α*> 1 remains the unresolved problem of relativistic quantum mechanics. Expanding the formula (7) over the order of powers *Zα*<sup>2</sup> in the Taylor series, with an accuracy of expansion up to the terms by the powers *Zα*6, we obtain

> <sup>2</sup>*n*<sup>2</sup> � ð Þ *<sup>Z</sup><sup>α</sup>* <sup>4</sup> 2*n*<sup>3</sup>

In 1925–1926, Schrödinger worked on the derivation of the equation for the wave function of a particle describing the De Broglie waves [6]. The derivation of the equation also was based on the relativistic relation (1) between the momentum **p** and the energy *E* of the particle, which he presented with the help of the operators of squares of energy and momentum in the form of an equation for the wave

> <sup>2</sup> �*i*<sup>ℏ</sup> *<sup>∂</sup> ∂***r** � �<sup>2</sup>

<sup>2</sup> �*i*<sup>ℏ</sup> *<sup>∂</sup> ∂***r** � *q c* **A**

Like Sommerfeld, Schrödinger used the following representation for a particle in

� �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>j</sup>*þ1*=*<sup>2</sup> <sup>2</sup>�*Z*2*α*<sup>2</sup>

1 *<sup>j</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>3</sup>

� �

<sup>Ψ</sup> <sup>¼</sup> *mc*<sup>2</sup> � �<sup>2</sup>

<sup>Ψ</sup> <sup>¼</sup> *mc*<sup>2</sup> � �<sup>2</sup>

4*n*

<sup>1</sup> <sup>þ</sup> *<sup>Z</sup>α*<sup>2</sup> *<sup>n</sup>*� *<sup>Z</sup>*2*α*<sup>2</sup> *j*þ1*=*2þ

The formula (6) perfectly described all the peculiarities of the structure of the spectrum of hydrogen and other similar atoms with the limiting for those years accuracy of measurements, and there was no doubt about the correctness of the formula itself. Therefore, the Sommerfeld formula was perceived as empirical, and instead of the quantum number l, a '*mysterious*' internal quantum number with halfinteger values *j* ¼ 1*=*2, 3*=*2, 5*=*2, … , *n* þ 1*=*2 was introduced, and formula (6)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>*þ<sup>1</sup> <sup>2</sup>�ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup> p � �<sup>2</sup> *:* (6)

, (7)

þ … (8)

Ψ (9)

Ψ (10)

where *c* is the speed of light.

In an external field with a four-dimensional potential (*φ*, **A**), it was supposed that for a particle with the charge *q* this relation can also be used if we subtract the components of the four-dimensional momentum of the field (*qφ*, *q***A**) from the expression for the generalized particle momentum:

$$(E - q\rho)^2 - (\mathbf{p}c - q\mathbf{A})^2 = \left(mc^2\right)^2. \tag{2}$$

In the case of the Coulomb potential *φ* ¼ *Z e*j j*=r*, where e is the charge of electron, *r* is the distance from the nucleus, and *Z* is an atomic number, we obtain in spherical coordinates

$$\left(p\_{\mathbf{r}}\,^2 + r^2 p\_{\boldsymbol{\varrho}}\,^2 = p\_{\mathbf{r}}\,^2 + \frac{L^2}{r^2} = \frac{\left(E + Z\boldsymbol{\hat{z}}^2/r\right)^2 - \left(mc^2\right)^2}{c^2} \tag{3}$$

where *L* is the angular momentum. The Bohr-Sommerfeld quantization conditions take the form

$$\begin{aligned} \oint p\_{\varphi} d\rho &= \hbar n\_{\varphi}, \\\\ \oint p\_{\mathbf{r}} dr &= \oint \sqrt{\frac{\left(E + Z\mathbf{z}^2/r\right)^2 - \left(mc^2\right)^2}{c^2} - \frac{L^2}{r^2}} dr = \hbar n\_{\mathbf{r}}, \end{aligned} \tag{4}$$

where *n<sup>φ</sup>* and *nr* are the orbital and radial quantum numbers, respectively. For the energy levels, Sommerfeld obtained the formula

$$E\_{n,l} = \frac{mc^2}{\sqrt{\mathbf{1} + \frac{(Za)^2}{\left(n - \frac{(Za)^2}{l + 1/2 + \sqrt{\left(l + 1/2\right)^2 - (Za)^2}}\right)^2}}},\tag{5}$$

where the principal quantum number *n* ¼ *n*<sup>r</sup> þ *l* þ 1 ¼ 1, 2, 3, … , *l* ¼ 0, 1 , 2, 3, … , *n* � 1, and *α* ¼ 1*=*137*:*036 is the fine structure constant. However, in a paper published in 1916 [3], Sommerfeld '*made a fortunate mistake*' [4] and the derived formula was presented in the following form

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

In 1913, Bohr, based on the Balmer empirical formulas, constructed a model of atom based on the quantization of the orbital momentum [1], which was subsequently supplemented by the more general Sommerfeld quantization rules. In those years, naturally, the presence of a spin or an intrinsic magnetic moment of the particle or, especially, spin-orbit interaction, or interaction with the nuclear spin,

In 1916, Sommerfeld, within the framework of relativistic approaches, derived a formula for the energy levels of a hydrogen-like atom, without taking into account the spin [2]. Sommerfeld proceeded from the model of the Bohr atom and used the relativistic relation between the momentum **p** and the energy *E* of a free particle

<sup>2</sup> <sup>¼</sup> *mc*<sup>2</sup> � �<sup>2</sup>

–ð Þ **<sup>p</sup>***c*–*q***<sup>A</sup>** <sup>2</sup> <sup>¼</sup> *mc*<sup>2</sup> � �<sup>2</sup>

*<sup>r</sup>*<sup>2</sup> <sup>¼</sup> *<sup>E</sup>* <sup>þ</sup> *Ze*<sup>2</sup>

*<sup>=</sup><sup>r</sup>* � �<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup>

*r*2

*dr* ¼ ℏ*n*r,

In an external field with a four-dimensional potential (*φ*, **A**), it was supposed that for a particle with the charge *q* this relation can also be used if we subtract the components of the four-dimensional momentum of the field (*qφ*, *q***A**) from the

In the case of the Coulomb potential *φ* ¼ *Z e*j j*=r*, where e is the charge of electron, *r* is the distance from the nucleus, and *Z* is an atomic number, we obtain in

where *L* is the angular momentum. The Bohr-Sommerfeld quantization

<sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where *n<sup>φ</sup>* and *nr* are the orbital and radial quantum numbers, respectively. For

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup> *<sup>n</sup>*� ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup> *l*þ1*=*2þ

p � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>*þ1*=*<sup>2</sup> <sup>2</sup>�ð Þ *<sup>Z</sup><sup>α</sup>* <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> � *<sup>L</sup>*<sup>2</sup>

*<sup>=</sup><sup>r</sup>* � �<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup>

*<sup>E</sup>* <sup>þ</sup> *Ze*<sup>2</sup>

*En*,*<sup>l</sup>* <sup>¼</sup> *mc*<sup>2</sup>

where the principal quantum number *n* ¼ *n*<sup>r</sup> þ *l* þ 1 ¼ 1, 2, 3, … , *l* ¼ 0, 1 , 2, 3, … , *n* � 1, and *α* ¼ 1*=*137*:*036 is the fine structure constant. However, in a paper published in 1916 [3], Sommerfeld '*made a fortunate mistake*'

vuuut

[4] and the derived formula was presented in the following form

, (1)

*:* (2)

*<sup>c</sup>*<sup>2</sup> (3)

, (5)

(4)

*E*2 –ð Þ **p***c*

was not supposed.

*Quantum Mechanics*

with the mass *m***.**

spherical coordinates

conditions take the form

**108**

where *c* is the speed of light.

*p*r <sup>2</sup> <sup>þ</sup> *<sup>r</sup>* 2 *pφ* <sup>2</sup> <sup>¼</sup> *<sup>p</sup>*<sup>r</sup> 2 þ *L*2

þ

þ

*pφdφ* ¼ ℏ*nφ*,

the energy levels, Sommerfeld obtained the formula

s

*p*r*dr* ¼

expression for the generalized particle momentum:

ð Þ *E*–*qφ* 2

$$E\_{n,l} = \frac{mc^2}{\sqrt{\mathbf{1} + \frac{(Za)^2}{\left(n - \frac{(Za)^2}{l + 1 + \sqrt{(l+1)^2 - (Za)^2}}\right)^2}}}} \,\tag{6}$$

The formula (6) perfectly described all the peculiarities of the structure of the spectrum of hydrogen and other similar atoms with the limiting for those years accuracy of measurements, and there was no doubt about the correctness of the formula itself. Therefore, the Sommerfeld formula was perceived as empirical, and instead of the quantum number l, a '*mysterious*' internal quantum number with halfinteger values *j* ¼ 1*=*2, 3*=*2, 5*=*2, … , *n* þ 1*=*2 was introduced, and formula (6) was used in the representation

$$E\_{n,j} = \frac{mc^2}{\sqrt{\mathbf{1} + \frac{\mathbf{1}\_{\alpha^2}}{\left(n - \frac{\mathbf{1}^2 \mathbf{a}^2}{(\mathbf{1}\_{\alpha^2} + \mathbf{1}\_{\alpha^2})^2 - \mathbf{1}^2 \mathbf{a}^2}\right)^2}}},\tag{7}$$

where *n* ¼ *n*<sup>r</sup> þ *j* þ 1*=*2 ¼ 1, 2, 3, … , *j* ¼ 1*=*2, 3*=*2, 5*=*2, … , *n* þ 1*=*2, and l possess the values *l* ¼ 0 at *j* ¼ 1*=*2 and *l* ¼ *j* � 1*=*2 for others. This formula coincides with the result of an exact solution of the relativistic Dirac equations in 1928 [5] for a particle with the spin 1*=*2 with the classical expression for the potential energy of an immobile charge in the Coulomb field of a nucleus with an atomic number *Z* in the form *U r*ð Þ¼ *Ze*<sup>2</sup> *=r*.

Formula (7) also indicated a strange limitation of value the charge of a nucleus with the atomic number *Z* <137, above which the formula is losing its meaning. It was also evident that within the framework of the approaches outlined, the strong and gravitational interactions, the motions of the planets are not described. The problem *Z* < 137 or *α*> 1 remains the unresolved problem of relativistic quantum mechanics. Expanding the formula (7) over the order of powers *Zα*<sup>2</sup> in the Taylor series, with an accuracy of expansion up to the terms by the powers *Zα*6, we obtain

$$E\_{n,j} = mc^2 - \frac{\left(Za\right)^2}{2n^2} - \frac{\left(Za\right)^4}{2n^3} \left(\frac{1}{j + 1/2} - \frac{3}{4n}\right) + \dots \tag{8}$$

In 1925–1926, Schrödinger worked on the derivation of the equation for the wave function of a particle describing the De Broglie waves [6]. The derivation of the equation also was based on the relativistic relation (1) between the momentum **p** and the energy *E* of the particle, which he presented with the help of the operators of squares of energy and momentum in the form of an equation for the wave function

$$\left(i\hbar\frac{\partial}{\partial t}\right)^2\Psi - c^2\left(-i\hbar\frac{\partial}{\partial \mathbf{r}}\right)^2\Psi = \left(mc^2\right)^2\Psi\tag{9}$$

Like Sommerfeld, Schrödinger used the following representation for a particle in an external field

$$\left(i\hbar\frac{\partial}{\partial t} - q\rho\boldsymbol{\rho}\right)^{2}\boldsymbol{\Psi} - c^{2}\left(-i\hbar\frac{\partial}{\partial \mathbf{r}} - \frac{q}{c}\mathbf{A}\right)^{2}\boldsymbol{\Psi} = \left(mc^{2}\right)^{2}\boldsymbol{\Psi}\tag{10}$$

In the case of stationary states of a charged particle in the field of the Coulomb potential for a hydrogen atom it was necessary to solve the equation

$$\frac{d^2\Psi}{d\mathbf{r}^2} + \frac{2m}{\hbar^2} \left( \frac{\mathbf{E}^2 - m^2 c^4}{2mc^2} - \frac{E}{mc^2} q\rho(\mathbf{r}) + \frac{q^2}{2mc^2} \rho^2(\mathbf{r}) \right) \psi = \mathbf{0} \tag{11}$$

As can be seen, the quadratic expression of potential energy *<sup>q</sup>*2*φ*2ð Þ**<sup>r</sup>** *<sup>=</sup>*2*mc*<sup>2</sup> is present in the equation with a positive sign and in the case of attracting fields, the solutions lead to certain difficulties. When approaching the singularity point, due to the negative sign, the attractive forces increase and the presence of the singularity leads to known limitations on the magnitude of the interactions (**Figure 1**).

Next, the wave vector *k* is represented as

$$k\_1 = \frac{1}{\hbar c} \sqrt{E^2 - (mc^2)^2}, \qquad k\_2 = \frac{1}{\hbar c} \sqrt{\left(E - U\right)^2 - \left(mc^2\right)^2} \tag{12}$$

and when considering the problem of the passage of a particle with energy *E* through a potential barrier *U* ¼ *qφ*ð Þ**r** (**Figure 2**), the height of which is greater than the doubled resting energy of the particle *U* >2*mc*2, the transmission coefficient becomes unity, regardless of the height of the barrier (Klein paradox) [7].

Another difficulty is that, as the solution of the particle problem in a potential well shows, at a sufficient depth, a particle with a wavelength *ƛ* ¼ ℏ*=mc* can have bound states (can be localized) in a well width narrower than the wavelength of the particle *d*< *ƛ=*2 (**Figure 3**), which contradicts the fundamental principle of quantum mechanics—the Heisenberg's uncertainty principle.

Also, the solution of the problem of a hydrogen-like atom is limited by the value of the ordinal number of the atomic nucleus *Z* ≤68 (for the Dirac equation, the restriction of the atomic number is *Z* ≤ 137). The same in relativistic mechanics when considering strong interactions, the solution of the Hamilton-Jacoby relativistic equation indicates the so-called "*particle fall on the center*" [8].

In order to get rid of the quadratic term or reverse its sign, in recent years it has been proposed to represent potential energy in the Klein-Gordon and Dirac equations as the difference of squares from the expressions of scalar and vector potentials

> (S-wave equation) [9–11]. Such a mathematical formalism corrects the situation, but from a physical point of view such representations are in no way justified, and the

> Things are even worse with the presence of a quadratic term of the vector field, because of the sign of which we obtain non-existent states in nature and solutions

> > *<sup>E</sup>*<sup>2</sup> � *<sup>m</sup>*<sup>2</sup>*c*<sup>4</sup> <sup>2</sup>*mc*<sup>2</sup> � *<sup>q</sup>*<sup>2</sup>

According to the solutions of the equations of quantum mechanics and Hamilton-Jacoby, it turns out that a charged particle in a magnetic field, in addition to rotating in a circle, also has radial vibrations—Landau levels [12] (even in the

> *z* <sup>2</sup>*<sup>M</sup>* � *<sup>M</sup>ω*<sup>2</sup>

*<sup>E</sup>* <sup>¼</sup> <sup>ℏ</sup>*ω<sup>H</sup> <sup>n</sup><sup>ρ</sup>* <sup>þ</sup> j j *<sup>m</sup>* <sup>þ</sup> *<sup>m</sup>* <sup>þ</sup> <sup>1</sup>

*H*

2 

<sup>þ</sup> *<sup>E</sup>* � *<sup>p</sup>*<sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> **<sup>A</sup>**<sup>2</sup>

<sup>8</sup> *<sup>ρ</sup>*<sup>2</sup> � <sup>ℏ</sup>*ωHm* 2 *<sup>R</sup>* <sup>¼</sup> <sup>0</sup>*:* (14)

> þ *p*2 *z* 2*M*

ð Þ**r** *<sup>ψ</sup>* <sup>¼</sup> 0 (13)

fields corresponding to such pseudo-potentials do not exist in nature.

that contradict experience.

ℏ*c*

case of zero orbital momentum).

*R*00 þ 1 *ρ* **A r**ð Þ� *<sup>d</sup><sup>ψ</sup> d***r** þ 2*m* ℏ2

*A particle with a wavelength ƛ can be localized in a well width d*<*ƛ=*2*.*

*<sup>R</sup>*<sup>0</sup> � *<sup>m</sup>*<sup>2</sup> *ρ*2

*d*2 *ψ <sup>d</sup>***r**<sup>2</sup> � <sup>2</sup>*<sup>i</sup> <sup>q</sup>*

**Figure 2.**

**Figure 3.**

*Passage of a particle through a potential barrier U.*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

ℏ2 2*M*

**111**

#### **Figure 1.**

*The sample dependency of the attractive field potential* �1*=r and potential interaction energy* �1*=<sup>r</sup>* � <sup>1</sup>*=*2*r*<sup>2</sup> *in the Klein-Gordon equations.*

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

**Figure 2.**

In the case of stationary states of a charged particle in the field of the Coulomb

� �

As can be seen, the quadratic expression of potential energy *<sup>q</sup>*2*φ*2ð Þ**<sup>r</sup>** *<sup>=</sup>*2*mc*<sup>2</sup> is present in the equation with a positive sign and in the case of attracting fields, the solutions lead to certain difficulties. When approaching the singularity point, due to the negative sign, the attractive forces increase and the presence of the singularity leads to known limitations on the magnitude of the interactions (**Figure 1**).

, *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

and when considering the problem of the passage of a particle with energy *E* through a potential barrier *U* ¼ *qφ*ð Þ**r** (**Figure 2**), the height of which is greater than the doubled resting energy of the particle *U* >2*mc*2, the transmission coefficient becomes unity, regardless of the height of the barrier (Klein paradox) [7].

Another difficulty is that, as the solution of the particle problem in a potential well shows, at a sufficient depth, a particle with a wavelength *ƛ* ¼ ℏ*=mc* can have bound states (can be localized) in a well width narrower than the wavelength of the particle *d*< *ƛ=*2 (**Figure 3**), which contradicts the fundamental principle of quan-

Also, the solution of the problem of a hydrogen-like atom is limited by the value

In order to get rid of the quadratic term or reverse its sign, in recent years it has been proposed to represent potential energy in the Klein-Gordon and Dirac equations as the difference of squares from the expressions of scalar and vector potentials

*The sample dependency of the attractive field potential* �1*=r and potential interaction energy* �1*=<sup>r</sup>* � <sup>1</sup>*=*2*r*<sup>2</sup> *in*

of the ordinal number of the atomic nucleus *Z* ≤68 (for the Dirac equation, the restriction of the atomic number is *Z* ≤ 137). The same in relativistic mechanics when considering strong interactions, the solution of the Hamilton-Jacoby relativ-

istic equation indicates the so-called "*particle fall on the center*" [8].

ℏ*c*

q

*mc*<sup>2</sup> *<sup>q</sup>φ*ð Þþ **<sup>r</sup>** *<sup>q</sup>*<sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

ð Þ**r**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>E</sup>* � *<sup>U</sup>* <sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup>

*ψ* ¼ 0 (11)

(12)

potential for a hydrogen atom it was necessary to solve the equation

*<sup>E</sup>*<sup>2</sup> � *<sup>m</sup>*2*c*<sup>4</sup> <sup>2</sup>*mc*<sup>2</sup> � *<sup>E</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup>

tum mechanics—the Heisenberg's uncertainty principle.

*d*2 *ψ d***r**<sup>2</sup> þ

*Quantum Mechanics*

2*m* ℏ2

Next, the wave vector *k* is represented as

q

*<sup>k</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ℏ*c*

**Figure 1.**

**110**

*the Klein-Gordon equations.*

*Passage of a particle through a potential barrier U.*

(S-wave equation) [9–11]. Such a mathematical formalism corrects the situation, but from a physical point of view such representations are in no way justified, and the fields corresponding to such pseudo-potentials do not exist in nature.

Things are even worse with the presence of a quadratic term of the vector field, because of the sign of which we obtain non-existent states in nature and solutions that contradict experience.

$$\frac{d^2\psi}{d\mathbf{r}^2} - 2i\frac{q}{\hbar c}\mathbf{A}(\mathbf{r}) \cdot \frac{d\psi}{d\mathbf{r}} + \frac{2m}{\hbar^2} \left(\frac{\mathbf{E}^2 - m^2c^4}{2mc^2} - \frac{q^2}{2mc^2}\mathbf{A}^2(\mathbf{r})\right)\psi = 0\tag{13}$$

According to the solutions of the equations of quantum mechanics and Hamilton-Jacoby, it turns out that a charged particle in a magnetic field, in addition to rotating in a circle, also has radial vibrations—Landau levels [12] (even in the case of zero orbital momentum).

$$\frac{\hbar^2}{2M}\left(R' + \frac{1}{\rho}R' - \frac{m^2}{\rho^2}\right) + \left(E - \frac{p\_x^2}{2M} - \frac{M\alpha\_H^2}{8}\rho^2 - \frac{\hbar\alpha\_H m}{2}\right)R = 0.\tag{14}$$

$$E = \hbar\omega\_H \left(n\_\rho + \frac{|m|+m+1}{2}\right) + \frac{p\_x^2}{2M}$$

Over these 90 years, especially in very accurate cyclotron resonance experiments, none has detected the electron radial vibrations and the Landau levels.

Solving this equation, Schrödinger, like Sommerfeld, received the formula (5), which described the structure of the hydrogen spectrum not exactly. Moreover, from the solution of the problem for a particle in a potential well, it turns out that a particle with a wavelength *ƛ* ¼ ℏ*=mc* has bound states (is placed) in a well of arbitrary size and, in particular, much smaller than *ƛ=*2. This fact contradicts the fundamental principle of the quantum (wave) theory, the principle of uncertainty.

In 1925 Schrödinger sent this work to the editors of 'Annalen der Physik' [13], but then took the manuscript, refused the relativistic approaches and in 1926 built a wave equation based on the classical Hamiltonian expression, the Schrödinger equation [14].

$$i\mathbf{H} = \frac{\mathbf{p}^2}{2m} + U; \quad \rightarrow \quad i\hbar \frac{\partial}{\partial t} \Psi = \left(\frac{1}{2m} \left(-i\hbar \frac{\partial}{\partial \mathbf{r}}\right)^2 + U\right) \Psi \tag{15}$$

Clifford-Lipschitz numbers [22]). In the standard representation the Dirac equation

^*ε ϕ* � **σ** � **p**^ χ ¼ *mc ϕ*,

1 0 , <sup>σ</sup>*<sup>y</sup>* <sup>¼</sup> <sup>0</sup> �*<sup>i</sup>*

are the Pauli matrices (the unit matrix in the formulas is omitted). For a particle in an external field, Eq. (16) is usually written in the form

> ^*<sup>ε</sup>* � *<sup>q</sup> c φ <sup>ϕ</sup>* � **<sup>σ</sup>** � **<sup>p</sup>**^ � *<sup>q</sup>*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

from the exact solution of the hydrogen atom problem.

than were laid down by the derivation of the Dirac equation.

� ^*<sup>ε</sup>* � *<sup>q</sup> c φ* <sup>χ</sup> <sup>þ</sup> **<sup>σ</sup>** � **<sup>p</sup>**^ � *<sup>q</sup>*

�^*<sup>ε</sup>* <sup>χ</sup> <sup>þ</sup> **<sup>σ</sup>** � **<sup>p</sup>**^ <sup>φ</sup> <sup>¼</sup> *m c* <sup>χ</sup>, (16)

, <sup>σ</sup>*<sup>z</sup>* <sup>¼</sup> 1 0

0 �1 

(17)

(18)

*i* 0

*c* **A** <sup>χ</sup> <sup>¼</sup> *mc <sup>ϕ</sup>*,

*c* **A** *<sup>ϕ</sup>* <sup>¼</sup> *mc* <sup>χ</sup>,

where for an invariant representation in the case of a free particle, the equations

In the case of the potential energy of an immobile charge in a Coulomb field, we obtain the Sommerfeld-Dirac formula as a result of an exact solution of this particular equation. There, again, although for a system with spin 1*=*2 the energy of the spin-orbit interaction is not taken into account initially, but the half is obtained

More accurate measurements of Lamb in 1947 and subsequent improvements in the spectrum of the hydrogen atom revealed that, in addition to the lines with the maximum j, all the others are also split and somewhat displaced (the Lamb shift). To harmonize the results of the theory with more accurate experimental data on the spectrum of the hydrogen atom, one had to propose other solutions and approaches

The new theoretical approaches had yield nothing and only supplemented the theory with the illogical and non-physical proposals to overcome the emerging singularity of solutions: the renormalization, the finite difference of infinities with the desired value of the difference, and so on. The accounting for the size of the nucleus corrected only the *Z* value into the bigger value, but did not solve the *Z* >137 problem. An incredible result was also obtained for the hydrogen atom problem that the electron is located, most likely, at the center of the atom, that is, in

The results of solution of the problem for a particle in a potential well both in the case of the Klein-Fock-Gordon equation and of the Dirac equation contradict to the basic principle of quantum mechanics, to the uncertainty principle. From the solutions, it turns out to be that a particle can be in a bound state in a well with any dimensions, in particular, with the size much smaller than the wavelength of the

Despite Dirac himself proposed a system of linear first-degree relativistic equa-

tions in the matrix representation that described the system with spin 1*=*2, the contradictions did not disappear, and he himself remained unhappy with the results of his theory. As Dirac wrote in 1956 [24], the development of relativistic electron theory can now be considered as an example of how incorrect arguments sometimes

are composed for the difference between the generalized momentum and the

for a free particle has the form [23].

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

0 1 , <sup>σ</sup>*<sup>x</sup>* <sup>¼</sup> 0 1

where

**<sup>1</sup>** <sup>¼</sup> 1 0

momentum of the field.

the nucleus.

**113**

particle itself, *A* ¼ ℏ*=mc* [23].

Equation described the spectrum of the hydrogen atom only qualitatively, however, it did not have any unreasonable restrictions or singular solutions in the form of the Sommerfeld-Dirac formula. Klein [15], Fock [16] and Gordon [17] published the relativistic equation based on the wave equation for a particle without spin in 1926; it is called the Klein-Fock-Gordon equation.

With the discovery of the spin, the situation changed drastically, and in 1926 Heisenberg and Jordan [18] showed that, within the Pauli description of the spin of an electron, half the energy of the spin-orbit interaction is equal to a term with a power of α<sup>4</sup> in the Taylor series expansion of the Sommerfeld formula *equation reference goes here*.

Why exactly the half, Thomas tried to explain this in 1927 by the presence of a relativistic precession of an electron in the reference frame of motion along the orbit [19]. The energy of the Thomas precession is exactly equal to half the value of the energy of the spin-orbit interaction with the inverse (positive) sign, which should be added to the energy of the spin-orbit interaction. However, the incorrect assumption that the Thomas precession frequency is identical in both frames of reference and the absence of a common and correct derivation for non-inertial (rotating) frames of reference raised doubts about the correctness of such approaches. The reason for the appearance of half the energy of the spin-orbit interaction in the Sommerfeld formula is still under investigation and is one of the unresolved problems in modern physics.

On the other hand, both in the derivation of the Sommerfeld formula and at the solution of the Klein-Fock-Gordon equation for the hydrogen atom problem [20], neither the spin nor the spin-orbit interaction energy was taken into account initially. Therefore, the obtained fine splitting can in no way be owing to the spin-orbit interaction. This is a relativistic but purely mechanical effect, when the mass (inertia) of a particle is already depends on the velocity of motion along the orbit (of the angular momentum), because of which the radial motion of the electron changes, and vice versa. Just this dependence, which results in the splitting of the energy levels of the electron, and to the impossibility of introducing only one, the principal quantum number. Nevertheless, even with this assumption, the order of splitting of the levels according to formula (8) contradicts to the logic; it turns out to be that the greater the orbital angular momentum, the lesser the energy of the split level.

The matrix representation of the second-order wave Eq. (9) by a system of equations of the first order is the Dirac construction of the relativistic electron equation [21] (the Dirac matrices are the particular representation of the

Clifford-Lipschitz numbers [22]). In the standard representation the Dirac equation for a free particle has the form [23].

$$\begin{aligned} \hat{\boldsymbol{\varepsilon}} \, \boldsymbol{\phi} - \boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \, \boldsymbol{\chi} &= m \boldsymbol{\varepsilon} \, \boldsymbol{\phi}, \\ -\hat{\boldsymbol{\varepsilon}} \, \boldsymbol{\chi} + \boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \, \boldsymbol{\phi} &= m \boldsymbol{\varepsilon} \, \boldsymbol{\chi}, \end{aligned} \tag{16}$$

where

Over these 90 years, especially in very accurate cyclotron resonance experiments, none has detected the electron radial vibrations and the Landau levels. Solving this equation, Schrödinger, like Sommerfeld, received the formula (5), which described the structure of the hydrogen spectrum not exactly. Moreover, from the solution of the problem for a particle in a potential well, it turns out that a particle with a wavelength *ƛ* ¼ ℏ*=mc* has bound states (is placed) in a well of arbitrary size and, in particular, much smaller than *ƛ=*2. This fact contradicts the fundamental principle of the quantum (wave) theory, the principle of uncertainty. In 1925 Schrödinger sent this work to the editors of 'Annalen der Physik' [13], but then took the manuscript, refused the relativistic approaches and in 1926 built a wave equation based on the classical Hamiltonian expression, the Schrödinger

equation [14].

*Quantum Mechanics*

*reference goes here*.

**112**

<sup>H</sup> <sup>¼</sup> **<sup>p</sup>**<sup>2</sup> 2*m*

1926; it is called the Klein-Fock-Gordon equation.

unresolved problems in modern physics.

<sup>þ</sup> *<sup>U</sup>*; ! *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup>*

*∂t*

<sup>Ψ</sup> <sup>¼</sup> <sup>1</sup>

Equation described the spectrum of the hydrogen atom only qualitatively, however, it did not have any unreasonable restrictions or singular solutions in the form of the Sommerfeld-Dirac formula. Klein [15], Fock [16] and Gordon [17] published the relativistic equation based on the wave equation for a particle without spin in

With the discovery of the spin, the situation changed drastically, and in 1926 Heisenberg and Jordan [18] showed that, within the Pauli description of the spin of an electron, half the energy of the spin-orbit interaction is equal to a term with a power of α<sup>4</sup> in the Taylor series expansion of the Sommerfeld formula *equation*

Why exactly the half, Thomas tried to explain this in 1927 by the presence of a relativistic precession of an electron in the reference frame of motion along the orbit [19]. The energy of the Thomas precession is exactly equal to half the value of the energy of the spin-orbit interaction with the inverse (positive) sign, which should be added to the energy of the spin-orbit interaction. However, the incorrect assumption that the Thomas precession frequency is identical in both frames of reference and the absence of a common and correct derivation for non-inertial (rotating) frames of reference raised doubts about the correctness of such approaches. The reason for the appearance of half the energy of the spin-orbit interaction in the Sommerfeld formula is still under investigation and is one of the

On the other hand, both in the derivation of the Sommerfeld formula and at the solution of the Klein-Fock-Gordon equation for the hydrogen atom problem [20], neither the spin nor the spin-orbit interaction energy was taken into account initially. Therefore, the obtained fine splitting can in no way be owing to the spin-orbit interaction. This is a relativistic but purely mechanical effect, when the mass (inertia) of a particle is already depends on the velocity of motion along the orbit (of the angular momentum), because of which the radial motion of the electron changes, and vice versa. Just this dependence, which results in the splitting of the energy levels of the electron, and to the impossibility of introducing only one, the principal quantum number. Nevertheless, even with this assumption, the order of splitting of the levels according to formula (8) contradicts to the logic; it turns out to be that the greater the orbital angular momentum, the lesser the energy of the split level. The matrix representation of the second-order wave Eq. (9) by a system of equations of the first order is the Dirac construction of the relativistic electron equation [21] (the Dirac matrices are the particular representation of the

<sup>2</sup>*<sup>m</sup>* �*i*<sup>ℏ</sup> *<sup>∂</sup>*

*∂***r** � �<sup>2</sup>

!

þ *U*

Ψ (15)

$$\mathbf{1} = \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{pmatrix}, \qquad \sigma\_{\mathbf{x}} = \begin{pmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{1} & \mathbf{0} \end{pmatrix}, \qquad \sigma\_{\mathbf{y}} = \begin{pmatrix} \mathbf{0} & -i \\ i & \mathbf{0} \end{pmatrix}, \qquad \sigma\_{\mathbf{z}} = \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{pmatrix} \tag{17}$$

are the Pauli matrices (the unit matrix in the formulas is omitted). For a particle in an external field, Eq. (16) is usually written in the form

$$\begin{aligned} \left(\hat{\boldsymbol{e}} - \frac{q}{c}\boldsymbol{\varrho}\right)\boldsymbol{\phi} - \boldsymbol{\sigma} \cdot \left(\hat{\mathbf{p}} - \frac{q}{c}\mathbf{A}\right)\boldsymbol{\chi} &= mc\,\boldsymbol{\phi}, \\ -\left(\hat{\boldsymbol{e}} - \frac{q}{c}\boldsymbol{\sigma}\right)\boldsymbol{\chi} + \boldsymbol{\sigma} \cdot \left(\hat{\mathbf{p}} - \frac{q}{c}\mathbf{A}\right)\boldsymbol{\phi} &= mc\,\boldsymbol{\chi}, \end{aligned} \tag{18}$$

where for an invariant representation in the case of a free particle, the equations are composed for the difference between the generalized momentum and the momentum of the field.

In the case of the potential energy of an immobile charge in a Coulomb field, we obtain the Sommerfeld-Dirac formula as a result of an exact solution of this particular equation. There, again, although for a system with spin 1*=*2 the energy of the spin-orbit interaction is not taken into account initially, but the half is obtained from the exact solution of the hydrogen atom problem.

More accurate measurements of Lamb in 1947 and subsequent improvements in the spectrum of the hydrogen atom revealed that, in addition to the lines with the maximum j, all the others are also split and somewhat displaced (the Lamb shift). To harmonize the results of the theory with more accurate experimental data on the spectrum of the hydrogen atom, one had to propose other solutions and approaches than were laid down by the derivation of the Dirac equation.

The new theoretical approaches had yield nothing and only supplemented the theory with the illogical and non-physical proposals to overcome the emerging singularity of solutions: the renormalization, the finite difference of infinities with the desired value of the difference, and so on. The accounting for the size of the nucleus corrected only the *Z* value into the bigger value, but did not solve the *Z* >137 problem. An incredible result was also obtained for the hydrogen atom problem that the electron is located, most likely, at the center of the atom, that is, in the nucleus.

The results of solution of the problem for a particle in a potential well both in the case of the Klein-Fock-Gordon equation and of the Dirac equation contradict to the basic principle of quantum mechanics, to the uncertainty principle. From the solutions, it turns out to be that a particle can be in a bound state in a well with any dimensions, in particular, with the size much smaller than the wavelength of the particle itself, *A* ¼ ℏ*=mc* [23].

Despite Dirac himself proposed a system of linear first-degree relativistic equations in the matrix representation that described the system with spin 1*=*2, the contradictions did not disappear, and he himself remained unhappy with the results of his theory. As Dirac wrote in 1956 [24], the development of relativistic electron theory can now be considered as an example of how incorrect arguments sometimes

#### *Quantum Mechanics*

lead to a valuable result. In the 70s, it became clear that the relativistic theory of quantum mechanics does not exist, and new, fundamental approaches and equations should be sought for constructing a consistent theory of relativistic quantum mechanics. And in the 80s, Dirac already spoke about the insuperable difficulties of the existing quantum theory and the need to create a new one [25].

The application of variational principles to construct the relativistic and quantum theory was based on the principles of construction the mechanics with the help of the Lagrangian of the system [27], which originally was not intended for relativistic approaches. The Lagrangian construction is parametric with the one time variable τ = ct, singled out from the variables of the four-dimensional space (the rest are represented by the dependence on this variable τ) and contains the total differential with respect to this variable, the velocity of the particle. Such a construction is unacceptable because of the impossibility to apply the principle of invariance of the representation of variables and the covariant representation of the action of the

*Equations of Relativistic and Quantum Mechanics (without Spin)*

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

In [28], to construct the relativistic theory on the basis of variational principles, the canonical (non- parametric) solutions of the variational problem for canonically defined integral functionals have been considered and the canonical solutions of the variational problems of mechanics in the Minkowski spaces are written. Because of unifying the variational principles of least action, flow, and hyperflow, the canonically invariant equations for the generalized momentum are obtained. From these equations, the expressions for the action function and the wave function are obtained as the general solution of the unified variational problem of mechanics. Below, we present the generalized invariance principle and the corresponding representation of the generalized momentum of the system, the equations of relativistic and quantum mechanics [29], give the solutions of the problems of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), the problems of charged particle in a magnetic field, and

The principle of invariance of the representation of a generalized pulse is applicable also in the case of motion of a particle with the velocity **v** and in the case of a

The four-dimensional momentum of a particle **P** with the rest mass *m* moving

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>**

If to consider the representation of the four-dimensional momentum of an immobile particle with a mass *m* by transition into the reference frame

moving with the velocity **β**<sup>0</sup> ¼ **V***=c*, for the four-dimensional particle momentum **P**

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>**<sup>0</sup>

This is a property of invariance of the representation of the four-dimensional

This is the property of invariance of the representation of the four-dimensional

, **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> ð Þ *mc*

, **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> ð Þ *mc*

2

<sup>2</sup> (24)

*:* (25)

also the problems of a hydrogen atom are given.

**2.1 Generalization of the principle of invariance**

with the velocity **β** ¼ **v***=c* is represented in the form.

*mc* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , *mc*

*mc* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , *mc*

momentum **P** in terms of the velocity of the particle **β** ¼ **v***=c*.

transition to a reference frame moving with the velocity **V**.

!

!

momentum **P** through the velocity of the reference system **β**<sup>0</sup> ¼ **V***=c*.

**2. Principle of invariance**

**P** ¼ ð Þ¼ *ε*, **p**

**P** ¼ ð Þ¼ *ε*, **p**

we have.

**115**

system.

The reason for the failure of these theories is quite simple—it is in the ignoring of the dependence of the interaction energy with the field on the velocity of the particle. The generalized momentum of the system, the particle plus the external field, is the sum of the relativistic expression for the mechanical momentum of the particle and the field momentum in the case of interaction with the immobile particle

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) = \frac{1}{c} \left( \frac{mc^2}{\sqrt{1-\rho^2}} + q\rho, \quad \frac{mc^2}{\sqrt{1-\rho^2}}\mathbf{\hat{p}} + q\mathbf{A}, \quad \mathbf{P}^2 \neq inv} \right), \tag{19}$$

which is not an invariant representation of the particle velocity. To construct some invariant from such a representation, an 'invariant' relation was used in all cases in the form of a difference between the generalized momentum of the system and the field momentum in the case of interaction with the immobile particle

$$(e - q\rho, \ \mathbf{p} - q\mathbf{A}) = \frac{1}{c} \left( \frac{mc^2}{\sqrt{1 - \rho^2}}, \quad \frac{mc^2}{\sqrt{1 - \rho^2}} \mathbf{p} \right), \quad (e - q\rho)^2 - (\mathbf{p} - q\mathbf{A})^2 = (mc)^2 \tag{20}$$

Obviously, the permutation of the components of the generalized momentum for the construction of the invariant does not solve the posed problem. The statement that the expression (20) is the mechanical momentum of a particle and therefore is an invariant is unproven and it is necessary to apprehend the formula (20) as an empirical. Therefore, at high velocities or strong interactions, an unaccounted dependence of the energy of particle interaction with the field on the velocity of the particle motion, which results to the erroneous results or the impossibility of calculations.

In [26], an invariant representation of the generalized momentum of the system was suggested, where the dependence of the interaction energy of the particle with the field on the velocity was taken into account:

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) = \frac{\mathbf{1}}{\varepsilon} \left( \frac{mc^2 + q\rho + q\mathbf{b} \cdot \mathbf{A}}{\sqrt{1 - \rho^2}}, \quad \frac{(mc^2 + q\rho)\mathbf{b} + q\mathbf{A}\_{\parallel}}{\sqrt{1 - \rho^2}} + q\mathbf{A}\_{\perp} \right) \tag{21}$$

$$\mathbf{P}^2 = \varepsilon^2 - \mathbf{p}^2 = \frac{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{c^2},\tag{22}$$

which is the four-dimensional representation of the generalized momentum of the system based on the expression for the generalized momentum of an immobile particle in a state of rest

$$\mathbf{P}\_0 = \begin{pmatrix} \varepsilon\_0, & \mathbf{p}\_0 \end{pmatrix} = \frac{1}{c} \begin{pmatrix} mc^2 + q\rho, & q\mathbf{A} \end{pmatrix} \tag{23}$$

whose invariant is always equal to the expression (19) regardless of the state of the system.

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

lead to a valuable result. In the 70s, it became clear that the relativistic theory of quantum mechanics does not exist, and new, fundamental approaches and equations should be sought for constructing a consistent theory of relativistic quantum mechanics. And in the 80s, Dirac already spoke about the insuperable difficulties of

The reason for the failure of these theories is quite simple—it is in the ignoring of the dependence of the interaction energy with the field on the velocity of the particle. The generalized momentum of the system, the particle plus the external field, is the sum of the relativistic expression for the mechanical momentum of the particle and the field momentum in the case of interaction with the immobile

ffiffiffiffiffiffiffiffiffiffiffiffiffi

which is not an invariant representation of the particle velocity. To construct some invariant from such a representation, an 'invariant' relation was used in all cases in the form of a difference between the generalized momentum of the system and the field momentum in the case of interaction with the immobile particle

> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>**

Obviously, the permutation of the components of the generalized momentum for the construction of the invariant does not solve the posed problem. The statement that the expression (20) is the mechanical momentum of a particle and therefore is an invariant is unproven and it is necessary to apprehend the formula (20) as an empirical. Therefore, at high velocities or strong interactions, an unaccounted dependence of the energy of particle interaction with the field on the velocity of the particle motion, which results to the erroneous results or the impos-

In [26], an invariant representation of the generalized momentum of the system was suggested, where the dependence of the interaction energy of the particle with

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>***A**<sup>∥</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

which is the four-dimensional representation of the generalized momentum of the system based on the expression for the generalized momentum of an immobile

*c*

whose invariant is always equal to the expression (19) regardless of the state of

!

!

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>***A**, **<sup>P</sup>**<sup>2</sup> 6¼ *inv*

, ð Þ *ε* � *qφ*

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> <sup>þ</sup> *<sup>q</sup>***A**<sup>⊥</sup>

*<sup>c</sup>*<sup>2</sup> , (22)

*mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>φ*, *<sup>q</sup>***<sup>A</sup>** � � (23)

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

, (19)

<sup>2</sup> � ð Þ **<sup>p</sup>** � *<sup>q</sup>***<sup>A</sup>** <sup>2</sup> <sup>¼</sup> ð Þ *mc*

2

(20)

(21)

the existing quantum theory and the need to create a new one [25].

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> <sup>þ</sup> *<sup>q</sup>φ*, *mc*<sup>2</sup>

!

particle

*Quantum Mechanics*

**P** ¼ ð Þ¼ *ε*, **p**

<sup>ð</sup>*<sup>ε</sup>* � *<sup>q</sup>φ*, **<sup>p</sup>** � *<sup>q</sup>***A**Þ ¼ <sup>1</sup>

sibility of calculations.

**P** ¼ ð Þ¼ *ε*, **p**

particle in a state of rest

the system.

**114**

1 *c*

*c*

the field on the velocity was taken into account:

*mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>***<sup>β</sup>** � **<sup>A</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffi

**P**<sup>0</sup> ¼ *ε*0, **p**<sup>0</sup>

**<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

� � <sup>¼</sup> <sup>1</sup>

1 *c* *mc*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

*mc*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , *mc*<sup>2</sup>

The application of variational principles to construct the relativistic and quantum theory was based on the principles of construction the mechanics with the help of the Lagrangian of the system [27], which originally was not intended for relativistic approaches. The Lagrangian construction is parametric with the one time variable τ = ct, singled out from the variables of the four-dimensional space (the rest are represented by the dependence on this variable τ) and contains the total differential with respect to this variable, the velocity of the particle. Such a construction is unacceptable because of the impossibility to apply the principle of invariance of the representation of variables and the covariant representation of the action of the system.

In [28], to construct the relativistic theory on the basis of variational principles, the canonical (non- parametric) solutions of the variational problem for canonically defined integral functionals have been considered and the canonical solutions of the variational problems of mechanics in the Minkowski spaces are written. Because of unifying the variational principles of least action, flow, and hyperflow, the canonically invariant equations for the generalized momentum are obtained. From these equations, the expressions for the action function and the wave function are obtained as the general solution of the unified variational problem of mechanics.

Below, we present the generalized invariance principle and the corresponding representation of the generalized momentum of the system, the equations of relativistic and quantum mechanics [29], give the solutions of the problems of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), the problems of charged particle in a magnetic field, and also the problems of a hydrogen atom are given.

#### **2. Principle of invariance**

#### **2.1 Generalization of the principle of invariance**

The principle of invariance of the representation of a generalized pulse is applicable also in the case of motion of a particle with the velocity **v** and in the case of a transition to a reference frame moving with the velocity **V**.

The four-dimensional momentum of a particle **P** with the rest mass *m* moving with the velocity **β** ¼ **v***=c* is represented in the form.

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) = \left(\frac{mc}{\sqrt{1-\beta^2}}, \quad \frac{mc}{\sqrt{1-\beta^2}}\mathfrak{F}\right), \mathbf{P}^2 = \varepsilon^2 - \mathbf{p}^2 = \left(mc\right)^2\tag{24}$$

This is the property of invariance of the representation of the four-dimensional momentum **P** in terms of the velocity of the particle **β** ¼ **v***=c*.

If to consider the representation of the four-dimensional momentum of an immobile particle with a mass *m* by transition into the reference frame moving with the velocity **β**<sup>0</sup> ¼ **V***=c*, for the four-dimensional particle momentum **P** we have.

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) = \left(\frac{mc}{\sqrt{1-\beta^2}}, \quad \frac{mc}{\sqrt{1-\beta^2}}\mathbf{p}'\right), \mathbf{P}^2 = c^2 - \mathbf{p}^2 = \left(mc\right)^2. \tag{25}$$

This is a property of invariance of the representation of the four-dimensional momentum **P** through the velocity of the reference system **β**<sup>0</sup> ¼ **V***=c*.

For an invariant of the system *I*, we have

$$I^2 = \mathbf{P}^2 = \varepsilon^2 - \mathbf{p}^2 = (\varepsilon\_0)^2 = (mc)^2. \tag{26}$$

If one represents the generalized momentum of the particle in the form

where *φ*<sup>0</sup> and **A**<sup>0</sup> already effective values of the interaction potentials of the particle moving with velocity **v** in a field with the potentials *φ* and **A**, we obtain

!

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>***A**<sup>∥</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

*mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>***<sup>β</sup>** � **<sup>A</sup>** *c* ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>*

!

, *mc*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>***A**<sup>0</sup>

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> <sup>þ</sup> *<sup>q</sup>***A**<sup>⊥</sup>

*c* **<sup>A</sup>** � *<sup>q</sup> c*

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> ð Þ **<sup>A</sup>** � **<sup>β</sup> <sup>β</sup>**

, **<sup>p</sup>**<sup>0</sup> f g¼ f gþ *<sup>ε</sup>*, **<sup>p</sup>** <sup>T</sup>^f g *<sup>ε</sup>*, **<sup>p</sup>** (33)

1 <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

> � � � � � � � � � � � �

The matrices of the invariant representation of a four-dimensional vector, which preserve the vector module in four-dimensional space, form the Poincare group (inhomogeneous Lorentz group). In addition to displacements and rotations, the group contains space-time reflection representations P, ^ T and inversion ^ <sup>P</sup>^T^ <sup>¼</sup> ^I. For the module *I* of the four-dimensional vector of the generalized momentum

þ ð Þ *γ* � 1

, (29)

*:* (30)

!

(31)

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> ð Þ **<sup>A</sup>** � **<sup>β</sup> <sup>β</sup>**

*:* (32)

00 0 0

*β*1*β*<sup>2</sup> *β*2

*β*2*β*<sup>2</sup> *β*2

*β*3*β*<sup>2</sup> *β*2

*β*1*β*<sup>3</sup> *β*2

� � � � � � � � � � � � � � � � �

*β*2*β*<sup>3</sup> *β*2

*β*3*β*<sup>3</sup> *β*2

(34)

<sup>0</sup> *<sup>β</sup>*1*β*<sup>1</sup> *β*2

� � � � � � � � � � � � � � � � �

<sup>0</sup> *<sup>β</sup>*2*β*<sup>1</sup> *β*2

<sup>0</sup> *<sup>β</sup>*3*β*<sup>1</sup> *β*2

1 <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

!

*mc*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> <sup>þ</sup> *<sup>q</sup>φ*<sup>0</sup>

*Equations of Relativistic and Quantum Mechanics (without Spin)*

**<sup>P</sup>** <sup>¼</sup> <sup>1</sup> *c*

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>***<sup>β</sup>** � **<sup>A</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffi

The expression (30) can be represented in the form

**<sup>P</sup>** <sup>¼</sup> *<sup>ε</sup>*, *<sup>ε</sup>***<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>*

This transformation can be presents in matrices form

*ε*0

� � � � � � � � � � � �

þ

� � � � � � � � � � � �

0 *β*<sup>1</sup> *β*<sup>2</sup> *β*<sup>3</sup> *β*<sup>1</sup> 000 *β*<sup>2</sup> 000 *β*<sup>3</sup> 000

where a Lorentz transformation have a form

� � � � � � � � � � � �

þ *γ*

*c* **<sup>A</sup>** � *<sup>q</sup> c*

**<sup>P</sup>** <sup>¼</sup> <sup>1</sup> *c*

**<sup>P</sup>** <sup>¼</sup> *mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>***<sup>β</sup>** � **<sup>A</sup>** *c* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> ,

or

^<sup>1</sup> <sup>þ</sup> <sup>T</sup>^ <sup>¼</sup>

ð Þ *γ* � 1

**P**, we have

**117**

� � � � � � � � � � � � � � � � � � � � � � � �

At **β** ¼ **β**<sup>0</sup> ¼ 0, we obtain

$$\mathbf{P} = (\varepsilon, \ \mathbf{p})|\_{\mathfrak{F} = \mathfrak{F}' = 0} = \varepsilon\_0(\mathbf{1}, \ \mathbf{0}) = mc(\mathbf{1}, \ \mathbf{0}).\tag{27}$$

Thus, the generalized momentum of a particle has an invariant representation on the particle velocity **v** and the velocity of the reference system **V**. This property should be considered because of the general principle of the relativity of motion. Accordingly, the generalized momentum of the particle **P** is an invariant regardless of the state of the system.

If a charged particle is in an external electromagnetic field with potentials ð Þ *φ*, **A** , then the stationary charge sees the field exactly with such potentials. If the charge has a nonzero velocity **v**, then it will interact with the field differently. To determine the interaction for a charge moving with the velocity **v**, one can start from the principle of the relativity of motion. The effective values of the force or interaction with the field of the charge moving with the velocity v are the same as in the case when the charge is immobile, and the field moves with the velocity �**v** (in the laboratory frame of reference).

The fact that the interaction of a charged particle with a field depends on the speed of motion is evidently represented in the formula for the Liénard-Wiechert potential [8].

More clearly, this can be demonstrated by an example of the Doppler effect for two atoms in the field of a resonant radiation, when one of the atoms is at rest and the other moves with the velocity **v** (**Figure 4**).

The atom, which is at rest, absorbs a photon, and the moving one does not absorb or interacts weakly with the field, because of the dependence of the interaction on the velocity of the atom. It is also known that the acting field for an atom moving with the velocity v corresponds to the interaction with the field moving with the velocity �**v**.

#### **2.2 Invariant representation of the generalized momentum**

Thus, for a moving charge, the effective values of the potentials *φ*<sup>0</sup> , **A**<sup>0</sup> ð Þ (in the laboratory frame of reference) can be written in the form [8]

$$\mathbf{A}(\boldsymbol{\varrho}',\mathbf{A}') = \left(\frac{\boldsymbol{\varrho} + \boldsymbol{\mathfrak{B}} \cdot \mathbf{A}}{\sqrt{\mathbf{1} - \boldsymbol{\beta}^2}}, \quad \mathbf{A}\_{\perp} + \frac{\mathbf{A}\_{\parallel} + \boldsymbol{\varrho}\,\mathbf{\mathfrak{B}}}{\sqrt{\mathbf{1} - \boldsymbol{\beta}^2}}\right). \tag{28}$$

**Figure 4.** *Two atoms in the field of a resonant radiation.*

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

If one represents the generalized momentum of the particle in the form

$$\mathbf{P} = \frac{1}{c} \left( \frac{mc^2}{\sqrt{1-\beta^2}} + q\rho', \quad \frac{mc^2}{\sqrt{1-\beta^2}}\mathbf{\hat{\theta}} + q\mathbf{A'} \right), \tag{29}$$

where *φ*<sup>0</sup> and **A**<sup>0</sup> already effective values of the interaction potentials of the particle moving with velocity **v** in a field with the potentials *φ* and **A**, we obtain

$$\mathbf{P} = \frac{1}{c} \left( \frac{mc^2 + q\rho + q\mathbf{\mathcal{B}} \cdot \mathbf{A}}{\sqrt{1 - \rho^2}}, \quad \frac{(mc^2 + q\rho)\mathbf{\mathcal{B}} + q\mathbf{A}\_{\parallel}}{\sqrt{1 - \rho^2}} + q\mathbf{A}\_{\perp} \right). \tag{30}$$

The expression (30) can be represented in the form

$$\mathbf{P} = \left(\frac{mc^2 + q\rho + q\mathbf{\mathfrak{f}} \cdot \mathbf{A}}{c\sqrt{1 - \rho^2}}, \frac{mc^2 + q\rho + q\mathbf{\mathfrak{f}} \cdot \mathbf{A}}{c\sqrt{1 - \rho^2}}\mathfrak{f} + \frac{q}{c}\mathbf{A} - \frac{q}{c}\frac{1}{1 + \sqrt{1 - \rho^2}}(\mathbf{A} \cdot \mathfrak{f})\mathfrak{f}\right) \tag{31}$$

or

For an invariant of the system *I*, we have

At **β** ¼ **β**<sup>0</sup> ¼ 0, we obtain

*Quantum Mechanics*

of the state of the system.

potential [8].

with the velocity �**v**.

**Figure 4.**

**116**

(in the laboratory frame of reference).

the other moves with the velocity **v** (**Figure 4**).

*φ*0

*Two atoms in the field of a resonant radiation.*

**2.2 Invariant representation of the generalized momentum**

laboratory frame of reference) can be written in the form [8]

, **<sup>A</sup>**<sup>0</sup> ð Þ¼ *<sup>φ</sup>* <sup>þ</sup> **<sup>β</sup>** � **<sup>A</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffi

Thus, for a moving charge, the effective values of the potentials *φ*<sup>0</sup>

<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> , **<sup>A</sup>**<sup>⊥</sup> <sup>þ</sup>

!

**<sup>A</sup>**<sup>k</sup> <sup>þ</sup> *<sup>φ</sup>***<sup>β</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup>

*I*

**P** ¼ ð Þ *ε*, **p <sup>β</sup>**¼**β**<sup>0</sup>

�

<sup>2</sup> <sup>¼</sup> **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> ð Þ *<sup>ε</sup>*<sup>0</sup>

¼0

Thus, the generalized momentum of a particle has an invariant representation on the particle velocity **v** and the velocity of the reference system **V**. This property should be considered because of the general principle of the relativity of motion. Accordingly, the generalized momentum of the particle **P** is an invariant regardless

If a charged particle is in an external electromagnetic field with potentials ð Þ *φ*, **A** , then the stationary charge sees the field exactly with such potentials. If the charge has a nonzero velocity **v**, then it will interact with the field differently. To determine the interaction for a charge moving with the velocity **v**, one can start from the principle of the relativity of motion. The effective values of the force or interaction with the field of the charge moving with the velocity v are the same as in the case when the charge is immobile, and the field moves with the velocity �**v**

The fact that the interaction of a charged particle with a field depends on the speed of motion is evidently represented in the formula for the Liénard-Wiechert

More clearly, this can be demonstrated by an example of the Doppler effect for two atoms in the field of a resonant radiation, when one of the atoms is at rest and

The atom, which is at rest, absorbs a photon, and the moving one does not absorb or interacts weakly with the field, because of the dependence of the interaction on the velocity of the atom. It is also known that the acting field for an atom moving with the velocity v corresponds to the interaction with the field moving

<sup>2</sup> <sup>¼</sup> ð Þ *mc*

2

� ¼ *ε*0ð Þ¼ 1, 0 *mc*ð Þ 1, 0 *:* (27)

*:* (26)

, **A**<sup>0</sup> ð Þ (in the

*:* (28)

$$\mathbf{P} = \left( \varepsilon, \ \varepsilon \mathfrak{P} + \frac{q}{c} \mathbf{A} - \frac{q}{c} \frac{1}{\mathbf{1} + \sqrt{\mathbf{1} - \boldsymbol{\beta}^2}} (\mathbf{A} \cdot \mathfrak{P}) \mathfrak{P} \right). \tag{32}$$

This transformation can be presents in matrices form

$$\{\varepsilon',\ \mathbf{p}'\} = \{\varepsilon,\ \mathbf{p}\} + \hat{\mathcal{T}}\{\varepsilon,\ \mathbf{p}\} \tag{33}$$

where a Lorentz transformation have a form

^<sup>1</sup> <sup>þ</sup> <sup>T</sup>^ <sup>¼</sup> 1000 0100 0010 0001 � � � � � � � � � � � � � � � � � � � � � � � � þ ð Þ *γ* � 1 1000 0000 0000 0000 � � � � � � � � � � � � � � � � � � � � � � � � þ *γ* 0 *β*<sup>1</sup> *β*<sup>2</sup> *β*<sup>3</sup> *β*<sup>1</sup> 000 *β*<sup>2</sup> 000 *β*<sup>3</sup> 000 � � � � � � � � � � � � � � � � � � � � � � � � þ ð Þ *γ* � 1 00 0 0 <sup>0</sup> *<sup>β</sup>*1*β*<sup>1</sup> *β*2 *β*1*β*<sup>2</sup> *β*2 *β*1*β*<sup>3</sup> *β*2 <sup>0</sup> *<sup>β</sup>*2*β*<sup>1</sup> *β*2 *β*2*β*<sup>2</sup> *β*2 *β*2*β*<sup>3</sup> *β*2 <sup>0</sup> *<sup>β</sup>*3*β*<sup>1</sup> *β*2 *β*3*β*<sup>2</sup> *β*2 *β*3*β*<sup>3</sup> *β*2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (34)

The matrices of the invariant representation of a four-dimensional vector, which preserve the vector module in four-dimensional space, form the Poincare group (inhomogeneous Lorentz group). In addition to displacements and rotations, the group contains space-time reflection representations P, ^ T and inversion ^ <sup>P</sup>^T^ <sup>¼</sup> ^I.

For the module *I* of the four-dimensional vector of the generalized momentum **P**, we have

$$I^2 = \mathbf{P}^2 = \varepsilon^2 - \mathbf{p}^2 = \frac{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{c^2},\tag{35}$$

*<sup>U</sup>* <sup>¼</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup>

*U* ¼ �*eφ* þ

*e*2

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

**A** ¼ ½ � **r** � **B** *=*2 is

<sup>2</sup>*mc*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> � **<sup>A</sup>**<sup>2</sup> � �, H <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> � *<sup>m</sup>*2*c*<sup>4</sup>

*Equations of Relativistic and Quantum Mechanics (without Spin)*

<sup>2</sup>*mc*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> � **<sup>A</sup>**<sup>2</sup> � � ¼ � *Ze*<sup>2</sup>

reflected in the expression for the potential energy of the particle.

<sup>2</sup>*mc*<sup>2</sup> ) *<sup>E</sup>* ¼ �*mc*<sup>2</sup>

*Z*2 *e*4 *<sup>r</sup>*<sup>2</sup> � *<sup>e</sup>*2*B*<sup>2</sup> <sup>8</sup>*mc*<sup>2</sup> *<sup>r</sup>*

For example, the potential energy *U* of the electron in the field of the Coulomb potential *φ* ¼ *Ze=r* and in a homogeneous magnetic field **B** with the vector potential

> *r* þ 1 2*mc*<sup>2</sup>

Many well-known expressions of the potential energy of interaction with attractive fields have a repulsive component in the form of half the square of these attractive potentials—Kratzer [30], Lennard-Jones [31], Morse [32], Rosen [33] and others. Expression (41) justifies this approach, which until now is phenomenological or the result of an appropriate selection for agreement with experimental data.

true values of the energy *E* and changes of its quantity. Thus, the classical

r

energy expression can be represented in the form

**3. Equations of relativistic mechanics**

ð Þ¼� *εdt* � **p** � *d***r**

tion of the generalized momentum **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*

**<sup>P</sup>**<sup>2</sup> � *<sup>I</sup>* 2

2*λ* � �*ds* <sup>¼</sup>

method of indefinite Lagrange coefficients in the form

*S* ¼ �

alized velocity.

*S* ¼ ð*s*1

**119**

*s*1

�**P** � **V** þ

*t*2, ð **r**2

*t*1, **r**<sup>1</sup>

*<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup>

**3.1 Canonical Lagrangian and Hamilton-Jacoby equation**

**R** ð2

**R**1

The Hamiltonian H can be called the energy and its value remains constant in the case of conservation of energy *E*, but the value of Hand its changes differ from the

approaches are permissible only in the case of low velocities, when H ≪ *mc*<sup>2</sup> and the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2H *mc*<sup>2</sup>

Let us use the parametric representation of the Hamilton action in the form [28].

where ds is the four-dimensional interval and **V** is the four-dimensional gener-

The functional that takes into account the condition of the invariant representa-

ð Þ **<sup>P</sup>** � *<sup>λ</sup>***<sup>V</sup>** <sup>2</sup>

2*λ*

þ

!

*<sup>λ</sup>*<sup>2</sup> � *<sup>I</sup>* 2

2*λ*

**R** ð2

**R**1 **P** � *d***R** *ds ds* ¼ �

**P** � *d***R** ¼ �

ð*s*1

*s*1

Note, whatever is the dependence of the potential *φ*, the possible minimum potential energy *<sup>U</sup>*min ¼ �*mc*2*=*2, and the potential energy as a function of the vector potential is always negative. The hard constraint of the classical potential energy value *<sup>U</sup>*min ¼ �*mc*2*=*2, which does not depend on the nature of the interactions, results in the fundamental changes in the description of interactions and the revision of the results of classical mechanics. At short distances, the origination of repulsion for attraction forces caused by the uncertainty principle is clearly

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2H *mc*<sup>2</sup>

*:* (41)

*θ:* (42)

r

<sup>2</sup> sin <sup>2</sup>

<sup>≈</sup> *mc*<sup>2</sup> <sup>þ</sup> <sup>H</sup>*:* (43)

**R** ð2

**R**1

ð Þ **P** � **V** *ds* ! min ,

<sup>2</sup> <sup>¼</sup> *inv*, can be composed by the

*ds* ! min , (45)

(44)

which is the four-dimensional representation of the generalized momentum of the system on the basis of the expression of the generalized momentum of a particle in the state of rest

$$\mathbf{P}\_0 = \begin{pmatrix} \varepsilon\_0, & \mathbf{p}\_0 \end{pmatrix} = \frac{1}{c} \begin{pmatrix} mc^2 + q\rho, \ q\mathbf{A} \end{pmatrix},\tag{36}$$

whose invariant is defined by the expression (30).

Thus, the generalized momentum of the particle in an external field is not only invariant relative to the transformations at the transition from one reference system to another but also has an invariant representation in terms of the velocity of motion of the particle (30); at each point of space, the value of the invariant *I* is determined by the expression (35). This property has not only the representation of the proper momentum of the particle (the mechanical part), but also the generalized momentum of the particle in general.

Let us generalize this result to the case of representation of the generalized momentum of any systems and interactions, arguing that, regardless of the state (the motion) of the system, the generalized four-dimensional momentum always has an invariant representation

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) \quad \Rightarrow \quad \mathbf{P}^2 = \varepsilon^2 - \mathbf{p}^2 = \varepsilon\_0^{-2} - \mathbf{p}\_0^{-2} = \pm I^2 = \text{inv},\tag{37}$$

where ε и p are the energy and momentum of the system, respectively, and the invariant is determined by the modulus of sum of the components of the generalized momentum of the system *ε*<sup>0</sup> and **p**<sup>0</sup> at rest. If the particles interact with the field in the form *ε*<sup>0</sup> þ *αφ*, the invariants of the generalized momentum of the system are represented by the expressions [25].

$$\mathbf{P}\_{+}{}^{2} = \left(\varepsilon\_{0} + a\rho\right)^{2} - \left(a\mathbf{A}\right)^{2} = \varepsilon\_{0}{}^{2} + 2\varepsilon\_{0}a\rho + \left(a\rho\right)^{2} - \left(a\mathbf{A}\right)^{2},$$

$$\mathbf{P}\_{-}{}^{2} = \left(a\rho\right)^{2} - \left(\varepsilon\_{0}\mathbf{n} + a\mathbf{A}\right)^{2} = -\varepsilon\_{0}{}^{2} - 2\varepsilon\_{0}a\mathbf{n}\cdot\mathbf{A} + \left(a\rho\right)^{2} - \left(a\mathbf{A}\right)^{2},\tag{38}$$

$$\mathbf{P}\_{0}{}^{2} = \left(\varepsilon\_{0} + a\rho\right)^{2} - \left(\varepsilon\_{0}\mathbf{n} + a\mathbf{A}\right)^{2} = 2\varepsilon\_{0}a\left(\rho - \mathbf{n}\cdot\mathbf{A}\right) + \left(a\rho\right)^{2} - \left(a\mathbf{A}\right)^{2}.$$

Let us represent the expression for the invariant *<sup>ε</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>2</sup> (35) in the following form

$$\epsilon^2 = \frac{E^2}{c^2} = \mathbf{p}^2 + \frac{(mc^2 + q\rho)^2 - (q\mathbf{A})^2}{c^2} = \mathbf{p}^2 + m^2 c^2 + 2mq\rho + \frac{q^2}{c^2}(\rho^2 - \mathbf{A}^2) \tag{39}$$

and divide it by 2*m*. Grouping, we obtain the Hamiltonian H of the system in the form

$$\mathbf{H} = \frac{\mathbf{c}^2 - m^2 c^2}{2m} = \frac{E^2 - m^2 c^4}{2mc^2} = \frac{\mathbf{p}^2}{2m} + q\rho + \frac{q^2}{2mc^2}(\rho^2 - \mathbf{A}^2),\tag{40}$$

that is, we obtain the formula for the correspondence between the energy of the system *E* and the energy of the system in the classical meaning H. The correspondence in the form H <sup>¼</sup> **<sup>p</sup>**<sup>2</sup>*=*2*<sup>m</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ *<sup>τ</sup>*,*<sup>r</sup>* [26] will be complete and accurate if we determine the potential energy of interaction *U* and the energy of system in the classical meaning as

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

*I*

ized momentum of the particle in general.

are represented by the expressions [25].

<sup>2</sup> <sup>¼</sup> ð Þ *<sup>ε</sup>*<sup>0</sup> <sup>þ</sup> *αφ*

<sup>2</sup> <sup>¼</sup> ð Þ *<sup>ε</sup>*<sup>0</sup> <sup>þ</sup> *αφ*

*<sup>c</sup>*<sup>2</sup> <sup>¼</sup> **<sup>p</sup>**<sup>2</sup> <sup>þ</sup>

classical meaning as

<sup>H</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � *<sup>m</sup>*<sup>2</sup>*c*<sup>2</sup>

**<sup>P</sup>**�<sup>2</sup> <sup>¼</sup> ð Þ *αφ*

**P**þ

**P**0

*<sup>ε</sup>*<sup>2</sup> <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

form

form

**118**

has an invariant representation

in the state of rest

*Quantum Mechanics*

<sup>2</sup> <sup>¼</sup> **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

<sup>¼</sup> <sup>1</sup>

**P**<sup>0</sup> ¼ *ε*0, **p**<sup>0</sup>

**<sup>P</sup>** <sup>¼</sup> ð Þ) *<sup>ε</sup>*, **<sup>p</sup> <sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup>

<sup>2</sup> � ð Þ *<sup>α</sup>***<sup>A</sup>** <sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup>

<sup>2</sup> � ð Þ *<sup>ε</sup>*0**<sup>n</sup>** <sup>þ</sup> *<sup>α</sup>***<sup>A</sup>** <sup>2</sup> ¼ �*ε*<sup>0</sup>

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>¼</sup> **<sup>p</sup>**<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> <sup>¼</sup> **<sup>p</sup>**<sup>2</sup>

*mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

<sup>2</sup>*<sup>m</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> � *<sup>m</sup>*<sup>2</sup>*c*<sup>4</sup>

whose invariant is defined by the expression (30).

which is the four-dimensional representation of the generalized momentum of the system on the basis of the expression of the generalized momentum of a particle

*c*

Thus, the generalized momentum of the particle in an external field is not only invariant relative to the transformations at the transition from one reference system to another but also has an invariant representation in terms of the velocity of motion of the particle (30); at each point of space, the value of the invariant *I* is determined by the expression (35). This property has not only the representation of the proper momentum of the particle (the mechanical part), but also the general-

Let us generalize this result to the case of representation of the generalized momentum of any systems and interactions, arguing that, regardless of the state (the motion) of the system, the generalized four-dimensional momentum always

where ε и p are the energy and momentum of the system, respectively, and the invariant is determined by the modulus of sum of the components of the generalized momentum of the system *ε*<sup>0</sup> and **p**<sup>0</sup> at rest. If the particles interact with the field in the form *ε*<sup>0</sup> þ *αφ*, the invariants of the generalized momentum of the system

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ε*0*αφ* <sup>þ</sup> ð Þ *αφ*

<sup>2</sup> � ð Þ *<sup>ε</sup>*0**<sup>n</sup>** <sup>þ</sup> *<sup>α</sup>***<sup>A</sup>** <sup>2</sup> <sup>¼</sup> <sup>2</sup>*ε*0*α φ*ð Þþ � **<sup>n</sup>** � **<sup>A</sup>** ð Þ *αφ*

Let us represent the expression for the invariant *<sup>ε</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>2</sup> (35) in the following

and divide it by 2*m*. Grouping, we obtain the Hamiltonian H of the system in the

that is, we obtain the formula for the correspondence between the energy of the system *E* and the energy of the system in the classical meaning H. The correspondence in the form H <sup>¼</sup> **<sup>p</sup>**<sup>2</sup>*=*2*<sup>m</sup>* <sup>þ</sup> *<sup>U</sup>*ð Þ *<sup>τ</sup>*,*<sup>r</sup>* [26] will be complete and accurate if we determine the potential energy of interaction *U* and the energy of system in the

2*m*

<sup>2</sup> � <sup>2</sup>*ε*0*α***<sup>n</sup>** � **<sup>A</sup>** <sup>þ</sup> ð Þ *αφ*

*c*

<sup>þ</sup> *<sup>q</sup><sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup>

<sup>2</sup> � **<sup>p</sup>**<sup>0</sup>

<sup>2</sup> ¼ �*<sup>I</sup>*

<sup>2</sup> � ð Þ *<sup>α</sup>***<sup>A</sup>** <sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*mq<sup>φ</sup>* <sup>þ</sup> *<sup>q</sup>*<sup>2</sup>

,

<sup>2</sup> � ð Þ *<sup>α</sup>***<sup>A</sup>** <sup>2</sup>

,

*:*

*<sup>c</sup>*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> � **<sup>A</sup>**<sup>2</sup> (39)

(38)

<sup>2</sup> � ð Þ *<sup>α</sup>***<sup>A</sup>** <sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> � **<sup>A</sup>**<sup>2</sup> , (40)

<sup>2</sup> <sup>¼</sup> inv, (37)

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> , (35)

*mc*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>φ*, *<sup>q</sup>***<sup>A</sup>** , (36)

$$\mathbf{U} = q\rho + \frac{q^2}{2mc^2} \left(\rho^2 - \mathbf{A}^2\right), \quad \mathbf{H} = \frac{E^2 - m^2c^4}{2mc^2} \quad \Rightarrow \quad E = \pm mc^2 \sqrt{1 + \frac{2\mathbf{H}}{mc^2}}.\tag{41}$$

For example, the potential energy *U* of the electron in the field of the Coulomb potential *φ* ¼ *Ze=r* and in a homogeneous magnetic field **B** with the vector potential **A** ¼ ½ � **r** � **B** *=*2 is

$$U = -e\rho + \frac{e^2}{2mc^2} \left(\rho^2 - \mathbf{A}^2\right) = -\frac{Ze^2}{r} + \frac{1}{2mc^2} \frac{Z^2e^4}{r^2} - \frac{e^2B^2}{8mc^2}r^2\sin^2\theta. \tag{42}$$

Note, whatever is the dependence of the potential *φ*, the possible minimum potential energy *<sup>U</sup>*min ¼ �*mc*2*=*2, and the potential energy as a function of the vector potential is always negative. The hard constraint of the classical potential energy value *<sup>U</sup>*min ¼ �*mc*2*=*2, which does not depend on the nature of the interactions, results in the fundamental changes in the description of interactions and the revision of the results of classical mechanics. At short distances, the origination of repulsion for attraction forces caused by the uncertainty principle is clearly reflected in the expression for the potential energy of the particle.

Many well-known expressions of the potential energy of interaction with attractive fields have a repulsive component in the form of half the square of these attractive potentials—Kratzer [30], Lennard-Jones [31], Morse [32], Rosen [33] and others. Expression (41) justifies this approach, which until now is phenomenological or the result of an appropriate selection for agreement with experimental data.

The Hamiltonian H can be called the energy and its value remains constant in the case of conservation of energy *E*, but the value of Hand its changes differ from the true values of the energy *E* and changes of its quantity. Thus, the classical approaches are permissible only in the case of low velocities, when H ≪ *mc*<sup>2</sup> and the energy expression can be represented in the form

$$E = mc^2 \sqrt{\mathbf{1} + \frac{2\mathbf{H}}{mc^2}} \approx mc^2 + \mathbf{H}.\tag{43}$$

#### **3. Equations of relativistic mechanics**

#### **3.1 Canonical Lagrangian and Hamilton-Jacoby equation**

Let us use the parametric representation of the Hamilton action in the form [28].

$$\mathcal{S} = -\int\_{t\_1}^{t\_2} (edt - \mathbf{p} \cdot d\mathbf{r}) = -\int\_{\mathbf{R}\_1} \mathbf{P} \cdot d\mathbf{R} = -\int\_{\mathbf{R}\_1} \mathbf{P} \cdot \frac{d\mathbf{R}}{ds} ds = -\int\_{\mathbf{R}\_1} (\mathbf{P} \cdot \mathbf{V}) ds \to \min,\tag{44}$$

where ds is the four-dimensional interval and **V** is the four-dimensional generalized velocity.

The functional that takes into account the condition of the invariant representation of the generalized momentum **<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>* <sup>2</sup> <sup>¼</sup> *inv*, can be composed by the method of indefinite Lagrange coefficients in the form

$$S = \int\_{s\_1}^{s\_1} (-\mathbf{P} \cdot \mathbf{V} + \frac{\mathbf{P}^2 - I^2}{2\lambda}) ds = \int\_{s\_1}^{s\_1} (\frac{(\mathbf{P} - \lambda \mathbf{V})^2 + \lambda^2 - I^2}{2\lambda}) ds \to \min,\tag{45}$$

where *λ* ¼ *λ*ð Þ*s* is the given parameter, determined by the condition of invariance of the representation. Because *λ* and *I* are given and they do not depend on the velocity, we have an explicit solution in the form

$$\mathbf{P} - \lambda \mathbf{V} = \mathbf{0}, \qquad \lambda = \pm I(\tau, \mathbf{r}), \tag{46}$$

Using the explicit form of the generalized momentum (32) with the accuracy of

*∂***r**

where the velocity-dependent components of the force are present. In particular, the velocity-dependent force is present in the Faraday law of electromagnetic induction [34], which is absent in the traditional expression for the Lorentz force.

*q c*

<sup>¼</sup> *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

and it reflects the invariance of the representation of the generalized momentum. The well-known representations of the Hamilton-Jacobi Eq. (8) also contain the differential forms of potentials—the components of the electric and the

Let us consider the motion of a charged particle with the mass m and charge –*q* in the constant electric field between the plane electrodes with the potential difference *U* and the distance *l* between them. For one-dimensional motion, taking the cathode location as the origin and anode at the point *x* ¼ *l*, from (56) we have

where *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> <sup>þ</sup> *qU* is an the electron energy at the origin on the surface of the

We find the solution from the condition *<sup>∂</sup>S=∂<sup>E</sup>* <sup>¼</sup> const. As a result of

ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> <sup>þ</sup> *qU* � *qU <sup>x</sup>*

1 þ *α*

*l c*

The well-known solution in the framework of the traditional theory [8] is the

1 þ *α α* � � arccos

*x l*

� �<sup>2</sup> <sup>r</sup>

<sup>¼</sup> *mc* ð Þ <sup>2</sup> <sup>þ</sup> *qU*ð Þ <sup>1</sup> � *<sup>x</sup>=<sup>l</sup>* <sup>2</sup>

, we obtain the equation of motion in the form

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> � **<sup>A</sup>**<sup>2</sup> � � � �, (55)

*<sup>c</sup>*<sup>2</sup> (56)

*<sup>c</sup>*<sup>2</sup> *:* (57)

*dx:* (59)

1 1 þ *α*

� �*:* (61)

*S* ¼ �*Et* þ *f x*ð Þ, (58)

*l*

� �, *<sup>α</sup>* <sup>¼</sup> *qU=mc*<sup>2</sup> (60)

**<sup>A</sup>** � **<sup>β</sup>** <sup>þ</sup> *<sup>q</sup>*<sup>2</sup>

the expansion to the power of *β*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

� �**<sup>β</sup>** <sup>¼</sup> *<sup>q</sup>***<sup>E</sup>** <sup>þ</sup> *<sup>q</sup>*½ �� **<sup>β</sup>** � **<sup>B</sup>** *<sup>∂</sup>*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

*∂S ∂τ* � �<sup>2</sup>

*∂S ∂τ* � �<sup>2</sup>

Let us represent the action *S* in the form

*S* ¼ �*Et* þ

integration, we obtain

or

following:

**121**

*x* ¼ *l*

1 þ *α*

*t* ¼ *l c* 1 þ 1 *α*

*<sup>α</sup>* <sup>1</sup> � cos *<sup>α</sup>*

The Hamilton-Jacobi equation is represented in the form

� *<sup>∂</sup><sup>S</sup> ∂***r** � �<sup>2</sup>

**3.2 Motion of a charged particle in a constant electric field**

� *<sup>∂</sup><sup>S</sup> ∂x* � �<sup>2</sup>

cathode under voltage –*U*; as a result, from (57) we obtain

1 *c*

� � arccos 1 � *<sup>α</sup>*

*ct l* � � � � , *<sup>t</sup>*<sup>≤</sup>

1 þ *α*

*d <sup>d</sup><sup>τ</sup> <sup>ε</sup>* � *<sup>q</sup>* 2*c* **A** � **β**

magnetic fields.

where the four-dimensional momentum is represented in the form

$$\mathbf{P} = I\mathbf{V} = \sqrt{\varepsilon^2 - \mathbf{p}^2} \left( \frac{\mathbf{1}}{\sqrt{1 - \eta^2}}, \qquad \frac{\eta}{\sqrt{1 - \eta^2}} \right). \tag{47}$$

Thus, the action is represented in the form

$$S = \int\_{s\_1}^{s\_2} I ds = \int\_{s\_1}^{s\_2} \sqrt{\varepsilon^2 - \mathbf{p}^2} ds = \int\_{\tau\_1}^{\tau\_2} \sqrt{\varepsilon^2 - \mathbf{p}^2} \sqrt{\mathbf{1} - \eta^2} d\tau \tag{48}$$

and the canonical Lagrangian of the system is given by

$$\mathbf{L} = I\sqrt{\mathbf{1} - \eta^2} = \sqrt{\epsilon^2 - \mathbf{p}^2}\sqrt{\mathbf{1} - \eta^2}. \tag{49}$$

The correctness of the presented parametrization is confirmed by the obtained expressions for the generalized momentum and energy from the Lagrangian of the system in the form

$$\begin{aligned} \varepsilon &= \mathfrak{n} \frac{\partial \mathcal{L}}{\partial \mathfrak{n}} - \mathcal{L} = \frac{I}{\sqrt{\mathfrak{1} - \eta^2}} = \frac{\sqrt{\varepsilon^2 - \mathfrak{p}^2}}{\sqrt{\mathfrak{1} - \eta^2}}, \\ \mathbf{p} &= \frac{\partial \mathcal{L}}{\partial \mathfrak{n}} = \frac{I}{\sqrt{\mathfrak{1} - \eta^2}} \mathfrak{n} = \varepsilon \mathfrak{n}, \end{aligned} \tag{50}$$

which coincide with the initial representations of the generalized momentum and energy. Accordingly, the Lagrange equation of motion takes the form

$$\frac{d\mathbf{p}}{d\tau} = -\frac{I}{\varepsilon} \frac{\partial I}{\partial \mathbf{r}}.\tag{51}$$

If we multiply Eq. (50) by **p** ¼ *ε***η** scalarly, after reduction to the total time differential, we obtain,

$$\frac{d\varepsilon^2}{d\tau} = \frac{\partial I^2}{\partial \tau}.\tag{52}$$

If the invariant is clearly independent of time, then the energy ε is conserved and the equation of motion is represented in the form of the Newtonian equation

$$\frac{d\eta}{d\tau} = -\frac{I}{\varepsilon^2} \frac{\partial I}{\partial \mathbf{r}}.\tag{53}$$

For a particle in an external field we have

$$\mathcal{L} = -\frac{1}{c}\sqrt{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}\sqrt{\mathbf{1} - \eta^2/c^2}.\tag{54}$$

where *λ* ¼ *λ*ð Þ*s* is the given parameter, determined by the condition of invariance

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> , **<sup>η</sup>**

*τ*1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>2</sup> � **p**<sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffi

**P** � *λ***V** ¼ 0, *λ* ¼ �*I*ð Þ *τ*, **r** , (46)

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>2</sup> � **p**<sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffi

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> ,

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> *:* (47)

(50)

<sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> *<sup>d</sup><sup>τ</sup>* (48)

<sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> *:* (49)

*:* (51)

*<sup>∂</sup><sup>τ</sup> :* (52)

*:* (53)

<sup>1</sup> � *<sup>η</sup>*<sup>2</sup>*=c*<sup>2</sup> <sup>p</sup> *:* (54)

of the representation. Because *λ* and *I* are given and they do not depend on the

where the four-dimensional momentum is represented in the form

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>2</sup> � **p**<sup>2</sup> q 1

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>2</sup> � **p**<sup>2</sup>

*ds* ¼ ð*τ*2

The correctness of the presented parametrization is confirmed by the obtained expressions for the generalized momentum and energy from the Lagrangian of the

> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> <sup>¼</sup>

which coincide with the initial representations of the generalized momentum

If we multiply Eq. (50) by **p** ¼ *ε***η** scalarly, after reduction to the total time

If the invariant is clearly independent of time, then the energy ε is conserved and the equation of motion is represented in the form of the Newtonian equation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup> q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

*dε*<sup>2</sup> *<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup><sup>I</sup>*

*d***η** *<sup>d</sup><sup>τ</sup>* ¼ � *<sup>I</sup> ε*2 *∂I ∂***r**

*mc* ð Þ <sup>2</sup> þ *qφ*

velocity, we have an explicit solution in the form

**P** ¼ *I***V** ¼

Thus, the action is represented in the form

*Ids* ¼

*ε* ¼ **η**

**<sup>p</sup>** <sup>¼</sup> <sup>∂</sup><sup>L</sup>

For a particle in an external field we have

<sup>L</sup> ¼ � <sup>1</sup> *c*

ð*s*2

q

*s*1

and the canonical Lagrangian of the system is given by

<sup>L</sup> <sup>¼</sup> *<sup>I</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> <sup>¼</sup>

∂L

*<sup>∂</sup>***<sup>η</sup>** <sup>¼</sup> *<sup>I</sup>*

*<sup>∂</sup>***<sup>η</sup>** � <sup>L</sup> <sup>¼</sup> *<sup>I</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> <sup>p</sup> **<sup>η</sup>** <sup>¼</sup> *<sup>ε</sup>***η**,

and energy. Accordingly, the Lagrange equation of motion takes the form

*d***p** *<sup>d</sup><sup>τ</sup>* ¼ � *<sup>I</sup> ε ∂I ∂***r**

*S* ¼ ð*s*2

system in the form

*Quantum Mechanics*

differential, we obtain,

**120**

*s*1

Using the explicit form of the generalized momentum (32) with the accuracy of the expansion to the power of *β*<sup>2</sup> , we obtain the equation of motion in the form

$$\frac{d}{d\tau}\left(\varepsilon-\frac{q}{2c}\mathbf{A}\cdot\mathfrak{H}\right)\mathfrak{H}=q\mathbf{E}+q[\mathfrak{H}\times\mathbf{B}]-\frac{\partial}{\partial\mathbf{r}}\left(\frac{q}{c}\mathbf{A}\cdot\mathfrak{H}+\frac{q^{2}}{2mc^{2}}(\varrho^{2}-\mathbf{A}^{2})\right),\tag{55}$$

where the velocity-dependent components of the force are present. In particular, the velocity-dependent force is present in the Faraday law of electromagnetic induction [34], which is absent in the traditional expression for the Lorentz force.

The Hamilton-Jacobi equation is represented in the form

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 = \frac{\left(m\mathbf{c}^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{c^2} \tag{56}$$

and it reflects the invariance of the representation of the generalized momentum. The well-known representations of the Hamilton-Jacobi Eq. (8) also contain the differential forms of potentials—the components of the electric and the magnetic fields.

#### **3.2 Motion of a charged particle in a constant electric field**

Let us consider the motion of a charged particle with the mass m and charge –*q* in the constant electric field between the plane electrodes with the potential difference *U* and the distance *l* between them. For one-dimensional motion, taking the cathode location as the origin and anode at the point *x* ¼ *l*, from (56) we have

$$\left(\frac{\partial \mathbb{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbb{S}}{\partial \mathbf{x}}\right)^2 = \frac{\left(mc^2 + qU(\mathbf{1} - \mathbf{x}/l)\right)^2}{c^2}.\tag{57}$$

Let us represent the action *S* in the form

$$S = -Et + f(\mathbf{x}),\tag{58}$$

where *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> <sup>þ</sup> *qU* is an the electron energy at the origin on the surface of the cathode under voltage –*U*; as a result, from (57) we obtain

$$\mathcal{S} = -Et + \frac{1}{c} \int \sqrt{E^2 - \left(mc^2 + qU - qU\frac{\mathcal{X}}{l}\right)^2} d\mathbf{x}.\tag{59}$$

We find the solution from the condition *<sup>∂</sup>S=∂<sup>E</sup>* <sup>¼</sup> const. As a result of integration, we obtain

$$t = \frac{l}{c} \left( 1 + \frac{1}{a} \right) \arccos \left( 1 - \frac{a}{1 + a} \frac{\varkappa}{l} \right), \qquad a = qU / mc^2 \tag{60}$$

or

$$\infty = l \frac{1+a}{a} \left( 1 - \cos\left(\frac{a}{1+a} \frac{ct}{l}\right) \right), \qquad t \le \frac{l}{c} \left( \frac{1+a}{a} \right) \arccos\left(\frac{1}{1+a}\right). \tag{61}$$

The well-known solution in the framework of the traditional theory [8] is the following:

$$t = \frac{l}{ac} \sqrt{\left(1 + a\frac{\infty}{l}\right)^2 - 1} \text{ or } \\ x = l \frac{\left(act/l\right)^2}{1 + \sqrt{1 + \left(act/l\right)^2}}, \qquad t \le \frac{l}{c} \sqrt{1 + \frac{2}{a}}.\tag{62}$$

In the ultrarelativistic limit *qU* ≫ *mc*2, the ratio of the flight time of the gap between the electrodes ð Þ *x* ¼ *l* is equal to *π=*2 according to formulas (60) and (62) (**Figure 5**). The electron velocity *v* ¼ *dx=dt* when reaching the anode is

$$v = c\sqrt{1 - 1/(1+a)^2}.\tag{63}$$

*S* ¼ �*Et* þ *Mφ* þ

1 *c*

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*φ* ¼

1 þ

which results in the solution

*<sup>r</sup>* <sup>¼</sup> ð Þ *Mc* <sup>2</sup> <sup>þ</sup> *Ze*<sup>2</sup> � �<sup>2</sup> *mc*2*Ze*<sup>2</sup>

*M*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>þ</sup> *Ze*<sup>2</sup> � �<sup>2</sup>

whence, we obtain

an electron.

s

*Equations of Relativistic and Quantum Mechanics (without Spin)*

<sup>ð</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

We find trajectories from the condition *<sup>∂</sup>S=∂<sup>M</sup>* <sup>¼</sup> const, with use of which we obtain,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*Ze*<sup>2</sup> � �<sup>2</sup> � � � *Mc*

The coefficient of the repulsive effective potential is essentially positive, that is,

The minimum radius *<sup>r</sup>*min <sup>¼</sup> *<sup>r</sup>*0ð Þ *<sup>Z</sup>* <sup>þ</sup> <sup>1</sup> , where *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*=mc*<sup>2</sup> is the classical radius of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *Ze*<sup>2</sup>

*=Mc* � �<sup>2</sup>

*<sup>=</sup>Mc* � �<sup>2</sup> <sup>q</sup> <sup>≈</sup> *<sup>π</sup> Ze*<sup>2</sup>

>0 therefore, any fall of the particle onto the center is impossible.

<sup>q</sup> *<sup>d</sup>* <sup>1</sup>

*<sup>r</sup>* � *<sup>M</sup>*<sup>2</sup>

*<sup>r</sup>* � *<sup>M</sup>*2*c*2þ*Ze*<sup>2</sup> *r*2

> *Ze*<sup>2</sup> � �<sup>2</sup>

1

*<sup>c</sup>*<sup>2</sup> <sup>þ</sup> *Ze*<sup>2</sup> � �<sup>2</sup> *r*2

*r*

cos *φ*

¼ 2*π*, (69)

<sup>2</sup> <sup>¼</sup> inv, (71)

Ψ ¼

þ div**p** � �, (72)

, (70)

*Mc* � �<sup>2</sup>

*dr:* (66)

, (67)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *Ze*<sup>2</sup> *Mc* � �<sup>2</sup> ! <sup>r</sup> *:*

(68)

*<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>2</sup>*mc*<sup>2</sup> *Ze*<sup>2</sup>

*<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>2</sup>*mc*<sup>2</sup> *Ze*<sup>2</sup>

ð *Mc*

*E mc*<sup>2</sup>

s

The secular precession is found from the condition

with the rotating field, that is, the spin-orbit interaction.

**4. Equations of the relativistic quantum mechanics**

corresponding operators ^*<sup>ε</sup>* <sup>¼</sup> *<sup>i</sup>*ℏ∂*=∂<sup>τ</sup>* and **<sup>p</sup>**^ ¼ �*i*ℏ∂*=∂***r**:

<sup>Ψ</sup> � ð Þ **<sup>p</sup>**^ <sup>2</sup>

*<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> � �<sup>Ψ</sup> <sup>þ</sup> *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup><sup>ε</sup>*

ð Þ^*ε* 2

**123**

*φ*

q

<sup>Δ</sup>*<sup>φ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>* � <sup>2</sup>*<sup>π</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> <sup>þ</sup> *Ze*<sup>2</sup>

that has the opposite sign as compared with the solution in [8]. The reason for the discrepancy of the sign is the unaccounted interaction of the self-momentum

Using the principle of the invariant representation of the generalized momentum

it is possible to compose the corresponding equation of the relativistic quantum

<sup>Ψ</sup> � �*i*<sup>ℏ</sup> *<sup>∂</sup>*

¼ *I* 2

*∂***r** � �<sup>2</sup>

> <sup>Ψ</sup> <sup>þ</sup> *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup><sup>ε</sup> ∂τ*

**<sup>P</sup>**<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*

mechanics by representing the energy and momentum variables by the

*∂τ* � �<sup>2</sup>

þ div**p** � �

<sup>Ψ</sup> <sup>¼</sup> *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup>*

*∂τ*

� �<sup>2</sup> <sup>1</sup> <sup>þ</sup> *Mc*

#### **3.3 Problem of the hydrogen-like atom**

Let us consider the motion of an electron with the mass m and charge –*e* in the field of an immobile nucleus with the charge *Ze*. Then the problem reduces to an investigation of the motion of the electron in the centrally symmetric electric field with the potential �*Ze*<sup>2</sup> *=r*.

Choosing the polar coordinates ð Þ *r*, *φ* in the plane of motion, we obtain the Hamilton-Jacobi equation in the form

$$\left(\frac{\partial \mathbf{S}}{\partial \tau}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \frac{\left(mc^2 - \mathbf{Z}c^2/r\right)^2}{c^2} = \mathbf{0}.\tag{64}$$

Let us represent the action *S* in the form

$$S = -Et + M\rho + f(r),\tag{65}$$

where *E* and *M* are the constant energy and angular momentum of the moving particle, respectively. As a result, we obtain

#### **Figure 5.**

*Dependence of the flight time of the gap between the electrodes on the applied voltage according to the formula (60) and (curve 1) and (62) (curve 2) in l=c units.*

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

$$S = -Et + M\rho + \frac{1}{c} \left[ \sqrt{E^2 - \left(mc^2\right)^2 + 2mc^2\frac{Ze^2}{r} - \frac{M^2c^2 + \left(Ze^2\right)^2}{r^2}} dr.\tag{66}$$

We find trajectories from the condition *<sup>∂</sup>S=∂<sup>M</sup>* <sup>¼</sup> const, with use of which we obtain,

$$\rho = \left[\frac{Mc}{\sqrt{E^2 - \left(mc^2\right)^2 + 2mc^2\frac{Z\omega^2}{r} - \frac{M^2c^2 + Z\omega^2}{r^2}}}d\frac{1}{r},\tag{67}$$

which results in the solution

*<sup>t</sup>* <sup>¼</sup> *<sup>l</sup> αc*

*Quantum Mechanics*

r

with the potential �*Ze*<sup>2</sup>

**Figure 5.**

**122**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>x</sup> l* � �<sup>2</sup>

**3.3 Problem of the hydrogen-like atom**

Hamilton-Jacobi equation in the form

*∂S ∂τ* � �<sup>2</sup> *=r*.

� *<sup>∂</sup><sup>S</sup> ∂r* � �<sup>2</sup>

Let us represent the action *S* in the form

particle, respectively. As a result, we obtain

*(60) and (curve 1) and (62) (curve 2) in l=c units.*

� 1

or *<sup>x</sup>* <sup>¼</sup> *<sup>l</sup>* ð Þ *<sup>α</sup>ct=<sup>l</sup>* <sup>2</sup> 1 þ

The electron velocity *v* ¼ *dx=dt* when reaching the anode is

*v* ¼ *c*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ *<sup>α</sup>ct=<sup>l</sup>* <sup>2</sup> <sup>q</sup> , *<sup>t</sup>*<sup>≤</sup>

In the ultrarelativistic limit *qU* ≫ *mc*2, the ratio of the flight time of the gap between the electrodes ð Þ *x* ¼ *l* is equal to *π=*2 according to formulas (60) and (62) (**Figure 5**).

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>1</sup>*=*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* <sup>2</sup>

Let us consider the motion of an electron with the mass m and charge –*e* in the field of an immobile nucleus with the charge *Ze*. Then the problem reduces to an investigation of the motion of the electron in the centrally symmetric electric field

Choosing the polar coordinates ð Þ *r*, *φ* in the plane of motion, we obtain the

*∂S ∂φ* � �<sup>2</sup>

where *E* and *M* are the constant energy and angular momentum of the moving

*Dependence of the flight time of the gap between the electrodes on the applied voltage according to the formula*

� *mc*<sup>2</sup> � *Ze*<sup>2</sup> *=r* � �<sup>2</sup>

*S* ¼ �*Et* þ *Mφ* þ *f r*ð Þ, (65)

� 1 *r*2 *l c* r

*:* (63)

*<sup>c</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (64)

ffiffiffiffiffiffiffiffiffiffiffi 1 þ 2 *α*

*:* (62)

$$r = \frac{\left(Mc\right)^2 + \left(Ze^2\right)^2}{mc^2Ze^2} \frac{1}{1 + \sqrt{\left(\frac{E}{mc^2}\right)^2 \left(1 + \left(\frac{Mc}{Zc^2}\right)^2\right) - \left(\frac{Mc}{Zc^2}\right)^2} \cos\left(\rho\sqrt{1 + \left(\frac{Ze^2}{Mc}\right)^2}\right)},\tag{68}$$

The coefficient of the repulsive effective potential is essentially positive, that is, *M*<sup>2</sup> *<sup>c</sup>*<sup>2</sup> <sup>þ</sup> *Ze*<sup>2</sup> � �<sup>2</sup> >0 therefore, any fall of the particle onto the center is impossible. The minimum radius *<sup>r</sup>*min <sup>¼</sup> *<sup>r</sup>*0ð Þ *<sup>Z</sup>* <sup>þ</sup> <sup>1</sup> , where *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*=mc*<sup>2</sup> is the classical radius of an electron.

The secular precession is found from the condition

$$
\rho \sqrt{\mathbf{1} + \left( Z \mathbf{c}^2 / \mathbf{M} \mathbf{c} \right)^2} = 2\pi,\tag{69}
$$

whence, we obtain

$$
\Delta \rho = 2\pi - \frac{2\pi}{\sqrt{1 + \left(Ze^2/Mc\right)^2}} \approx \pi \left(\frac{Ze^2}{Mc}\right)^2,\tag{70}
$$

that has the opposite sign as compared with the solution in [8]. The reason for the discrepancy of the sign is the unaccounted interaction of the self-momentum with the rotating field, that is, the spin-orbit interaction.

#### **4. Equations of the relativistic quantum mechanics**

Using the principle of the invariant representation of the generalized momentum

$$\mathbf{P}^2 = \boldsymbol{\varepsilon}^2 - \mathbf{p}^2 = I^2 = \text{inv},\tag{71}$$

it is possible to compose the corresponding equation of the relativistic quantum mechanics by representing the energy and momentum variables by the corresponding operators ^*<sup>ε</sup>* <sup>¼</sup> *<sup>i</sup>*ℏ∂*=∂<sup>τ</sup>* and **<sup>p</sup>**^ ¼ �*i*ℏ∂*=∂***r**:

$$\begin{split} \left(\hat{\boldsymbol{e}}\right)^{2}\Psi - \left(\hat{\mathbf{p}}\right)^{2}\Psi &= \left(i\hbar\frac{\partial}{\partial\mathbf{r}}\right)^{2}\Psi - \left(-i\hbar\frac{\partial}{\partial\mathbf{r}}\right)^{2}\Psi = \\ \left(\boldsymbol{e}^{2} - \mathbf{p}^{2}\right)\Psi + i\hbar\left(\frac{\partial\boldsymbol{e}}{\partial\boldsymbol{\tau}} + \mathrm{div}\mathbf{p}\right) &= I^{2}\Psi + i\hbar\left(\frac{\partial\boldsymbol{e}}{\partial\boldsymbol{\tau}} + \mathrm{div}\mathbf{p}\right), \end{split} \tag{72}$$

and

$$\left(\left(\hat{\varepsilon}\Psi\right)^{2} - \left(\hat{\mathbf{p}}\Psi\right)^{2} = \left(i\hbar\frac{\partial\Psi}{\partial\mathbf{r}}\right)^{2} - \left(-i\hbar\frac{\partial\Psi}{\partial\mathbf{r}}\right)^{2} = \left(e^{2} - \mathbf{p}^{2}\right)\Psi^{2} = I^{2}\Psi^{2}.\tag{73}$$

The case of conservative systems, when any energy losses or sources in space are absent, corresponds to the relation *<sup>∂</sup>ε=∂<sup>τ</sup>* <sup>þ</sup> div**<sup>p</sup>** <sup>¼</sup> 0. In this way,

$$\begin{cases} \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} - \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} = -\frac{I^2}{\hbar^2} \Psi\\ \left(\frac{\partial \Psi}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \Psi}{\partial \mathbf{r}}\right)^2 = -\frac{I^2}{\hbar^2} \Psi^2. \end{cases} \tag{74}$$

**4.1 Particle in the one-dimensional potential well**

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

From the first equation of system (70) we have

*d*2 Ψ *dx*<sup>2</sup> <sup>þ</sup>

2

<sup>2</sup>*ma*<sup>2</sup> *<sup>n</sup>*<sup>2</sup> ¼ �*V*<sup>0</sup> <sup>þ</sup>

*λ* <sup>2</sup>*<sup>a</sup> <sup>n</sup>* � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>λ</sup>*

**4.2 Penetration of a particle through a potential barrier**

0 s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>π</sup>*<sup>2</sup>ℏ<sup>2</sup> *<sup>m</sup>*<sup>2</sup>*c*<sup>2</sup>*a*<sup>2</sup> *<sup>n</sup>*<sup>2</sup>

well of the form

Then, *U*<sup>0</sup> ¼ –*V*<sup>0</sup> þ *V*<sup>0</sup>

*<sup>U</sup>*<sup>0</sup> ¼ � *<sup>π</sup>*<sup>2</sup>ℏ<sup>2</sup>

*En* <sup>¼</sup> *mc*<sup>2</sup> <sup>1</sup> �

<sup>¼</sup> *mc*<sup>2</sup>

relation

the particle.

**125**

meaning, and H <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

@

1 þ

depth *V*<sup>0</sup> with the *d* ¼ *λn=*2, *n* ¼ 1, 2, 3 … .

through the potential barrier at *<sup>E</sup>* <sup>&</sup>gt; *<sup>V</sup>*<sup>0</sup> <sup>þ</sup> *mc*<sup>2</sup> �

Let us consider the particle of mass *m* in a one-dimensional rectangular potential

�*V*0, 0≤*x*≤*a:*

*<sup>c</sup>*<sup>2</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>*:* (80)

*<sup>n</sup>* <sup>¼</sup> *<sup>λ</sup>* 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>λ</sup>* 2*a n* � �<sup>2</sup>

*n*, (81)

2 *<sup>=</sup>* <sup>2</sup>*mc*<sup>2</sup> ð Þ

1 A

*<sup>=</sup>* <sup>2</sup>*mc*<sup>2</sup> ð Þ corresponds to the potential energy of the parti-

<sup>2</sup> , *<sup>a</sup>*<sup>≥</sup> *<sup>π</sup>*<sup>ℏ</sup>

0 s

<sup>2</sup>*<sup>a</sup> <sup>n</sup>* � �<sup>2</sup> <sup>q</sup> , (82)

*a* ¼ *λn=*2, *n* ¼ 1, 2, 3 … (83)

*mc*

(79)

*V x*ð Þ¼ 0, 0 <sup>≥</sup>*x*≥*<sup>a</sup>*

*<sup>E</sup>*<sup>2</sup> � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *V x*ð Þ <sup>2</sup> ℏ2

cle in the well in the classical meaning. In the latter case, it is known [12] that the

bound state with the energy *<sup>H</sup>* <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ arises under the conditions

*V*<sup>0</sup> 2 <sup>2</sup>*mc*<sup>2</sup> <sup>≥</sup> � *mc*<sup>2</sup>

1

<sup>A</sup> <sup>¼</sup> *mc*<sup>2</sup> <sup>1</sup> �

where *λ* ¼ 2*πƛ* ¼ 2*π*ℏ*=mc* ¼ *h=mc:* Maximum depth of the classic well is equal to *<sup>U</sup>*<sup>0</sup> <sup>¼</sup> –*mc*<sup>2</sup>*=*2 at *<sup>V</sup>*<sup>0</sup> <sup>¼</sup> *mc*<sup>2</sup>*:* The condition for the existence of the bound state with an energy H <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ in a potential well of size a is expressed by the

In the three-dimensional case, the bound state with the energy H <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ arises under the same conditions [23] for a spherical well with a diameter *d* and

The solution of this simple example is fundamental and accurately represents the uncertainty principle Δ*x*Δ*p* ≥ℏ*=*2. It clearly represents the wave property of the particle, clearly showing that the standing wave exists only at the condition *a*≥*λ=*2 when the geometric dimensions of the well are greater than half the wavelength of

Let us consider the problem of penetration of a particle through the rectangular

expressions into the solution of the Schrödinger equation for the rectangular potential barrier, we obtain for the transmission coefficient *D* of the particle penetrating

� �

�

–*m*<sup>2</sup>*c*<sup>4</sup> � �*=*2*mc*<sup>2</sup> corresponds to the energy. Substituting these

potential barrier [23] with the height *V*<sup>0</sup> and width *a*. Then, *U*<sup>0</sup> ¼ *V*<sup>0</sup> þ *V*<sup>0</sup>

corresponds to the potential energy of the particle in the well in the classical

@

�

For the charged particle in an external field with an invariant in the form of (30), the equations will take the form

$$\begin{cases} \frac{\partial^2 \Psi}{\partial \tau^2} - \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} = -\frac{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{\hbar^2 c^2} \Psi\\ \left(\frac{\partial \Psi}{\partial \tau}\right)^2 - \left(\frac{\partial \Psi}{\partial \mathbf{r}}\right)^2 = -\frac{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{\hbar^2 c^2} \Psi^2. \end{cases} \tag{75}$$

For stationary states we obtain

$$\begin{cases} \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} + \frac{E^2 - (mc^2 + q\rho)^2 + (q\mathbf{A})^2}{\hbar^2 c^2} \Psi = \mathbf{0} \\\\ \left(\frac{\partial \Psi}{\partial \mathbf{r}}\right)^2 + \frac{E^2 - (mc^2 + q\rho)^2 + (q\mathbf{A})^2}{\hbar^2 c^2} \Psi^2 = \mathbf{0}. \end{cases} \tag{76}$$

Rewriting the equations taking into account the formulas of the classical correspondence (40), we will obtain the equations for the wave function in the traditional representation

$$\begin{aligned} \Delta \Psi + \frac{2m}{\hbar^2} (\mathbf{H} - U) \Psi &= \mathbf{0}, \\ \left(\frac{\partial \Psi}{\partial \mathbf{r}}\right)^2 + \frac{2m}{\hbar^2} (\mathbf{H} - U) \Psi^2 &= \mathbf{0}, \end{aligned} \tag{77}$$

the first of which formally coincides with the Schrödinger equation for the wave function of stationary states.

For the action function S associated with the wave function by the representation Ψ ¼ *A exp* ð Þ –*iS=*ℏ or *S* ¼ *i*ℏ ln Ψ þ *i*ℏ ln *A*, we will obtain

$$\begin{cases} \frac{\partial^2 \mathbf{S}}{\partial \mathbf{r}^2} = \mathbf{0} \\\\ \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \frac{E^2 - \left(mc^2 + q\rho\right)^2 + \left(q\mathbf{A}\right)^2}{c^2} = \mathbf{0} \end{cases} \Rightarrow \begin{cases} \frac{\partial^2 \mathbf{S}}{\partial \mathbf{r}^2} = \mathbf{0} \\\\ \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - 2m(\mathbf{H} - U) = \mathbf{0}, \end{cases} \tag{78}$$

which represents the exact classical correspondence instead of the quasiclassical approximation [12]. Note, the equations similar to (78) also follow from the Eq. (46) in [12] if we demand for an exact correspondence and equate to zero the real and imaginary parts.

#### **4.1 Particle in the one-dimensional potential well**

and

*Quantum Mechanics*

ð Þ ^*ε*<sup>Ψ</sup> <sup>2</sup> � ð Þ **<sup>p</sup>**^<sup>Ψ</sup> <sup>2</sup> <sup>¼</sup> *<sup>i</sup>*<sup>ℏ</sup> <sup>∂</sup><sup>Ψ</sup>

(30), the equations will take the form

8 >>><

>>>:

For stationary states we obtain

*∂*2 Ψ *<sup>∂</sup>τ*<sup>2</sup> � *<sup>∂</sup>*<sup>2</sup>

∂Ψ *∂τ* � �<sup>2</sup> � <sup>∂</sup><sup>Ψ</sup> *∂***r**

*∂*2 Ψ *∂***r**<sup>2</sup> þ

8 >>><

>>>:

tional representation

function of stationary states.

� �<sup>2</sup> � *<sup>E</sup>*<sup>2</sup> � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

*∂*2 *S <sup>∂</sup>***r**<sup>2</sup> <sup>¼</sup> <sup>0</sup>

8 >><

>>:

**124**

*∂S ∂***r**

real and imaginary parts.

∂Ψ *∂***r** � �<sup>2</sup> <sup>þ</sup>

*Δ*Ψ þ

∂Ψ *∂***r** � �<sup>2</sup>

tion Ψ ¼ *A exp* ð Þ –*iS=*ℏ or *S* ¼ *i*ℏ ln Ψ þ *i*ℏ ln *A*, we will obtain

2*m*

þ 2*m*

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup> *<sup>c</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>

approximation [12]. Note, the equations similar to (78) also follow from the Eq. (46) in [12] if we demand for an exact correspondence and equate to zero the

*∂τ* � �<sup>2</sup>

absent, corresponds to the relation *<sup>∂</sup>ε=∂<sup>τ</sup>* <sup>þ</sup> div**<sup>p</sup>** <sup>¼</sup> 0. In this way,

*∂*2 Ψ *<sup>∂</sup>τ*<sup>2</sup> � *<sup>∂</sup>*<sup>2</sup>

8 >><

>>:

∂Ψ *∂τ* � �<sup>2</sup> � <sup>∂</sup><sup>Ψ</sup> *∂***r** � �<sup>2</sup> ¼ � *<sup>I</sup>*

Ψ

� �*i*<sup>ℏ</sup> <sup>∂</sup><sup>Ψ</sup> *∂***r** � �<sup>2</sup>

> Ψ *<sup>∂</sup>***r**<sup>2</sup> ¼ � *<sup>I</sup>*

For the charged particle in an external field with an invariant in the form of

*<sup>∂</sup>***r**<sup>2</sup> ¼ � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

*<sup>E</sup>*<sup>2</sup> � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

*<sup>E</sup>*<sup>2</sup> � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

� �<sup>2</sup> ¼ � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

ℏ2

ℏ2

Rewriting the equations taking into account the formulas of the classical correspondence (40), we will obtain the equations for the wave function in the tradi-

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>H</sup> � *<sup>U</sup>* <sup>Ψ</sup> <sup>¼</sup> 0,

the first of which formally coincides with the Schrödinger equation for the wave

For the action function S associated with the wave function by the representa-

which represents the exact classical correspondence instead of the quasiclassical

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>H</sup> � *<sup>U</sup>* <sup>Ψ</sup><sup>2</sup> <sup>¼</sup> 0,

)

*∂*2 *S <sup>∂</sup>***r**<sup>2</sup> <sup>¼</sup> <sup>0</sup> *∂S ∂***r**

8 ><

>:

� �<sup>2</sup> � <sup>2</sup>*m*ð Þ¼ <sup>H</sup> � *<sup>U</sup>* 0,

The case of conservative systems, when any energy losses or sources in space are

2 <sup>ℏ</sup><sup>2</sup> <sup>Ψ</sup>

ℏ2 *c*2

> ℏ2 *c*2

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>Ψ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*

2 <sup>ℏ</sup><sup>2</sup> <sup>Ψ</sup><sup>2</sup> *:*

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

Ψ

Ψ2 *:*

<sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> � **<sup>p</sup>**<sup>2</sup> � �Ψ<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*

2 Ψ2

*:* (73)

(74)

(75)

(76)

(77)

(78)

Let us consider the particle of mass *m* in a one-dimensional rectangular potential well of the form

$$V(\mathbf{x}) = \begin{cases} \mathbf{0}, & \mathbf{0} \ge \mathbf{x} \ge a \\ -V\_0, & \mathbf{0} \le \mathbf{x} \le a. \end{cases} \tag{79}$$

From the first equation of system (70) we have

$$\frac{d^2\Psi}{d\mathfrak{x}^2} + \frac{E^2 - \left(mc^2 + V(\mathfrak{x})\right)^2}{\hbar^2 c^2} \Psi = \mathbf{0}.\tag{80}$$

Then, *U*<sup>0</sup> ¼ –*V*<sup>0</sup> þ *V*<sup>0</sup> 2 *<sup>=</sup>* <sup>2</sup>*mc*<sup>2</sup> ð Þ corresponds to the potential energy of the particle in the well in the classical meaning. In the latter case, it is known [12] that the bound state with the energy *<sup>H</sup>* <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ arises under the conditions

$$U\_0 = -\frac{\pi^2 \hbar^2}{2ma^2} n^2 = -V\_0 + \frac{V\_0}{2mc^2} \ge -\frac{mc^2}{2}, \quad a \ge \frac{\pi \hbar}{mc} n = \frac{\lambda}{2}n,\tag{81}$$

$$\begin{split} E\_n &= mc^2 \left( 1 - \sqrt{1 - \frac{\pi^2 \hbar^2}{m^2 c^2 a^2} n^2} \right) = mc^2 \left( 1 - \sqrt{1 - \left(\frac{\lambda}{2a} n\right)^2} \right) \\ &= mc^2 \frac{\left(\frac{\lambda}{2a} n\right)^2}{1 + \sqrt{1 - \left(\frac{\lambda}{2a} n\right)^2}}, \end{split} \tag{82}$$

where *λ* ¼ 2*πƛ* ¼ 2*π*ℏ*=mc* ¼ *h=mc:* Maximum depth of the classic well is equal to *<sup>U</sup>*<sup>0</sup> <sup>¼</sup> –*mc*<sup>2</sup>*=*2 at *<sup>V</sup>*<sup>0</sup> <sup>¼</sup> *mc*<sup>2</sup>*:* The condition for the existence of the bound state with an energy H <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ in a potential well of size a is expressed by the relation

$$
\mathfrak{a} = \lambda \mathfrak{n}/2, \qquad \mathfrak{n} = \mathfrak{1}, \ \mathfrak{2}, \ \mathfrak{3} \dots \tag{83}
$$

In the three-dimensional case, the bound state with the energy H <sup>¼</sup> <sup>0</sup> *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> ð Þ arises under the same conditions [23] for a spherical well with a diameter *d* and depth *V*<sup>0</sup> with the *d* ¼ *λn=*2, *n* ¼ 1, 2, 3 … .

The solution of this simple example is fundamental and accurately represents the uncertainty principle Δ*x*Δ*p* ≥ℏ*=*2. It clearly represents the wave property of the particle, clearly showing that the standing wave exists only at the condition *a*≥*λ=*2 when the geometric dimensions of the well are greater than half the wavelength of the particle.

#### **4.2 Penetration of a particle through a potential barrier**

Let us consider the problem of penetration of a particle through the rectangular potential barrier [23] with the height *V*<sup>0</sup> and width *a*. Then, *U*<sup>0</sup> ¼ *V*<sup>0</sup> þ *V*<sup>0</sup> 2 *<sup>=</sup>* <sup>2</sup>*mc*<sup>2</sup> ð Þ corresponds to the potential energy of the particle in the well in the classical meaning, and H <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> –*m*<sup>2</sup>*c*<sup>4</sup> � �*=*2*mc*<sup>2</sup> corresponds to the energy. Substituting these expressions into the solution of the Schrödinger equation for the rectangular potential barrier, we obtain for the transmission coefficient *D* of the particle penetrating through the potential barrier at *<sup>E</sup>* <sup>&</sup>gt; *<sup>V</sup>*<sup>0</sup> <sup>þ</sup> *mc*<sup>2</sup> � � � �

$$D = \left[ 1 + \frac{\left(\left(\frac{V\_0}{mc^2} + 1\right)^2 - 1\right)^2}{4\left(\left(\frac{E}{mc^2}\right)^2 - 1\right)\left(\left(\frac{E}{mc^2}\right)^2 - \left(\frac{V\_0}{mc^2} + 1\right)^2\right)} \sin^2\left(\frac{a}{\hbar}\sqrt{\left(\frac{E}{mc^2}\right)^2 - \left(\frac{V\_0}{mc^2} + 1\right)^2}\right) \right] \tag{84}$$

*<sup>W</sup>* <sup>¼</sup> *Mc*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

and from (88) we obtain

or

*<sup>φ</sup>*ð Þ¼� *<sup>x</sup> <sup>φ</sup>*0*e*�*x=<sup>d</sup>*.

*V x*ð Þ¼�*qφ*0*e*

**Figure 6.**

**127**

�*x=<sup>d</sup>* <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� *<sup>ρ</sup>* 2*ρ<sup>H</sup>* � �<sup>2</sup>

<sup>þ</sup> *pz Mc* � �<sup>2</sup>

<sup>8</sup> *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 0, (91)

*<sup>e</sup> :* (93)

*qφ*<sup>0</sup> *mc*<sup>2</sup> *<sup>e</sup>*

�*x=<sup>d</sup>* � �<sup>2</sup> � �*:*

*:* (90)

(92)

(94)

<sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>ƛ</sup> ρ* � �<sup>2</sup>

where *ρ<sup>H</sup>* ¼ *c=ω<sup>H</sup>* (magnetic event horizon), and *ρ* as a constant parameter. If an electron is excited by a magnetic field from a state of rest, then *<sup>W</sup>* <sup>¼</sup> *Mc*<sup>2</sup>

> *Mω*<sup>2</sup> *H*

> > 2

We determine the energy levels for a particle moving in a field with a potential

<sup>¼</sup> *mc*<sup>2</sup> �*qφ*0*<sup>e</sup>*

*mc*<sup>2</sup>

�*x=d* þ 1 2

According to (41), for the potential energy of interaction *V x*ð Þ with the field

*<sup>=</sup>*<sup>2</sup> <sup>¼</sup> *Mc*<sup>2</sup> 2

*hc* <sup>Φ</sup> <sup>¼</sup> *<sup>m</sup>*, ΔΦ <sup>¼</sup> *hc*

*ρ ƛH* � �<sup>2</sup>

s

*Equations of Relativistic and Quantum Mechanics (without Spin)*

� ℏ2 *m*<sup>2</sup> 2*Mρ*<sup>2</sup> þ

*m*ℏ*ω<sup>H</sup>* ¼ *M*ð Þ *ρω<sup>H</sup>*

From (92) for a magnetic flux quantum we have

**4.4 Particle in the field with Morse potential energy**

1 <sup>2</sup>*mc*<sup>2</sup> *<sup>q</sup>φ*0*<sup>e</sup>*

*hc <sup>H</sup>πρ*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>*

We get the same results when solving the Hamilton-Jacobi equation.

*φ*ð Þ *x* we obtain the expression of the potential Morse energy (**Figure 6**)

�*x=<sup>d</sup>* � �<sup>2</sup>

*The exponential potential of the field φ* ð Þ *x and Morse potential energy of interaction V x*ð Þ*.*

*e*

and at *<sup>E</sup>* <sup>&</sup>lt; *<sup>V</sup>*<sup>0</sup> <sup>þ</sup> *mc*<sup>2</sup> � � � �

$$D = \left[ \mathbf{1} + \frac{\left( \left( \frac{V\_0}{mc^2} + \mathbf{1} \right)^2 + \mathbf{1} - 2 \left( \frac{E}{mc^2} \right)^2 \right)^2}{\mathbf{4} \left( \left( \frac{E}{mc^2} \right)^2 - \mathbf{1} \right) \left( \left( \frac{V\_0}{mc^2} + \mathbf{1} \right)^2 - \left( \frac{E}{mc^2} \right)^2 \right)} \sinh^2 \left( \frac{a}{\lambda} \sqrt{\left( \frac{V\_0}{mc^2} + \mathbf{1} \right)^2 - \left( \frac{E}{mc^2} \right)^2} \right) \right]^{-1} \tag{85}$$

where *ƛ* ¼ ℏ*=mc* is the de Broglie wavelength of the particle. As can be seen, the barrier is formed only in the energy range �2*mc*<sup>2</sup> <sup>&</sup>gt;*V*<sup>0</sup> <sup>&</sup>gt; *mc*2.

For the problem of the passage of a particle with energy *E* through a potential barrier *U* (**Figure 2**) the wave vector *k* is represented as

$$k\_1 = \frac{1}{\hbar c} \sqrt{E^2 - \left(mc^2\right)^2}, \qquad k\_2 = \frac{1}{\hbar c} \sqrt{E^2 - \left(mc^2 + U\right)^2} \tag{86}$$

and if the particle energy does not exceed the potential barrier, then the transmission coefficient is zero, regardless of the height of the barrier and not have. In this case, there is no contradiction similar to the Klein paradox.

#### **4.3 Charged particle in a magnetic field**

The vector potential of a uniform magnetic field **A** along the **z** axis direction in the cylindrical coordinate system ð Þ *ρ*, *φ*, *z* has components *A<sup>φ</sup>* ¼ *Hρ=*2, *A<sup>ρ</sup>* ¼ *Az* ¼ 0 and Eq. (76) takes the form

$$\frac{\hbar^2}{2\mathbf{M}}\left(\mathbf{R}^{\prime\prime} + \frac{\mathbf{1}}{\rho}\mathbf{R}^{\prime}\right) + \left(E - \frac{\hbar^2 m^2}{2\mathbf{M}}\frac{\mathbf{1}}{\rho^2} + \frac{\mathbf{M}m\_H^2}{8}\rho^2 - \frac{p\_x^2}{2\mathbf{M}}\right)\mathbf{R} = \mathbf{0},\tag{87}$$

where *m* – angular quantum number, *M* – mass of electron, *H*– magnetic field value, *ω<sup>H</sup>* ¼ *eH=Mc*. In this case, the equation below differs from the known [12] one by the absence of the field linear term ℏ*ωHm=*2 and the sign of a quadratic term *Mω*<sup>2</sup> *<sup>H</sup>ρ*<sup>2</sup>*=*8.

In this form, the Eq. (87) does not have a finite solution depending on the variable *ρ* and, provided R = const, we have

$$\begin{aligned} R'' + \frac{1}{\rho} R' &= 0, \\\\ \left( E - \frac{\hbar^2 m^2}{2M\rho^2} + \frac{M o\_H^2}{8} \rho^2 - \frac{p\_x^2}{2M} \right) R &= 0. \end{aligned} \tag{88}$$

Or

$$E - \frac{\hbar^2 m^2}{2M} \frac{1}{\rho^2} + \frac{M o\_H^2}{8} \rho^2 - \frac{p\_z^2}{2M} \equiv \begin{array}{c} 0. \end{array} \tag{89}$$

From (89) we have for the energy levels

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

$$\mathcal{W} = \mathcal{M}c^2 \sqrt{1 + m^2 \left(\frac{\mathcal{N}}{\rho}\right)^2 - \left(\frac{\rho}{2\rho\_H}\right)^2 + \left(\frac{p\_x}{\mathcal{M}c}\right)^2}. \tag{90}$$

where *ρ<sup>H</sup>* ¼ *c=ω<sup>H</sup>* (magnetic event horizon), and *ρ* as a constant parameter.

If an electron is excited by a magnetic field from a state of rest, then *<sup>W</sup>* <sup>¼</sup> *Mc*<sup>2</sup> and from (88) we obtain

$$-\frac{\hbar^2 m^2}{2\mathbf{M}\rho^2} + \frac{\mathbf{M}o\_H^2}{8}\rho^2 = \mathbf{0},\tag{91}$$

or

*D* ¼ 1 þ

*D* ¼ 1 þ

2 6 4

2 6 4

*Quantum Mechanics*

4 *<sup>E</sup> mc*<sup>2</sup> � �<sup>2</sup> � <sup>1</sup> � � *<sup>E</sup>*

and at *<sup>E</sup>* <sup>&</sup>lt; *<sup>V</sup>*<sup>0</sup> <sup>þ</sup> *mc*<sup>2</sup> �

4 *<sup>E</sup> mc*<sup>2</sup> � �<sup>2</sup> � <sup>1</sup> � � *<sup>V</sup>*<sup>0</sup>

� �

*V*<sup>0</sup> *mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

*<sup>k</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ℏ*c*

**4.3 Charged particle in a magnetic field**

variable *ρ* and, provided R = const, we have

*R*<sup>00</sup> þ 1 *ρ*

*<sup>E</sup>* � <sup>ℏ</sup><sup>2</sup>

*<sup>E</sup>* � <sup>ℏ</sup><sup>2</sup>

From (89) we have for the energy levels

and Eq. (76) takes the form

*R*00 þ 1 *ρ R*0 � �

ℏ2 2*M*

Or

**126**

*V*<sup>0</sup> *mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

� �<sup>2</sup>

*mc*<sup>2</sup> � �<sup>2</sup> � *<sup>V</sup>*<sup>0</sup>

� �<sup>2</sup> � �<sup>2</sup>

�

� 1

<sup>þ</sup> <sup>1</sup> � <sup>2</sup> *<sup>E</sup> mc*<sup>2</sup>

*mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

barrier is formed only in the energy range �2*mc*<sup>2</sup> <sup>&</sup>gt;*V*<sup>0</sup> <sup>&</sup>gt; *mc*2.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> ð Þ<sup>2</sup>

this case, there is no contradiction similar to the Klein paradox.

<sup>þ</sup> *<sup>E</sup>* � <sup>ℏ</sup><sup>2</sup>

*R*<sup>0</sup> ¼ 0,

*m*<sup>2</sup> 2*Mρ*<sup>2</sup> þ

*m*<sup>2</sup> 2*M*

1 *ρ*<sup>2</sup> þ

*m*<sup>2</sup> 2*M*

1 *ρ*<sup>2</sup> þ

In this form, the Eq. (87) does not have a finite solution depending on the

*Mω*<sup>2</sup> *H* <sup>8</sup> *<sup>ρ</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>2</sup>

*Mω*<sup>2</sup> *H* <sup>8</sup> *<sup>ρ</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>2</sup>

� �

*Mω*<sup>2</sup> *H* <sup>8</sup> *<sup>ρ</sup>*<sup>2</sup> � *<sup>p</sup>*<sup>2</sup>

> *z* 2*M*

> > *z*

*R* ¼ 0*:*

<sup>2</sup>*<sup>M</sup>* � <sup>0</sup>*:* (89)

� �

where *m* – angular quantum number, *M* – mass of electron, *H*– magnetic field value, *ω<sup>H</sup>* ¼ *eH=Mc*. In this case, the equation below differs from the known [12] one by the absence of the field linear term ℏ*ωHm=*2 and the sign of a quadratic term *Mω*<sup>2</sup>

barrier *U* (**Figure 2**) the wave vector *k* is represented as

q

*mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup> � � sin <sup>2</sup> *<sup>a</sup>*

� *<sup>E</sup> mc*<sup>2</sup> � �<sup>2</sup> � � sinh <sup>2</sup> *<sup>a</sup>*

where *ƛ* ¼ ℏ*=mc* is the de Broglie wavelength of the particle. As can be seen, the

For the problem of the passage of a particle with energy *E* through a potential

, *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

and if the particle energy does not exceed the potential barrier, then the transmission coefficient is zero, regardless of the height of the barrier and not have. In

The vector potential of a uniform magnetic field **A** along the **z** axis direction in the cylindrical coordinate system ð Þ *ρ*, *φ*, *z* has components *A<sup>φ</sup>* ¼ *Hρ=*2, *A<sup>ρ</sup>* ¼ *Az* ¼ 0

ℏ*c*

q

*ƛ*

*ƛ*

0 s @

0 s @

*E mc*<sup>2</sup> � �<sup>2</sup>

> *V*<sup>0</sup> *mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>2</sup> � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>U</sup>* <sup>2</sup>

> *z* 2*M*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� *<sup>V</sup>*<sup>0</sup> *mc*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� *<sup>E</sup> mc*<sup>2</sup> � �<sup>2</sup>

*R* ¼ 0, (87)

*<sup>H</sup>ρ*<sup>2</sup>*=*8.

(88)

(84)

1 A

(85)

(86)

3 7 5

�1

3 7 5

�1

$$m\hbar\alpha\_{H} = M(\rho\alpha\_{H})^{2}/2 = \frac{Mc^{2}}{2}\left(\frac{\rho}{\hbar\_{H}}\right)^{2} \tag{92}$$

From (92) for a magnetic flux quantum we have

$$
\frac{e}{hc}H\pi\rho^2 = \frac{e}{hc}\Phi = m, \qquad \Delta\Phi = \frac{hc}{e}.\tag{93}
$$

We get the same results when solving the Hamilton-Jacobi equation.

#### **4.4 Particle in the field with Morse potential energy**

We determine the energy levels for a particle moving in a field with a potential *<sup>φ</sup>*ð Þ¼� *<sup>x</sup> <sup>φ</sup>*0*e*�*x=<sup>d</sup>*.

According to (41), for the potential energy of interaction *V x*ð Þ with the field *φ*ð Þ *x* we obtain the expression of the potential Morse energy (**Figure 6**)

$$V(\mathbf{x}) = -q\rho\_0 e^{-\mathbf{x}/d} + \frac{1}{2mc^2} \left( q\rho\_0 e^{-\mathbf{x}/d} \right)^2 = mc^2 \left( -\frac{q\rho\_0 e^{-\mathbf{x}/d}}{mc^2} + \frac{1}{2} \left( \frac{q\rho\_0}{mc^2} e^{-\mathbf{x}/d} \right)^2 \right). \tag{94}$$

**Figure 6.** *The exponential potential of the field φ* ð Þ *x and Morse potential energy of interaction V x*ð Þ*.*

Schrödinger equation takes the form

$$\left(\frac{d^2\psi}{d\mathbf{x}^2} + \frac{2m}{\hbar^2}\left(E - mc^2\left(-\frac{q\rho\_0}{mc^2}e^{-\mathbf{x}/d} + \frac{1}{2}\left(\frac{q\rho\_0}{mc^2}e^{-\mathbf{x}/d}\right)^2\right)\right)\psi = 0. \tag{95}$$

For the binding energy in the ground state *W*<sup>0</sup> for *n* ¼ 0 of (100) we have

s

Because parameter *s* is determined to be positive (96) *s* ¼ *d=ƛ* � *n* � 1*=*2 ≥0 and

The interaction constant *qφ*0*=mc*<sup>2</sup> (97) does not have any limitation on the value and is not included in the expression for energy levels (100) and only determines the spatial properties of the wave function (99) through variable *ξ* (**Figures 9** and **10**). We emphasize that despite the fact that the potential energy for a stationary particle *V x*ð Þ has a depth of *mc*<sup>2</sup>*=*2, the maximum binding energy for a moving

*The dependence of the binding energy of the ground state mc*<sup>2</sup> � *<sup>W</sup>*<sup>0</sup> *(101) on the size d* <sup>≥</sup>*ƛ=*<sup>2</sup> *in units of mc*<sup>2</sup>*.*

*n*≤ *d=ƛ* � 1*=*2, then at *n* ¼ 0 the minimum value is *d* ¼ *ƛ=*2, which reflects the Heisenberg uncertainty principle. The maximum binding energy of a particle *mc*<sup>2</sup> � *<sup>W</sup>* is limited from above by a value *mc*<sup>2</sup> regardless of the nature and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>1</sup> � *<sup>ƛ</sup>*

2*d* � �<sup>2</sup>

(101)

*<sup>W</sup>*<sup>0</sup> <sup>¼</sup> *mc*<sup>2</sup> � *mc*<sup>2</sup>

*Equations of Relativistic and Quantum Mechanics (without Spin)*

magnitude of the interaction (**Figure 9**).

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

(**Figure 8**).

**Figure 9.**

**129**

**Figure 8.**

*Dependency of function* j j *ψ ξ*ð Þ <sup>2</sup> *at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> and n* <sup>¼</sup> <sup>0</sup>*.*

Following the procedure for solving Eq. (95) in [12], introducing a variable (taking values in the interval [0,∞]) and the notation

$$\xi = 2d \frac{q \rho\_0}{mc^2} \frac{\lambda}{d} e^{-\varkappa/d}, \qquad \mathfrak{s} = \frac{d}{\lambda} \sqrt{-\frac{2E}{mc^2}}, \qquad n = \frac{d}{\lambda} - \left(\mathfrak{s} + \frac{1}{2}\right), \tag{96}$$

We get

$$\frac{d^2\psi}{d\xi^2} + \frac{\mathbf{1}}{\xi}\frac{d\psi}{d\xi} + \left(-\frac{\mathbf{1}}{4} + \frac{n + \mathfrak{s} + \mathfrak{1}/2}{\xi} - \frac{\mathfrak{s}^2}{\xi^2}\right)\psi = \mathbf{0}.\tag{97}$$

Given the asymptotic behavior of function *ψ* for *ξ* ! ∞ and *ξ* ! 0, after substituting *<sup>ψ</sup>* <sup>¼</sup> *<sup>e</sup>*�*ξ=*<sup>2</sup>*ξ<sup>s</sup> w*ð Þ*ξ* we obtain

$$
\xi w' + (2\mathfrak{s} + \mathfrak{1} - \mathfrak{f})w' + nw = \mathfrak{0} \tag{98}
$$

equation of degenerate hypergeometric function (Kummer function).

$$w = {}\_{1}F\_{1}(-n, 2s+1, \xi) \tag{99}$$

A solution satisfying the finiteness condition for *ξ* ¼ 0 and when *ξ* ! ∞ the*w* turns to infinity no faster than a finite degree *ξ* is obtained for a generally positive *n*. Moreover, the Kummer function <sup>1</sup>*F*<sup>1</sup> reduces to a polynomial.

In accordance with (96) and (99), we obtain values for energy levels *W* (**Figure 7**)

$$\mathcal{W} = mc^2 \sqrt{\mathbf{1} - \left(\mathbf{1} - \frac{\lambda}{2d}(2n+1)\right)^2}.\tag{100}$$

**Figure 7.** *The dependence of the energy of particle W on the quantum number n(100) at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> in units of mc*<sup>2</sup>*.*

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

For the binding energy in the ground state *W*<sup>0</sup> for *n* ¼ 0 of (100) we have (**Figure 8**).

$$\mathcal{W}\_0 = mc^2 - mc^2 \sqrt{1 - \left(1 - \frac{\lambda}{2d}\right)^2} \tag{101}$$

Because parameter *s* is determined to be positive (96) *s* ¼ *d=ƛ* � *n* � 1*=*2 ≥0 and *n*≤ *d=ƛ* � 1*=*2, then at *n* ¼ 0 the minimum value is *d* ¼ *ƛ=*2, which reflects the Heisenberg uncertainty principle. The maximum binding energy of a particle *mc*<sup>2</sup> � *<sup>W</sup>* is limited from above by a value *mc*<sup>2</sup> regardless of the nature and magnitude of the interaction (**Figure 9**).

The interaction constant *qφ*0*=mc*<sup>2</sup> (97) does not have any limitation on the value and is not included in the expression for energy levels (100) and only determines the spatial properties of the wave function (99) through variable *ξ* (**Figures 9** and **10**).

We emphasize that despite the fact that the potential energy for a stationary particle *V x*ð Þ has a depth of *mc*<sup>2</sup>*=*2, the maximum binding energy for a moving

**Figure 8.**

Schrödinger equation takes the form

<sup>ℏ</sup><sup>2</sup> *<sup>E</sup>* � *mc*<sup>2</sup> � *<sup>q</sup>φ*<sup>0</sup>

(taking values in the interval [0,∞]) and the notation

1 *ξ dψ dξ* þ �

�*x=d*, *<sup>s</sup>* <sup>¼</sup> *<sup>d</sup>*

*w*ð Þ*ξ* we obtain

Moreover, the Kummer function <sup>1</sup>*F*<sup>1</sup> reduces to a polynomial.

s

*<sup>W</sup>* <sup>¼</sup> *mc*<sup>2</sup>

*ξw*00

*mc*<sup>2</sup> *<sup>e</sup>*

*ƛ*

1 4 þ

Given the asymptotic behavior of function *ψ* for *ξ* ! ∞ and *ξ* ! 0, after

equation of degenerate hypergeometric function (Kummer function).

A solution satisfying the finiteness condition for *ξ* ¼ 0 and when *ξ* ! ∞ the*w* turns to infinity no faster than a finite degree *ξ* is obtained for a generally positive *n*.

In accordance with (96) and (99), we obtain values for energy levels *W* (**Figure 7**)

<sup>1</sup> � <sup>1</sup> � *<sup>ƛ</sup>*

*The dependence of the energy of particle W on the quantum number n(100) at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> in units of mc*<sup>2</sup>*.*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup>*<sup>d</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �<sup>2</sup>

r

�*x=<sup>d</sup>* <sup>þ</sup> 1 2

Following the procedure for solving Eq. (95) in [12], introducing a variable

ffiffiffiffiffiffiffiffiffiffiffiffi � <sup>2</sup>*<sup>E</sup> mc*<sup>2</sup>

�*x=<sup>d</sup>* � �<sup>2</sup> � � � �

*qφ*<sup>0</sup> *mc*<sup>2</sup> *<sup>e</sup>*

, *<sup>n</sup>* <sup>¼</sup> *<sup>d</sup>*

*n* þ *s* þ 1*=*2 *<sup>ξ</sup>* � *<sup>s</sup>*

� �

*<sup>ƛ</sup>* � *<sup>s</sup>* <sup>þ</sup>

2 *ξ*2

þ ð Þ 2*s* þ 1 � *ξ w*<sup>0</sup> þ *nw* ¼ 0 (98)

*w* ¼ <sup>1</sup>*F*1ð Þ �*n*, 2*s* þ 1, *ξ* (99)

1 2 � �

*ψ* ¼ 0*:* (95)

, (96)

*ψ* ¼ 0*:* (97)

*:* (100)

2*m*

*<sup>ξ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>d</sup> <sup>q</sup>φ*<sup>0</sup> *mc*<sup>2</sup> *ƛ d e*

substituting *<sup>ψ</sup>* <sup>¼</sup> *<sup>e</sup>*�*ξ=*<sup>2</sup>*ξ<sup>s</sup>*

*d*2 *ψ <sup>d</sup>ξ*<sup>2</sup> <sup>þ</sup>

*d*2 *ψ dx*<sup>2</sup> þ

*Quantum Mechanics*

We get

**Figure 7.**

**128**

*The dependence of the binding energy of the ground state mc*<sup>2</sup> � *<sup>W</sup>*<sup>0</sup> *(101) on the size d* <sup>≥</sup>*ƛ=*<sup>2</sup> *in units of mc*<sup>2</sup>*.*

**Figure 9.** *Dependency of function* j j *ψ ξ*ð Þ <sup>2</sup> *at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> and n* <sup>¼</sup> <sup>0</sup>*.*

Separating the variables

*<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> ℏ*c*

<sup>H</sup>*<sup>n</sup>* <sup>¼</sup> *En*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

we obtain the energy levels

*<sup>Е</sup><sup>n</sup>*,*<sup>j</sup>* <sup>¼</sup> *mc*<sup>2</sup>

q

**131**

and introducing the notations [12]

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

<sup>2</sup> � *<sup>m</sup>*2*c*<sup>4</sup>

*s s*ð Þ¼ <sup>þ</sup> <sup>1</sup> *l l*ð Þþ <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*<sup>2</sup>

*d*2 Φ *<sup>d</sup>φ*<sup>2</sup> ¼ �*m*<sup>2</sup>

*d*2 *R dρ*<sup>2</sup> þ

*α*2

.

1 sin *θ*

*d <sup>d</sup><sup>θ</sup>* sin *<sup>θ</sup>*

> 2 *ρ dR*

for the molecular Kratzer potential in the form *<sup>U</sup>* <sup>¼</sup> *<sup>A</sup>*

<sup>H</sup>*<sup>n</sup>*,*<sup>j</sup>* ¼ �*mc*<sup>2</sup> *<sup>Z</sup>*<sup>2</sup>

*En*,*<sup>j</sup>* <sup>¼</sup> *mc*<sup>2</sup>

*<sup>E</sup>*<sup>0</sup> <sup>¼</sup> *mc*<sup>2</sup>

1*=*2 þ

For the ground state with the *l* ¼ 0 and *n* ¼ 1, we have

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>*=*<sup>4</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

*α*2 <sup>r</sup> <sup>q</sup> , *<sup>s</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 *n*<sup>r</sup> þ 1*=*2 þ

<sup>1</sup> � *<sup>Z</sup>*<sup>2</sup>

*n*<sup>r</sup> þ 1*=*2 þ

quantum number *n* ¼ *n*<sup>r</sup> þ *l* þ 1*=*2, *l*< *n n*ð Þ ¼ 1, 2, 3, … , we finally obtain

, *<sup>ρ</sup>* <sup>¼</sup> *mZe*<sup>2</sup>

<sup>2</sup>*mc*<sup>2</sup> ¼ � *mZ*<sup>2</sup>

ℏ2

*Equations of Relativistic and Quantum Mechanics (without Spin)*

2*r*

*<sup>N</sup>* <sup>¼</sup> *<sup>Z</sup><sup>α</sup> mc* ℏ 2*r*

1

(only the positive root is taken for *s*), for stationary states we have

*dY dθ* � �

*<sup>d</sup><sup>ρ</sup>* � *s s*ð Þ <sup>þ</sup> <sup>1</sup>

Φ,

*<sup>α</sup>*<sup>2</sup> ) *<sup>s</sup>* ¼ �1*=*<sup>2</sup> <sup>þ</sup>

<sup>2</sup>*N*<sup>2</sup> ¼ �*mc*<sup>2</sup>

� *<sup>m</sup>*<sup>2</sup> sin <sup>2</sup> *θ*

*<sup>ρ</sup>*<sup>2</sup> *<sup>R</sup>* ¼ � *<sup>n</sup>*<sup>r</sup>

where *m* ¼ �0, � 1, � 2, … , *l* ¼ 0, 1, 2, 3, … , j j *m* <*l* and *s* ¼ �1*=*2 þ

The solution of Eq. (88) formally coincides with the well-known Fuse solution

condition, that *n* � *s* � 1 ¼ *n*<sup>r</sup> must be a positive integer or zero. According to (87),

*α*2

� � <sup>q</sup> <sup>2</sup> ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where the radial quantum number *n*<sup>r</sup> ¼ 0, 1, 2, … . Introducing the principal

<sup>1</sup> � *<sup>Z</sup>*<sup>2</sup>

*α*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p � �<sup>2</sup>

1*=*2 þ

*<sup>n</sup>* <sup>þ</sup> *<sup>Z</sup>α*<sup>2</sup> *l*þ1*=*2þ

*α*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>*þ1*=*<sup>2</sup> <sup>2</sup>

vuuuut *:* (107)

*α*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>*=*<sup>4</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

*α*2 <sup>q</sup> (108)

<sup>þ</sup>*Z*2*α*<sup>2</sup>

� � q <sup>2</sup> vuuuut ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

*e*4 ℏ2

Ψ ¼ Φ*m*ð Þ *φ Yl*, *<sup>m</sup>*ð Þ*θ R nR*, *<sup>l</sup>*ð Þ*r* (103)

*l l*ð Þ þ 1 ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

*Y* ¼ �*l l*ð Þ þ 1 *Y*,

*<sup>r</sup>*<sup>2</sup> � *<sup>B</sup>*

*α*2

*α*2

*<sup>r</sup>* <sup>¼</sup> *<sup>Z</sup>*2*e*<sup>4</sup> 2*mc*<sup>2</sup> 1 *<sup>r</sup>*<sup>2</sup> � *Ze*<sup>2</sup> <sup>1</sup>

*<sup>ρ</sup>* � <sup>1</sup> 4 � �*R*, *α*2

(104)

(105)

*<sup>r</sup>* at the

(106)

*<sup>N</sup>* , *<sup>M</sup>*<sup>2</sup> <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>

*Z*2 *<sup>α</sup>*<sup>2</sup> <sup>1</sup> <sup>2</sup>*N*<sup>2</sup> ,

q

**Figure 10.**

*Dependency of function* j j *ψ ξ*ð Þ <sup>2</sup> *at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> and n* <sup>¼</sup> <sup>5</sup>*.*

**Figure 11.**

*The dependency of the potential energy of the interaction of Morse V x*ð Þ *and energy levels of the particle W* � *mc*<sup>2</sup> *at d* <sup>¼</sup> <sup>27</sup>*<sup>ƛ</sup> in units of mc*<sup>2</sup>*.*

particle in the ground state is equal to *mc*<sup>2</sup> (**Figure 11**), which is a relativistic effect of the particle's motion in the ground state - in the ground state, the particle not at rest.

#### **4.5 Problem of the hydrogen-like atom**

The motion of a charged particle in the Coulomb field can be described as a motion in the field of an atomic nucleus (without the spin and magnetic moment) with the potential energy –*Ze*<sup>2</sup> *=r*.

In spherical coordinates, Eq. (70) for the wave function takes the form

$$\begin{split} &\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial \Psi}{\partial r}\right) + \frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \Psi}{\partial \theta}\right) + \\ &\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}\Psi}{\partial\phi^{2}} + \frac{1}{\hbar^{2}c^{2}}\left(E^{2} - \left(mc^{2} - \frac{Ze^{2}}{r}\right)^{2}\right)\Psi = 0. \end{split} \tag{102}$$

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

Separating the variables

$$\Psi = \Phi\_m(\rho) Y\_{l,\ \ m}(\theta) R\_{\ \ n\_{\text{R}},\ \ l}(r) \tag{103}$$

and introducing the notations [12]

$$\begin{aligned} a &= \frac{e^2}{\hbar c}, & \rho &= \frac{mZ e^2}{\hbar^2} \frac{2r}{N} = Za \frac{mc}{\hbar} \frac{2r}{N}, & M^2 &= \hbar^2 l(l+1), \\ \mathcal{H}\_n &= \frac{E\_n^2 - m^2 c^4}{2mc^2} = -\frac{mZ^2 e^4}{\hbar^2} \frac{1}{2N^2} = -mc^2 Z^2 a^2 \frac{1}{2N^2}, & \text{(104)} \\ s(s+1) &= l(l+1) + Z^2 a^2 \quad \Rightarrow \quad s = -1/2 + \sqrt{\left(l + 1/2\right)^2 + Z^2 a^2} \end{aligned} \tag{104}$$

(only the positive root is taken for *s*), for stationary states we have

$$\begin{aligned} \frac{d^2 \Phi}{d\rho^2} &= -m^2 \Phi, \\ \frac{1}{\sin \theta} \frac{d}{d\theta} \left( \sin \theta \frac{dY}{d\theta} \right) - \frac{m^2}{\sin^2 \theta} Y &= -l(l+1)Y, \\ \frac{d^2 R}{d\rho^2} + \frac{2}{\rho} \frac{dR}{d\rho} - \frac{s(s+1)}{\rho^2} R &= -\left(\frac{n\_r}{\rho} - \frac{1}{4}\right)R, \end{aligned} \tag{105}$$

where *m* ¼ �0, � 1, � 2, … , *l* ¼ 0, 1, 2, 3, … , j j *m* <*l* and *s* ¼ �1*=*2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>l</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *α*2 q .

The solution of Eq. (88) formally coincides with the well-known Fuse solution for the molecular Kratzer potential in the form *<sup>U</sup>* <sup>¼</sup> *<sup>A</sup> <sup>r</sup>*<sup>2</sup> � *<sup>B</sup> <sup>r</sup>* <sup>¼</sup> *<sup>Z</sup>*2*e*<sup>4</sup> 2*mc*<sup>2</sup> 1 *<sup>r</sup>*<sup>2</sup> � *Ze*<sup>2</sup> <sup>1</sup> *<sup>r</sup>* at the condition, that *n* � *s* � 1 ¼ *n*<sup>r</sup> must be a positive integer or zero. According to (87), we obtain the energy levels

$$\begin{aligned} \mathcal{H}\_{n,j} &= -mc^2 \frac{Z^2 a^2}{2\left(n\_\mathrm{r} + 1/2 + \sqrt{(l+1/2)^2 + Z^2 a^2}\right)^2}, \\ E\_{n,j} &= mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n\_\mathrm{r} + 1/2 + \sqrt{(l+1/2)^2 + Z^2 a^2}\right)^2}}, \end{aligned} \tag{106}$$

where the radial quantum number *n*<sup>r</sup> ¼ 0, 1, 2, … . Introducing the principal quantum number *n* ¼ *n*<sup>r</sup> þ *l* þ 1*=*2, *l*< *n n*ð Þ ¼ 1, 2, 3, … , we finally obtain

$$E\_{nj} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n + \frac{z\_{\alpha^2}}{l + 1/2 + \sqrt{\left(l + 1/2\right)^2 + Z^2 a^2}}\right)^2}} \tag{107}$$

For the ground state with the *l* ¼ 0 and *n* ¼ 1, we have

$$E\_0 = \frac{mc^2}{\sqrt{1/2 + \sqrt{1/4 + Z^2a^2}}}, \quad s = \frac{Z^2a^2}{1/2 + \sqrt{1/4 + Z^2a^2}}\tag{108}$$

particle in the ground state is equal to *mc*<sup>2</sup> (**Figure 11**), which is a relativistic effect of the particle's motion in the ground state - in the ground state, the particle

*The dependency of the potential energy of the interaction of Morse V x*ð Þ *and energy levels of the particle W* �

The motion of a charged particle in the Coulomb field can be described as a motion in the field of an atomic nucleus (without the spin and magnetic moment)

> *∂ <sup>∂</sup><sup>θ</sup>* sin *<sup>θ</sup>*

∂Ψ *∂θ* � �

*r*

*<sup>E</sup>*<sup>2</sup> � *mc*<sup>2</sup> � *Ze*<sup>2</sup>

� �<sup>2</sup> !

þ

Ψ ¼ 0*:*

(102)

In spherical coordinates, Eq. (70) for the wave function takes the form

1 *r*<sup>2</sup> sin *θ*

*=r*.

þ

1 ℏ2 *c*2

not at rest.

**130**

**Figure 11.**

**Figure 10.**

*Quantum Mechanics*

*mc*<sup>2</sup> *at d* <sup>¼</sup> <sup>27</sup>*<sup>ƛ</sup> in units of mc*<sup>2</sup>*.*

**4.5 Problem of the hydrogen-like atom**

*Dependency of function* j j *ψ ξ*ð Þ <sup>2</sup> *at d* <sup>¼</sup> <sup>10</sup>*<sup>ƛ</sup> and n* <sup>¼</sup> <sup>5</sup>*.*

with the potential energy –*Ze*<sup>2</sup>

1 *r*2 *∂ ∂r r* <sup>2</sup> ∂Ψ *∂r* � �

1 *r*<sup>2</sup> sin <sup>2</sup> *θ ∂*2 Ψ *∂φ*<sup>2</sup> þ

without any restrictions for the value of *Z*. In this case, 1–*s*> 0 and there is no fall of the particle on the center [8], and the probability of finding the particle at the center (in the nucleus) is always equal to zero.

In this case, the obtained fine splitting is in no way connected with the spin-orbit interaction and is due to the relativistic dependence of the mass on the orbital and radial velocity of motion, which results to the splitting of the levels.

#### **4.6 Dirac equations**

In the standard representation, the Dirac equations in compact notation for a particle have the form [21].

$$
\begin{aligned}
\hat{\boldsymbol{\varepsilon}}\,\boldsymbol{\phi} - \boldsymbol{\sigma} \cdot \hat{\mathbf{p}}\,\boldsymbol{\chi} &= m\boldsymbol{\varepsilon}\boldsymbol{\phi}, \\
\hat{\boldsymbol{\varepsilon}}\boldsymbol{\chi} + \boldsymbol{\sigma} \cdot \hat{\mathbf{p}}\,\boldsymbol{\phi} &= m\boldsymbol{\varepsilon}\boldsymbol{\chi}.
\end{aligned}
\tag{109}
$$

After substituting (96) into (95), we obtain

*c* 1 *r* � �*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>

*Equations of Relativistic and Quantum Mechanics (without Spin)*

*j* ¼ j j *l* � 1*=*2 , *j*

8 ><

>:

*<sup>ρ</sup><sup>γ</sup> <sup>Q</sup>*<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>2</sup> ð Þ,

*<sup>ρ</sup><sup>γ</sup> <sup>Q</sup>*<sup>1</sup> � *<sup>Q</sup>*<sup>2</sup> ð Þ,

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>χ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

<sup>0</sup> <sup>þ</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> *<sup>Z</sup>αmc=<sup>λ</sup>* � *<sup>ρ</sup> <sup>Q</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>χ</sup>* <sup>þ</sup> *<sup>Z</sup>αε=<sup>λ</sup> <sup>Q</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:* (118)

*γ* þ *Zαmc=λ*

<sup>0</sup> � ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>Z</sup>αmc=<sup>λ</sup> <sup>Q</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (121)

*<sup>F</sup>*ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>Z</sup>αmc=λ*, 2*<sup>γ</sup>* <sup>þ</sup> 1, *<sup>ρ</sup>* , (122)

*<sup>λ</sup>* ¼ �*n*r*:* (123)

<sup>0</sup> � ð Þ *γ* � *Zαmc=λ Q*<sup>1</sup> ¼ 0,

*α*2

, *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>

ℏ*c*

*:* (119)

*Q*1*:* (120)

�*ρ=*2

�*ρ=*2

, *γ* ¼

Substituting (116) into the Eq. (117), for the sum and difference of the equations

<sup>0</sup> þ ð Þ *γ* � *Zαmc=λ Q*<sup>1</sup> þ ð Þ *χ* � *Zαε=λ Q*<sup>2</sup> ¼ 0,

*<sup>γ</sup>*<sup>2</sup> � ð Þ *<sup>Z</sup>αmc=<sup>λ</sup>* <sup>2</sup> <sup>¼</sup> *<sup>χ</sup>*<sup>2</sup> � ð Þ *<sup>Z</sup>αε=<sup>λ</sup>* <sup>2</sup>

*<sup>χ</sup>* � *<sup>Z</sup>αε=<sup>λ</sup> <sup>Q</sup>*<sup>1</sup> ¼ � *<sup>χ</sup>* <sup>þ</sup> *<sup>Z</sup>αε=<sup>λ</sup>*

Forming equations of the second order and solving with respect to *Q*<sup>1</sup> and *Q*2,

where *F*ð Þ *α*, *β*, *z* is the degenerate hypergeometric function and A is the normalization constant of the wave function. The function *F*ð Þ *α*, *β*, *z* reduces to a polynomial, if the parameter α is equal to an integer negative number or zero. Therefore,

*<sup>γ</sup>* � *<sup>Z</sup>αmc*

Close to *ρ* ¼ 0, the system of equations always has a solution, because

*χ* ¼ �1, *l* ¼ 0 *χ* ¼ �ð Þ *j* þ 1*=*2 *:*

max ¼ *l*max þ 1*=*2

(115)

(116)

*:* (117)

*Ze*<sup>2</sup> *c* 1 *r* � �*<sup>f</sup>* <sup>¼</sup> 0,

Let us represent the functions f and *g* in the form

*<sup>f</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *mc* <sup>þ</sup> *<sup>ε</sup>* <sup>p</sup> *<sup>e</sup>*

*<sup>g</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *mc* � *<sup>ε</sup>* <sup>p</sup> *<sup>e</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *mc*

*<sup>Q</sup>*<sup>2</sup> ¼ � *<sup>γ</sup>* � *<sup>Z</sup>αmc=<sup>λ</sup>*

<sup>00</sup> þ ð Þ 2*γ* þ 1 � *ρ Q*<sup>1</sup>

<sup>00</sup> þ ð Þ 2*γ* þ 1 � *ρ Q*<sup>2</sup>

*<sup>Q</sup>*<sup>2</sup> ¼ �*<sup>A</sup> <sup>γ</sup>* � *<sup>Z</sup>αmc=<sup>λ</sup>*

finite solutions for the functions f and *g* are

With allowance for (121), the solution of these equations is

*Q*<sup>1</sup> ¼ *AF*ð Þ *γ* � *Zαmc=λ*, 2*γ* þ 1, *ρ* ,

*χ* � *Zαε=λ*

q

<sup>2</sup> � *<sup>ε</sup>*<sup>2</sup>

*<sup>f</sup>* � *<sup>ε</sup>* <sup>þ</sup> *mc* � *Ze*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*g* þ *ε* � *mc* þ

*ρ* ¼ 2*λr=*ℏ, *λ* ¼

*ρQ*<sup>1</sup>

*ρQ*<sup>2</sup>

*ρQ*<sup>1</sup>

*ρQ*<sup>2</sup>

*f* 0 þ *χ r*

where

we have

Then

we obtain

**133**

8 >>><

>>>:

*<sup>g</sup>*<sup>0</sup> � *<sup>χ</sup> r*

In addition, for the particle in an external field they can be represented in theorm

$$\begin{aligned} \hat{\boldsymbol{\varepsilon}} \, \boldsymbol{\phi} - \boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \, \boldsymbol{\chi} &= \left( \boldsymbol{m} \boldsymbol{c} + \frac{q}{c} \boldsymbol{\rho} \right) \boldsymbol{\phi} + \frac{q}{c} \boldsymbol{\sigma} \cdot \mathbf{A} \boldsymbol{\chi}, \\ \hat{\boldsymbol{\varepsilon}} \boldsymbol{\chi} + \boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \, \boldsymbol{\phi} &= \left( \boldsymbol{m} \boldsymbol{c} + \frac{q}{c} \boldsymbol{\rho} \right) \boldsymbol{\chi} - \frac{q}{c} \boldsymbol{\sigma} \cdot \mathbf{A} \boldsymbol{\phi}. \end{aligned} \tag{110}$$

By writing the wave equations for the wave functions, we obtain

$$\begin{split} \left(\frac{\partial^2}{\partial \mathbf{r}^2} - \frac{\partial^2}{\partial \mathbf{r}^2}\right) \phi &= -\frac{\left(mc^2 + q\rho\right)^2 - (q\mathbf{A})^2}{\hbar^2 c^2} \phi - \frac{q}{\hbar c} \boldsymbol{\sigma} \cdot (\mathbf{B} - i\mathbf{E})\boldsymbol{\chi}, \\ \left(\frac{\partial^2}{\partial \mathbf{r}^2} - \frac{\partial^2}{\partial \mathbf{r}^2}\right) \boldsymbol{\chi} &= -\frac{\left(mc^2 + q\rho\right)^2 - (q\mathbf{A})^2}{\hbar^2 c^2} \boldsymbol{\chi} + \frac{q}{\hbar c} \boldsymbol{\sigma} \cdot (\mathbf{B} - i\mathbf{E})\phi, \end{split} \tag{111}$$

where we used the properties of the Pauli matrices. It is easy to verify that the functions *<sup>ϕ</sup>* and *<sup>χ</sup>* differ only in the constant phase *<sup>ϕ</sup>* <sup>¼</sup> *<sup>χ</sup>e*�*i<sup>π</sup>* ¼ �*<sup>χ</sup>* and the equations can be completely separated and only one equation can be used, bearing in mind that (111) can be of a variable sign

$$\left(\frac{\partial^2}{\partial \mathbf{r}^2} - \frac{\partial^2}{\partial \mathbf{r}^2}\right)\Psi = -\frac{\left(mc^2 + q\rho\right)^2 - \left(q\mathbf{A}\right)^2}{\hbar^2 c^2} \Psi \pm \frac{q}{\hbar c} \boldsymbol{\sigma} \cdot (\mathbf{B} - i\mathbf{E})\Psi. \tag{112}$$

In the case of a stationary state, the standard representation of the wave Eq. (110) has the form

$$\begin{aligned} \left(\boldsymbol{\varepsilon} - m\boldsymbol{\varepsilon} - \frac{q}{c}\boldsymbol{\varrho}\right) \boldsymbol{\upvarrho} &= \boldsymbol{\sigma} \cdot \left(\mathbf{p} + \frac{q}{c}\mathbf{A}\right) \boldsymbol{\upchi}, \\ \left(\boldsymbol{\varepsilon} + m\boldsymbol{\varepsilon} + \frac{q}{c}\boldsymbol{\uprho}\right) \boldsymbol{\upchi} &= \boldsymbol{\sigma} \cdot \left(\mathbf{p} - \frac{q}{c}\mathbf{A}\right) \boldsymbol{\uprho}. \end{aligned} \tag{113}$$

#### **4.7 Dirac equations solution for a hydrogen-like atom**

For a charge in a potential field with the central symmetry [23], we have

$$
\begin{pmatrix} \varrho \\\\ \chi \end{pmatrix} = \begin{pmatrix} \frac{f(r)}{r} \Omega\_{jlm} \\\\ (-\mathbf{1})^{1+l-l'} \frac{\mathbf{g}(r)}{r} \Omega\_{jl'm} \end{pmatrix} . \tag{114}
$$

*Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

After substituting (96) into (95), we obtain

$$\begin{cases} f' + \frac{\chi}{r} f - \left( \varepsilon + mc - \frac{Ze^2}{c} \frac{\mathbf{1}}{r} \right) \mathbf{g} = \mathbf{0} \\\ \mathbf{g'} - \frac{\chi}{r} \mathbf{g} + \left( \varepsilon - mc + \frac{Ze^2}{c} \frac{\mathbf{1}}{r} \right) \mathbf{f} = \mathbf{0}, \end{cases} \qquad \begin{cases} j = |l \pm 1/2|, \qquad j\_{\text{max}} = l\_{\text{max}} + 1/2 \\\ \chi = -\mathbf{1}, \qquad l = \mathbf{0} \\\ \chi = \pm (|j + 1/2). \end{cases} \tag{115}$$

Let us represent the functions f and *g* in the form

$$\begin{aligned} f &= \sqrt{mc + \epsilon}e^{-\rho/2}\rho^{\mathbb{Y}}(\mathbf{Q}\_1 + \mathbf{Q}\_2), \\ \mathbf{g} &= \sqrt{mc - \epsilon}e^{-\rho/2}\rho^{\mathbb{Y}}(\mathbf{Q}\_1 - \mathbf{Q}\_2), \end{aligned} \tag{116}$$

where

without any restrictions for the value of *Z*. In this case, 1–*s*> 0 and there is no fall of the particle on the center [8], and the probability of finding the particle at the

In this case, the obtained fine splitting is in no way connected with the spin-orbit interaction and is due to the relativistic dependence of the mass on the orbital and

In the standard representation, the Dirac equations in compact notation for a

^*εϕ* � **σ** � **p**^ χ ¼ *mcϕ*,

In addition, for the particle in an external field they can be represented in theorm

*c φ* � �

*c φ* � �

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

where we used the properties of the Pauli matrices. It is easy to verify that the functions *<sup>ϕ</sup>* and *<sup>χ</sup>* differ only in the constant phase *<sup>ϕ</sup>* <sup>¼</sup> *<sup>χ</sup>e*�*i<sup>π</sup>* ¼ �*<sup>χ</sup>* and the equations can be completely separated and only one equation can be used, bearing in

<sup>2</sup> � ð Þ *<sup>q</sup>***<sup>A</sup>** <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>Ψ</sup> � *<sup>q</sup>*

<sup>φ</sup> <sup>¼</sup> **<sup>σ</sup>** � **<sup>p</sup>** <sup>þ</sup> *<sup>q</sup>*

<sup>χ</sup> <sup>¼</sup> **<sup>σ</sup>** � **<sup>p</sup>** � *<sup>q</sup>*

*<sup>c</sup>*<sup>2</sup> *<sup>ϕ</sup>* � *<sup>q</sup>*

*<sup>c</sup>*<sup>2</sup> *<sup>χ</sup>* <sup>þ</sup> *<sup>q</sup>*

*<sup>ϕ</sup>* <sup>þ</sup> *<sup>q</sup> c* **σ** � **A**χ,

<sup>χ</sup> � *<sup>q</sup> c* **σ** � **A***ϕ:*

ℏ*c*

ℏ*c*

ℏ*c*

*c* **A** � �

*c* **A** � � χ,

φ*:*

1

**σ** � ð Þ **B** � *i***E** *χ*,

**σ** � ð Þ **B** � *i***E** *ϕ*,

**σ** � ð Þ **B** � *i***E** Ψ*:* (112)

CCA*:* (114)

^*ε*<sup>χ</sup> <sup>þ</sup> **<sup>σ</sup>** � **<sup>p</sup>**^ *<sup>ϕ</sup>* <sup>¼</sup> *mc*χ*:* (109)

(110)

(111)

(113)

radial velocity of motion, which results to the splitting of the levels.

^*εϕ* � **<sup>σ</sup>** � **<sup>p</sup>**^ <sup>χ</sup> <sup>¼</sup> *mc* <sup>þ</sup> *<sup>q</sup>*

^*ε*<sup>χ</sup> <sup>þ</sup> **<sup>σ</sup>** � **<sup>p</sup>**^ *<sup>ϕ</sup>* <sup>¼</sup> *mc* <sup>þ</sup> *<sup>q</sup>*

*<sup>ϕ</sup>* ¼ � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

*<sup>χ</sup>* ¼ � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

<sup>Ψ</sup> ¼ � *mc* ð Þ <sup>2</sup> <sup>þ</sup> *<sup>q</sup><sup>φ</sup>*

*<sup>ε</sup>* � *mc* � *<sup>q</sup>*

*<sup>ε</sup>* <sup>þ</sup> *mc* <sup>þ</sup> *<sup>q</sup>*

**4.7 Dirac equations solution for a hydrogen-like atom**

*φ χ* � �

¼

� �

� �

mind that (111) can be of a variable sign

By writing the wave equations for the wave functions, we obtain

ℏ2

ℏ2

ℏ2

*c φ*

*c φ*

0

BB@

In the case of a stationary state, the standard representation of the wave

For a charge in a potential field with the central symmetry [23], we have

*f r*ð Þ *r*

ð Þ �<sup>1</sup> <sup>1</sup>þ*l*�*<sup>l</sup>*

Ω*jlm*

<sup>0</sup> *g r*ð Þ *r* Ω *jl*<sup>0</sup> *m*

center (in the nucleus) is always equal to zero.

**4.6 Dirac equations**

*Quantum Mechanics*

particle have the form [21].

*∂*2 *<sup>∂</sup>τ*<sup>2</sup> � *<sup>∂</sup>*<sup>2</sup> *∂***r**<sup>2</sup> � �

*∂*2 *<sup>∂</sup>τ*<sup>2</sup> � *<sup>∂</sup>*<sup>2</sup> *∂***r**<sup>2</sup> � �

*∂*2 *<sup>∂</sup>τ*<sup>2</sup> � *<sup>∂</sup>*<sup>2</sup> *∂***r**<sup>2</sup> � �

Eq. (110) has the form

**132**

$$\rho = 2\lambda r/\hbar, \qquad \lambda = \sqrt{\left(mc\right)^2 - \epsilon^2}, \qquad \gamma = \sqrt{\chi^2 + Z^2 a^2}, \qquad a = \frac{\epsilon^2}{\hbar c}.\tag{117}$$

Substituting (116) into the Eq. (117), for the sum and difference of the equations we have

$$\begin{aligned} \rho \mathbf{Q}\_1' + (\boldsymbol{\chi} - \mathbf{Z}\boldsymbol{\alpha}\mathbf{m}\boldsymbol{\epsilon}/\lambda)\mathbf{Q}\_1 + (\boldsymbol{\chi} - \mathbf{Z}\boldsymbol{\alpha}\boldsymbol{\epsilon}/\lambda)\mathbf{Q}\_2 &= \mathbf{0}, \\ \rho \mathbf{Q}\_2' + (\boldsymbol{\chi} + \mathbf{Z}\boldsymbol{\alpha}\mathbf{m}\boldsymbol{\epsilon}/\lambda - \rho)\mathbf{Q}\_2 + (\boldsymbol{\chi} + \mathbf{Z}\boldsymbol{\alpha}\boldsymbol{\epsilon}/\lambda)\mathbf{Q}\_1 &= \mathbf{0}. \end{aligned} \tag{118}$$

Close to *ρ* ¼ 0, the system of equations always has a solution, because

$$
\chi^2 - (\mathsf{Zamc}/\lambda)^2 = \chi^2 - (\mathsf{Zam}/\lambda)^2. \tag{119}
$$

Then

$$\mathbf{Q}\_2 = -\frac{\chi - Z\alpha mc/\lambda}{\chi - Z\alpha c/\lambda} \mathbf{Q}\_1 = -\frac{\chi + Z\alpha c/\lambda}{\chi + Z\alpha mc/\lambda} \mathbf{Q}\_1. \tag{120}$$

Forming equations of the second order and solving with respect to *Q*<sup>1</sup> and *Q*2, we obtain

$$\begin{aligned} \rho \mathbf{Q}\_1'' + (2\gamma + \mathbf{1} - \rho) \mathbf{Q}\_1' - (\mathbf{y} - \mathbf{Z}\rho\mathbf{m}c/\lambda) \mathbf{Q}\_1 &= \mathbf{0}, \\ \rho \mathbf{Q}\_2'' + (2\gamma + \mathbf{1} - \rho) \mathbf{Q}\_2' - (\mathbf{y} + \mathbf{1} - \mathbf{Z}\rho\mathbf{m}c/\lambda) \mathbf{Q}\_2 &= \mathbf{0}. \end{aligned} \tag{121}$$

With allowance for (121), the solution of these equations is

$$\begin{aligned} \mathbf{Q}\_1 &= AF(\mathbf{y} - \mathbf{Z}amc/\lambda, \ 2\gamma + 1, \ \rho), \\ \mathbf{Q}\_2 &= -A \frac{\chi - \mathbf{Z}amc/\lambda}{\chi - \mathbf{Z}ae/\lambda} F(\mathbf{y} + \mathbf{1} - \mathbf{Z}amc/\lambda, \ 2\gamma + 1, \ \rho), \end{aligned} \tag{122}$$

where *F*ð Þ *α*, *β*, *z* is the degenerate hypergeometric function and A is the normalization constant of the wave function. The function *F*ð Þ *α*, *β*, *z* reduces to a polynomial, if the parameter α is equal to an integer negative number or zero. Therefore, finite solutions for the functions f and *g* are

$$
\eta - \frac{Zamc}{\lambda} = -n\_{\text{r.}}\tag{123}
$$

From expressions (117), we obtain

$$f = A\sqrt{mc + \epsilon\epsilon}e^{-\rho/2}\rho^{r-1}\left(F(-n\_{\rm r}, \quad 2\gamma + 1, \quad \rho) + \frac{n\_{\rm r}}{\chi - Zae/\lambda}F(1 - n\_{\rm r}, \quad 2\gamma + 1, \quad \rho)\right),$$

$$g = A\sqrt{mc - \epsilon\epsilon}e^{-\rho/2}\rho^{r-1}\left(F(-n\_{\rm r}, \quad 2\gamma + 1, \quad \rho) - \frac{n\_{\rm r}}{\chi - Zae/\lambda}F(1 - n\_{\rm r}, \quad 2\gamma + 1, \quad \rho)\right),\tag{124}$$

where *nr* ¼ 0, 1, 2, … is the radial quantum number. For the energy levels, we obtain from the condition (117)

$$\frac{\varepsilon\_{p,\varkappa}}{mc} = \sqrt{1 - \frac{Z^2 a^2}{\left(n\_\mathrm{r} + \sqrt{\chi^2 + Z^2 a^2}\right)^2}}\tag{125}$$

where the last term is the expression for the spin-orbit interaction energy. Thus, to obtain the true value of the energy levels of the hydrogen atom, it is necessary to add the energy of the spin-orbit interaction in formula (126) in the form (130). This is completely justified, because such an interaction was not initially included in

The principle of invariance is generalized and the corresponding representation of the generalized momentum of the system is proposed; the equations of relativistic and quantum mechanics are proposed, which are devoid of the above-mentioned shortcomings and contradictions. The equations have solutions for any values of the interaction constant of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. The equations

Based on the parametric representation of the action and the canonical equations, the corresponding relativistic mechanics based on the canonical Lagrangian is constructed and the equations of motion and expression are derived for the force

The matrix representation of equations of the characteristics for the action function and the wave function results in the Dirac equation with the correct enabling of the interaction. In this form, the solutions of the Dirac equations are not restricted by the value of the interaction constant and have a spinor representation by scalar solutions of the equations for the action function and the wave function. The analysis of the solutions shows the full compliance with the principles of the relativistic and quantum mechanics, and the solutions are devoid of any restrictions

The theory of spin fields and equations for spin systems will be described in

Institute for Physical Research, Armenian National Academy, Ashtarak, Armenia

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Eq. (115) and was not reflected in the final result.

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

are applicable for different types of particles and interactions.

acting on the charge moving in an external electromagnetic field.

on the nature and magnitude of the interactions.

\*Address all correspondence to: vm@ipr.sci.am

provided the original work is properly cited.

**5. Conclusion**

subsequent works.

**Author details**

**135**

Vahram Mekhitarian

and taking into account the obtained values of χ, we finally have

$$E\_{n,j} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n\_\mathrm{r} + \sqrt{\left(\left(\frac{j}{j} + 1/2\right)^2 + Z^2 a^2\right)}\right)^2}},\tag{126}$$

$$mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n + \frac{Z\_a a^2}{j + 1/2 + \sqrt{\left(\left(\frac{j}{j} + 1/2\right)^2 + Z^2 a^2\right)}}\right)^2}},\tag{127}$$

where the principal quantum number *n* ¼ *nr* þ *j* þ 1*=*2. Besides *j* ¼ *n* � 1*=*2, all other levels with *j*<*n* � 1*=*2 are degenerated twice in the orbital angular momentum *l* ¼ j j *j* � 1*=*2 . The ground state energy for *n* ¼ 1 and *j* ¼ 1*=*2 is

$$E\_0 = \frac{mc^2}{\sqrt{1 + Z^2 a^2}}\tag{127}$$

without any limitations for the value of *<sup>Z</sup>*. In this case *<sup>γ</sup>* � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *α*2 p > 0, and no falling of particle on the center is observed, and the probability to find the particle in the center (in the nucleus) is always equal to zero.

In the resulting formula (126), the order of sequence of the fine splitting levels is inverse relative to the order of sequence in the well-known Sommerfeld-Dirac formula. If to compare the expansions in a series in the degree of the fine-structure constant of two formulas

$$\frac{E\_n}{mc^2} = \frac{1}{2} \left( 1 - \frac{1}{n^2} \right) a^2 + \left( \frac{1}{8} + \frac{3}{8n^4} - \frac{1}{2n^3(\ j + 1/2)} \right) a^4,\\ \frac{\Delta E\_{3/2, 1/2}}{mc^2} = \frac{a^4}{32},\tag{128}$$

$$\frac{E\_n}{mc^2} = \frac{1}{2} \left( 1 - \frac{1}{n^2} \right) a^2 + \left( -\frac{3}{8} - \frac{1}{8n^4} + \frac{1}{2n^3(\frac{1}{2} + 1/2)} \right) a^4,\\ \frac{\Delta E\_{3/2, 1/2}}{mc^2} = -\frac{a^4}{32}, \quad \text{(129)}$$

then the difference will be equal to

$$\frac{\Delta E\_n}{mc^2} = \frac{a^4}{2} - \frac{a^4}{2n^4} - \frac{a^4}{n^3((j+1/2))},\tag{130}$$

where the last term is the expression for the spin-orbit interaction energy. Thus, to obtain the true value of the energy levels of the hydrogen atom, it is necessary to add the energy of the spin-orbit interaction in formula (126) in the form (130). This is completely justified, because such an interaction was not initially included in Eq. (115) and was not reflected in the final result.

#### **5. Conclusion**

From expressions (117), we obtain

*ργ*�<sup>1</sup> *<sup>F</sup>* �*n*<sup>r</sup> <sup>ð</sup> , 2*<sup>γ</sup>* <sup>þ</sup> 1, *<sup>ρ</sup>*Þ þ *nr*

*ργ*�<sup>1</sup> *<sup>F</sup>* �*n*<sup>r</sup> <sup>ð</sup> , 2*<sup>γ</sup>* <sup>þ</sup> 1, *<sup>ρ</sup>*Þ � *nr*

where *nr* ¼ 0, 1, 2, … is the radial quantum number. For the energy levels, we

<sup>1</sup> � *<sup>Z</sup>*<sup>2</sup>

<sup>1</sup> � *<sup>Z</sup>*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p � �<sup>2</sup> vuuuut ,

*<sup>E</sup>*<sup>0</sup> <sup>¼</sup> *mc*<sup>2</sup>

without any limitations for the value of *<sup>Z</sup>*. In this case *<sup>γ</sup>* � <sup>1</sup> ¼ �<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

bility to find the particle in the center (in the nucleus) is always equal to zero.

inverse relative to the order of sequence in the well-known Sommerfeld-Dirac formula. If to compare the expansions in a series in the degree of the fine-structure

<sup>8</sup>*n*<sup>4</sup> � <sup>1</sup>

� �

<sup>2</sup> � *<sup>α</sup>*<sup>4</sup>

� �

*α*2

where the principal quantum number *n* ¼ *nr* þ *j* þ 1*=*2. Besides *j* ¼ *n* � 1*=*2, all other levels with *j*<*n* � 1*=*2 are degenerated twice in the orbital angular momentum

*n*<sup>r</sup> þ

*<sup>n</sup>* <sup>þ</sup> *<sup>Z</sup>α*<sup>2</sup> *j*þ1*=*2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*α*2

*α*2

*<sup>n</sup>*<sup>r</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>χ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>j</sup>*þ1*=*<sup>2</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *α*2

> 0, and no falling of particle on the center is observed, and the proba-

In the resulting formula (126), the order of sequence of the fine splitting levels is

2*n*<sup>3</sup>ð Þ *j* þ 1*=*2

1 2*n*<sup>3</sup>ð Þ *j* þ 1*=*2

<sup>2</sup>*n*<sup>4</sup> � *<sup>α</sup>*<sup>4</sup>

*α*4,

*α*4,

*ΔE*3*=*2,1*=*<sup>2</sup> *mc*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>4</sup>

*ΔE*3*=*2,1*=*<sup>2</sup>

*<sup>n</sup>*<sup>3</sup>ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> , (130)

*mc*<sup>2</sup> ¼ � *<sup>α</sup>*<sup>4</sup>

<sup>32</sup> , (128)

<sup>32</sup> , (129)

<sup>þ</sup>*Z*2*α*<sup>2</sup>

*α*2

� � q <sup>2</sup> vuuuut <sup>¼</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup>

*α*2

<sup>p</sup> (127)

� � p <sup>2</sup>

*χ* � *Zαε=λ*

� �

*χ* � *Zαε=λ*

� �

*F* 1 � *n*<sup>r</sup> ð Þ , 2*γ* þ 1, *ρ*

*F* 1 � *n*<sup>r</sup> ð Þ , 2*γ* þ 1, *ρ*

,

,

(124)

(125)

(126)

�*ρ=*2

�*ρ=*2

obtain from the condition (117)

*εp*,*<sup>χ</sup> mc* ¼

*En*,*<sup>j</sup>* <sup>¼</sup> *mc*<sup>2</sup>

*mc*<sup>2</sup>

vuuut

<sup>1</sup> � *<sup>Z</sup>*<sup>2</sup>

*l* ¼ j j *j* � 1*=*2 . The ground state energy for *n* ¼ 1 and *j* ¼ 1*=*2 is

and taking into account the obtained values of χ, we finally have

*<sup>f</sup>* <sup>¼</sup> *<sup>A</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *mc* <sup>þ</sup> *<sup>ε</sup>* <sup>p</sup> *<sup>e</sup>*

*Quantum Mechanics*

*<sup>g</sup>* <sup>¼</sup> *<sup>A</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *mc* � *<sup>ε</sup>* <sup>p</sup> *<sup>e</sup>*

> <sup>1</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>2</sup> *α*2

> > *Еn mc*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

*Еn mc*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

**134**

constant of two formulas

<sup>1</sup> � <sup>1</sup> *n*2 � �

<sup>1</sup> � <sup>1</sup> *n*2 � � *<sup>α</sup>*<sup>2</sup> <sup>þ</sup>

*<sup>α</sup>*<sup>2</sup> þ �

then the difference will be equal to

1 8 þ

3 <sup>8</sup> � <sup>1</sup> 8*n*<sup>4</sup> þ

*ΔЕ<sup>n</sup> mc*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>4</sup>

3

p

The principle of invariance is generalized and the corresponding representation of the generalized momentum of the system is proposed; the equations of relativistic and quantum mechanics are proposed, which are devoid of the above-mentioned shortcomings and contradictions. The equations have solutions for any values of the interaction constant of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. The equations are applicable for different types of particles and interactions.

Based on the parametric representation of the action and the canonical equations, the corresponding relativistic mechanics based on the canonical Lagrangian is constructed and the equations of motion and expression are derived for the force acting on the charge moving in an external electromagnetic field.

The matrix representation of equations of the characteristics for the action function and the wave function results in the Dirac equation with the correct enabling of the interaction. In this form, the solutions of the Dirac equations are not restricted by the value of the interaction constant and have a spinor representation by scalar solutions of the equations for the action function and the wave function.

The analysis of the solutions shows the full compliance with the principles of the relativistic and quantum mechanics, and the solutions are devoid of any restrictions on the nature and magnitude of the interactions.

The theory of spin fields and equations for spin systems will be described in subsequent works.

#### **Author details**

Vahram Mekhitarian Institute for Physical Research, Armenian National Academy, Ashtarak, Armenia

\*Address all correspondence to: vm@ipr.sci.am

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Bohr N. On the constitution of atoms and molecules. Philosophical Magazine. 1913;**26**:1-25. DOI: 10.1080/ 14786441308634955

[2] Uhlenbeck GE, Goudsmit S. Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons. Naturwissenschaften. 1925;**13**:953-954. DOI: 10.1007/BF01558878

[3] Sommerfeld A. Zur Quantentheorie der Spektrallinien. Ann. Phys. 1916;**356**: 1-94. DOI: 10.1002/andp.19163561702

[4] Granovskii YI. Sommerfeld formula and Dirac's theory. Physics-Uspekhi. 2004;**47**:523-524. DOI: 10.1070/ PU2004v047n05ABEH001885

[5] Dirac PAM. The quantum theory of the electron. Proceedings of the Royal Society. 1928;**117**:610-624. DOI: 10.1098/rspa.1928.0023

[6] De Broglie ML. Ondes et quanta. Comptes Rendus. 1923;**177**:507-510

[7] Klein O. Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von Dirac. Zeitschrift für Physik. 1929;**53**:157-165. DOI: 10.1007/BF01339716

[8] Landau LD, Lifshitz EM. The Classical Theory of Fields. Vol. 2. 4th ed. Oxford: Butterworth-Heinemann; 1980. p. 444

[9] Talukar B, Yunus A, Amin MR. Continuum states of the Klein-Gordon equation for vector and scalar interactions. Physics Letters A. 1989; **141**:326-330. DOI: 10.1016/0375-9601 (89)90058-3

[10] Chetouani L, Guechi L, Lecheheb A, Hammann TF, Messouber A. Path integral for Klein-Gordon particle in

vector plus scalar Hulthén-type potentials. Physica A. 1996;**234**:529-544. DOI: 10.1016/S0378-4371(96)00288-9

Relativistic Electrons. New York: American Institute of Physics; 1986.

[21] Berestetskii VB, Lifshitz EM,

[22] Madelung E. Mathematische

[23] Akhiezer AI, Berestetskii VB. Quantum Electrodynamic. New York: John Wiley & Sons; 1965. p. 875

[24] Dirac PAM. The current state of the relativistic theory of the electron. Proceedings of the Institute of History of Science and Technology

estestvoznaniya i tekhniki, in Russian).

[25] The Birth of Particle Physics. Laurie B and Lillian H, edittors. Cambridge University Press; 1983. p. 432. Origin of quantum field theory, Paul A. M. Dirac, pp. 39-55. International Symposium on

the History of Particle Physics, Fermilab, May 1980. "Some new relativistic equations are needed; new kinds of interactions must be brought into play. When these new equations and new interactions are thought out, the problems that are now bewildering to us will get automatically explained, and we should no longer have to make use of such illogical processes as infinite renormalization. This is quite nonsense physically, and I have always been opposed to it. It is just a rule of thumb that gives results. In spite of its successes, one should be prepared to abandon it completely and look on all the successes that have been obtained by using the usual forms of quantum electrodynamics with the infinities removed by artificial processes as just accidents when they give the right answers, in the same way as the successes of the Bohr theory are

Electrodynamics. Oxford: Butterworth-

*DOI: http://dx.doi.org/10.5772/intechopen.93336*

*Equations of Relativistic and Quantum Mechanics (without Spin)*

considered merely as accidents when

momentum. Journal of Contemporary Physics. 2012;**47**:249-256. DOI: 10.3103/

[27] Landau LD, Lifshitz EM. Mechanics. 3rd ed. Vol. 1. Oxford: Butterworth–

[28] Mekhitarian V. Canonical solutions of variational problems and canonical equations of mechanics. Journal of Contemporary Physics. 2013;**48**:1-11. DOI: 10.3103/S1068337213010015

[29] Mekhitarian V. Equations of Relativistic and Quantum Mechanics and Exact Solutions of Some Problems. Journal of Contemporary Physics. 2018;

**53**:1-21. DOI: 10.3103/ S1068337218020123

BF01327754

[30] Kratzer A. Die ultraroten Rotationsspektren der

[31] Lennard-Jones JE. On the determination of molecular fields. Proceedings of the Royal Society of London A. 1924;**106**:463-477. DOI:

10.1098/rspa.1924.0082

Halogenwasserstoffe. Zeitschrift für Physik. 1920;**3**:289-307. DOI: 10.1007/

[32] Morse PM. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Physics Review. 1929; **34**:57-64. DOI: 10.1103/PhysRev.34.57

[33] Rosen N, Morse PM. On the vibrations of polyatomic molecules. Physics Review. 1932;**42**:210-217. DOI:

[34] Mekhitarian V. The Faraday law of induction for an arbitrarily moving charge. Journal of Contemporary Physics. 2016;**51**:108-126. DOI: 10.3103/

10.1103/PhysRev.42.210

S1068337216020031

[26] Mekhitarian V. The invariant representation of generalized

they tum out to be correct"

S1068337212060011

Heinemann; 1982. p. 224

Hilfsmittel des Physikers. 3rd ed. Berlin:

Pitaevski LP. Quantum

Heinemann; 2012. p. 667

Springer; 1936. p. 359

(Trudy instituta istorii

1959;**22**:32-33

**137**

p. 312

[11] Dominguez-Adame F. Bound states of the Klein-Gordon equation with vector and scalar Hulthén-type potentials. Physics Letters A. 1989;**136**:175-177. DOI: 10.1016/0375-9601(89)90555-0

[12] Landau LD, Lifshitz EM. Quantum Mechanics: Non-Relativistic Theory. 3rd ed. Vol. 3. Oxford: Butterworth-Heinemann; 1981. p. 689

[13] Hoffman D. Erwin Schrödinger. Leipzig: Teubner; 1984. p. 97. DOI: 10.1007/978-3-322-92064-5

[14] Schrödinger E. Quantisierung als Eigenwertproblem (Erste Mitteilung.). Ann. Phys. 1926;**79**:361-376. DOI: 10.1002/andp.19263840404

[15] Klein O. Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik. 1926;**37**:895-906. DOI: 10.1007/BF01397481

[16] Fock V. Zur Schrödingerschen Wellenmechanik. Zeitschrift für Physik. 1926;**38**:242-250. DOI: 10.1007/ BF01399113

[17] Gordon W. Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik. 1926;**40**:117-133. DOI: 10.1007/BF01390840

[18] Heisenberg W, Jordan P. Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte. Zeitschrift für Physiotherapie. 1926;**37**:263-277. DOI: 10.1007/BF01397100

[19] Thomas LH. The motion of the spinning electron. Nature. 1926;**117**:514. DOI: 10.1038/117514a0

[20] Sokolov AA, Ternov IM, Kilmister CW. Radiation from

#### *Equations of Relativistic and Quantum Mechanics (without Spin) DOI: http://dx.doi.org/10.5772/intechopen.93336*

Relativistic Electrons. New York: American Institute of Physics; 1986. p. 312

**References**

*Quantum Mechanics*

[1] Bohr N. On the constitution of atoms and molecules. Philosophical Magazine.

vector plus scalar Hulthén-type

10.1016/0375-9601(89)90555-0

ed. Vol. 3. Oxford: Butterworth-Heinemann; 1981. p. 689

10.1007/978-3-322-92064-5

[13] Hoffman D. Erwin Schrödinger. Leipzig: Teubner; 1984. p. 97. DOI:

[14] Schrödinger E. Quantisierung als Eigenwertproblem (Erste Mitteilung.). Ann. Phys. 1926;**79**:361-376. DOI: 10.1002/andp.19263840404

[15] Klein O. Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik. 1926;**37**:895-906.

[16] Fock V. Zur Schrödingerschen Wellenmechanik. Zeitschrift für Physik.

[17] Gordon W. Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik. 1926;**40**:117-133.

[18] Heisenberg W, Jordan P. Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte. Zeitschrift für Physiotherapie. 1926;**37**:263-277. DOI:

[19] Thomas LH. The motion of the spinning electron. Nature. 1926;**117**:514.

1926;**38**:242-250. DOI: 10.1007/

DOI: 10.1007/BF01390840

10.1007/BF01397100

DOI: 10.1038/117514a0

[20] Sokolov AA, Ternov IM, Kilmister CW. Radiation from

DOI: 10.1007/BF01397481

BF01399113

potentials. Physica A. 1996;**234**:529-544. DOI: 10.1016/S0378-4371(96)00288-9

[11] Dominguez-Adame F. Bound states of the Klein-Gordon equation with vector and scalar Hulthén-type potentials. Physics Letters A. 1989;**136**:175-177. DOI:

[12] Landau LD, Lifshitz EM. Quantum Mechanics: Non-Relativistic Theory. 3rd

1913;**26**:1-25. DOI: 10.1080/

[2] Uhlenbeck GE, Goudsmit S. Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons. Naturwissenschaften. 1925;**13**:953-954.

DOI: 10.1007/BF01558878

10.1098/rspa.1928.0023

DOI: 10.1007/BF01339716

p. 444

(89)90058-3

**136**

[8] Landau LD, Lifshitz EM. The

[9] Talukar B, Yunus A, Amin MR. Continuum states of the Klein-Gordon

equation for vector and scalar interactions. Physics Letters A. 1989; **141**:326-330. DOI: 10.1016/0375-9601

[3] Sommerfeld A. Zur Quantentheorie der Spektrallinien. Ann. Phys. 1916;**356**: 1-94. DOI: 10.1002/andp.19163561702

[4] Granovskii YI. Sommerfeld formula and Dirac's theory. Physics-Uspekhi. 2004;**47**:523-524. DOI: 10.1070/ PU2004v047n05ABEH001885

[5] Dirac PAM. The quantum theory of the electron. Proceedings of the Royal Society. 1928;**117**:610-624. DOI:

[6] De Broglie ML. Ondes et quanta. Comptes Rendus. 1923;**177**:507-510

[7] Klein O. Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von Dirac. Zeitschrift für Physik. 1929;**53**:157-165.

Classical Theory of Fields. Vol. 2. 4th ed. Oxford: Butterworth-Heinemann; 1980.

[10] Chetouani L, Guechi L, Lecheheb A, Hammann TF, Messouber A. Path integral for Klein-Gordon particle in

14786441308634955

[21] Berestetskii VB, Lifshitz EM, Pitaevski LP. Quantum Electrodynamics. Oxford: Butterworth-Heinemann; 2012. p. 667

[22] Madelung E. Mathematische Hilfsmittel des Physikers. 3rd ed. Berlin: Springer; 1936. p. 359

[23] Akhiezer AI, Berestetskii VB. Quantum Electrodynamic. New York: John Wiley & Sons; 1965. p. 875

[24] Dirac PAM. The current state of the relativistic theory of the electron. Proceedings of the Institute of History of Science and Technology (Trudy instituta istorii estestvoznaniya i tekhniki, in Russian). 1959;**22**:32-33

[25] The Birth of Particle Physics. Laurie B and Lillian H, edittors. Cambridge University Press; 1983. p. 432. Origin of quantum field theory, Paul A. M. Dirac, pp. 39-55. International Symposium on the History of Particle Physics, Fermilab, May 1980. "Some new relativistic equations are needed; new kinds of interactions must be brought into play. When these new equations and new interactions are thought out, the problems that are now bewildering to us will get automatically explained, and we should no longer have to make use of such illogical processes as infinite renormalization. This is quite nonsense physically, and I have always been opposed to it. It is just a rule of thumb that gives results. In spite of its successes, one should be prepared to abandon it completely and look on all the successes that have been obtained by using the usual forms of quantum electrodynamics with the infinities removed by artificial processes as just accidents when they give the right answers, in the same way as the successes of the Bohr theory are

considered merely as accidents when they tum out to be correct"

[26] Mekhitarian V. The invariant representation of generalized momentum. Journal of Contemporary Physics. 2012;**47**:249-256. DOI: 10.3103/ S1068337212060011

[27] Landau LD, Lifshitz EM. Mechanics. 3rd ed. Vol. 1. Oxford: Butterworth– Heinemann; 1982. p. 224

[28] Mekhitarian V. Canonical solutions of variational problems and canonical equations of mechanics. Journal of Contemporary Physics. 2013;**48**:1-11. DOI: 10.3103/S1068337213010015

[29] Mekhitarian V. Equations of Relativistic and Quantum Mechanics and Exact Solutions of Some Problems. Journal of Contemporary Physics. 2018; **53**:1-21. DOI: 10.3103/ S1068337218020123

[30] Kratzer A. Die ultraroten Rotationsspektren der Halogenwasserstoffe. Zeitschrift für Physik. 1920;**3**:289-307. DOI: 10.1007/ BF01327754

[31] Lennard-Jones JE. On the determination of molecular fields. Proceedings of the Royal Society of London A. 1924;**106**:463-477. DOI: 10.1098/rspa.1924.0082

[32] Morse PM. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Physics Review. 1929; **34**:57-64. DOI: 10.1103/PhysRev.34.57

[33] Rosen N, Morse PM. On the vibrations of polyatomic molecules. Physics Review. 1932;**42**:210-217. DOI: 10.1103/PhysRev.42.210

[34] Mekhitarian V. The Faraday law of induction for an arbitrarily moving charge. Journal of Contemporary Physics. 2016;**51**:108-126. DOI: 10.3103/ S1068337216020031

**Chapter 8**

**Abstract**

we embrace.

**1. Introduction**

virtual.

**139**

*Francis T.S. Yu*

Nature of Temporal (t > 0)

It is our science governs the mathematics and it is "not" our mathematics governs our science. One of the very important aspects is that every science has to comply with the boundary condition of our universe; dimensionality and temporal (t > 0) or causality. In which I have shown that time is real and it is not an illusion, since every aspect within our universe is coexisted with time. Since our universe is a temporal (t > 0) subspace, everything within our universe is temporal. Science is mathematics but mathematics is not science, we have shown that any analytic solution has to be temporal (t > 0); otherwise, it cannot be implemented within our universe. Which includes all the laws, principles, and theories have to be temporal? Uncertainty principle is one of the most fascinated principles in quantum mechanics, yet Heisenberg principle was based on diffraction limited observation, it is not due to the nature of time. We have shown it is the temporal (t > 0) uncertainty that changes with time. We have introduced a certainty principle as in contrast with uncertainty principle. Of which certainty subspace can be created within our universe; which can be exploited for application. Overall of this chapter is to show that; it is not how rigorous the mathematics is, it is the physical realizable paradigm that

**Keywords:** temporal universe, timeless space, physical realizable, uncertainty

Strictly speaking every scientific solution has to be proven whether it is physical realizable before considering for experimentation, since analytical solution is mathematics. For example, if an elementary particle has proven not a temporal (t > 0) or a timeless (t = 0) particle, it has no reason to spend that big a budget for experimentally searching a timeless (t = 0) particle since timeless particle does not exist within our universe. Similarly, a mathematician discovers a 10-dimensional subspace, would not you want to prove that his 10-dimensional subspace is a temporal (t > 0) subspace, before experimentally search for it since mathematical solution is

Nevertheless at the dawn of science, scientists have been using a piece or pieces

of papers; drawn models and paradigms in it and using mathematics as a tool analyzing for possible solution. But never occurs to them the back ground of that piece of paper represented a mathematical subspace that is "not" existed within our universe, for which practically all the laws, principles, and theories were developed

principle, certainty principle, quantum mechanics

Quantum Theory: Part I

#### **Chapter 8**

## Nature of Temporal (t > 0) Quantum Theory: Part I

*Francis T.S. Yu*

#### **Abstract**

It is our science governs the mathematics and it is "not" our mathematics governs our science. One of the very important aspects is that every science has to comply with the boundary condition of our universe; dimensionality and temporal (t > 0) or causality. In which I have shown that time is real and it is not an illusion, since every aspect within our universe is coexisted with time. Since our universe is a temporal (t > 0) subspace, everything within our universe is temporal. Science is mathematics but mathematics is not science, we have shown that any analytic solution has to be temporal (t > 0); otherwise, it cannot be implemented within our universe. Which includes all the laws, principles, and theories have to be temporal? Uncertainty principle is one of the most fascinated principles in quantum mechanics, yet Heisenberg principle was based on diffraction limited observation, it is not due to the nature of time. We have shown it is the temporal (t > 0) uncertainty that changes with time. We have introduced a certainty principle as in contrast with uncertainty principle. Of which certainty subspace can be created within our universe; which can be exploited for application. Overall of this chapter is to show that; it is not how rigorous the mathematics is, it is the physical realizable paradigm that we embrace.

**Keywords:** temporal universe, timeless space, physical realizable, uncertainty principle, certainty principle, quantum mechanics

#### **1. Introduction**

Strictly speaking every scientific solution has to be proven whether it is physical realizable before considering for experimentation, since analytical solution is mathematics. For example, if an elementary particle has proven not a temporal (t > 0) or a timeless (t = 0) particle, it has no reason to spend that big a budget for experimentally searching a timeless (t = 0) particle since timeless particle does not exist within our universe. Similarly, a mathematician discovers a 10-dimensional subspace, would not you want to prove that his 10-dimensional subspace is a temporal (t > 0) subspace, before experimentally search for it since mathematical solution is virtual.

Nevertheless at the dawn of science, scientists have been using a piece or pieces of papers; drawn models and paradigms in it and using mathematics as a tool analyzing for possible solution. But never occurs to them the back ground of that piece of paper represented a mathematical subspace that is "not" existed within our universe, for which practically all the laws, principles, and theories were developed from a piece or pieces of papers, which are timeless (t = 0) and strictly speaking are virtual.

where ›E(t)/›t is the rate of increasing energy conversion, ›m/›t is the corresponding rate of mass reduction, *c* is the speed of light, and t > 0 denotes a forward time-variable equation. In which we see Eq. (2) is a time-dependent equation exists at time t > 0, which represents a forwarded time variable function that only occurs after time excitation at t = 0. Incidentally, this is the well-known "causality" constraint (i.e., t > 0) [2] as imposed by our temporal (t > 0) universe. Nevertheless in mathematical, a postulation is first needed to proof that there is solution existed before we search for the solution, although it is not guarantee that we can find it. But it seems to me it does not have a criterion to proof that a

hypothetical science is existed within our universe, before we search for the science. For example, an analytically solution indicates that it exists an "angle particle" from a complicated mathematical analyses, will not you want to find out first is the solution existed within our temporal (t > 0) universe before experimentally to search for it. And this is precisely that we shall know first before experimentation is taken place, since it is a very costly in time and in revenue to find a physical particle. Although science needs mathematics, but without simplicity mathematically approximation, science would be very difficult to learn and to facilitate. And this is precisely the reason practically all the fundamental laws are point-singularity approximated. In which we see precisely, science is a "law of approximation" and mathematics is "an axiom of certainty". Again we take Einstein's energy equation of Eq. (1) as an example, no dimension and size and it is a typical point-singularity approximated equation. It is discernible; if we include all the negligibly terms, "physical significances" of this equation would be over whelmed by the terms of mathematics. For which we see that an ounce of good approximation worth more

Let me stress that the essence of simplicity in science is that without the symbolic substitution and approximation, it will be extremely difficult or even impossible to develop science since science itself is already very complicated. Yet

simplicity representation of science has also been misinterpreted as referred them as "classical and deterministic (i.e., classical physics)." The implication of deterministic or classical is a totally misled by our part, since our predecessors who developed those fundamental laws and principles were "precisely" understood the deficiency of approximation. Yet without the approximated presentation, how can we develop science? Instead of ignoring our predecessors' wisdom, turns around we had treated them "deterministic" or classical, which were "never" been our predecessors intention. Again without the point-singularity approximated science, please tell me how we can develop those simple and elegant laws, principles, and theories. Although those laws, principles, and theories were timeless (t = 0), most of them were and "still" are the foundation and corner stone of our science. Nevertheless, mathemat-

Since all laws, principles, and theories were made to be broken or revised or even to replace, as science advances into sub-subatomic scale regime and moving closer to near real time processing, those timeless (t = 0) laws, principles, and theories could produce incomprehensible consequences; particularly as applied them directly confronting the temporal (t > 0) constraint of our universe. For example, as applying superposition principle to quantum computing and commu-

In this section, I will show several subspaces that have been used by the scientists, in the past as depicted in **Figure 1**. It is reasonable to stress that why subspace

ics is a "symbolic" langue of science, but mathematics is not science.

nication, since superposition is a timeless (t = 0) principle [3].

**3. Temporal (t > 0) subspace**

**141**

than tons of mathematical calculation!

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

Since science is mathematics but mathematics is "not" equaled to science, it is vitally important for us to understand what science really is. In order to understand science, firstly we have to understand what supported the science? For which the supporter must be the subspace within our universe. In other words, any scientific solution has to be proven existed within our universe; otherwise, it may be fictitious and virtual as mathematics is, since science is mathematics. In which we see that, our universe is a physical subspace that supports every physical realizable aspect within her space, "if and only if " the scientific postulation complies within the existent condition of our universe; dimensionality and causality or temporal (t > 0).

The essence of our temporal (t > 0) universe is that; if a mathematical solution is "not" complied within the temporal (t > 0) condition of our universe, it cannot exist within our universe. Since quantum mechanics is one of the pillars in modern science, I will start with one of the most intriguing principles in quantum mechanics; uncertainty principle. I will carry on the principle onto a newly found "certainty" principle. In which I will show Heisenberg's principle was based on diffraction limited observation, instead upon on "nature" of time, developing his principle. I will also show the mystery of coherence theory can be understood with principle of certainty. In which I will show that; certainty subspace can be created within our temporal (t > 0) universe. Samples as applied to synthetic aperture imaging and wave front reconstruction will be included.

#### **2. Science and mathematics**

There is a profound relationship between science and mathematics, in which we have seen that without mathematics there would be no science. In other words, science needs mathematics but mathematics does not need science. Although science is mathematics but mathematics is not science. For example, any mathematical solution if it cannot be proven it exists within our universe, then her solution is "not" a "physical realizable" solution that can be "directly" implemented within our temporal (t > 0) universe.

But this is by no means to say that; the solutions are not temporal (t > 0) or timeless (t = 0) solutions there are not science. In fact practically all the fundamental laws, principles, and theories are timeless (t = 0) or time-independent. And these timeless (t = 0) laws, principles, and theories were and "still" are the corner stone and foundation of our science, as I will call them timeless (t = 0) or timeindependent science; a topic I will elaborate in a different occasion. For simplicity, let me take one of the simplest examples; Einstein's energy Eq. (1) as given by;

$$\mathbf{E} = \mathbf{m}\mathbf{c}^2\tag{1}$$

where E is the energy, m is the mass and c is the velocity of light. This equation is one of the most famous equations in science, yet it is timeless (t = 0). Although this equation has been repeatedly used and applied in practice, but strictly speaking; it cannot be directly implemented within our temporal (t > 0) universe, since it is not a time variable function. Let us transform Einstein's equation into a time variable equation as given by [1].

$$\frac{\partial E(t)}{\partial t} = -c^2 \frac{\partial m(t)}{\partial t}, \text{t} > 0 \tag{2}$$

#### *Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

from a piece or pieces of papers, which are timeless (t = 0) and strictly speaking are

Since science is mathematics but mathematics is "not" equaled to science, it is vitally important for us to understand what science really is. In order to understand science, firstly we have to understand what supported the science? For which the supporter must be the subspace within our universe. In other words, any scientific solution has to be proven existed within our universe; otherwise, it may be fictitious and virtual as mathematics is, since science is mathematics. In which we see that, our universe is a physical subspace that supports every physical realizable aspect within her space, "if and only if " the scientific postulation complies within the existent condition of our universe; dimensionality and causality or temporal (t > 0). The essence of our temporal (t > 0) universe is that; if a mathematical solution is "not" complied within the temporal (t > 0) condition of our universe, it cannot exist within our universe. Since quantum mechanics is one of the pillars in modern

science, I will start with one of the most intriguing principles in quantum

imaging and wave front reconstruction will be included.

**2. Science and mathematics**

temporal (t > 0) universe.

equation as given by [1].

**140**

mechanics; uncertainty principle. I will carry on the principle onto a newly found "certainty" principle. In which I will show Heisenberg's principle was based on diffraction limited observation, instead upon on "nature" of time, developing his principle. I will also show the mystery of coherence theory can be understood with principle of certainty. In which I will show that; certainty subspace can be created within our temporal (t > 0) universe. Samples as applied to synthetic aperture

There is a profound relationship between science and mathematics, in which we have seen that without mathematics there would be no science. In other words, science needs mathematics but mathematics does not need science. Although science is mathematics but mathematics is not science. For example, any mathematical solution if it cannot be proven it exists within our universe, then her solution is "not" a "physical realizable" solution that can be "directly" implemented within our

But this is by no means to say that; the solutions are not temporal (t > 0) or timeless (t = 0) solutions there are not science. In fact practically all the fundamental laws, principles, and theories are timeless (t = 0) or time-independent. And these timeless (t = 0) laws, principles, and theories were and "still" are the corner stone

independent science; a topic I will elaborate in a different occasion. For simplicity, let me take one of the simplest examples; Einstein's energy Eq. (1) as given by;

where E is the energy, m is the mass and c is the velocity of light. This equation is one of the most famous equations in science, yet it is timeless (t = 0). Although this equation has been repeatedly used and applied in practice, but strictly speaking; it cannot be directly implemented within our temporal (t > 0) universe, since it is not a time variable function. Let us transform Einstein's equation into a time variable

<sup>2</sup> *<sup>∂</sup>m t*ð Þ

<sup>E</sup> <sup>¼</sup> mc2 (1)

*<sup>∂</sup><sup>t</sup>* , t><sup>0</sup> (2)

and foundation of our science, as I will call them timeless (t = 0) or time-

*<sup>∂</sup>E t*ð Þ *<sup>∂</sup><sup>t</sup>* ¼ �*<sup>c</sup>*

virtual.

*Quantum Mechanics*

where ›E(t)/›t is the rate of increasing energy conversion, ›m/›t is the corresponding rate of mass reduction, *c* is the speed of light, and t > 0 denotes a forward time-variable equation. In which we see Eq. (2) is a time-dependent equation exists at time t > 0, which represents a forwarded time variable function that only occurs after time excitation at t = 0. Incidentally, this is the well-known "causality" constraint (i.e., t > 0) [2] as imposed by our temporal (t > 0) universe.

Nevertheless in mathematical, a postulation is first needed to proof that there is solution existed before we search for the solution, although it is not guarantee that we can find it. But it seems to me it does not have a criterion to proof that a hypothetical science is existed within our universe, before we search for the science. For example, an analytically solution indicates that it exists an "angle particle" from a complicated mathematical analyses, will not you want to find out first is the solution existed within our temporal (t > 0) universe before experimentally to search for it. And this is precisely that we shall know first before experimentation is taken place, since it is a very costly in time and in revenue to find a physical particle.

Although science needs mathematics, but without simplicity mathematically approximation, science would be very difficult to learn and to facilitate. And this is precisely the reason practically all the fundamental laws are point-singularity approximated. In which we see precisely, science is a "law of approximation" and mathematics is "an axiom of certainty". Again we take Einstein's energy equation of Eq. (1) as an example, no dimension and size and it is a typical point-singularity approximated equation. It is discernible; if we include all the negligibly terms, "physical significances" of this equation would be over whelmed by the terms of mathematics. For which we see that an ounce of good approximation worth more than tons of mathematical calculation!

Let me stress that the essence of simplicity in science is that without the symbolic substitution and approximation, it will be extremely difficult or even impossible to develop science since science itself is already very complicated. Yet simplicity representation of science has also been misinterpreted as referred them as "classical and deterministic (i.e., classical physics)." The implication of deterministic or classical is a totally misled by our part, since our predecessors who developed those fundamental laws and principles were "precisely" understood the deficiency of approximation. Yet without the approximated presentation, how can we develop science? Instead of ignoring our predecessors' wisdom, turns around we had treated them "deterministic" or classical, which were "never" been our predecessors intention. Again without the point-singularity approximated science, please tell me how we can develop those simple and elegant laws, principles, and theories. Although those laws, principles, and theories were timeless (t = 0), most of them were and "still" are the foundation and corner stone of our science. Nevertheless, mathematics is a "symbolic" langue of science, but mathematics is not science.

Since all laws, principles, and theories were made to be broken or revised or even to replace, as science advances into sub-subatomic scale regime and moving closer to near real time processing, those timeless (t = 0) laws, principles, and theories could produce incomprehensible consequences; particularly as applied them directly confronting the temporal (t > 0) constraint of our universe. For example, as applying superposition principle to quantum computing and communication, since superposition is a timeless (t = 0) principle [3].

#### **3. Temporal (t > 0) subspace**

In this section, I will show several subspaces that have been used by the scientists, in the past as depicted in **Figure 1**. It is reasonable to stress that why subspace

Nevertheless what temporal (t > 0) space means is that any subspace is coexisted with time, where time is a forward dependent variable with respect to its subspace and its speed has been well settled when our universe was created. This means that before the creation of our temporal universe, there is a "larger" temporal space that our universe is embedded in; otherwise, our universe will "not" be existed. Nevertheless every subspace within our universe is a time varying stochastic [7] subspace, in which every substance or subspace changes with time. Strictly speaking our universe is a "temporal (t > 0) stochastic expanding subspace." For which we see that; any postulated law, principle, and theory has to comply with the

temporal (t > 0) condition within our universe; otherwise, it is virtual as

Let me show what mathematicians can do within a virtual subspace as depicted in **Figure 2**. Since quantum mechanists are also mathematicians, they can implant coordinate system within an empty space as they wishes, regardless whether the

The basic difference between **Figure 2(a)** and **(b)** is that there is a virtual coordinate system that has been added in **Figure 2(b)** by quantum mechanists. Once the coordinate system is implanted, dimensionality of the sub-atomic particles cannot be ignored. The reason is that for the atomic model to be existed within the subspace, the atomic model has to "comply" with the existence conditions within the subspace, since it is the subspace affects the solution and not the solution changes the subspace. In which we see that neither **Figure 2(a)** nor **Figure 2(b)** are "not" physical realizable paradigms. For which solutions obtained from these empty

Aside the non-physical realizable paradigms of **Figure 2**, I will show what a timeless (t = 0) subspace can do for substances within the subspace. Let me assume we have three particles situated within an empty space, as normally do on a "piece

*A set of atomic models embedded within virtual empty subspaces. (a) shows a singularity approximated atomic model is situated within an empty space, which has no coordinate system. (b) shows an atomic model is*

*embedded within empty space that has a coordinate system drawn into it.*

mathematics.

**4. Timeless (t = 0) space**

model is physically realizable or not.

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

subspace models will be timeless (t = 0).

of paper", shown in **Figure 3**.

**Figure 2.**

**143**

#### **Figure 1.**

*(a) Shows an absolute-empty space, (b) a virtual mathematical space, (c) a Newtonian space, and (d) a temporal (t > 0) space, respectively.*

of a scientific model embedded is crucially important is that any analytical solution produced follows the "limitation" of the subspace, because it is the subspace dictates the science but "not" the mathematics changes the subspace.

For example, when you are designing a submarine, the subspace that the submarine is supposed to be situated within is vitally important; otherwise, your submarine will very "likely" not to survive thousands of feet underwater pressure. Therefore, it is necessary to know the subspace that a postulated science to be implementing into it; otherwise, the postulated science is very likely "cannot" be existed within the subspace.

In view of **Figure 1**, we see that; there is an absolute-empty space, a mathematical virtual space, a Newtonian's space [4], and a temporal (t > 0) space. An absolute-empty space or just empty space has no substance and has no time. A mathematical virtual space is an empty space which has no substance in it, but mathematicians and theoretical scientists can implant coordinate system in it, since mathematics is virtual and theoretical scientists are also mathematicians.

We note that mathematical virtual space has been used over centuries by scientists at the dawn of science, but this is a virtual space that does "not" exist within our temporal (i.e., t > 0) universe. The next subspace is known as Newtonian space [4]; it has substance and coordinates in it, but treated time as an "independent" variable, for which Newtonian and mathematical spaces are virtual the "same." Since Newtonian space is time independent, it "cannot" be exist within our temporal (t > 0) space since time and substance has to be "mutually coexisted" within our temporal (t > 0) universe. Yet scientists have been using Newtonian space for their analyses over centuries and not knowingly it is a virtual space.

The last subspace is known as temporal (t > 0) space [5], where time and substance are interdependently "coexisted" and time is a forward "dependent variable" runs at a "constant speed". We stress that this temporal (t > 0) subspace is currently "only" physical realizable space, where the space was created by Einstein energy Eq. (2).

Physical reality is that any scientific hypothesis that deviates "away" the boundary condition that imposed by our temporal (t > 0) universe is "not" a physically realizable solution. But this is by no means that the virtual mathematical empty space and Newtonian space are useless. The fact is that all the physical sciences were developed within timeless (t = 0) or Newtonian subspaces "inadvertently," at the dawn of science. Practically all the fundamental laws, principles, and theories were derived from a timeless (t = 0) subspace, which was from the background subspace of a piece of paper although not intentionally [6]. In which we see that practically all the laws, principle, and theories are timeless (t = 0).

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

Nevertheless what temporal (t > 0) space means is that any subspace is coexisted with time, where time is a forward dependent variable with respect to its subspace and its speed has been well settled when our universe was created. This means that before the creation of our temporal universe, there is a "larger" temporal space that our universe is embedded in; otherwise, our universe will "not" be existed. Nevertheless every subspace within our universe is a time varying stochastic [7] subspace, in which every substance or subspace changes with time. Strictly speaking our universe is a "temporal (t > 0) stochastic expanding subspace." For which we see that; any postulated law, principle, and theory has to comply with the temporal (t > 0) condition within our universe; otherwise, it is virtual as mathematics.

#### **4. Timeless (t = 0) space**

of a scientific model embedded is crucially important is that any analytical solution produced follows the "limitation" of the subspace, because it is the subspace dic-

*(a) Shows an absolute-empty space, (b) a virtual mathematical space, (c) a Newtonian space, and*

For example, when you are designing a submarine, the subspace that the submarine is supposed to be situated within is vitally important; otherwise, your submarine will very "likely" not to survive thousands of feet underwater pressure. Therefore, it is necessary to know the subspace that a postulated science to be implementing into it; otherwise, the postulated science is very likely "cannot" be

In view of **Figure 1**, we see that; there is an absolute-empty space, a mathemat-

We note that mathematical virtual space has been used over centuries by scientists at the dawn of science, but this is a virtual space that does "not" exist within our temporal (i.e., t > 0) universe. The next subspace is known as Newtonian space [4]; it has substance and coordinates in it, but treated time as an "independent" variable, for which Newtonian and mathematical spaces are virtual the "same." Since Newtonian space is time independent, it "cannot" be exist within our temporal (t > 0) space since time and substance has to be "mutually coexisted" within our temporal (t > 0) universe. Yet scientists have been using Newtonian space for their

The last subspace is known as temporal (t > 0) space [5], where time and substance are interdependently "coexisted" and time is a forward "dependent variable" runs at a "constant speed". We stress that this temporal (t > 0) subspace is currently "only" physical realizable space, where the space was created by Einstein

Physical reality is that any scientific hypothesis that deviates "away" the boundary condition that imposed by our temporal (t > 0) universe is "not" a physically realizable solution. But this is by no means that the virtual mathematical empty space and Newtonian space are useless. The fact is that all the physical sciences were developed within timeless (t = 0) or Newtonian subspaces "inadvertently," at the dawn of science. Practically all the fundamental laws, principles, and theories were derived from a timeless (t = 0) subspace, which was from the background subspace of a piece of paper although not intentionally [6]. In which we

see that practically all the laws, principle, and theories are timeless (t = 0).

ical virtual space, a Newtonian's space [4], and a temporal (t > 0) space. An absolute-empty space or just empty space has no substance and has no time. A mathematical virtual space is an empty space which has no substance in it, but mathematicians and theoretical scientists can implant coordinate system in it, since

mathematics is virtual and theoretical scientists are also mathematicians.

analyses over centuries and not knowingly it is a virtual space.

tates the science but "not" the mathematics changes the subspace.

existed within the subspace.

*(d) a temporal (t > 0) space, respectively.*

**Figure 1.**

*Quantum Mechanics*

energy Eq. (2).

**142**

Let me show what mathematicians can do within a virtual subspace as depicted in **Figure 2**. Since quantum mechanists are also mathematicians, they can implant coordinate system within an empty space as they wishes, regardless whether the model is physically realizable or not.

The basic difference between **Figure 2(a)** and **(b)** is that there is a virtual coordinate system that has been added in **Figure 2(b)** by quantum mechanists. Once the coordinate system is implanted, dimensionality of the sub-atomic particles cannot be ignored. The reason is that for the atomic model to be existed within the subspace, the atomic model has to "comply" with the existence conditions within the subspace, since it is the subspace affects the solution and not the solution changes the subspace. In which we see that neither **Figure 2(a)** nor **Figure 2(b)** are "not" physical realizable paradigms. For which solutions obtained from these empty subspace models will be timeless (t = 0).

Aside the non-physical realizable paradigms of **Figure 2**, I will show what a timeless (t = 0) subspace can do for substances within the subspace. Let me assume we have three particles situated within an empty space, as normally do on a "piece of paper", shown in **Figure 3**.

#### **Figure 2.**

*A set of atomic models embedded within virtual empty subspaces. (a) shows a singularity approximated atomic model is situated within an empty space, which has no coordinate system. (b) shows an atomic model is embedded within empty space that has a coordinate system drawn into it.*

**Figure 3.** *A hypothetical scenario shows three particles are embedded within an empty subspace.*

Since empty subspace has "no time," all particles within the subspace collapse or "superimposing" instantly all together at t = 0, because time is distance and distance is time. This is precisely the "simultaneous and instantaneous" superposition principle does in quantum mechanics [3]. The reason particles collapsed at t = 0, it is because the subspace has "no time." And the other reason that particles superimposed together, since within a timeless (t = 0) space, it has "no distance" or no space.

incomplete even though correct" [8]. In view of preceding illustration, we see that Schrödinger's superposition principle is "correct" but only within a timeless (t = 0) subspace and it is "incorrect" within our temporal (t > 0) space," since timeless

As we accepted subspace and time are coexisted within our temporal (t > 0) universe, time has to be real and it cannot be virtual, since we are physically real. And every physical existence within our universe is real. The reason some scientists believed time is virtual or illusion is that; it has no mass, no weight, no coordinate, no origin, and it cannot be detected or even be seen. Yet time is an everlasting existed real variable within our known universe. Without time there would be no physical matter, no physical space, and no life. The fact is that every physical matter is associated with time which including our universe. Therefore, when one is dealing with science, time is one of the most enigmatic variables that ever presence and cannot be simply ignored. Strictly speaking, all the laws of science as well every physical substance cannot be existed without the existence of time. For which we see that time "cannot" be a dimension or an illusion. In other words, if time is an illusion, then time will be "independent" from physical reality or from our universe. And this is precisely that many scientists have treated time as an "independent" variable such as Murkowski's space [9], for which the space can be "curved" or time-space can be changed by gravity [10]. If time-space can be curved, then we can change the "speed" of time. In other words, is our universe exists with time, or time exists with universe? The answer is our universe exists with time, although space and time are interdependent but is not time exists with our universe.

As time is coexisted with subspace, we see that any subspace within our temporal (t > 0) universe cannot be empty and speed of time is the same everywhere within our universe. This means that the speed of time within a subspace is "relatively" with respect to the different subspaces, as based on Einstein's special theory of relativity [9]. For example, subspaces closer to the edge of our universe, their time runs faster "relative" to ours, but the speed of time within the subspaces near the edge as well within our subspace are the "same," which has been determined by the speed of light as our universe was created by a big bang theory using Einstein

space cannot exist within our temporal universe.

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

**5. Time is not an illusion but real**

**Figure 5.**

*A 1935 New York times' article.*

equation as given by [5];

**145**

By virtue of energy conservation, we see that superimposed particles has a mass equals to the sum of entire superimposed particles, but it has "no size." In view of timelessness space, we see that the superimposed particles can be found everywhere within the entire timeless (t = 0) subspace, since timeless (t = 0) subspace has "no" distance, as depicted hypothetically in **Figure 4**. In which we see that Schrödinger's fundamental principle of superposition is existed within a virtual timeless (t = 0) subspace, and it cannot be existed within our temporal (t > 0) universe, since timeless and temporal are "mutually exclusive."

By the way, this is precisely the superposition principle that Einstein was objecting to, which he called it spooky. As I quote from a 1935 The New York Times' article (i.e., **Figure 5**), "Einstein and two scientists found quantum theory is

**Figure 4.** *Superimposed particle existed "simultaneously and instantaneously" all over the entire timeless (t = 0) subspace.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

**Figure 5.** *A 1935 New York times' article.*

Since empty subspace has "no time," all particles within the subspace collapse or "superimposing" instantly all together at t = 0, because time is distance and distance is time. This is precisely the "simultaneous and instantaneous" superposition principle does in quantum mechanics [3]. The reason particles collapsed at t = 0, it is

superimposed together, since within a timeless (t = 0) space, it has "no distance" or

By the way, this is precisely the superposition principle that Einstein was objecting to, which he called it spooky. As I quote from a 1935 The New York Times' article (i.e., **Figure 5**), "Einstein and two scientists found quantum theory is

*Superimposed particle existed "simultaneously and instantaneously" all over the entire timeless (t = 0) subspace.*

By virtue of energy conservation, we see that superimposed particles has a mass equals to the sum of entire superimposed particles, but it has "no size." In view of timelessness space, we see that the superimposed particles can be found everywhere within the entire timeless (t = 0) subspace, since timeless (t = 0) subspace has "no" distance, as depicted hypothetically in **Figure 4**. In which we see that Schrödinger's fundamental principle of superposition is existed within a virtual timeless (t = 0) subspace, and it cannot be existed within our temporal (t > 0) universe, since

because the subspace has "no time." And the other reason that particles

*A hypothetical scenario shows three particles are embedded within an empty subspace.*

timeless and temporal are "mutually exclusive."

no space.

**Figure 4.**

**144**

**Figure 3.**

*Quantum Mechanics*

incomplete even though correct" [8]. In view of preceding illustration, we see that Schrödinger's superposition principle is "correct" but only within a timeless (t = 0) subspace and it is "incorrect" within our temporal (t > 0) space," since timeless space cannot exist within our temporal universe.

#### **5. Time is not an illusion but real**

As we accepted subspace and time are coexisted within our temporal (t > 0) universe, time has to be real and it cannot be virtual, since we are physically real. And every physical existence within our universe is real. The reason some scientists believed time is virtual or illusion is that; it has no mass, no weight, no coordinate, no origin, and it cannot be detected or even be seen. Yet time is an everlasting existed real variable within our known universe. Without time there would be no physical matter, no physical space, and no life. The fact is that every physical matter is associated with time which including our universe. Therefore, when one is dealing with science, time is one of the most enigmatic variables that ever presence and cannot be simply ignored. Strictly speaking, all the laws of science as well every physical substance cannot be existed without the existence of time. For which we see that time "cannot" be a dimension or an illusion. In other words, if time is an illusion, then time will be "independent" from physical reality or from our universe. And this is precisely that many scientists have treated time as an "independent" variable such as Murkowski's space [9], for which the space can be "curved" or time-space can be changed by gravity [10]. If time-space can be curved, then we can change the "speed" of time. In other words, is our universe exists with time, or time exists with universe? The answer is our universe exists with time, although space and time are interdependent but is not time exists with our universe.

As time is coexisted with subspace, we see that any subspace within our temporal (t > 0) universe cannot be empty and speed of time is the same everywhere within our universe. This means that the speed of time within a subspace is "relatively" with respect to the different subspaces, as based on Einstein's special theory of relativity [9]. For example, subspaces closer to the edge of our universe, their time runs faster "relative" to ours, but the speed of time within the subspaces near the edge as well within our subspace are the "same," which has been determined by the speed of light as our universe was created by a big bang theory using Einstein equation as given by [5];

$$\frac{\partial E(t)}{\partial t} = -c^2 \frac{\partial m(t)}{\partial t}, \mathbf{t} > \mathbf{0} \tag{3}$$

cannot exist within our universe. In other words, our time is the same as our planet and the universe but the velocity of our planet is different from other subspaces. For example, subspaces near the edge of our universe are moving faster than us, for which it has "relative" speed of time between us and a subspace closer to the edge of our universe. On the other hand, if we assume that we are timeless (t = 0), we could "not" have existed within our universe, since time and timelessness are mutually

*A schematic diagram of our temporal (t > 0) universe. c is the speed of light, m(t) is the temporal mass, and v is*

I further note that any subspace within our universe cannot empty, since subspace is coexisted with time. Although subspace is coexisted with time, but time is neither equaled to subspace. Yet, space is time and time is space since time and space are mutually inclusive. For example, substance has dimension (or space), but time has no dimension and no mass. In which we see that time is "not" a dimension but it is "dependently" existed with respect to subspace. In which we stress that it is our universe governs the science and it is not the science changes our universe. Once again, we have shown that time is "not "an illusion or virtual, time is physically real because everything existed within our living space is physical real; otherwise, it will not be existed within our temporal universe. In other words, everything within our universe is temporal (t > 0), of which I have discovered that practically all the laws, principles, theories, and paradoxes of science were developed from a timeless (t = 0) platform (i.e., a pieces or pieces of papers) for

Nevertheless, one of the important aspects within our universe is that every subspace has a price, an amount of energy ΔE, and a section of time Δt to create (i.e., ΔE and Δt), and it is "not free." For example, a simple facial tissue takes a huge amount of energy ΔE and a section of time Δt to create. It is, however, a "necessary" but not sufficient condition, because it also needs an amount of information

In short, I would stress that if there is a beginning then there is an end. Since time and space are coexisted, then time and space have no beginning and no end. In which we see that time-space [or temporal (t > 0) space] is ever existed, since existence and non-existence are mutually exclusive. In other words, emptiness and non-emptiness are mutually excluded, then time "always" exists with space. Thus, time is real because the space is real, for which time-space has no beginning and has

centuries, at the dawn of science "inadvertently" [6].

ΔI to make it happen (i.e., ΔE, Δt, and ΔI) [12].

no end. And this must be the art of temporal (t > 0) universe.

exclusive.

**147**

**Figure 7.**

*the radial velocity.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

where ›E/›t is the rate of increasing energy conversion, �›m/›t is the corresponding rate of mass reduction, *c* is the speed of light and t > 0 represents a forward time-variable. In which we see that it a "time-dependent" equation exists at time t > 0; a well-known causality constraint (i.e., t > 0) [2] as imposed by our universe. Similarly preceding equation can be written as:

$$\frac{\partial \mathbf{E}}{\partial t} = -\mathbf{c}^2 \frac{\partial m}{\partial t} = [\nabla \cdot \mathbf{S}(\boldsymbol{\nu})] = -\frac{\partial}{\partial t} \left[ \frac{1}{2} \mathbf{e} \mathbf{E}^2(\boldsymbol{\nu}) + \frac{1}{2} \mu \mathbf{H}^2(\boldsymbol{\nu}) \right], \mathbf{t} > \mathbf{0} \tag{4}$$

where ε and μ are the permittivity and the permeability of the deep space, respectively, υ is the radian frequency variable, E2 (υ) and H<sup>2</sup> (υ) are the respective electric and magnetic field intensities, the negative sign represents the "out-flow" energy per unit time from an unit volume, ð Þ ∇� is the divergent operator, and *S* is known as the Poynting Vector or "Energy Vector" of an electro-magnetic radiator as can be shown by S(υ) = E(υ) � H(υ) [11].

In view of this equation, we see how our universe was created as depicted by a composited diagram in **Figure 6**, in which we see that radian energy (i.e., radiation) diverges from the mass, as mass reduces with time. In which we see that our universe enlarges and her boundary expands at speed of speed of light.

**Figure 7** shows a schematic diagram of our temporal (t > 0) universe, which depicts approximately the behavior of subspace changes as her boundary expands with speed of light. In which we see that, subspace enlarges faster closer toward the boundary, but solid substance m (t) changes little within the subspace. We also see that the out-ward speed of particle (or subspace) increases "linearly" as boundary increases with light speed. For example; out-ward speed of particle 2 is somewhat faster than particle 1 (i.e., v2 > v1). For which we see that our universe is a dynamic temporal (t > 0) "stochastic" universe that simple geometrical equation or mathematical abstract space can describe. One of the important aspects of our universe is that every subspace, no matter how small it is, "cannot" be empty and it has time.

For instance, in order for us to be existed within our planet, we must be temporal (t > 0): that is we have time and must change with time; otherwise, we

#### **Figure 6.**

*Composite temporal (t > 0) universe diagrams. r = ct, r is the radius of our universe, t is time, c is the velocity of light, and ε<sup>0</sup> and μ<sup>0</sup> are the permittivity and permeability of the space.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

*<sup>∂</sup>E t*ð Þ

universe. Similarly preceding equation can be written as:

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> ½ �¼� **<sup>∇</sup>** � *S v*ð Þ *<sup>∂</sup>*

**<sup>2</sup>** *<sup>∂</sup><sup>m</sup>*

as can be shown by S(υ) = E(υ) � H(υ) [11].

respectively, υ is the radian frequency variable, E2

*∂E <sup>∂</sup><sup>t</sup>* ¼ �*<sup>c</sup>*

*Quantum Mechanics*

be empty and it has time.

**Figure 6.**

**146**

*<sup>∂</sup><sup>t</sup>* ¼ �*c***<sup>2</sup>** *<sup>∂</sup>m t*ð Þ

where ›E/›t is the rate of increasing energy conversion, �›m/›t is the corresponding rate of mass reduction, *c* is the speed of light and t > 0 represents a forward time-variable. In which we see that it a "time-dependent" equation exists at time t > 0; a well-known causality constraint (i.e., t > 0) [2] as imposed by our

> *∂t* **1 <sup>2</sup>** *<sup>ε</sup>E***<sup>2</sup>**

where ε and μ are the permittivity and the permeability of the deep space,

electric and magnetic field intensities, the negative sign represents the "out-flow" energy per unit time from an unit volume, ð Þ ∇� is the divergent operator, and *S* is known as the Poynting Vector or "Energy Vector" of an electro-magnetic radiator

In view of this equation, we see how our universe was created as depicted by a composited diagram in **Figure 6**, in which we see that radian energy (i.e., radiation) diverges from the mass, as mass reduces with time. In which we see that our universe enlarges and her boundary expands at speed of speed of light.

**Figure 7** shows a schematic diagram of our temporal (t > 0) universe, which depicts approximately the behavior of subspace changes as her boundary expands with speed of light. In which we see that, subspace enlarges faster closer toward the boundary, but solid substance m (t) changes little within the subspace. We also see that the out-ward speed of particle (or subspace) increases "linearly" as boundary increases with light speed. For example; out-ward speed of particle 2 is somewhat faster than particle 1 (i.e., v2 > v1). For which we see that our universe is a dynamic temporal (t > 0) "stochastic" universe that simple geometrical equation or mathematical abstract space can describe. One of the important aspects of our universe is that every subspace, no matter how small it is, "cannot"

For instance, in order for us to be existed within our planet, we must be temporal (t > 0): that is we have time and must change with time; otherwise, we

*Composite temporal (t > 0) universe diagrams. r = ct, r is the radius of our universe, t is time, c is the velocity of*

*light, and ε<sup>0</sup> and μ<sup>0</sup> are the permittivity and permeability of the space.*

ð Þþ *v* **1 <sup>2</sup>** *<sup>μ</sup>H***<sup>2</sup>** ð Þ *v*

(υ) and H<sup>2</sup>

*<sup>∂</sup><sup>t</sup>* ,**<sup>t</sup> <sup>&</sup>gt; <sup>0</sup>** (3)

,**t > 0** (4)

(υ) are the respective

**Figure 7.** *A schematic diagram of our temporal (t > 0) universe. c is the speed of light, m(t) is the temporal mass, and v is the radial velocity.*

cannot exist within our universe. In other words, our time is the same as our planet and the universe but the velocity of our planet is different from other subspaces. For example, subspaces near the edge of our universe are moving faster than us, for which it has "relative" speed of time between us and a subspace closer to the edge of our universe. On the other hand, if we assume that we are timeless (t = 0), we could "not" have existed within our universe, since time and timelessness are mutually exclusive.

I further note that any subspace within our universe cannot empty, since subspace is coexisted with time. Although subspace is coexisted with time, but time is neither equaled to subspace. Yet, space is time and time is space since time and space are mutually inclusive. For example, substance has dimension (or space), but time has no dimension and no mass. In which we see that time is "not" a dimension but it is "dependently" existed with respect to subspace. In which we stress that it is our universe governs the science and it is not the science changes our universe.

Once again, we have shown that time is "not "an illusion or virtual, time is physically real because everything existed within our living space is physical real; otherwise, it will not be existed within our temporal universe. In other words, everything within our universe is temporal (t > 0), of which I have discovered that practically all the laws, principles, theories, and paradoxes of science were developed from a timeless (t = 0) platform (i.e., a pieces or pieces of papers) for centuries, at the dawn of science "inadvertently" [6].

Nevertheless, one of the important aspects within our universe is that every subspace has a price, an amount of energy ΔE, and a section of time Δt to create (i.e., ΔE and Δt), and it is "not free." For example, a simple facial tissue takes a huge amount of energy ΔE and a section of time Δt to create. It is, however, a "necessary" but not sufficient condition, because it also needs an amount of information ΔI to make it happen (i.e., ΔE, Δt, and ΔI) [12].

In short, I would stress that if there is a beginning then there is an end. Since time and space are coexisted, then time and space have no beginning and no end. In which we see that time-space [or temporal (t > 0) space] is ever existed, since existence and non-existence are mutually exclusive. In other words, emptiness and non-emptiness are mutually excluded, then time "always" exists with space. Thus, time is real because the space is real, for which time-space has no beginning and has no end. And this must be the art of temporal (t > 0) universe.

#### **6. Law of uncertainty**

One of the most intriguing principles in quantum mechanics [13] must be the Heisenberg's Uncertainty Principle [14], as shown by the following equation:

$$
\Delta \mathbf{p} \,\,\Delta \mathbf{x} \ge \mathbf{h} \,\,\tag{5}
$$

see that the position error Δx is "not" due to particle in motion, but based on the diffraction limited aperture. This is precisely why Heisenberg's position error Δx has been interpreted as an "observation error" which is independent with time. But uncertainty changes naturally with time, since without time it has no uncertainty. Secondly, the momentum error Δp as I quote [15]: after collision the particle being observed, the photon's path is only to lie within a cone having semi-vertical angle α in which momentum of the particle is uncertain by the amount as given by:

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

where λ is the wavelength of the quantum leap of hυ. In which we see that; momentum error Δp is "not" due to band width Δυ of quantum leap since any physical radiator has to be band limited. In other words, the momentum error Δp of preceding Eq. (8) is a singularity approximated λ, which is "not" a band limited Δλ

As we look back at the subspace that Heisenberg's principle developed from, it was an "inadvertently" timeless (t = 0) subspace as shown in **Figure 8**. Aside the timeless (t = 0) subspace, it is the uncertainty mainly due to diffraction limited observation, which is a "secondary cause" by human intervention, but not due to naturally change with time. This is similar to entropy theory of Boltzmann [16]: entropy increases naturally with time within an enclosed subspace. In which we see that uncertainty should be increasing with time, without human intervention. As I

Since it is our universe governs the science and it is not the science governs our universe. Therefore, every principle within our universe has to comply with the temporal (t > 0) condition within our universe; otherwise, the principle cannot be existed within our universe. Which includes all the laws, principles and theories; such as Maxwell's Electro-Magnetic theory, Boltzmann's entropy theory, Einstein's relativity theory, Bohr's atomic model, Schrödinger's superposition principle, and

Let us now assumed a temporal (t > 0) particle m(t) is situated within a temporal (t > 0) subspace as depicted in **Figure 9**. Strictly speaking any particle existed within a temporal subspace must be a temporal (t > 0) particle; otherwise,

others. Of which uncertainty principle cannot be the exception?

the particle cannot be existed within our temporal (t > 0) universe.

have noted, without time, there would be no entropy and no uncertainty. Nevertheless, momentum error Δp and position error Δx are mutually "coexisted." In principle they can be traded. But the trading cannot without constraint, since time is a dependent forward variable. But Heisenberg uncertainty; Δp and Δx are "not" mutually dependent, since his position error Δx is due to diffraction limited observation, which is nothing to do with time. For which it poses a physical "inconsistency" within our universe, although Heisenberg principle has been widely used without any abnormality. But it is from the "physical consistency" standpoint, Heisenberg's position error Δx was based on diffraction limited observation has "nothing" to do with time. And also added and his momentum error Δp was based on singularity wavelength λ which is "not" a band limited reality. Yet, uncertainty principle can be made temporal (t > 0), similar to entropy theory of Boltzmann. For which we have a "law of uncertainty" as stated: uncertainty of an isolated particle increases naturally with time. Or more specific: uncertainty of an isolated particle within an isolated subspace, increases with time and eventually reaches to a maximum amount within the isolated subspace. For which we see that there it exists a profound connection between uncertainty and entropy.

of physical reality.

**7. Temporal (t > 0) uncertainty**

**149**

Δp ¼ h sin ð αÞ*=*λ (8)

where Δp and Δx are the momentum and position errors, respectively, and h is the Planck's constant. As reference to "wave-particle dynamics," the momentum p of a "photonic particle" is presented by a "quanta" of energy hυ as given by:

$$\mathbf{p} = \mathbf{h}/\lambda = \mathbf{h}\mathbf{v}/\mathbf{c} \tag{6}$$

where h is the Planck's constant, λ is the wavelength, υ is the frequency, and c is the velocity of light.

In which we see that Heisenberg's principle was based on "wave-particle duality" existed within an "empty space." The essence of the Heisenberg's uncertainty principle is that one cannot precisely determine the position x and the momentum p of a particle "simultaneously under observation", as illustrated in **Figure 8**. In which we see that; it is "independent" of time, since Heisenberg's principle was based on "observation" stand point which has nothing to do with changing naturally with time. Yet we know that if there is "no" time there is "no" uncertainty.

In view of **Figure 8**, Heisenberg principle was derived on an empty timeless (t = 0) subspace and it has "nothing to do or independent" with the "underneath subspace" that the particle is situated. Strictly speaking, it is "not" a physical realizable paradigm should be used in the first place, since particle and empty subspace are "mutual exclusive." Secondly, the position error Δx of Heisenberg was based on a "diffraction limited" microscopic observation, where the "spatial" ambiguity of Δx is given by [15]:

$$
\Delta \mathbf{x} = \mathbf{0}.6 \,\lambda/\sin\,\alpha \tag{7}
$$

where λ is the observation wavelength, 2(sin α) is the "numerical aperture" of the microscope and α is subtended half-angle of observation aperture. In which we

#### **Figure 8.**

*A particle in motion within an "empty" subspace. v is the velocity. Note that background paper has been treated as an "empty" subspace for centuries.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

**6. Law of uncertainty**

*Quantum Mechanics*

the velocity of light.

ambiguity of Δx is given by [15]:

**Figure 8.**

**148**

*as an "empty" subspace for centuries.*

One of the most intriguing principles in quantum mechanics [13] must be the Heisenberg's Uncertainty Principle [14], as shown by the following equation:

where Δp and Δx are the momentum and position errors, respectively, and h is the Planck's constant. As reference to "wave-particle dynamics," the momentum p of a "photonic particle" is presented by a "quanta" of energy hυ as given by:

where h is the Planck's constant, λ is the wavelength, υ is the frequency, and c is

In which we see that Heisenberg's principle was based on "wave-particle duality"

existed within an "empty space." The essence of the Heisenberg's uncertainty principle is that one cannot precisely determine the position x and the momentum p of a particle "simultaneously under observation", as illustrated in **Figure 8**. In which we see that; it is "independent" of time, since Heisenberg's principle was based on "observation" stand point which has nothing to do with changing naturally

with time. Yet we know that if there is "no" time there is "no" uncertainty.

In view of **Figure 8**, Heisenberg principle was derived on an empty timeless (t = 0) subspace and it has "nothing to do or independent" with the "underneath subspace" that the particle is situated. Strictly speaking, it is "not" a physical realizable paradigm should be used in the first place, since particle and empty subspace are "mutual exclusive." Secondly, the position error Δx of Heisenberg was based on a "diffraction limited" microscopic observation, where the "spatial"

where λ is the observation wavelength, 2(sin α) is the "numerical aperture" of the microscope and α is subtended half-angle of observation aperture. In which we

*A particle in motion within an "empty" subspace. v is the velocity. Note that background paper has been treated*

Δp Δx≥h (5)

p ¼ h*=*λ ¼ hυ*=*c (6)

Δx ¼ 0*:*6 λ*=* sin α (7)

see that the position error Δx is "not" due to particle in motion, but based on the diffraction limited aperture. This is precisely why Heisenberg's position error Δx has been interpreted as an "observation error" which is independent with time. But uncertainty changes naturally with time, since without time it has no uncertainty.

Secondly, the momentum error Δp as I quote [15]: after collision the particle being observed, the photon's path is only to lie within a cone having semi-vertical angle α in which momentum of the particle is uncertain by the amount as given by:

$$
\Delta \mathbf{p} = \mathbf{h} (\sin \alpha) / \lambda \tag{8}
$$

where λ is the wavelength of the quantum leap of hυ. In which we see that; momentum error Δp is "not" due to band width Δυ of quantum leap since any physical radiator has to be band limited. In other words, the momentum error Δp of preceding Eq. (8) is a singularity approximated λ, which is "not" a band limited Δλ of physical reality.

As we look back at the subspace that Heisenberg's principle developed from, it was an "inadvertently" timeless (t = 0) subspace as shown in **Figure 8**. Aside the timeless (t = 0) subspace, it is the uncertainty mainly due to diffraction limited observation, which is a "secondary cause" by human intervention, but not due to naturally change with time. This is similar to entropy theory of Boltzmann [16]: entropy increases naturally with time within an enclosed subspace. In which we see that uncertainty should be increasing with time, without human intervention. As I have noted, without time, there would be no entropy and no uncertainty.

Nevertheless, momentum error Δp and position error Δx are mutually "coexisted." In principle they can be traded. But the trading cannot without constraint, since time is a dependent forward variable. But Heisenberg uncertainty; Δp and Δx are "not" mutually dependent, since his position error Δx is due to diffraction limited observation, which is nothing to do with time. For which it poses a physical "inconsistency" within our universe, although Heisenberg principle has been widely used without any abnormality. But it is from the "physical consistency" standpoint, Heisenberg's position error Δx was based on diffraction limited observation has "nothing" to do with time. And also added and his momentum error Δp was based on singularity wavelength λ which is "not" a band limited reality.

Yet, uncertainty principle can be made temporal (t > 0), similar to entropy theory of Boltzmann. For which we have a "law of uncertainty" as stated: uncertainty of an isolated particle increases naturally with time. Or more specific: uncertainty of an isolated particle within an isolated subspace, increases with time and eventually reaches to a maximum amount within the isolated subspace. For which we see that there it exists a profound connection between uncertainty and entropy.

#### **7. Temporal (t > 0) uncertainty**

Since it is our universe governs the science and it is not the science governs our universe. Therefore, every principle within our universe has to comply with the temporal (t > 0) condition within our universe; otherwise, the principle cannot be existed within our universe. Which includes all the laws, principles and theories; such as Maxwell's Electro-Magnetic theory, Boltzmann's entropy theory, Einstein's relativity theory, Bohr's atomic model, Schrödinger's superposition principle, and others. Of which uncertainty principle cannot be the exception?

Let us now assumed a temporal (t > 0) particle m(t) is situated within a temporal (t > 0) subspace as depicted in **Figure 9**. Strictly speaking any particle existed within a temporal subspace must be a temporal (t > 0) particle; otherwise, the particle cannot be existed within our temporal (t > 0) universe.

**Figure 9.**

*A temporal (t > 0) particle m(t) within a temporal (t > 0) subspace. r is the radial direction. Note: it is a "physical realizable" paradigm, since a temporal particle m(t) is embedded within a temporal subspace.*

For simplicity, we further assume m(t) has no time or "pseudo-timeless," after all science is a law of approximation. The same as Heisenberg's assumption, the particle is a photonic particle (i.e., a photon), as from wave particle-duality standpoint [17] momentum of a photon is given by:

$$\mathbf{p} = \mathbf{h}/\lambda = \mathbf{h} \,\mathrm{v}/\mathbf{c} \tag{9}$$

where (t > 0) denotes that uncertainty principle is complied with the temporal (t > 0) condition within our universe. In view of either conservation of momentum or energy conservation, we see that position error Δr increases naturally with time. Which shows that momentum error Δp "decreases" naturally with bandwidth Δυ, as in contrast with Heisenberg's assumption; momentum error Δp has "nothing" to do with the changes of Δυ. This is precisely the "law of uncertainty" as I have described earlier, uncertainty of an isolated particle increases naturally with time. Since the increase in position error Δr is due to time, it must be due to the dynamic expansion of our universe [5]. For example, as the boundary of our universe constantly expanding at the speed of light, by virtue of energy conservation, it changes every dynamic aspect within our universe. As time moves on naturally, the larger the position error Δr increases with respect to that starting point, as illus-

*Position error Δr (i.e., sphere of Δr) enlarges naturally with time within a temporal (t > 0) subspace: Δr*

*represents a position error of the particle, at various locations as time moves constantly.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

Therefore we see that uncertainty is "not" a static process it is a temporal (t > 0) dynamic principle, as in contrast with Heisenberg's position error Δr is "independent" with time and his momentum error Δp is "independent" with Δυ. In which we see that if there is no time, there is no uncertainty and no probability. Nevertheless, each of the uncertainty unit or cell, such as (Δp, Δx), (ΔE, Δt) and (Δυ, Δt) is self-contained. In other words, ΔE and Δt are coexisted which they can be bilateral traded, but under the constraint of time as a forward moving dependent variable. In other words, if a section of Δt has been used, we cannot get the "same" section back, but can exchange for a different section of Δt. In which we see that we can trade for a narrower Δt with a wider ΔE or wider Δt with a narrower ΔE. But we "cannot" trade Δt for ΔE, since Δt is a real dependent variable has "no"

One of the important aspects of "temporal uncertainty" is that subspace within

our universe is a temporal (t > 0) uncertain "subspace." In other words, any subspace is a temporal (t > 0) stochastic subspace, such that the dynamic behavior of the subspace changes "dependently" with time. In which any change within our universe has a profound connection with the constant expanding universe. In which we have shown that uncertainty increases naturally with time, even though without

trated in **Figure 10**.

**Figure 10.**

substance to manipulate.

**8. Certainty principle**

**151**

where h is the Planck's constant, λ is the wavelength and υ is the frequency of the photonic particle. As I have mentioned earlier, within our universe any radiator has to be band limited. Thus the momentum error is naturally due changes of bandwidth Δυ, as given by;

$$
\Delta \mathbf{p} = \mathbf{h} \,\Delta \mathbf{v} / \mathbf{c} \tag{10}
$$

Instead of using a cone of light as Heisenberg had postulated. By virtue of timebandwidth product Δυ Δt = 1, Δυ "decreases" with time. For which position error can be written as:

$$
\Delta \mathbf{r} = \mathbf{c} \,\Delta \mathbf{t} \tag{11}
$$

where r is the radial distance, we have the following uncertainty relationship;

$$
\Delta \mathbf{p} \,\,\Delta \mathbf{r} = \begin{bmatrix} \mathbf{h} \,\,\Delta \mathbf{u} / \mathbf{c} \end{bmatrix} \begin{bmatrix} \mathbf{c} \,\,\Delta \mathbf{t} \end{bmatrix} = \mathbf{h} \,\,\Delta \mathbf{u} \,\,\Delta \mathbf{t} \tag{12}
$$

In which we see that; Δυ � Δt is the "time-bandwidth" product. As we imposed the optimum energy transfer criterion on time-bandwidth product [12], as given by:

$$
\Delta \mathbf{v} \,\,\Delta \mathbf{t} \ge \mathbf{1} \,\tag{13}
$$

Since lower bound for a photonic particle is limited by Planck's constant, we have the following equivalent form as given by:

$$
\Delta \mathbf{E} \,\,\Delta \mathbf{t} \ge \mathbf{h} \tag{14}
$$

Nevertheless, in view of Eq. (13), momentum uncertainty principle can be shown as:

$$
\Delta \mathbf{p} \,\,\Delta \mathbf{r} \ge \mathbf{h}, \mathbf{t} > \mathbf{0} \tag{15}
$$

#### **Figure 10.**

For simplicity, we further assume m(t) has no time or "pseudo-timeless," after all science is a law of approximation. The same as Heisenberg's assumption, the particle is a photonic particle (i.e., a photon), as from wave particle-duality stand-

*A temporal (t > 0) particle m(t) within a temporal (t > 0) subspace. r is the radial direction. Note: it is a "physical realizable" paradigm, since a temporal particle m(t) is embedded within a temporal subspace.*

where h is the Planck's constant, λ is the wavelength and υ is the frequency of the photonic particle. As I have mentioned earlier, within our universe any radiator has to be band limited. Thus the momentum error is naturally due changes of band-

Instead of using a cone of light as Heisenberg had postulated. By virtue of timebandwidth product Δυ Δt = 1, Δυ "decreases" with time. For which position error

where r is the radial distance, we have the following uncertainty relationship;

In which we see that; Δυ � Δt is the "time-bandwidth" product. As we imposed the optimum energy transfer criterion on time-bandwidth product [12], as given by:

Since lower bound for a photonic particle is limited by Planck's constant, we

Nevertheless, in view of Eq. (13), momentum uncertainty principle can be

p ¼ h*=*λ ¼ h υ*=*c (9)

Δp ¼ h Δυ*=*c (10)

Δr ¼ c Δt (11)

Δυ Δt≥1 (13)

ΔE Δt≥h (14)

Δp Δr≥h, t >0 (15)

Δp Δr ¼ ½h Δυ*=*c� ½c Δt� ¼ h Δυ Δt (12)

point [17] momentum of a photon is given by:

have the following equivalent form as given by:

width Δυ, as given by;

**Figure 9.**

*Quantum Mechanics*

can be written as:

shown as:

**150**

*Position error Δr (i.e., sphere of Δr) enlarges naturally with time within a temporal (t > 0) subspace: Δr represents a position error of the particle, at various locations as time moves constantly.*

where (t > 0) denotes that uncertainty principle is complied with the temporal (t > 0) condition within our universe. In view of either conservation of momentum or energy conservation, we see that position error Δr increases naturally with time. Which shows that momentum error Δp "decreases" naturally with bandwidth Δυ, as in contrast with Heisenberg's assumption; momentum error Δp has "nothing" to do with the changes of Δυ. This is precisely the "law of uncertainty" as I have described earlier, uncertainty of an isolated particle increases naturally with time.

Since the increase in position error Δr is due to time, it must be due to the dynamic expansion of our universe [5]. For example, as the boundary of our universe constantly expanding at the speed of light, by virtue of energy conservation, it changes every dynamic aspect within our universe. As time moves on naturally, the larger the position error Δr increases with respect to that starting point, as illustrated in **Figure 10**.

Therefore we see that uncertainty is "not" a static process it is a temporal (t > 0) dynamic principle, as in contrast with Heisenberg's position error Δr is "independent" with time and his momentum error Δp is "independent" with Δυ. In which we see that if there is no time, there is no uncertainty and no probability. Nevertheless, each of the uncertainty unit or cell, such as (Δp, Δx), (ΔE, Δt) and (Δυ, Δt) is self-contained. In other words, ΔE and Δt are coexisted which they can be bilateral traded, but under the constraint of time as a forward moving dependent variable. In other words, if a section of Δt has been used, we cannot get the "same" section back, but can exchange for a different section of Δt. In which we see that we can trade for a narrower Δt with a wider ΔE or wider Δt with a narrower ΔE. But we "cannot" trade Δt for ΔE, since Δt is a real dependent variable has "no" substance to manipulate.

#### **8. Certainty principle**

One of the important aspects of "temporal uncertainty" is that subspace within our universe is a temporal (t > 0) uncertain "subspace." In other words, any subspace is a temporal (t > 0) stochastic subspace, such that the dynamic behavior of the subspace changes "dependently" with time. In which any change within our universe has a profound connection with the constant expanding universe. In which we have shown that uncertainty increases naturally with time, even though without any other perturbation or human intervention. Similar to the myth of Boltzmann's entropy theory [16], entropy increases naturally with time within an enclosed subspace, which has been shown is related to the expanding universe [5].

Similarly, there is a profound "connection" between coherence theory [18] and "certainty" principle as I shall address. Nevertheless, it is always a myth of coherence, as refer to **Figure 11**, where coherence theory can be easily understood by Young's experiment. In which degree of coherence can be determined by the "visibility" equation as given by:

$$\nu = \frac{\text{Imax-Imin}}{\text{Imax} + \text{Imin}} \tag{16}$$

In view of this plot, we see that when bandwidth Δυ decreases, a larger certainty subspace enlarges "exponentially" since the volume of the subspace is given by:

<sup>3</sup> (21)

Certainty subspace ¼ ð4π*=*3ÞðΔrÞ

In which we see that, a very "large" certainty subspace can be realized within our universe which is within limited Planck's constant h as depicted in **Figure 13**, where we see a steady state radiator A emits a continuous band limited Δυ electromagnetic wave as illustrated. A "certainty subspace" with respect to an assumed "photonic particle" A for a give Δt can be defined as illustrated within r = c Δt, where Δt = 1/Δυ. In other words, it has a high degree of certainty to relocate particle A within the certainty subspace. Nevertheless, from electro-magnetic disturbance standpoint; within the certainty subspace provides a high "degree of certainty" (i.e.,

As from coherence theory stand point, any other disturbances away from point A but within the certainty subspace (i.e., within r < c Δt) are mutual coherence (i.e., certainty) with respect A; where r = c Δt is the radius of the "certainty subspace" of A. In other words, any point-pair within d < c Δt, where Δt = 1/Δυ, are "mutual coherence" within a radiation subspace. On the other hand, distance

*A certainty subspace is embedded within uncertainty subspace. A is assumed a steady state photonic particle emits a band limited Δυ radiation, r is the radius with respect to the emitter A; and B represents the boundary of*

degree of coherence) as with respect to point A.

*A plot of position error Δr versus bandwidth Δυ.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

**Figure 12.**

**Figure 13.**

**153**

*certainty subspace of A.*

where Imax and Imin are the maximum and minimum intensities of the fringes. But the theory does not tell us where the physics comes from. For which, it can be understood from "certainty principle," as I shall address.

It is trivial that if there is an uncertainty principle, it is inevitable not to have a certainty principle. This means that, as photonic particle we are looking for is "likely" to be found within a "certainty" subspace. Since "perfect certainty" (or absolute uncertainty) occurs at t = 0, which is a timeless (t = 0) virtual subspace not exist within our universe. Nevertheless, "certainty principle" can be written in the following equivalent forms;

$$\Delta \mathbf{p} \,\,\Delta \mathbf{r} < \mathbf{h}, (\mathbf{t} > \mathbf{0}) \tag{17}$$

$$\Delta \mathbf{E} \,\Delta \mathbf{t} < \mathbf{h}, \left(\mathbf{t} > \mathbf{0}\right) \tag{18}$$

$$
\Delta \mathbf{v} \,\,\Delta \mathbf{t} < \mathbf{1}, \left(\mathbf{t} > \mathbf{0}\right) \tag{19}
$$

where (t > 0) denotes that equation is subjected to temporal (t > 0) constrain. In view of the position error Δr in Eq. (17), it means that it is "likely" the photonic particle can be found within the certainty subspace. Since the size of the subspace is limited by Planck constant h, it is normally used as limited boundary "not" to be violated. Yet within this limited boundary, certainty subspace had been exploited by Dennis Gabor for his discovery of wave front reconstruction in 1948 [19] and as well it was applied to synthetic aperture radar imaging in 1950s [20].

Since the size of certainty subspace is exponentially enlarging as the position error Δr increases, for which the "radius" of the certainty sub-sphere is given by:

$$
\Delta \mathbf{r} = \mathbf{c} \,\Delta \mathbf{t} = \mathbf{c}/(\Delta \mathbf{v}) \tag{20}
$$

where c is the speed of light, Δt is the time error, and Δυ is the bandwidth of a light source or a quantum leap hυ. Thus we see that position error Δr is inversely proportional to bandwidth Δυ, as plotted in **Figure 12**.

**Figure 11.**

*Young's experiment. Σ represents an extended monochromatic source, Q1 and Q2 are the pinholes, and "I" represents the irradiance distribution.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

any other perturbation or human intervention. Similar to the myth of Boltzmann's entropy theory [16], entropy increases naturally with time within an enclosed subspace, which has been shown is related to the expanding universe [5].

coherence, as refer to **Figure 11**, where coherence theory can be easily understood by Young's experiment. In which degree of coherence can be determined by the

<sup>ν</sup> <sup>¼</sup> Imax–Imin

where Imax and Imin are the maximum and minimum intensities of the fringes. But the theory does not tell us where the physics comes from. For which, it can be

It is trivial that if there is an uncertainty principle, it is inevitable not to have a

where (t > 0) denotes that equation is subjected to temporal (t > 0) constrain. In view of the position error Δr in Eq. (17), it means that it is "likely" the photonic particle can be found within the certainty subspace. Since the size of the subspace is limited by Planck constant h, it is normally used as limited boundary "not" to be violated. Yet within this limited boundary, certainty subspace had been exploited by Dennis Gabor for his discovery of wave front reconstruction in 1948 [19] and as

Since the size of certainty subspace is exponentially enlarging as the position error Δr increases, for which the "radius" of the certainty sub-sphere is given by:

where c is the speed of light, Δt is the time error, and Δυ is the bandwidth of a light source or a quantum leap hυ. Thus we see that position error Δr is inversely

*Young's experiment. Σ represents an extended monochromatic source, Q1 and Q2 are the pinholes, and "I"*

well it was applied to synthetic aperture radar imaging in 1950s [20].

proportional to bandwidth Δυ, as plotted in **Figure 12**.

certainty principle. This means that, as photonic particle we are looking for is "likely" to be found within a "certainty" subspace. Since "perfect certainty" (or absolute uncertainty) occurs at t = 0, which is a timeless (t = 0) virtual subspace not exist within our universe. Nevertheless, "certainty principle" can be written in the

Imax <sup>þ</sup> Imin (16)

Δp Δr<h, tð Þ >0 (17) ΔE Δt<h, tð Þ >0 (18) Δυ Δt<1, tð Þ >0 (19)

Δr ¼ c Δt ¼ c*=*ðΔυÞ (20)

"certainty" principle as I shall address. Nevertheless, it is always a myth of

understood from "certainty principle," as I shall address.

"visibility" equation as given by:

*Quantum Mechanics*

following equivalent forms;

**Figure 11.**

**152**

*represents the irradiance distribution.*

Similarly, there is a profound "connection" between coherence theory [18] and

**Figure 12.** *A plot of position error Δr versus bandwidth Δυ.*

In view of this plot, we see that when bandwidth Δυ decreases, a larger certainty subspace enlarges "exponentially" since the volume of the subspace is given by:

$$\text{Certainty subspace} = \left(4\pi/3\right) \left(\Delta \mathbf{r}\right)^{\mathfrak{3}} \tag{21}$$

In which we see that, a very "large" certainty subspace can be realized within our universe which is within limited Planck's constant h as depicted in **Figure 13**, where we see a steady state radiator A emits a continuous band limited Δυ electromagnetic wave as illustrated. A "certainty subspace" with respect to an assumed "photonic particle" A for a give Δt can be defined as illustrated within r = c Δt, where Δt = 1/Δυ. In other words, it has a high degree of certainty to relocate particle A within the certainty subspace. Nevertheless, from electro-magnetic disturbance standpoint; within the certainty subspace provides a high "degree of certainty" (i.e., degree of coherence) as with respect to point A.

As from coherence theory stand point, any other disturbances away from point A but within the certainty subspace (i.e., within r < c Δt) are mutual coherence (i.e., certainty) with respect A; where r = c Δt is the radius of the "certainty subspace" of A. In other words, any point-pair within d < c Δt, where Δt = 1/Δυ, are "mutual coherence" within a radiation subspace. On the other hand, distance

#### **Figure 13.**

*A certainty subspace is embedded within uncertainty subspace. A is assumed a steady state photonic particle emits a band limited Δυ radiation, r is the radius with respect to the emitter A; and B represents the boundary of certainty subspace of A.*

greater than r > c Δt from point A is a mutual "uncertainty" subspace with respect to A. In other words, any point-pair distance is larger than d > c Δt within the radiation space are mutually "incoherent." In which we see that; it is more "unlikely" to relocate a photonic particle, after it has been seen at point A, within a "certainty subspace."

This means that any point disturbance within in the certainty subspace has a strong certainty (or coherence) with respect to point B disturbance. Similarly if we pick an arbitrary point A, then a certainty subspace of A can be defined as illustrated in the figure, of which we see that a portion is overlapped with certainty subspace of B. Any other disturbances outside the corresponding subspaces of certainty A, B, and C are the uncertainty subspace. It is trivial to see that a number of configurations of certainty subspaces can be designed for application. In which we see that multi wavelengths, such as Δυ1, Δυ2, and Δυ3, can also be simultaneously implemented to create various certainty subspace configurations, such as for multi spectral imaging

One of the commonly used for producing certainty subspaces for complex wave

∣d1–d2 ∣< c Δt ¼ c*=*Δυ (22)

Δr ¼ ∣d1–d2 ∣<c Δt ¼ c*=*Δυ (23)

front reconstruction is depicted in **Figure 15** [21]. In which we see that a band limited Δυ laser is employed, where a beam of light is split-up by a splitter BS. One beam B2 is directly impinging on a photographic plate at plane P and other beam B1 diverted by a mirror and then is combined with beam B2 at the same spot on the photographic plate P. It is trivial to know that if the difference in distances between these two beams is within the certainty subspace, then B1 and B2 are "mutually" coherence (or certainty); otherwise, they are mutually incoherence (or uncertain). In which we see that the distance between B1 and B2 is required as given by:

where d1 and d2 are the distances of bean B2 and B2, respectively, from the splitter BS. In which we see that radius of certainty subspace of BS is written by;

where |d1 – d2 | = c/Δυ is the "coherent length" of the laser. In which we see that by simply reducing the bandwidth Δυ, a lager certainty subspace can be created

*An example of exploiting certainty subspace for wave front reconstruction. BS, beam splitter; P, photographic*

or information processing application.

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

within a temporal (t > 0) subspace.

**Figure 15.**

*plate.*

**155**

Since certainty subspace represents a "global" probabilistic distribution of a particle's location as from particle physicists stand point, which means that it is "very likely" the particle can be found within the certainty subspace. In which we see that a postulated particle firstly is temporal (t > 0) or has time; otherwise, there is no reason to search for it. Then after it has been proven it is a temporal (i.e., m(t)) particle, it is more favorable to search the particle, within a certainty subspace.

The essence of "wave-particle duality" is a mathematical simplistic assumption to equivalence a package of wavelet energy as a particle in motion from statistical mechanics stand point, in which the momentum p = h/λ is conserved. However one should "not" treated wave as particle or particle as wave. It is the package of wavelet energy "equivalent" to a particle dynamics (i.e., photon), but they are "not" equaled. Similar to Einstein's energy equation, mass is equivalent to energy and energy is equivalent to mass, but they are not equaled. Therefore as from energy conservation, bandwidth Δυ "decreases" with time is the physical reality instead of treating a package of wavelet as a particle (i.e., photon), which was due to the classical mechanics standpoint, treats quantum leap momentum p = h/λ. In which we see that photon is a "virtual" particle although many quantum scientists have been regarded photon as a physical particle?

We further note that any point-pair within the certainty subspace exhibits some degree of certainty or coherence, which has been known as "mutual coherence" [18]. And the mutual coherence can be easily understood as depicted in **Figure 14**, in which a steady state band limited Δυ electro-magnetic wave is assumed existed within a temporal (t > 0) subspace. As we pick an arbitrary disturbance at point B, a certainty subspace of B can be determined within r ≤ c Δt, as shown in the figure.

#### **Figure 14.**

*Various certainty subspace configurations, as with respect to various disturbances within a steady state band limited Δυ electro-magnetic environment within a temporal (t > 0) subspace.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

greater than r > c Δt from point A is a mutual "uncertainty" subspace with respect to A. In other words, any point-pair distance is larger than d > c Δt within the radiation space are mutually "incoherent." In which we see that; it is more

"unlikely" to relocate a photonic particle, after it has been seen at point A, within a

Since certainty subspace represents a "global" probabilistic distribution of a particle's location as from particle physicists stand point, which means that it is "very likely" the particle can be found within the certainty subspace. In which we see that a postulated particle firstly is temporal (t > 0) or has time; otherwise, there is no reason to search for it. Then after it has been proven it is a temporal (i.e., m(t)) particle, it is more favorable to search the particle, within a certainty

The essence of "wave-particle duality" is a mathematical simplistic assumption to equivalence a package of wavelet energy as a particle in motion from statistical mechanics stand point, in which the momentum p = h/λ is conserved. However one should "not" treated wave as particle or particle as wave. It is the package of wavelet energy "equivalent" to a particle dynamics (i.e., photon), but they are "not" equaled. Similar to Einstein's energy equation, mass is equivalent to energy and energy is equivalent to mass, but they are not equaled. Therefore as from energy conservation, bandwidth Δυ "decreases" with time is the physical reality instead of treating a package of wavelet as a particle (i.e., photon), which was due to the classical mechanics standpoint, treats quantum leap momentum p = h/λ. In which we see that photon is a "virtual" particle although many quantum scientists have

We further note that any point-pair within the certainty subspace exhibits some degree of certainty or coherence, which has been known as "mutual coherence" [18]. And the mutual coherence can be easily understood as depicted in **Figure 14**, in which a steady state band limited Δυ electro-magnetic wave is assumed existed within a temporal (t > 0) subspace. As we pick an arbitrary disturbance at point B, a certainty subspace of B can be determined within r ≤ c Δt, as shown in the figure.

*Various certainty subspace configurations, as with respect to various disturbances within a steady state band*

*limited Δυ electro-magnetic environment within a temporal (t > 0) subspace.*

"certainty subspace."

*Quantum Mechanics*

been regarded photon as a physical particle?

subspace.

**Figure 14.**

**154**

This means that any point disturbance within in the certainty subspace has a strong certainty (or coherence) with respect to point B disturbance. Similarly if we pick an arbitrary point A, then a certainty subspace of A can be defined as illustrated in the figure, of which we see that a portion is overlapped with certainty subspace of B. Any other disturbances outside the corresponding subspaces of certainty A, B, and C are the uncertainty subspace. It is trivial to see that a number of configurations of certainty subspaces can be designed for application. In which we see that multi wavelengths, such as Δυ1, Δυ2, and Δυ3, can also be simultaneously implemented to create various certainty subspace configurations, such as for multi spectral imaging or information processing application.

One of the commonly used for producing certainty subspaces for complex wave front reconstruction is depicted in **Figure 15** [21]. In which we see that a band limited Δυ laser is employed, where a beam of light is split-up by a splitter BS. One beam B2 is directly impinging on a photographic plate at plane P and other beam B1 diverted by a mirror and then is combined with beam B2 at the same spot on the photographic plate P. It is trivial to know that if the difference in distances between these two beams is within the certainty subspace, then B1 and B2 are "mutually" coherence (or certainty); otherwise, they are mutually incoherence (or uncertain). In which we see that the distance between B1 and B2 is required as given by:

$$|\mathbf{d}\_1 \mathbf{-d}\_2| < \mathbf{c} \,\,\Delta \mathbf{t} = \mathbf{c}/\Delta \mathbf{v} \tag{22}$$

where d1 and d2 are the distances of bean B2 and B2, respectively, from the splitter BS. In which we see that radius of certainty subspace of BS is written by;

$$
\Delta \mathbf{r} = |\mathbf{d}\_1 \mathbf{-d}\_2| < \mathbf{c} \,\Delta \mathbf{t} = \mathbf{c}/\Delta \mathbf{v} \tag{23}
$$

where |d1 – d2 | = c/Δυ is the "coherent length" of the laser. In which we see that by simply reducing the bandwidth Δυ, a lager certainty subspace can be created within a temporal (t > 0) subspace.

**Figure 15.** *An example of exploiting certainty subspace for wave front reconstruction. BS, beam splitter; P, photographic plate.*

#### **9. Essence of certainty principle**

Since every substance or subspace within our universe was created by an amount of energy ΔE and a section of time Δt [i.e., (ΔE, Δt)], any changes of ΔE changes the size of certainty subspace Δr. This is a topic that astrophysicists may be interested. Similarly to particle physicists, subatomic particle has to be temporal (t > 0); otherwise, the particle must be a virtual particle cannot exist within our universe. Secondly, it is more "likely" a temporal (t > 0) particle to be found within its certainty subspace; otherwise, it will be searching a timeless (t = 0) particle "forever" within our temporal (t > 0) universe. In view of the certainty unit: ΔE and Δt are mutually coexisted in which time is a forward dependent variable. Any changes of ΔE can "only" happen with an expenditure of a section time Δt, but it "cannot" change the speed of time. Since the energy is "conserved," Δt is a section of time required to have the amount of ΔE within a certainty unit of (ΔE, Δt). In other words, ΔE and Δt can be traded; for example, a wider variance of ΔE is traded for a narrower Δt.

high degree of certainty within a certainty subspace between points. This means that, if a photonic particle as it has been started at point u1, then it has a high degree of certainty that the particle to be found at the next instantly Δt at u2, since distance

For example, given any two arbitrary complex disturbances u1(r1; t) and u2(r2; t), as long the separation between them is shorter than the radius Δr of the

ally coherence). For which the "degree of certainty" (i.e., degree of coherence)

between u1 and u2 can be determined by the following equation:

Γ12ð Þ¼ *Δt* lim *T*!∞

Γ11ð Þ¼ *Δt* lim *T*!∞

Γ22ð Þ¼ *Δt* lim *T*!∞

γ12ð Þ¼ *Δt*

1 *T* ð*T* 0

1 *T* ð*T* 0

1 *T* ð*T* 0

the disturbances between u1(r1; t) and u2(r2; t) are "certainly" related (or mutu-

where, "mutual certainty" (or mutual coherence) function between u1 and u2

Similarly, the respective "self certainty" (or self coherence) functions are,

*Mutual certainty within a certainty subspace. u1(r1; t) and u2(r2; t) represent two arbitrary disturbances*

*<sup>u</sup>*1ð Þ *<sup>t</sup>*;*r*<sup>1</sup> *<sup>u</sup>*<sup>∗</sup>

*<sup>u</sup>*1ð Þ *<sup>t</sup>*;*r*<sup>1</sup> *<sup>u</sup>*<sup>∗</sup>

*<sup>u</sup>*2ð Þ *<sup>t</sup>*;*r*<sup>2</sup> *<sup>u</sup>*<sup>∗</sup>

Γ12ð Þ *Δt*

d≤c*=*ðΔυÞ (24)

<sup>Γ</sup>11ð Þ <sup>0</sup> <sup>Γ</sup>22ð Þ *<sup>Δ</sup>*<sup>0</sup> (25)

<sup>2</sup> ð Þ *t* � Δ*t*;*r*<sup>1</sup> *dt* (26)

<sup>1</sup> ð Þ *t* � Δ*t*;*r*<sup>1</sup> *dt* (27)

<sup>2</sup> ð Þ *t* � Δ*t*;*r*<sup>2</sup> *dt* (28)

is time within a temporal (t > 0) subspace.

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

certainty subspace as given by:

can be written as:

respectively, given by:

**Figure 17.**

**157**

*separated at distance d.*

Nevertheless, time has been treated as an "independent" variable for decades, as normally assumed by scientists. But whenever a section of time Δt has been used, it is not possible to bring back the "original" moment of Δt, even though it is possible to reproduce the same section of Δt. This similar as we reconstructed a damaged car, but we cannot bring back the "original" car that has been crashed. And this is precisely the "price of time" to pay for everything within our universe. Then my question is that if time is a forward dependent variable with respect to its subspace, how can we "curve" the space with time? Similarly, we are coexisted with time, how can we get back the moment of time that has passed by?

Since certainty subspace changes with bandwidth Δυ as illustrated in **Figure 16**, in which we see that as bandwidth Δυ decreases a very large certainty subspace can be created within our universe as depicted in **Figure 16(a)**–**(c)**.

High resolution observation requires shorter wavelength but shorter wavelength inherently has broader bandwidth Δυ that creates a smaller certainty subspace, which can be used for high resolution wave front reconstruction [21]. On the other hand, for a larger certainty subspace, it required a narrower bandwidth of Δυ which has a larger certainty subspace for exploitation, such as applied to side looking radar imaging [20]. In which we see that the size of the certainty subspace can be manipulated by the bandwidth Δυ as will be shown in the following:

Since narrower bandwidth Δυ offers a huge certainty subspace that can be exploited for long distance communication, in which I have found that the certainty subspace is "in fact" the coherence subspace as I have discussed in the preceding. In other words, within a certainty subspace it exhibits a "point-pair certainty" or coherent property among them as illustrated in **Figure 17**. In other words, it has a

#### **Figure 16.**

*Size of certainty subspace enlarges rapidly as band width Δυ narrows. (a) shows a very small size of certainty subspace as the result of Δυ approaching to very wide. (b) shows the size of certainty subspace reduces as Δυ continues to reduce. And (c) shows a huge size certainty subspace can be created as band width Δυ narrows.*

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

**9. Essence of certainty principle**

for a narrower Δt.

*Quantum Mechanics*

**Figure 16.**

**156**

Since every substance or subspace within our universe was created by an amount of energy ΔE and a section of time Δt [i.e., (ΔE, Δt)], any changes of ΔE changes the size of certainty subspace Δr. This is a topic that astrophysicists may be interested. Similarly to particle physicists, subatomic particle has to be temporal (t > 0); otherwise, the particle must be a virtual particle cannot exist within our universe. Secondly, it is more "likely" a temporal (t > 0) particle to be found within its certainty subspace; otherwise, it will be searching a timeless (t = 0) particle "forever" within our temporal (t > 0) universe. In view of the certainty unit: ΔE and Δt are mutually coexisted in which time is a forward dependent variable. Any changes of ΔE can "only" happen with an expenditure of a section time Δt, but it "cannot" change the speed of time. Since the energy is "conserved," Δt is a section of time required to have the amount of ΔE within a certainty unit of (ΔE, Δt). In other words, ΔE and Δt can be traded; for example, a wider variance of ΔE is traded

Nevertheless, time has been treated as an "independent" variable for decades, as normally assumed by scientists. But whenever a section of time Δt has been used, it is not possible to bring back the "original" moment of Δt, even though it is possible to reproduce the same section of Δt. This similar as we reconstructed a damaged car, but we cannot bring back the "original" car that has been crashed. And this is precisely the "price of time" to pay for everything within our universe. Then my question is that if time is a forward dependent variable with respect to its subspace, how can we "curve" the space with time? Similarly, we are coexisted with time, how

Since certainty subspace changes with bandwidth Δυ as illustrated in **Figure 16**, in which we see that as bandwidth Δυ decreases a very large certainty subspace can

High resolution observation requires shorter wavelength but shorter wavelength

inherently has broader bandwidth Δυ that creates a smaller certainty subspace, which can be used for high resolution wave front reconstruction [21]. On the other hand, for a larger certainty subspace, it required a narrower bandwidth of Δυ which has a larger certainty subspace for exploitation, such as applied to side looking radar

imaging [20]. In which we see that the size of the certainty subspace can be manipulated by the bandwidth Δυ as will be shown in the following:

Since narrower bandwidth Δυ offers a huge certainty subspace that can be exploited for long distance communication, in which I have found that the certainty subspace is "in fact" the coherence subspace as I have discussed in the preceding. In other words, within a certainty subspace it exhibits a "point-pair certainty" or coherent property among them as illustrated in **Figure 17**. In other words, it has a

*Size of certainty subspace enlarges rapidly as band width Δυ narrows. (a) shows a very small size of certainty subspace as the result of Δυ approaching to very wide. (b) shows the size of certainty subspace reduces as Δυ continues to reduce. And (c) shows a huge size certainty subspace can be created as band width Δυ narrows.*

can we get back the moment of time that has passed by?

be created within our universe as depicted in **Figure 16(a)**–**(c)**.

high degree of certainty within a certainty subspace between points. This means that, if a photonic particle as it has been started at point u1, then it has a high degree of certainty that the particle to be found at the next instantly Δt at u2, since distance is time within a temporal (t > 0) subspace.

For example, given any two arbitrary complex disturbances u1(r1; t) and u2(r2; t), as long the separation between them is shorter than the radius Δr of the certainty subspace as given by:

$$\mathbf{d} \le \mathbf{c} / (\Delta \mathbf{v}) \tag{24}$$

the disturbances between u1(r1; t) and u2(r2; t) are "certainly" related (or mutually coherence). For which the "degree of certainty" (i.e., degree of coherence) between u1 and u2 can be determined by the following equation:

$$\gamma\_{12}(\Delta t) = \frac{\Gamma\_{12}(\Delta t)}{\Gamma\_{11}(\mathbf{0})\Gamma\_{22}(\Delta \mathbf{0})} \tag{25}$$

where, "mutual certainty" (or mutual coherence) function between u1 and u2 can be written as:

$$\Gamma\_{12}(\Delta t) = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T u\_1(t; r\_1) u\_2^\*\left(t - \Delta t; r\_1\right) dt \tag{26}$$

Similarly, the respective "self certainty" (or self coherence) functions are, respectively, given by:

$$\Gamma\_{11}(\Delta t) = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T u\_1(t; r\_1) u\_1^\* \left(t - \Delta t; r\_1 \right) dt \tag{27}$$

$$\Gamma\_{22}(\Delta t) = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T u\_2(t; r\_2) u\_2^\*\left(t - \Delta t; r\_2\right) dt \tag{28}$$

**Figure 17.**

*Mutual certainty within a certainty subspace. u1(r1; t) and u2(r2; t) represent two arbitrary disturbances separated at distance d.*

and fictitious, and many of their animations are "not" physically real; for example such as the "instantaneous and simultaneous" superimposing principle for quantum computing is "not" actually existed within our universe. One of the important aspects within our universe is that, one cannot get something from nothing there is always a price to pay, an amount of energy ΔE and a section of time Δt. The

Since any science existed within our universe has time or temporal (t > 0), in which we see that any scientific law, principle, theory, and paradox has to comply with temporal (t > 0) aspect within our universe; otherwise, it may not be science. As we know that science is mathematics but mathematics is not equaled to science. In which we have shown that any analytic solution has to be temporal (t > 0); otherwise, it cannot be implemented within our universe. Which includes all the

Since it is our universe governs our science and it is not our science changes our universe. In which we have shown every hypothetical science, law, principle, and theory has be temporal (t > 0); otherwise, they are virtual and fictitious which cannot exist within our universe. Since time is a dependent variable coexisted with space, we have concluded that time is not an illusion but real, since we are real. As in contrast with most of the scientists, they believe that time is an independent

Uncertainty principle is one of the most fascinating principles in quantum mechanics, yet Heisenberg principle was based on diffraction limited observation, it is not due to the nature of time or temporal (t > 0) nature of our universe. We have shown uncertainty increases with time, as in contrast with Heisenberg's principle. We have also introduced a certainty principle, in which we have shown high degree of certainty within a certainty subspace can be exploited. For which we have shown that certainty subspace can be created within our temporal subspace for complex amplitude communication and imaging. Yet the important aspect of this chapter is that it is not how rigorous the mathematics is, but it is the physical realizably of

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

important is that they are not free!

*Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

laws, principles, and theories have to be temporal (t > 0)?

variable and some of them even believe that time is an illusion?

science is, since mathematics is not science.

Penn State University, University Park, PA, USA

\*Address all correspondence to: fty1@psu.edu

provided the original work is properly cited.

**Author details**

Francis T.S. Yu

**159**

**Figure 18.** *Side-looking radar imaging within certainty subspace: (a) shows a side-looking radar scanning flight path; (b) shows an example of synthetic aperture radar imagery.*

One of the interesting applications for certainty principle must be to synthetic aperture radar imaging as I have mentioned earlier is shown in **Figure 18**. In which we see an aircraft carried a side looking synthetic radar system shown in **Figure 18(a)**, emitting a sequence of radar pulses scanned across the flight path of the terrain. The returned pulses are combined with local radar pulses, which are "mutual coherence" (i.e., high degree of certainty), to construct a recording format that can be used for imaging the terrain, for which a synthetic imagery is shown in **Figure 18(b)**. In which we see a variety of scatters, including city streets, wooded areas, and farmlands and lake with some broken ice floes can also be identified on the right of this image. Since microwave antenna has a very narrow carrier bandwidth (i.e., Δυ) and its certainty radius (i.e., d = cΔt) or the coherence length can be easily reached to hundreds of thousand feet. In other words, a very large certainty subspace for complex-amplitude imaging (or for communication) can be realized.

Finally I would address again within the certainty unite (Δp, Δr) [i.e., equivalently for (ΔE, Δt) and (Δυ, t) unit] can be mutually traded. But it is the trading of Δp for Δr (or ΔE for Δt and Δυ for Δt) is physically visible, since time is not a physical substance but a forward constant dependent "variable" that we "cannot" manipulate. For which we see that the "section" of Δt that has been "used" cannot get it back. In other words, we can get back the same amount Δt, but "not" the same moment of Δt, that has been expensed. As I have shown earlier, everything within our universe has a price, an amount of energy ΔE, and a section of time Δt. Aside ΔE we can physically change, it is the moment of time Δt which has been expensed that is "preventing" us to get it back, because that moment of Δt is the "same moment" of time of our temporal (t > 0) universe that has been passed. And this is the "moment of time" Δt within our temporal (t > 0) universe, once the "moment" passes by and we can never able to get it back.

#### **10. Conclusion**

In conclusion, I would point out that quantum scientists used amazing mathematical analyses added with their fantastic computer simulations provide very convincing results. But mathematical analyses and computer animations are virtual *Nature of Temporal (t > 0) Quantum Theory: Part I DOI: http://dx.doi.org/10.5772/intechopen.93561*

and fictitious, and many of their animations are "not" physically real; for example such as the "instantaneous and simultaneous" superimposing principle for quantum computing is "not" actually existed within our universe. One of the important aspects within our universe is that, one cannot get something from nothing there is always a price to pay, an amount of energy ΔE and a section of time Δt. The important is that they are not free!

Since any science existed within our universe has time or temporal (t > 0), in which we see that any scientific law, principle, theory, and paradox has to comply with temporal (t > 0) aspect within our universe; otherwise, it may not be science. As we know that science is mathematics but mathematics is not equaled to science. In which we have shown that any analytic solution has to be temporal (t > 0); otherwise, it cannot be implemented within our universe. Which includes all the laws, principles, and theories have to be temporal (t > 0)?

Since it is our universe governs our science and it is not our science changes our universe. In which we have shown every hypothetical science, law, principle, and theory has be temporal (t > 0); otherwise, they are virtual and fictitious which cannot exist within our universe. Since time is a dependent variable coexisted with space, we have concluded that time is not an illusion but real, since we are real. As in contrast with most of the scientists, they believe that time is an independent variable and some of them even believe that time is an illusion?

Uncertainty principle is one of the most fascinating principles in quantum mechanics, yet Heisenberg principle was based on diffraction limited observation, it is not due to the nature of time or temporal (t > 0) nature of our universe. We have shown uncertainty increases with time, as in contrast with Heisenberg's principle. We have also introduced a certainty principle, in which we have shown high degree of certainty within a certainty subspace can be exploited. For which we have shown that certainty subspace can be created within our temporal subspace for complex amplitude communication and imaging. Yet the important aspect of this chapter is that it is not how rigorous the mathematics is, but it is the physical realizably of science is, since mathematics is not science.

#### **Author details**

One of the interesting applications for certainty principle must be to synthetic

*Side-looking radar imaging within certainty subspace: (a) shows a side-looking radar scanning flight path;*

Finally I would address again within the certainty unite (Δp, Δr) [i.e., equivalently for (ΔE, Δt) and (Δυ, t) unit] can be mutually traded. But it is the trading of Δp for Δr (or ΔE for Δt and Δυ for Δt) is physically visible, since time is not a physical substance but a forward constant dependent "variable" that we "cannot" manipulate. For which we see that the "section" of Δt that has been "used" cannot get it back. In other words, we can get back the same amount Δt, but "not" the same moment of Δt, that has been expensed. As I have shown earlier, everything within our universe has a price, an amount of energy ΔE, and a section of time Δt. Aside ΔE we can physically change, it is the moment of time Δt which has been expensed that is "preventing" us to get it back, because that moment of Δt is the "same moment" of time of our temporal (t > 0) universe that has been passed. And this is the "moment of time" Δt within our temporal (t > 0) universe, once the "moment"

In conclusion, I would point out that quantum scientists used amazing mathematical analyses added with their fantastic computer simulations provide very convincing results. But mathematical analyses and computer animations are virtual

aperture radar imaging as I have mentioned earlier is shown in **Figure 18**. In which we see an aircraft carried a side looking synthetic radar system shown in **Figure 18(a)**, emitting a sequence of radar pulses scanned across the flight path of the terrain. The returned pulses are combined with local radar pulses, which are "mutual coherence" (i.e., high degree of certainty), to construct a recording format that can be used for imaging the terrain, for which a synthetic imagery is shown in **Figure 18(b)**. In which we see a variety of scatters, including city streets, wooded areas, and farmlands and lake with some broken ice floes can also be identified on the right of this image. Since microwave antenna has a very narrow carrier bandwidth (i.e., Δυ) and its certainty radius (i.e., d = cΔt) or the coherence length can be easily reached to hundreds of thousand feet. In other words, a very large certainty subspace for

complex-amplitude imaging (or for communication) can be realized.

passes by and we can never able to get it back.

*(b) shows an example of synthetic aperture radar imagery.*

**10. Conclusion**

**158**

**Figure 18.**

*Quantum Mechanics*

Francis T.S. Yu Penn State University, University Park, PA, USA

\*Address all correspondence to: fty1@psu.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Yu FTS. From relativity to discovery of temporal (t > 0) universe. In: Origin of Temporal (t > 0) Universe: Correcting with Relativity, Entropy, Communication and Quantum Mechanics, Chapter 1. New York: CRC Press; 2019. pp. 1-26

[2] Bunge M. Causality: The Place of the Causal Principle in Modern Science. Cambridge: Harvard University Press; 1959

[3] Yu FTS. The fate of Schrodinger's cat. Asian Journal of Physics. 2019;**28**(1): 63-70

[4] Knudsen JM, Hjorth P. Elements of Newtonian Mechanics. Heidelberg: Springer Science & Business Media; 2012

[5] Yu FTS. Time: The enigma of space. Asian Journal of Physics. 2017;**26**(3): 143-158

[6] Yu FTS. What is "wrong" with current theoretical physicists? In: Bulnes F, Stavrou VN, Morozov O, Bourdine AV, editors. Advances in Quantum Communication and Information, Chapter 9. London: IntechOpen; 2020. pp. 123-143

[7] Parzen E. Stochastic Processes. San Francisco: Holden Day, Inc.; 1962

[8] Einstein Attacks Quantum Theory. Scientist and Two Colleagues Find It Is Not 'Complete' Even though 'Correct'. New York City: The New York Times; 1935

[9] Einstein A. Relativity, the Special and General Theory. New York: Crown Publishers; 1961

[10] Hawking S, Penrose R. The Nature of Space and Time. New Jersey: Princeton University Press; 1996

[11] Kraus JD. Electro-Magnetics. New York: McGraw-Hill Book Company; 1953. p. 370

**Chapter 9**

**Abstract**

*Francis T.S. Yu*

(t = 0) as mathematic does.

Schrödinger's cat

**1. Introduction**

**161**

Nature of Temporal (t > 0)

Since Schrödinger's quantum mechanics developed from Hamiltonian, I will show that his quantum machine is a timeless (t = 0) mechanics, which includes his fundamental principle of superposition. Since one of the most controversial paradoxes in science must be Schrödinger's cat. We will show that the myth of his hypothesis is "not" a physical realizable postulation. The most important aspect in quantum theory must be the probabilistic implication of science, a set of most elegant and simple laws and principles, which will be discussed. Since information and entropy have a profound connection, we will show that information is one of very important science in quantum theory, for which several significant aspects of information transmission will be stressed. Nevertheless, the myth of quantum theory turns out to be not Schrodinger's cat but the nature of a section of time Δt. Since time is a quantity that we cannot physically manipulate, we could change the section Δt but not the speed of time. Although we can squeeze a section of Δt, but we cannot squeeze Δt to zero. And this is the ultimate quantum limit of "instantaneous" response we can never be able to obtain. Since time traveling is one of the very interesting topics in science, I will show that time traveling is impossible even at the speed of light. Nevertheless, I will show quantum mechanics is a temporal (t > 0) physical realizable mechanics, and it should "not" be as virtual and timeless

**Keywords:** quantum mechanics, Hamiltonian mechanics, timeless mechanics, temporal mechanic, temporal universe, timeless space, physical realizable,

Two of the most important discoveries in the twentieth century in modern science must be the Einstein's relativity theory [1] and Schrödinger's quantum mechanics [2]; one is dealing with very large objects and the other is dealing with very small particles. Yet they were connected by means of Heisenberg's uncertainty principle [3] and Boltzmann's entropy theory [4]. Yet, practically, all the laws, principles, and theories of science were developed from an absolute empty space, and their solutions are all timeless (t = 0) or time-independent. Since our universe is a temporal (t > 0) space, timeless (t = 0) solution cannot be "directly" implemented

within our universe, because timeless and temporal are mutually exclusive.

Quantum Theory: Part II

[12] Yu FTS. Optics and Information Theory. New York: Wiley-Interscience; 1976

[13] Schrödinger E. An Undulatory theory of the mechanics of atoms and molecules. Physics Review. 1926;**28**(6): 1049

[14] Heisenberg W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik. 1927; **43**(3–4):172

[15] Lawden DF. The Mathematical Principles of Quantum Mechanics. London: Methuen & Co Ltd.; 1967

[16] Boltzmann L. Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte. 1866; **53**:195-220

[17] MacKinnon E. De Broglie's thesis: A critical retrospective. American Journal of Physics. 1976;**44**:1047-1055

[18] Yu FTS. Introduction to Diffraction, Information Processing and Holography, Chapter 10. Cambridge, Mass: MIT Press; 1973. pp. 91-98

[19] Gabor D. A new microscope principle. Nature. 1948;**161**:777

[20] Cultrona LJ, Leith EN, Porcello LJ, Vivian WE. On the application of coherent optical processing techniques to synthetic-aperture radar. Proceedings of the IEEE. 1966;**54**:1026

[21] Leith EN, Upatniecks J. Reconstructed wavefront and communication theory. Journal of the Optical Society of America. 1962;**52**:1123

#### **Chapter 9**

**References**

*Quantum Mechanics*

[1] Yu FTS. From relativity to discovery of temporal (t > 0) universe. In: Origin

[11] Kraus JD. Electro-Magnetics. New York: McGraw-Hill Book Company;

[12] Yu FTS. Optics and Information Theory. New York: Wiley-Interscience;

[13] Schrödinger E. An Undulatory theory of the mechanics of atoms and molecules. Physics Review. 1926;**28**(6):

[14] Heisenberg W. Über den anschaulichen Inhalt der

quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik. 1927;

[15] Lawden DF. The Mathematical Principles of Quantum Mechanics. London: Methuen & Co Ltd.; 1967

[16] Boltzmann L. Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte. 1866;

[17] MacKinnon E. De Broglie's thesis: A critical retrospective. American Journal

[18] Yu FTS. Introduction to Diffraction,

Holography, Chapter 10. Cambridge, Mass: MIT Press; 1973. pp. 91-98

[20] Cultrona LJ, Leith EN, Porcello LJ, Vivian WE. On the application of coherent optical processing techniques to synthetic-aperture radar. Proceedings

communication theory. Journal of the Optical Society of America. 1962;**52**:1123

[19] Gabor D. A new microscope principle. Nature. 1948;**161**:777

of Physics. 1976;**44**:1047-1055

Information Processing and

of the IEEE. 1966;**54**:1026

[21] Leith EN, Upatniecks J. Reconstructed wavefront and

1953. p. 370

1976

1049

**43**(3–4):172

**53**:195-220

Mechanics, Chapter 1. New York: CRC

[2] Bunge M. Causality: The Place of the Causal Principle in Modern Science. Cambridge: Harvard University Press;

[3] Yu FTS. The fate of Schrodinger's cat. Asian Journal of Physics. 2019;**28**(1):

[4] Knudsen JM, Hjorth P. Elements of Newtonian Mechanics. Heidelberg: Springer Science & Business Media;

[5] Yu FTS. Time: The enigma of space. Asian Journal of Physics. 2017;**26**(3):

[6] Yu FTS. What is "wrong" with current theoretical physicists? In: Bulnes F, Stavrou VN, Morozov O, Bourdine AV, editors. Advances in Quantum Communication and Information, Chapter 9. London: IntechOpen; 2020. pp. 123-143

[7] Parzen E. Stochastic Processes. San Francisco: Holden Day, Inc.; 1962

[8] Einstein Attacks Quantum Theory. Scientist and Two Colleagues Find It Is Not 'Complete' Even though 'Correct'. New York City: The New York Times;

[9] Einstein A. Relativity, the Special and General Theory. New York: Crown

[10] Hawking S, Penrose R. The Nature

of Space and Time. New Jersey: Princeton University Press; 1996

of Temporal (t > 0) Universe: Correcting with Relativity, Entropy, Communication and Quantum

Press; 2019. pp. 1-26

1959

63-70

2012

143-158

1935

**160**

Publishers; 1961

## Nature of Temporal (t > 0) Quantum Theory: Part II

*Francis T.S. Yu*

#### **Abstract**

Since Schrödinger's quantum mechanics developed from Hamiltonian, I will show that his quantum machine is a timeless (t = 0) mechanics, which includes his fundamental principle of superposition. Since one of the most controversial paradoxes in science must be Schrödinger's cat. We will show that the myth of his hypothesis is "not" a physical realizable postulation. The most important aspect in quantum theory must be the probabilistic implication of science, a set of most elegant and simple laws and principles, which will be discussed. Since information and entropy have a profound connection, we will show that information is one of very important science in quantum theory, for which several significant aspects of information transmission will be stressed. Nevertheless, the myth of quantum theory turns out to be not Schrodinger's cat but the nature of a section of time Δt. Since time is a quantity that we cannot physically manipulate, we could change the section Δt but not the speed of time. Although we can squeeze a section of Δt, but we cannot squeeze Δt to zero. And this is the ultimate quantum limit of "instantaneous" response we can never be able to obtain. Since time traveling is one of the very interesting topics in science, I will show that time traveling is impossible even at the speed of light. Nevertheless, I will show quantum mechanics is a temporal (t > 0) physical realizable mechanics, and it should "not" be as virtual and timeless (t = 0) as mathematic does.

**Keywords:** quantum mechanics, Hamiltonian mechanics, timeless mechanics, temporal mechanic, temporal universe, timeless space, physical realizable, Schrödinger's cat

#### **1. Introduction**

Two of the most important discoveries in the twentieth century in modern science must be the Einstein's relativity theory [1] and Schrödinger's quantum mechanics [2]; one is dealing with very large objects and the other is dealing with very small particles. Yet they were connected by means of Heisenberg's uncertainty principle [3] and Boltzmann's entropy theory [4]. Yet, practically, all the laws, principles, and theories of science were developed from an absolute empty space, and their solutions are all timeless (t = 0) or time-independent. Since our universe is a temporal (t > 0) space, timeless (t = 0) solution cannot be "directly" implemented within our universe, because timeless and temporal are mutually exclusive.

Although timeless laws and principles have been the foundation and cornerstone of our science, there are also scores of virtual solutions that are "not" physical realizable within our temporal (t > 0) space.

Yet, it is the major topic of the current state of science, fictitious and virtual as mathematics is. Added with very convincing computer simulation, fictitious science becomes "irrationally" real? As a scientist, I felt, in part, my obligation to point out where those fictitious solutions come from, since science is also mathematics.

Since Schrödinger's quantum mechanics is a legacy of Hamiltonian classical mechanics, I will first show that Hamiltonian was developed on a timeless (t = 0) platform, for which Schrödinger's quantum machine is also timeless (t = 0); this includes his quantum world as well his fundamental principle of superposition. I will further show that where Schrödinger's superposition principle is timeless (t = 0), it is from the adaption of Bohr's quantum state energy E = hυ, which is essentially time unlimited singularity approximated. I will also show that nonphysical realizable wave function can be reconfigured to becoming temporal (t > 0), since we knew a physical realizable wave function is supposed to be. And I will show that superposition principle existed "if and only if" within a timeless (t = 0) virtual mathematical subspace but not existed within our temporal (t > 0) space.

When dealing with quantum mechanics, it is unavoidable not to mention Schrödinger's cat, which is one of the most elusive cats in science, since Schrödinger disclosed the hypothesis in 1935? And the interesting part is that the paradox of Schrödinger's cat has been debated by score of world renounced scientists such as Einstein, Bohr, Schrödinger, and many others for over eight decades, and it is still under debate. Yet I will show that Schrödinger's hypothesis is "not" a physical realizable hypothesis, for which his half-life cat should "not" have had used as a physical postulated hypothesis.

model since particle and empty space are mutually exclusive. Notice that total energy of a "Hamiltonian particle" in motion is equal to its kinetic energy plus the

which is the well-known Hamiltonian equation, where p and m represent the particle's momentum and mass, respectively, and V is the particle's potential energy. Equivalently, Hamiltonian equation can be written in the following form as applied

*=* 8π<sup>2</sup>

<sup>∇</sup><sup>2</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> ∂xi ∂xj

We note that Eq. (2) is the well-known "Hamiltonian Operator" in classical

where ψ is the wave function that remains to be determined and E and V are the energy factor and potential energy that need to be incorporated within the equation. And this is precisely where Schrödinger's equation was derived from; by using the energy factor E = hυ (i.e., a quanta of light energy) adopted from Bohr's atomic

By virtue of "energy conservation", Hamiltonian equation is written as

*=*ð8π 2 mÞ

8*π*2m

where h is Planck's constant, m and V are the mass and potential energy of the

*=*ð Þþ 2m V (1)

<sup>m</sup> � � � � <sup>∇</sup><sup>2</sup> <sup>þ</sup> <sup>V</sup> (2)

h i <sup>∇</sup><sup>2</sup> <sup>þ</sup> <sup>V</sup>g<sup>ψ</sup> <sup>¼</sup> <sup>E</sup> <sup>ψ</sup> (3)

*<sup>h</sup>*<sup>2</sup> ð Þ *<sup>E</sup>* � *<sup>V</sup> <sup>ψ</sup>* <sup>¼</sup> <sup>0</sup> (4)

**<sup>H</sup>** <sup>¼</sup> <sup>p</sup><sup>2</sup>

*A particle in motion within a timeless (t = 0) subspace. v is the velocity of the particle.*

**<sup>H</sup>** ¼ � <sup>h</sup><sup>2</sup>

**<sup>H</sup>**<sup>ψ</sup> ¼ f� <sup>h</sup><sup>2</sup>

model [9], Schrödinger equation can be written as [7]

*∂*2 *ψ ∂x*<sup>2</sup> þ

particle's potential energy as given by [7];

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

particle, and ∇<sup>2</sup> is a Laplacian operator;

for a "subatomic particle";

mechanics.

**163**

**Figure 1.**

In short, the art of a quantum mechanics is all about temporal (t > 0) subspace, in which we see that everything existed within our universe; no matter how small it is, it has to be temporal (t > 0), otherwise it cannot exist within our universe.

#### **2. Hamiltonian to temporal (t > 0) quantum mechanics**

In modern physics, there are two most important pillars of disciplines: It seems to me one is dealing with macroscale objects of Einstein [1] and the other is dealing with microscale particle of Schrödinger [2]. Instead of speculating micro- and macro objects behave differently, they share a common denominator, temporal (t > 0) subspace. In other words, regardless of how small the particle is, it has to be temporal (t > 0), otherwise it cannot exist within our temporal (t > 0) universe.

As science progresses from Newtonian [5] to statistical mechanics [6], "time" has always been regarded as an "independent" variable with respect to substance or subspace. And this is precisely what modern physics has had been used the same timeless (t = 0) platform, for which they have treated time as an "independent" variable. Since Heisenberg [3] was one of the earlier starters in quantum mechanics, I have found that his principle was derived on the same timeless (t = 0) platform as depicted in **Figure 1**. And this is the "same" platform used in developing Hamiltonian classical mechanics [7]. Precisely, this is the reason why Schrödinger's quantum mechanics is "timeless (t = 0)" [8], since quantum mechanics is the legacy of Hamiltonian.

In view of **Figure 1**, we see that the background of the paradigm is a piece of paper, which represents a timeless (t = 0) subspace; it is "not" a physical realizable *Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

Although timeless laws and principles have been the foundation and cornerstone of our science, there are also scores of virtual solutions that are "not" physical realiz-

Yet, it is the major topic of the current state of science, fictitious and virtual as mathematics is. Added with very convincing computer simulation, fictitious science becomes "irrationally" real? As a scientist, I felt, in part, my obligation to point out where those fictitious solutions come from, since science is also mathematics. Since Schrödinger's quantum mechanics is a legacy of Hamiltonian classical mechanics, I will first show that Hamiltonian was developed on a timeless (t = 0) platform, for which Schrödinger's quantum machine is also timeless (t = 0); this includes his quantum world as well his fundamental principle of superposition. I will further show that where Schrödinger's superposition principle is timeless (t = 0), it is from the adaption of Bohr's quantum state energy E = hυ, which is essentially time unlimited singularity approximated. I will also show that nonphysical realizable wave function can be reconfigured to becoming temporal (t > 0), since we knew a physical realizable wave function is supposed to be. And I will show that superposition principle existed "if and only if" within a timeless (t = 0) virtual mathematical subspace

When dealing with quantum mechanics, it is unavoidable not to mention Schrödinger's cat, which is one of the most elusive cats in science, since Schrödinger disclosed the hypothesis in 1935? And the interesting part is that the paradox of Schrödinger's cat has been debated by score of world renounced scientists such as Einstein, Bohr, Schrödinger, and many others for over eight decades, and it is still under debate. Yet I will show that Schrödinger's hypothesis is "not" a physical realizable hypothesis, for which his half-life cat should "not" have had used as a

In short, the art of a quantum mechanics is all about temporal (t > 0) subspace, in which we see that everything existed within our universe; no matter how small it is, it has to be temporal (t > 0), otherwise it cannot exist within our universe.

In modern physics, there are two most important pillars of disciplines: It seems to me one is dealing with macroscale objects of Einstein [1] and the other is dealing with microscale particle of Schrödinger [2]. Instead of speculating micro- and macro objects behave differently, they share a common denominator, temporal (t > 0) subspace. In other words, regardless of how small the particle is, it has to be temporal (t > 0), otherwise it cannot exist within our temporal (t > 0) universe. As science progresses from Newtonian [5] to statistical mechanics [6], "time" has always been regarded as an "independent" variable with respect to substance or subspace. And this is precisely what modern physics has had been used the same timeless (t = 0) platform, for which they have treated time as an "independent" variable. Since Heisenberg [3] was one of the earlier starters in quantum mechanics, I have found that his principle was derived on the same timeless (t = 0) platform as depicted in **Figure 1**. And this is the "same" platform used in developing Hamiltonian classical mechanics [7]. Precisely, this is the reason why Schrödinger's quantum mechanics is "timeless (t = 0)" [8], since quantum mechanics is the legacy of

In view of **Figure 1**, we see that the background of the paradigm is a piece of paper, which represents a timeless (t = 0) subspace; it is "not" a physical realizable

**2. Hamiltonian to temporal (t > 0) quantum mechanics**

able within our temporal (t > 0) space.

*Quantum Mechanics*

but not existed within our temporal (t > 0) space.

physical postulated hypothesis.

Hamiltonian.

**162**

**Figure 1.** *A particle in motion within a timeless (t = 0) subspace. v is the velocity of the particle.*

model since particle and empty space are mutually exclusive. Notice that total energy of a "Hamiltonian particle" in motion is equal to its kinetic energy plus the particle's potential energy as given by [7];

$$\mathcal{H} = \mathbf{p}^2/(2\mathbf{m}) + \mathbf{V} \tag{1}$$

which is the well-known Hamiltonian equation, where p and m represent the particle's momentum and mass, respectively, and V is the particle's potential energy. Equivalently, Hamiltonian equation can be written in the following form as applied for a "subatomic particle";

$$\mathcal{H} = -\left[\mathbf{h}^2/(8\pi^2\mathbf{m})\right]\nabla^2 + \mathbf{V} \tag{2}$$

where h is Planck's constant, m and V are the mass and potential energy of the particle, and ∇<sup>2</sup> is a Laplacian operator;

$$\nabla^2 = \frac{\partial^2}{\partial \mathbf{x} \mathbf{i} \ \partial \mathbf{x} \mathbf{j}}$$

We note that Eq. (2) is the well-known "Hamiltonian Operator" in classical mechanics.

By virtue of "energy conservation", Hamiltonian equation is written as

$$\mathcal{H}\Psi = \{-\left[\mathbf{h}^2/(8\pi^2\mathbf{m})\right]\nabla^2 + \mathbf{V}\}\Psi = \mathbf{E}\,\Psi\tag{3}$$

where ψ is the wave function that remains to be determined and E and V are the energy factor and potential energy that need to be incorporated within the equation. And this is precisely where Schrödinger's equation was derived from; by using the energy factor E = hυ (i.e., a quanta of light energy) adopted from Bohr's atomic model [9], Schrödinger equation can be written as [7]

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} + \frac{8\pi^2 \mathbf{m}}{h^2} (E - V)\varphi = \mathbf{0} \tag{4}$$

In view of this Schrödinger's equation, we see that it is essentially "identical" to the Hamiltonian equation, where ψ is the wave function that has to be determined, m is the mass of a photonic-particle (i.e., photon), E and V are the dynamic quantum state energy and potential energy of the particle, x is the spatial variable, and h is Planck's constant.

scientific laws, principles, and theories were mostly developed on a piece or pieces of papers, since science is mathematics. This is by no means that timeless (t = 0) laws, principles, and theories were wrong [11], yet they were and "still" are the foundation and cornerstone of our science. However, it is their direct implementation within our temporal (t > 0) universe and also added a score of their solutions are irrational and virtual as "pretending" existed within our temporal (t > 0) subspace, for example, superposition principle of quantum mechanics, paradox of

Nevertheless as we refer to **Figure 1**, immediately we see that it is "not" a physically realizable model that should be used in the first place. Secondly, even though we pretend that the particle in motion within can exist in an empty space, a question is being asked: how can a particle-wave dynamic propagate within an empty space? Thirdly, even though we assumed wave can be exited within an empty space, why it has to be time unlimited? From all these physical reasons, we see that time unlimited Hamiltonian wave equation of Eq. (5) is "not" a physically realizable solution, since it only existed within a timeless (t = 0) virtual mathematical space, which is similar within a Newtonian space, where time has been treated

Since Schrödinger's mechanics is the legacy of Hamiltonian mechanics, firstly we see that Schrödinger's quantum "mechanics" is a solution as obtained from Hamiltonian's mechanics. Secondly, the reason why Schrödinger's quantum mechanics is timeless (t = 0) is the same reason as Hamiltonian, because its subspace is empty. Nevertheless, the major differences between Schrödinger's mechanics and

Hamiltonian mechanics must be the name sake of "quantum", where comes Bohr's

Schrödinger has used for the development of his mechanics. This is precisely since Schrödinger's solution is very similar to Hamiltonian of Eq. (3) as given by [7],

atomic quantum leap E = hυ, a quanta of light as shown in **Figure 2**, that

*Bohr atomic model embedded in a timeless (t = 0) platform (i.e., a piece of paper).*

Schrödinger's cat, time traveling, and many others.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

as an "independent" variable.

**Figure 2.**

**165**

Since Schrödinger's equation is the "core" of quantum mechanics, but without Hamiltonian's mechanics, it seems to me that we would "not" have the quantum mechanics. The "fact" is that quantum mechanics is essentially "identical" to Hamiltonian mechanics. The major difference between them is that Schrödinger used the dynamic quantum energy E = hυ as adapted from a quantum leap energy of Bohr's hypothesis, which changes from classical mechanics to quantum "leap" mechanics or quantum mechanics. In other words, Schrödinger used a package of wavelet quantum leap energy hυ to equivalent a particle (or photon) as from "wave-particle dynamics" of de Broglie's hypothesis [10], although photon is "not" actually a real particle. Nevertheless, where the mass m for a photonic particle in the Schrödinger's equation remains to be "physically reconciled", after all science is a law of approximation. Furthermore, without the adaptation of Bohr's quantum leap hυ, quantum physics would not have started. It seems to me that quantum leap energy E = hυ has played a "viable" role as transforming from Hamiltonian classical mechanics to quantum mechanics, which Schrödinger had done to his quantum theory.

Although Schrödinger equation has given scores of viable solutions for practical applications, at the "same time", it has also produced a number of fictitious and irrational results which are not existed within our universe, such as his Fundamental Principle of Superposition, the paradox of Schrödinger's Cat [8], and others.

In view of Schrödinger's equation as given by Eq. (4), we see that it is a timeless (t = 0) or time-independent equation. Since the equation is the "core" of Schrödinger's quantum mechanics, it needs a special mention. Let me stress the essence of energy factor E in the Hamiltonian equation. Since Schrödinger equation is the legacy of Hamiltonian, any wave solution ψ emerges from Schrödinger equation depends upon the E factor. In other words aside the embedded subspace, solution comes out from Schrödinger equation whether is it a physical realizable; it depends upon the E factor that we introduced into the equation. As referring to the conventional Hamiltonian mechanics, if we let the energy factor E be a "constant" quantity that exists at time t = t0, which is "exactly" the classical mechanics of Hamiltonian, this means that the Hamiltonian will take this value of E at t = t0 and evaluates the wave function ψ as has been given by [7]:

$$
\Psi = \Psi\_0 \,\,\exp\left[-\mathbf{i}\,\,2\pi\,\mathbf{E}(\mathbf{t}-\mathbf{t}\_0)/\mathbf{h}\right] \tag{5}
$$

which is the Hamiltonian wave equation, where ψ<sup>0</sup> is an arbitrary constant, h is Planck's constant, and a constant energy factor E(t – t0) occurs at t = t0. Although Hamiltonian wave equation is a time-variable function, it is "not" a time-limited solution, for which we see that it "cannot" be implemented within our temporal (t > 0) universe, since time unlimited solution cannot exist within our universe. This means that, wave solution ψ of Eq. (5) is "not" a physical realizable solution.

Then a question is being raised, why the Hamiltonian wave solution is time unlimited? The answer is trivial that Hamiltonian is mathematics and his mechanics was developed on an empty timeless (t = 0) platform as can be seen in **Figure 1**. Since it is the subspace that governs the mechanics, we see that particle-wave dynamics cannot exist within a timeless (t = 0) subspace. But Hamiltonian is mathematics and Hamilton himself is a theoretician; he could have had implanted a particle-wave dynamic into a timeless (t = 0) subspace, although timeless (t = 0) subspace and physical particle cannot coexist. Of which this is precisely all the

#### *Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

In view of this Schrödinger's equation, we see that it is essentially "identical" to the Hamiltonian equation, where ψ is the wave function that has to be determined, m is the mass of a photonic-particle (i.e., photon), E and V are the dynamic quantum state energy and potential energy of the particle, x is the spatial variable,

Since Schrödinger's equation is the "core" of quantum mechanics, but without Hamiltonian's mechanics, it seems to me that we would "not" have the quantum mechanics. The "fact" is that quantum mechanics is essentially "identical" to Hamiltonian mechanics. The major difference between them is that Schrödinger used the dynamic quantum energy E = hυ as adapted from a quantum leap energy of Bohr's hypothesis, which changes from classical mechanics to quantum "leap" mechanics or quantum mechanics. In other words, Schrödinger used a package of wavelet quantum leap energy hυ to equivalent a particle (or photon) as from "wave-particle dynamics" of de Broglie's hypothesis [10], although photon is "not" actually a real particle. Nevertheless, where the mass m for a photonic particle in the Schrödinger's equation remains to be "physically reconciled", after all science is a law of approximation. Furthermore, without the adaptation of Bohr's quantum leap hυ, quantum physics would not have started. It seems to me that quantum leap energy E = hυ has played a "viable" role as transforming from Hamiltonian classical mechanics to quantum mechanics, which Schrödinger had done to his quantum theory.

Although Schrödinger equation has given scores of viable solutions for practical applications, at the "same time", it has also produced a number of fictitious and irrational results which are not existed within our universe, such as his Fundamental Principle of Superposition, the paradox of Schrödinger's Cat [8], and others.

In view of Schrödinger's equation as given by Eq. (4), we see that it is a timeless

which is the Hamiltonian wave equation, where ψ<sup>0</sup> is an arbitrary constant, h is Planck's constant, and a constant energy factor E(t – t0) occurs at t = t0. Although Hamiltonian wave equation is a time-variable function, it is "not" a time-limited solution, for which we see that it "cannot" be implemented within our temporal (t > 0) universe, since time unlimited solution cannot exist within our universe. This means that, wave solution ψ of Eq. (5) is "not" a physical realizable solution. Then a question is being raised, why the Hamiltonian wave solution is time unlimited? The answer is trivial that Hamiltonian is mathematics and his mechanics was developed on an empty timeless (t = 0) platform as can be seen in **Figure 1**. Since it is the subspace that governs the mechanics, we see that particle-wave dynamics cannot exist within a timeless (t = 0) subspace. But Hamiltonian is mathematics and Hamilton himself is a theoretician; he could have had implanted a particle-wave dynamic into a timeless (t = 0) subspace, although timeless (t = 0) subspace and physical particle cannot coexist. Of which this is precisely all the

ψ ¼ ψ<sup>0</sup> exp ½ Þ �i 2π E tð Þ –t0 *=*h (5)

(t = 0) or time-independent equation. Since the equation is the "core" of Schrödinger's quantum mechanics, it needs a special mention. Let me stress the essence of energy factor E in the Hamiltonian equation. Since Schrödinger equation is the legacy of Hamiltonian, any wave solution ψ emerges from Schrödinger equation depends upon the E factor. In other words aside the embedded subspace, solution comes out from Schrödinger equation whether is it a physical realizable; it depends upon the E factor that we introduced into the equation. As referring to the conventional Hamiltonian mechanics, if we let the energy factor E be a "constant" quantity that exists at time t = t0, which is "exactly" the classical mechanics of Hamiltonian, this means that the Hamiltonian will take this value of E at t = t0 and

evaluates the wave function ψ as has been given by [7]:

**164**

and h is Planck's constant.

*Quantum Mechanics*

scientific laws, principles, and theories were mostly developed on a piece or pieces of papers, since science is mathematics. This is by no means that timeless (t = 0) laws, principles, and theories were wrong [11], yet they were and "still" are the foundation and cornerstone of our science. However, it is their direct implementation within our temporal (t > 0) universe and also added a score of their solutions are irrational and virtual as "pretending" existed within our temporal (t > 0) subspace, for example, superposition principle of quantum mechanics, paradox of Schrödinger's cat, time traveling, and many others.

Nevertheless as we refer to **Figure 1**, immediately we see that it is "not" a physically realizable model that should be used in the first place. Secondly, even though we pretend that the particle in motion within can exist in an empty space, a question is being asked: how can a particle-wave dynamic propagate within an empty space? Thirdly, even though we assumed wave can be exited within an empty space, why it has to be time unlimited? From all these physical reasons, we see that time unlimited Hamiltonian wave equation of Eq. (5) is "not" a physically realizable solution, since it only existed within a timeless (t = 0) virtual mathematical space, which is similar within a Newtonian space, where time has been treated as an "independent" variable.

Since Schrödinger's mechanics is the legacy of Hamiltonian mechanics, firstly we see that Schrödinger's quantum "mechanics" is a solution as obtained from Hamiltonian's mechanics. Secondly, the reason why Schrödinger's quantum mechanics is timeless (t = 0) is the same reason as Hamiltonian, because its subspace is empty. Nevertheless, the major differences between Schrödinger's mechanics and Hamiltonian mechanics must be the name sake of "quantum", where comes Bohr's atomic quantum leap E = hυ, a quanta of light as shown in **Figure 2**, that Schrödinger has used for the development of his mechanics. This is precisely since Schrödinger's solution is very similar to Hamiltonian of Eq. (3) as given by [7],

**Figure 2.** *Bohr atomic model embedded in a timeless (t = 0) platform (i.e., a piece of paper).*

$$\Psi \left( \mathbf{t} \right) = \psi\_0 \, \exp \left[ -\mathbf{i} \, 2\pi \, \text{v} \left( \mathbf{t} \, \text{-t}\_0 \right) / \text{h} \right] \tag{6}$$

which is the well-known Schrödinger wave equation, where ψ<sup>0</sup> is an arbitrary constant, hυ is the frequency of the quantum leap, and h is Planck's constant.

As anticipated, Schrödinger wave equation is also a "time unlimited" solution with "no" bandwidth. For the same reason as Hamiltonian, Schrödinger wave equation is "not" a physically realizable solution that can be implemented within our temporal (t > 0) universe, since any physically realizable wave equation has to be "time and band limited". Yet, many quantum scientists have been using this time unlimited solution to pursuing their dream for quantum supremacy computing [12] and communication [13] but "not" knowing the dream they are pursuing is "not" a physically realizable dream.

Since quantum mechanics is a "linear "system machine, similar to Hamiltonian mechanics, for a multi-quantum state energies atomic particle, the energy E factor to be applied in the Schrödinger's equation is a "linear" combination of those quantum state energies as given by

$$\mathbf{E} = \boldsymbol{\Sigma} \mathbf{h} \mathbf{u}\_{\mathbf{n},} \mathbf{n} = \mathbf{1}, \mathbf{2}, \dots \mathbf{N} \tag{7}$$

In order to mitigate the temporal (t > 0) requirement or the causality condition of those wave functions ψN(t), we can "reconfigure" each of the wave function to becoming temporal (t > 0). In other words, we can reconfigure each of the wave function to "comply" with the temporal (t > 0) condition within our universe. For example, as illustrated in **Figure 3** we see that each of the quantum leap hΔυ is

� cos ð2πυntÞ, t> 0, n ¼ 1, 2, 3*:* (9)

ψ ðÞ¼ t 0, t ≤0 (10)

by which it can be shown that "reconfigured" wave functions are

where t > 0 denotes equation is subjected to temporal (t > 0) condition, in words exited only in positive time domain. In view of these equations we see that the packages of quantum leaps are "likely" temporal separated, in which we see that all the wavelets are very "unlikely" to be "simultaneous and instantaneous" superposing together. Once again, we have proven that Schrödinger's fundamental prin-

On the other hand, if we take the preceding physical realizable wave functions of Eq. (9) and implement them within a timeless (t = 0) subspace, then it is trivial to see that how a timeless (t = 0) subspace can do to all the wave-particle dynamics within a timeless (t = 0) subspace. Since within a timeless (t = 0) space it has no time and no dimension, all wave-particles (i.e., package of wavelets) will be col-

Before this goes on, I would say that the wave-particle duality is a "nonphysical" reality assumption to "equivalence" a package wavelet of energy to a particle in motion, which is strictly from a statistical mechanics point of view, where momentum of a particle p = h/λ is conserved [7]. However, one should "not" be treated wave or a package of wavelet energy hΔυ as a particle or particle as wave. It is the

ciple of superposition "fails" to exist within our temporal (t > 0) universe.

represented by "time limited" wavelets.

<sup>ψ</sup> ðÞ¼ <sup>t</sup> Σ ψon exp ½�αon ð Þ <sup>t</sup> � ton <sup>2</sup>

*A multi-quantum state atomic model embedded within a temporal subspace.*

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

**3. Timeless (t = 0) space do to particles**

lapsed at t = 0, as can be seen in **Figure 4**.

approximated by

**167**

**Figure 3.**

where υ<sup>n</sup> is the frequency for the nth quantum leap, and h is Planck's constant. Therefore, the overall wave equation is a linear combination of all the wave functions as given by

$$\boldsymbol{\Psi}\boldsymbol{\Psi}\_{\rm N}\left(\mathbf{t}\right) = \boldsymbol{\Sigma}\,\boldsymbol{\Psi}\_{\rm 0n}\,\,\exp\left[-\mathbf{i}\,\,2\pi\,\,\mathbf{u}\_{\rm n}\left(\mathbf{t}\,\mathrm{-t}\_{\rm 0n}\right)/\mathrm{h}\right], \mathrm{n} = \mathbf{1}, \mathrm{2}, \dots \mathrm{N}\tag{8}$$

in which we see that all the wave functions are "super-imposing" together. This is precisely the Fundamental Principle of Superposition of Schrodinger. Yet, this is the principle that Einstein "opposed" the most as he commended as I quote: "mathematics is correct, but incomplete", published in The New York Times newspaper in 1935 [14]. And it is also the fundamental principle that quantum computing scientists are depending on the "simultaneous and instantaneous" superposition that quantum theory can offer to develop a quantum supremacy computer. But I will show that the superposition is a timeless (t = 0) principle and it does "not" exist within our universe.

Before I get started, it is interesting to show a hypothetical scenario of "superposition in life". If we assumed our life-expectancy can last for about 500 years, then we would have very good chance to coexist with Isaac Newton and possibly with Galileo Galilei somewhere in "time". Furthermore, if our universe is a "static" universe or timeless (t = 0), then we are also very likely to coexist with Galileo and Newton not only in "time" but superimposing with them everywhere in a timeless (t = 0) space. And this is precisely what "simultaneous and instantaneous" superposition can do for us, if our universe is timeless (t = 0) subspace.

As we understood from the preceding illustration, we know that any empty (i.e., timeless) subspace cannot be found within our universe. And we have also learned that within our universe, every quantum leap hυ has to be temporal (t > 0), that is time- and band-limited; otherwise it cannot be existed within our universe.

In view of Eqs. (7) and (8)), we see that they are time "unlimited" wave functions, and it is trivial to see that all of those wave functions, ψN(t), n = 1, 2 … N, are superimposing together at all times. Similar to an example that I had postulated earlier, if our life expectancy can be extended to 500 years, we would be coexist with Einstein and may be with Newton somewhere in time, although 500 years of life-expectancy is time limited. But again, time unlimited wave function is "not" a physical real function, since it cannot exist within our temporal (t > 0) universe.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

ψ ðÞ¼ t ψ<sup>0</sup> exp ½ � �i 2π υ ð Þ t–t0 *=*h (6)

E ¼ Σhυn, n ¼ 1, 2, … N (7)

which is the well-known Schrödinger wave equation, where ψ<sup>0</sup> is an arbitrary constant, hυ is the frequency of the quantum leap, and h is Planck's constant. As anticipated, Schrödinger wave equation is also a "time unlimited" solution with "no" bandwidth. For the same reason as Hamiltonian, Schrödinger wave equation is "not" a physically realizable solution that can be implemented within our temporal (t > 0) universe, since any physically realizable wave equation has to be "time and band limited". Yet, many quantum scientists have been using this time unlimited solution to pursuing their dream for quantum supremacy computing [12] and communication [13] but "not" knowing the dream they are pursuing is "not" a

Since quantum mechanics is a "linear "system machine, similar to Hamiltonian mechanics, for a multi-quantum state energies atomic particle, the energy E factor to be applied in the Schrödinger's equation is a "linear" combination of those

where υ<sup>n</sup> is the frequency for the nth quantum leap, and h is Planck's constant. Therefore, the overall wave equation is a linear combination of all the wave func-

in which we see that all the wave functions are "super-imposing" together. This is precisely the Fundamental Principle of Superposition of Schrodinger. Yet, this is the principle that Einstein "opposed" the most as he commended as I quote: "mathematics is correct, but incomplete", published in The New York Times newspaper in 1935 [14]. And it is also the fundamental principle that quantum computing scientists are depending on the "simultaneous and instantaneous" superposition that quantum theory can offer to develop a quantum supremacy computer. But I will show that the superposition is a timeless (t = 0) principle and it does "not" exist within our universe. Before I get started, it is interesting to show a hypothetical scenario of "superposition in life". If we assumed our life-expectancy can last for about 500 years, then we would have very good chance to coexist with Isaac Newton and possibly with Galileo Galilei somewhere in "time". Furthermore, if our universe is a "static" universe or timeless (t = 0), then we are also very likely to coexist with Galileo and Newton not only in "time" but superimposing with them everywhere in a timeless (t = 0) space. And this is precisely what "simultaneous and instantaneous" super-

As we understood from the preceding illustration, we know that any empty (i.e., timeless) subspace cannot be found within our universe. And we have also learned that within our universe, every quantum leap hυ has to be temporal (t > 0), that is time- and band-limited; otherwise it cannot be existed within our universe. In view of Eqs. (7) and (8)), we see that they are time "unlimited" wave functions, and it is trivial to see that all of those wave functions, ψN(t), n = 1, 2 … N, are superimposing together at all times. Similar to an example that I had postulated earlier, if our life expectancy can be extended to 500 years, we would be coexist with Einstein and may be with Newton somewhere in time, although 500 years of life-expectancy is time limited. But again, time unlimited wave function is "not" a physical real function, since it cannot exist within our temporal

position can do for us, if our universe is timeless (t = 0) subspace.

ψ<sup>N</sup> ðÞ¼ t Σ ψ0n exp ½ � �i 2π υ<sup>n</sup> ð Þ t–t0n *=*h , n ¼ 1, 2, … N (8)

physically realizable dream.

*Quantum Mechanics*

tions as given by

(t > 0) universe.

**166**

quantum state energies as given by

**Figure 3.** *A multi-quantum state atomic model embedded within a temporal subspace.*

In order to mitigate the temporal (t > 0) requirement or the causality condition of those wave functions ψN(t), we can "reconfigure" each of the wave function to becoming temporal (t > 0). In other words, we can reconfigure each of the wave function to "comply" with the temporal (t > 0) condition within our universe. For example, as illustrated in **Figure 3** we see that each of the quantum leap hΔυ is represented by "time limited" wavelets.

by which it can be shown that "reconfigured" wave functions are approximated by

$$\Psi\left(\mathbf{t}\right) = \Sigma\,\Psi\_{\rm on}\,\exp\left[-\mathbf{a}\_{\rm on}\left(\mathbf{t} - \mathbf{t}\_{\rm on}\right)^{2}\right]\cos\left(2\pi\mathbf{u}\_{\rm n}\mathbf{t}\right), \mathbf{t} > \mathbf{0}, \mathbf{n} = \mathbf{1}, 2, 3. \tag{9}$$

$$\Psi\left(\mathbf{t}\right) = \mathbf{0}, \mathbf{t} \le \mathbf{0} \tag{10}$$

where t > 0 denotes equation is subjected to temporal (t > 0) condition, in words exited only in positive time domain. In view of these equations we see that the packages of quantum leaps are "likely" temporal separated, in which we see that all the wavelets are very "unlikely" to be "simultaneous and instantaneous" superposing together. Once again, we have proven that Schrödinger's fundamental principle of superposition "fails" to exist within our temporal (t > 0) universe.

#### **3. Timeless (t = 0) space do to particles**

On the other hand, if we take the preceding physical realizable wave functions of Eq. (9) and implement them within a timeless (t = 0) subspace, then it is trivial to see that how a timeless (t = 0) subspace can do to all the wave-particle dynamics within a timeless (t = 0) subspace. Since within a timeless (t = 0) space it has no time and no dimension, all wave-particles (i.e., package of wavelets) will be collapsed at t = 0, as can be seen in **Figure 4**.

Before this goes on, I would say that the wave-particle duality is a "nonphysical" reality assumption to "equivalence" a package wavelet of energy to a particle in motion, which is strictly from a statistical mechanics point of view, where momentum of a particle p = h/λ is conserved [7]. However, one should "not" be treated wave or a package of wavelet energy hΔυ as a particle or particle as wave. It is the

package of wavelet energy "equivalent" to particle dynamics (i.e., photon), but they are "not" equaled [15]. Similar to Einstein's energy equation, mass is equivalent to energy and energy is equivalent to mass, but mass is not equal to energy and energy is not mass, for which quanta of light hΔυ or a "photon" is a "virtual" particle, in which we see that a photon has a momentum p = h/λ but no mass, although many quantum scientists regard a photon as a physical real particle.

From this illustration, we have shown once again that; it is not how rigorous the

mathematics is, it is the physical realizable paradigm determines her analytical solution is physical realizable or not? For which we see that; the wave functions as

Schrödinger's quantum mechanics was developed on an empty subspace platform,

When we are dealing with quantum mechanics, it is inevitable not to mention

Schrödinger's disclosed it in 1935 at a Copenhagen forum. Since then his half-life cat has intrigued by a score of scientists and has been debated by Einstein, Bohr, Schrödinger, and many others as soon as Schrödinger disclosed his hypothesis. And the debates have been persisted for over eight decades, and still debating. For example, I may quote one of the late Richard Feynman quotations as: "After you have leaned quantum mechanics, you really "do not" understand quantum

It is however not the art of the Schrödinger's half-life cat; it is the paradox that quantum scientists have treated it as a physical "real paradox". In other words, many scientists believed the paradox of Schrödinger's cat actually existed within our universe, without any hesitation. Or literally "accepted" superposition is a physical reality, although fictitious and irrational solutions have emerged; it seems like looking into the Alice wonderland. In order to justify some of their believing some quantum scientists even come up with their believing; particle behaves weird within a microenvironment as in contrast within a macro space. Yet, some of their potential applications such as quantum computing and quantum entanglement communication are in fact in macro subspace environment. Nevertheless, I have found many of those micro behaviors are "not" existed within our universe; and the paradox of Schrödinger's cat is one of them, as I shall discuss briefly in the

Let us start with the Schrödinger's box as shown in **Figure 5**; inside the box we have equipped a bottle of poison gas and a device (i.e., a hammer) to break the

*Inside the box we equipped a bottle of poison gas and a device (i.e., hammer) to break the bottle, triggered by the*

obtained from Schrödinger equation is virtual as mathematics is, because

Schrödinger's cat since it is one of the most elusive cats in the science since

the same platform as Hamiltonian classical mechanics.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

**4. Schrödinger's cat**

mechanics … ".

following:

**Figure 5.**

**169**

*decaying of a radio-active particle, to kill the cat.*

In view of **Figure 4** we see that within a timeless (t = 0) space, it has no time and no space; every particle exists anywhere within a timeless (t = 0) space but only exited at t = 0. This is precisely what the "simultaneous and instantaneous" superposition of Schrödinger's principle is anticipated for, since this is the fundamental principle that quantum scientists are aiming for, to build a quantum supremacy computer. This is as well applied to quantum entanglement communication, but unfortunately, the "simultaneous and instantaneous" superposition does "not" exist within our universe, of which we have had shown that superposition principle exists "if and only if" in a mathematical virtual timeless (t = 0) space, and it cannot exist within our temporal (t > 0) universe.

The reason that superposition principle "fails" to exist is coming from a nonphysical realizable paradigm used in the analysis, which can be traced back to the development of Hamiltonian mechanics, since quantum mechanics is an extension of Hamiltonian. I have found that it is the background subspace (i.e., a piece of paper) used in quantum mechanical analysis. Since the background represents an "inadvertently" empty timeless (t = 0) subspace, where a photonic particle in motion was embedded, it is also that piece of paper that Bohr's atomic model was used, added his quantum state energy hυ is not a time limited physical reality.

Aside the substance and emptiness are mutually excluded; it is the subspace that governs the behavior of each wave functions ψn(t). In which within a timeless (t = 0) subspace, we have shown all the wave functions ψN(t), regardless time limited or time unlimited, collapse all together at t = 0. In other words, all the quantum state wavelets superimposed at a "singularity" t = 0. This is the reason that superposed quantum state energies can be found anywhere and everywhere within a virtual mathematical timeless (t = 0) space, since a timeless (t = 0) space has no distance.

**Figure 4.** *All the particles within a timeless (t > 0) subspace actually have done; converges all the particles at t = 0.*

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

From this illustration, we have shown once again that; it is not how rigorous the mathematics is, it is the physical realizable paradigm determines her analytical solution is physical realizable or not? For which we see that; the wave functions as obtained from Schrödinger equation is virtual as mathematics is, because Schrödinger's quantum mechanics was developed on an empty subspace platform, the same platform as Hamiltonian classical mechanics.

#### **4. Schrödinger's cat**

package of wavelet energy "equivalent" to particle dynamics (i.e., photon), but they are "not" equaled [15]. Similar to Einstein's energy equation, mass is equivalent to energy and energy is equivalent to mass, but mass is not equal to energy and energy is not mass, for which quanta of light hΔυ or a "photon" is a "virtual" particle, in which we see that a photon has a momentum p = h/λ but no mass, although many

In view of **Figure 4** we see that within a timeless (t = 0) space, it has no time and no space; every particle exists anywhere within a timeless (t = 0) space but only exited at t = 0. This is precisely what the "simultaneous and instantaneous" superposition of Schrödinger's principle is anticipated for, since this is the fundamental principle that quantum scientists are aiming for, to build a quantum supremacy computer. This is as well applied to quantum entanglement communication, but unfortunately, the "simultaneous and instantaneous" superposition does "not" exist within our universe, of which we have had shown that superposition principle exists "if and only if" in a mathematical virtual timeless (t = 0) space, and it cannot

The reason that superposition principle "fails" to exist is coming from a nonphysical realizable paradigm used in the analysis, which can be traced back to the development of Hamiltonian mechanics, since quantum mechanics is an extension of Hamiltonian. I have found that it is the background subspace (i.e., a piece of paper) used in quantum mechanical analysis. Since the background represents an "inadvertently" empty timeless (t = 0) subspace, where a photonic particle in motion was embedded, it is also that piece of paper that Bohr's atomic model was used, added his quantum state energy hυ is not a time limited physical reality.

Aside the substance and emptiness are mutually excluded; it is the subspace that governs the behavior of each wave functions ψn(t). In which within a timeless (t = 0) subspace, we have shown all the wave functions ψN(t), regardless time limited or time unlimited, collapse all together at t = 0. In other words, all the quantum state wavelets superimposed at a "singularity" t = 0. This is the reason that superposed quantum state energies can be found anywhere and everywhere within a virtual mathematical timeless (t = 0) space, since a timeless (t = 0) space has no distance.

*All the particles within a timeless (t > 0) subspace actually have done; converges all the particles at t = 0.*

quantum scientists regard a photon as a physical real particle.

exist within our temporal (t > 0) universe.

*Quantum Mechanics*

**Figure 4.**

**168**

When we are dealing with quantum mechanics, it is inevitable not to mention Schrödinger's cat since it is one of the most elusive cats in the science since Schrödinger's disclosed it in 1935 at a Copenhagen forum. Since then his half-life cat has intrigued by a score of scientists and has been debated by Einstein, Bohr, Schrödinger, and many others as soon as Schrödinger disclosed his hypothesis. And the debates have been persisted for over eight decades, and still debating. For example, I may quote one of the late Richard Feynman quotations as: "After you have leaned quantum mechanics, you really "do not" understand quantum mechanics … ".

It is however not the art of the Schrödinger's half-life cat; it is the paradox that quantum scientists have treated it as a physical "real paradox". In other words, many scientists believed the paradox of Schrödinger's cat actually existed within our universe, without any hesitation. Or literally "accepted" superposition is a physical reality, although fictitious and irrational solutions have emerged; it seems like looking into the Alice wonderland. In order to justify some of their believing some quantum scientists even come up with their believing; particle behaves weird within a microenvironment as in contrast within a macro space. Yet, some of their potential applications such as quantum computing and quantum entanglement communication are in fact in macro subspace environment. Nevertheless, I have found many of those micro behaviors are "not" existed within our universe; and the paradox of Schrödinger's cat is one of them, as I shall discuss briefly in the following:

Let us start with the Schrödinger's box as shown in **Figure 5**; inside the box we have equipped a bottle of poison gas and a device (i.e., a hammer) to break the

#### **Figure 5.**

*Inside the box we equipped a bottle of poison gas and a device (i.e., hammer) to break the bottle, triggered by the decaying of a radio-active particle, to kill the cat.*

bottle, triggered by the decaying of a radio-active particle, to kill the cat. Since the box is assumed totally opaque of which no one knows that the cat will be killed or not, as imposed by the Schrödinger's superposition principle until we open his box. with mathematics, which is an axiom of "certainty", of which I state these laws and

Law of entropy; entropy within an enclosed subspace increases naturally "with

Theory of information; the higher the amount the information, the more uncer-

Principle of uncertainty; uncertainty of an isolated particle increases naturally

Theory of special relativity; when a subspace moves faster "relatively" than the other subspace; there is a "relativistic" time speed between them, although time

Nature of universe; every isolated subspace was created by amount of energy ΔE and a section of time Δt and it is a dynamic temporal (t > 0) stochastic subspace

Nevertheless, it is easier to facilitate these laws and principles in mathematical

<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup><sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where S, I, and U are entropy, information and universe respectively, k is the Boltzmann's constant, h is the Planck's constant, p is the probability, Δt is a section of time, Δt' is the dilated section of time, v is the velocity, m is the mass and c is the speed of light. k = 1.38 � <sup>10</sup>�<sup>16</sup> ergs per degree centigrade and h = 6.624 � <sup>10</sup>�<sup>27</sup>

In this we see that our universe was created by means a "huge" amount of energy ΔE and a "long" section of time Δt. And Δt is "still" extending rapidly, since

As we got back from Eq. (11) to Eq. (15) we see that; they are all pointsingularity approximated; otherwise it will be very difficult to write in simple mathematical forms. As the laws and principles stated, there are all associated with time, by which they are all space-time variable laws and principles, since time is space and space is time within our temporal (t > 0) universe. In short, they are all connected to a unit of (Δt, ΔE) which is the basic building blocks of our universe. For which I envision that; every existence within our universe has a beginning and

has an end. But it is time; it has "no" beginning and has "no" end!

In view of these laws and principles, they must be the most "elegant and simple" science equations that existed today in which these equations either attached or associated with a section of time Δt, except Eq. (12) since information theory is mathematics. But as soon information is recognized as related to entropy, information is equivalent to an amount of entropy; this makes an amount of information a physical quantity which is acceptable in science. For which we will show that a section of Δt will be associated with the theory of information, otherwise information will be very difficult to apply in science. Since Δt is coexisted with ΔE, we will further see that; every bit of information takes an amount of energy ΔE and a section of time Δt to transmit, to create, to process, to store, to process and to

the boundary of our universe is still expanding at the speed of light [17].

S ¼ �k ln p (11) I ¼ � log <sup>2</sup> p (12) ΔE Δt≥h (13)

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup>*=c*<sup>2</sup> <sup>p</sup> (14)

Þ*Δ*t, *Δ*E *Δ*t≥h (15)

principles "approximately" as follows:

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

speed within the subspaces remains the same.

forms, since mathematics is a "language", as given by

*Δt*

<sup>U</sup> : *<sup>Δ</sup>*<sup>E</sup> *<sup>Δ</sup>*<sup>t</sup> <sup>≥</sup>ðΔmc2

time" or remains constant.

changes naturally with time.

tain the information is.

"with time".

erg-second.

"tangle".

**171**

As we investigate Schrödinger's hypothesis of **Figure 5**, immediately we see that; it is "not" a physical realizable postulation at all, since within the box it has a timeless (t = 0) or time independent radioactive particle in it. As we know that; any particle within a temporal (t > 0) subspace has to be a temporal (t > 0) particle or has time with it, otherwise the proposed radioactive particle cannot be existed within Schrödinger's temporal (t > 0) box. It is therefore, the paradox of Schrödinger's cat is "not" a physical realizable hypothesis and we should "not" have had treated Schrödinger's cat as a physically real paradox.

Since every problem has multi solutions, I can change the scenarios of Schrödinger's box a little bit, such as allow a small group of individuals take turn to open the box. After each observation, close the box before passing on to the next observer. My question is that how many times the superposition has to collapse? With all those apparent contradicted logics, we see that Schrödinger's cat is "not" a paradox after all! And the root of timeless (t = 0) superposition principle as based on Bohr's quantum leap hυ, represents a time "unlimited" radiator, which is a singularity approximated wave solution. For which we should "not" have treated quantum leap hυ a physical real radiator, since any quantum leap has to be time and band limited within our universe.

Finally I would address that; all the laws, principles, theories and paradoxes were made to be broken, revised and replaced, it is not they were all approximated, because they all changes with time or temporal (t > 0). Yet, without approximated science, then there would be no science in which we have shown that a simple hypothetical paradox takes decades to resolve! And this is the nature of quantum mechanics and is all about temporal (t > 0) subspace.

#### **5. Nature of Δt**

Since our universe was assumed created with a huge energy explosion with time situated within a "non-empty" space. Every subspace "no" matter how small is created by an amount of energy ΔE and a section of time Δt for which every subspace is temporal (t > 0) (i.e., existed with time).

In view of modern science, there is a set of simple, yet elegant laws and principles that are profoundly associated with a unit of (ΔE Δt). The objective of this section is to explore the relationship between these laws and principles as related with the unit of (ΔE, Δt). Since time is a dependent forward variable moves at a constant speed, we see that Δt is one of the most "esoteric" variable existed within our universe. We will show that once a moment of Δt is used, we "cannot" get it back although ΔE and Δt can be traded. In which I will show that; there it is a physical limit for Δt to approaching to "none" (i.e., Δt⟶0), that "prevents" us to reach; even though we have the all the price to pay. And this must be the nature of Δt?

Nevertheless, there is a set of "simple and elegant" laws and principles that are profoundly associated with a section of time Δt. These are laws and principle of entropy of Boltzmann [4], information of Shannon [16], uncertainty of Heisenberg [3], relativity of Einstein [1] and temporal (t > 0) universe [17]. Each of them has associated with a section of time Δt which changes naturally with time. And all these evidences tell us science has to be temporal (t > 0) and dynamics, which cannot be "static" or timeless (t = 0). In other words, if there has no time, then there has no science. Nevertheless, science is a law of "approximation", as in contrast

bottle, triggered by the decaying of a radio-active particle, to kill the cat. Since the box is assumed totally opaque of which no one knows that the cat will be killed or not, as imposed by the Schrödinger's superposition principle until we open his box. As we investigate Schrödinger's hypothesis of **Figure 5**, immediately we see that;

Schrödinger's cat is "not" a physical realizable hypothesis and we should "not" have

Finally I would address that; all the laws, principles, theories and paradoxes were made to be broken, revised and replaced, it is not they were all approximated, because they all changes with time or temporal (t > 0). Yet, without approximated science, then there would be no science in which we have shown that a simple hypothetical paradox takes decades to resolve! And this is the nature of quantum

Since our universe was assumed created with a huge energy explosion with time

In view of modern science, there is a set of simple, yet elegant laws and principles that are profoundly associated with a unit of (ΔE Δt). The objective of this section is to explore the relationship between these laws and principles as related with the unit of (ΔE, Δt). Since time is a dependent forward variable moves at a constant speed, we see that Δt is one of the most "esoteric" variable existed within our universe. We will show that once a moment of Δt is used, we "cannot" get it back although ΔE and Δt can be traded. In which I will show that; there it is a physical limit for Δt to approaching to "none" (i.e., Δt⟶0), that "prevents" us to reach; even though we have the all the price to pay. And this must be the

Nevertheless, there is a set of "simple and elegant" laws and principles that are profoundly associated with a section of time Δt. These are laws and principle of entropy of Boltzmann [4], information of Shannon [16], uncertainty of Heisenberg [3], relativity of Einstein [1] and temporal (t > 0) universe [17]. Each of them has associated with a section of time Δt which changes naturally with time. And all these evidences tell us science has to be temporal (t > 0) and dynamics, which cannot be "static" or timeless (t = 0). In other words, if there has no time, then there has no science. Nevertheless, science is a law of "approximation", as in contrast

situated within a "non-empty" space. Every subspace "no" matter how small is created by an amount of energy ΔE and a section of time Δt for which every

it is "not" a physical realizable postulation at all, since within the box it has a timeless (t = 0) or time independent radioactive particle in it. As we know that; any particle within a temporal (t > 0) subspace has to be a temporal (t > 0) particle or has time with it, otherwise the proposed radioactive particle cannot be existed within Schrödinger's temporal (t > 0) box. It is therefore, the paradox of

Since every problem has multi solutions, I can change the scenarios of Schrödinger's box a little bit, such as allow a small group of individuals take turn to open the box. After each observation, close the box before passing on to the next observer. My question is that how many times the superposition has to collapse? With all those apparent contradicted logics, we see that Schrödinger's cat is "not" a paradox after all! And the root of timeless (t = 0) superposition principle as based on Bohr's quantum leap hυ, represents a time "unlimited" radiator, which is a singularity approximated wave solution. For which we should "not" have treated quantum leap hυ a physical real radiator, since any quantum leap has to be time and band

had treated Schrödinger's cat as a physically real paradox.

mechanics and is all about temporal (t > 0) subspace.

subspace is temporal (t > 0) (i.e., existed with time).

limited within our universe.

*Quantum Mechanics*

**5. Nature of Δt**

nature of Δt?

**170**

with mathematics, which is an axiom of "certainty", of which I state these laws and principles "approximately" as follows:

Law of entropy; entropy within an enclosed subspace increases naturally "with time" or remains constant.

Theory of information; the higher the amount the information, the more uncertain the information is.

Principle of uncertainty; uncertainty of an isolated particle increases naturally "with time".

Theory of special relativity; when a subspace moves faster "relatively" than the other subspace; there is a "relativistic" time speed between them, although time speed within the subspaces remains the same.

Nature of universe; every isolated subspace was created by amount of energy ΔE and a section of time Δt and it is a dynamic temporal (t > 0) stochastic subspace changes naturally with time.

Nevertheless, it is easier to facilitate these laws and principles in mathematical forms, since mathematics is a "language", as given by

$$\mathbf{S} = -\mathbf{k} \text{ ln } \mathbf{p} \tag{11}$$

$$\mathbf{I} = -\log\_2 \mathbf{p} \tag{12}$$

$$
\Delta \mathbf{E} \,\,\Delta \mathbf{t} \ge \mathbf{h} \,\tag{13}
$$

$$
\Delta t' = \frac{\Delta t}{\sqrt{1 - \nu^2/c^2}}\tag{14}
$$

$$\mathbf{U}: \Delta \mathbf{E} \, \Delta \mathbf{t} \ge (\Delta \mathbf{m} \mathbf{c}^2) \Delta \mathbf{t}, \,\, \Delta \mathbf{E} \, \Delta \mathbf{t} \ge \mathbf{h} \tag{15}$$

where S, I, and U are entropy, information and universe respectively, k is the Boltzmann's constant, h is the Planck's constant, p is the probability, Δt is a section of time, Δt' is the dilated section of time, v is the velocity, m is the mass and c is the speed of light. k = 1.38 � <sup>10</sup>�<sup>16</sup> ergs per degree centigrade and h = 6.624 � <sup>10</sup>�<sup>27</sup> erg-second.

In this we see that our universe was created by means a "huge" amount of energy ΔE and a "long" section of time Δt. And Δt is "still" extending rapidly, since the boundary of our universe is still expanding at the speed of light [17].

In view of these laws and principles, they must be the most "elegant and simple" science equations that existed today in which these equations either attached or associated with a section of time Δt, except Eq. (12) since information theory is mathematics. But as soon information is recognized as related to entropy, information is equivalent to an amount of entropy; this makes an amount of information a physical quantity which is acceptable in science. For which we will show that a section of Δt will be associated with the theory of information, otherwise information will be very difficult to apply in science. Since Δt is coexisted with ΔE, we will further see that; every bit of information takes an amount of energy ΔE and a section of time Δt to transmit, to create, to process, to store, to process and to "tangle".

As we got back from Eq. (11) to Eq. (15) we see that; they are all pointsingularity approximated; otherwise it will be very difficult to write in simple mathematical forms. As the laws and principles stated, there are all associated with time, by which they are all space-time variable laws and principles, since time is space and space is time within our temporal (t > 0) universe. In short, they are all connected to a unit of (Δt, ΔE) which is the basic building blocks of our universe. For which I envision that; every existence within our universe has a beginning and has an end. But it is time; it has "no" beginning and has "no" end!

Since our temporal (t > 0) universe was created based on a commonly accepted Big Bang Theory [17], we see that our universe as is a temporal (t > 0) dynamic "stochastic" subspace [18]. The boundary of our universe increases at the speed of light, we see that; every subspace within our universe is a "nonempty" temporal (t > 0) stochastic subspace. By the way, any one or two dimensional subspaces "cannot" be existed within our universe, since one or two dimensional subspaces are volume-less for which any independent Euclidian subspace "cannot" be simply applied to describe a temporal (t > 0) subspace. Because all the dimensional coordinates (e.g., x y z coordinates) of a temporal space are all "interdependent" with time, where time is a forward variable with respect to the subspace. In other words, every substance no matter how small it is, has to have time and temporal (t > 0).

In view of the time dilation of Einstein's relativity of Eq. (14) and Heisenberg's uncertainty principle of Eq. (13); we see that they are associated with a section of time Δt; which represents a "temporal (t > 0)" subspace, as given by;

$$
\Delta \mathbf{r} = \mathbf{c} \,\Delta \mathbf{t}
$$

where r is the radius of a spherical subspace and c is the velocity of light. In which we see that subspace enlarges rapidly as Δt increases is given by

$$\mathbf{V} = \left(\%\right) \pi \left(\mathbf{c} \,\Delta \mathbf{t}\right)^{\mathbf{\hat{3}}} \tag{16}$$

bits? Similarly, an amount of information in bits is not given us the actual information, but it is a "necessary cost" but "not sufficient" to obtain the precise information. In which we see that; the amount of entropy ΔS is a "necessary cost" needed to

where ΔE=hΔυ is the quantum leap energy and T = C + 273 is the absolute temperature in Kelvin, C is the temperature in degree Celsius. In which we see that; higher the thermal noise requires higher energy to transmit a of bit information.

Thus, we see that an amount of entropy is equivalent to an amount of information, but it is "not" the information. But an amount of information is equivalent to an amount of entropy that makes information a very "viable" physical quantity can be applied in science. In which we see that; information and entropy can be simply

Nevertheless, we have shown that; either information or entropy has to be a

where k is the Boltzmann's constant. In which we see that either information or entropy "increases" with time, and (t > 0) denotes imposition by temporal (t > 0) constraint. The amount of entropy for I(t) bits of information can be written as

where I(t) is in bits and k is the Boltzmann's constant. In view of preceding equation, it shows that entropy increases as amount information increases. In which we see that "every bit" of information ΔI takes an amount of energy ΔE and a

Since "every bit" of information is equivalent to an amount of entropy ΔS,

Thus, every quantity of entropy ΔS is "equivalently" equaled to an amount of

where T = C + 273 is the absolute thermal noise temperature in Kelvin, C is the temperature in degree Celsius, h is the Planck's constant. Since ΔE is "coexisted" with Δt, it is reasonable to say that; every ΔS is also associated with a section of time

temporal (t > 0) or time dependent law, as given by respectively;

section of time Δt to "create" or to transmit as given by

energy ΔE and a section of time Δt to produce as shown by

ΔS ¼ ΔE*=*T ¼ hΔυ*=*T (19)

ΔE ¼ T k ln 2 (20)

ΔS()ΔI (21)

I tðÞ¼� log <sup>2</sup> p tð Þ, t> 0 (22) S tðÞ¼�k ln p tð Þ, t> 0 (23)

S tðÞ¼ kI tð Þ ln2, t>0 (24)

ΔI � ΔE Δt ¼ h, per bit of information (25)

ΔS ¼ k ln 2, per bit of information (26)

ΔS ¼ ΔE*=*T (27)

Since entropy is a "physical quantity" similar to energy, as given by

obtain an equivalent number of information in bits.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

traded as given by

Δt as given by

**173**

This shows precisely our universe is expanding with a section of time Δt. Since ΔE is a physical quantity equivalent to a subspace that "cannot" be empty and coexisted with Δt, then every unit (ΔE, Δt) is a temporal (t > 0) subspace, in which we see that time and space "cannot" be separated. In other words, time and space are "interdependent" although ΔE is a physical quantity but Δt is an invisible "real" variable.

#### **6. Entropy and information**

As we look back at Boltzmann entropy Eq. (11), we see that it is a typical timeless (t = 0) point-singularity approximated equation. But the law described; entropy increases with "time", implies that entropy is associated with a section of time Δt, although it is "not" shown in the equation. Nevertheless, law of entropy is essentially identical to the law of information as can be seen by their logarithmic expressions of Eq. (11) and Eq. (12), for which we have the following relationships as given by [19];

$$\mathbf{S} = \mathbf{k} \text{ I } \ln \text{ 2} \tag{17}$$

where I is an amount of information in "bit" and k is Boltzmann's constant in which we see that, "every bit" of information is equal to an amount of entropy ΔS which is given by

$$
\Delta \mathbf{S} = \mathbf{k} \text{ \(n \) 2, per bit of information} \tag{18}
$$

Although an amount of information can be "traded" for a quantity of entropy, but entropy is a "cost" in energy "equivalents" to an amount of information, but "not" the "actual" information. In other words, it is a "necessary cost" of an amount of entropy to pay for an amount of information in bits. For example, if an amount of entropy ΔS is equivalent to 1000 bits of information of a specific book. Then how many books have the same 1000 bits or how many different items has also 1000

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

Since our temporal (t > 0) universe was created based on a commonly accepted Big Bang Theory [17], we see that our universe as is a temporal (t > 0) dynamic "stochastic" subspace [18]. The boundary of our universe increases at the speed of light, we see that; every subspace within our universe is a "nonempty" temporal (t > 0) stochastic subspace. By the way, any one or two dimensional subspaces "cannot" be existed within our universe, since one or two dimensional subspaces are volume-less for which any independent Euclidian subspace "cannot" be simply applied to describe a temporal (t > 0) subspace. Because all the dimensional coordinates (e.g., x y z coordinates) of a temporal space are all "interdependent" with time, where time is a forward variable with respect to the subspace. In other words, every substance no matter how small it is, has to have time and temporal (t > 0). In view of the time dilation of Einstein's relativity of Eq. (14) and Heisenberg's uncertainty principle of Eq. (13); we see that they are associated with a section of

time Δt; which represents a "temporal (t > 0)" subspace, as given by;

which we see that subspace enlarges rapidly as Δt increases is given by

although ΔE is a physical quantity but Δt is an invisible "real" variable.

**6. Entropy and information**

as given by [19];

*Quantum Mechanics*

which is given by

**172**

Δr ¼ c Δt

where r is the radius of a spherical subspace and c is the velocity of light. In

V ¼ ð Þ ¾ π ðc ΔtÞ

As we look back at Boltzmann entropy Eq. (11), we see that it is a typical timeless (t = 0) point-singularity approximated equation. But the law described; entropy increases with "time", implies that entropy is associated with a section of time Δt, although it is "not" shown in the equation. Nevertheless, law of entropy is essentially identical to the law of information as can be seen by their logarithmic expressions of Eq. (11) and Eq. (12), for which we have the following relationships

where I is an amount of information in "bit" and k is Boltzmann's constant in which we see that, "every bit" of information is equal to an amount of entropy ΔS

Although an amount of information can be "traded" for a quantity of entropy, but entropy is a "cost" in energy "equivalents" to an amount of information, but "not" the "actual" information. In other words, it is a "necessary cost" of an amount of entropy to pay for an amount of information in bits. For example, if an amount of entropy ΔS is equivalent to 1000 bits of information of a specific book. Then how many books have the same 1000 bits or how many different items has also 1000

This shows precisely our universe is expanding with a section of time Δt. Since ΔE is a physical quantity equivalent to a subspace that "cannot" be empty and coexisted with Δt, then every unit (ΔE, Δt) is a temporal (t > 0) subspace, in which we see that time and space "cannot" be separated. In other words, time and space are "interdependent"

<sup>3</sup> (16)

S ¼ k I ln 2 (17)

ΔS ¼ k ln 2, per bit of information (18)

bits? Similarly, an amount of information in bits is not given us the actual information, but it is a "necessary cost" but "not sufficient" to obtain the precise information. In which we see that; the amount of entropy ΔS is a "necessary cost" needed to obtain an equivalent number of information in bits.

Since entropy is a "physical quantity" similar to energy, as given by

$$
\Delta \mathbf{S} = \Delta \mathbf{E}/\mathbf{T} = \mathbf{h} \Delta \mathbf{u}/\mathbf{T} \tag{19}
$$

where ΔE=hΔυ is the quantum leap energy and T = C + 273 is the absolute temperature in Kelvin, C is the temperature in degree Celsius. In which we see that; higher the thermal noise requires higher energy to transmit a of bit information.

$$
\Delta \mathbf{E} = \mathbf{T} \,\mathbf{k} \,\, \ln \, \mathbf{2} \tag{20}
$$

Thus, we see that an amount of entropy is equivalent to an amount of information, but it is "not" the information. But an amount of information is equivalent to an amount of entropy that makes information a very "viable" physical quantity can be applied in science. In which we see that; information and entropy can be simply traded as given by

$$
\Delta \mathbf{S} \Longleftrightarrow \Delta \mathbf{I} \tag{21}
$$

Nevertheless, we have shown that; either information or entropy has to be a temporal (t > 0) or time dependent law, as given by respectively;

$$\mathbf{I(t)} = -\log\_2 \mathbf{p(t)}, \mathbf{t} > \mathbf{0} \tag{22}$$

$$\mathbf{S(t)} = -\mathbf{k} \text{ ln } \mathbf{p(t)}, \mathbf{t} > \mathbf{0} \tag{23}$$

where k is the Boltzmann's constant. In which we see that either information or entropy "increases" with time, and (t > 0) denotes imposition by temporal (t > 0) constraint. The amount of entropy for I(t) bits of information can be written as

$$\mathbf{S(t)} = \mathbf{k} \,\mathrm{I(t)} \,\mathrm{ln2}, \mathrm{t} > \mathbf{0} \tag{24}$$

where I(t) is in bits and k is the Boltzmann's constant. In view of preceding equation, it shows that entropy increases as amount information increases. In which we see that "every bit" of information ΔI takes an amount of energy ΔE and a section of time Δt to "create" or to transmit as given by

$$
\Delta \mathbf{I} \sim \Delta \mathbf{E} \text{ } \Delta \mathbf{t} = \mathbf{h} \text{, per bit of information} \tag{25}
$$

Since "every bit" of information is equivalent to an amount of entropy ΔS,

$$
\Delta \mathbf{S} = \mathbf{k} \text{ [n 2, per bit of information]} \tag{26}
$$

Thus, every quantity of entropy ΔS is "equivalently" equaled to an amount of energy ΔE and a section of time Δt to produce as shown by

$$
\Delta \mathbf{S} = \Delta \mathbf{E}/\mathbf{T} \tag{27}
$$

where T = C + 273 is the absolute thermal noise temperature in Kelvin, C is the temperature in degree Celsius, h is the Planck's constant. Since ΔE is "coexisted" with Δt, it is reasonable to say that; every ΔS is also associated with a section of time Δt as given by

$$
\Delta \mathbf{S} \sim \mathbf{E} \,\Delta \mathbf{t}/\mathbf{T} = \mathbf{h}/\mathbf{T}, \text{per bit of information} \tag{28}
$$

In which we see that information is connected with the law of uncertainty, where "every bit" of information is profoundly associated with ΔE and Δt.

Since every subspace within our universe is created by an amount of energy ΔE and a section of time Δt, we see that; Boltzmann's entropy, Shannon's information, Heisenberg's uncertainty and Einstein's relativity has a profound association with a section of Δt and of ΔE since they are coexisted. In other words, all the laws, principles, and theories as well the paradoxes have to comply with the "coexistence" of ΔE and Δt, otherwise those laws and principles cannot guarantee to be existed within our universe.

Nevertheless, increasing entropy is regarded as a "degradation" of energy by Kelvin [19], although entropy was originated by Clausius [19]. But he might have intended it to be used as a "negative" of entropy (i.e., neg-entropy) in which we see that as entropy or amount of information increases means that there is "energy degradation". This is also meant that entropy or amount of information "degrades with time". Let me stress again "energy degradation" within our universe is due to boundary expansion of our universe at the speed of light [17]. For which I see it entropy increases with time is "no longer" a myth, as most scientists believed it is.

Since all the laws and principles are attached with a price-tag of (ΔE, Δt), but it is the Δt⟶0 that "cannot" be reached, even though we assumed having all the energy of ΔE to pay for! This is precisely the "physical" limit of a temporal (t > 0) subspace, by which the "instantaneous" moment of time (i.e., t = 0) can be approached but can "never" be able to attend, regardless how much of energy ΔE we willing to pay for. And this is the nature of Δt!

#### **7. Uncertainty and information**

Every substance or subspace has a piece of information which includes all the elementary particles, basic building blocks of the subspaces, atoms, papers, our planet, solar system, galaxy, and even our universe! In other words, the universe is flooded with information (i.e., spatial and temporal), or information fills up the whole universe. Strictly speaking, when one is dealing with the origin of the universe, the aspect of information has never been absence. Then, one would ask: What would be the amount of information, aside the needed energy ΔE, is required to create a specific substance? Or equivalently, what would be the "cost" of entropy to create it? To answer this question is to let me start with the law of uncertainty, in equivalent form, as given by

$$
\Delta \mathbf{u} \,\Delta \mathbf{t} = \mathbf{1} \tag{29}
$$

to Δt or from ΔE to Δt, since Δυ and ΔE are physical quantities. For which we see that; once a section of Δt is "used", we "cannot" get back the same moment of Δt, although we can create the same section of Δt, since time is a forward dependent

*Various (Δυ, Δt) information cells, where Δυ<sup>n</sup> and Δtn are the bandwidths and time-limited sections, and*

Nevertheless, there are basically two types of information transmission; one

Since every subspace within our universe is a temporal (t > 0) subspace, the radius of any subspace can be described by a time-dependent variable as given by

r ¼ *c* �Δ*t* (30)

is limited by uncertainty Principle and the other is constrained within the "certainty subspace". And the boundary between these two regimes is given by Δυ�Δt = 1 (or ΔE�Δt = h) as I called this limit a Quantum Unit [21]. In which we see that Δυ can be traded for Δt. But under uncertainty regime, information is carried by means of intensity (i.e., amplitude square) variation. Yet, information can also be transmitted within the certainty regime, such as applied to complexamplitude communication [22, 23]. As limited by the law of uncertainty, a quantum unit subspace QLS, for (ΔE, Δt) and (Δυ, Δt), are shown in **Figure 7**

variable.

**Figure 6.**

*υ<sup>1</sup> > υ<sup>2</sup> >υ<sup>3</sup> > … >υ<sup>n</sup> are the frequencies.*

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

for reference.

**175**

where υ is the bandwidth, in which there exists a profound relationship of an "information cell" [20], as illustrated in **Figure 6**. In which we see that, the shape of (Δυ, Δt) or equivalently (ΔE, Δt) can be "mutually" exchanged. Since every bit of information can be efficiently transmitted, if and only if it is transmitting within the constraint of the uncertainty principle (i.e., Δυ�Δt ≥ 1). This relationship implies that the signal bandwidth should be either equal or smaller than the system bandwidth (i.e., 1/Δt ≤ Δυ). In which we see that Δt and Δυ can be "traded".

It is however the unit region but not the shape of the information cell that determines the limit, as illustrated in **Figure 6**, we see that; within each unit cell, that is (Δυ, Δt) [or equivalently (ΔE, Δt)] can be mutually traded. But it is from Δυ *Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

#### **Figure 6.**

ΔS � E Δt*=*T ¼ h*=*T, per bit of information (28)

In which we see that information is connected with the law of uncertainty, where "every bit" of information is profoundly associated with ΔE and Δt.

within our universe.

*Quantum Mechanics*

we willing to pay for. And this is the nature of Δt!

**7. Uncertainty and information**

equivalent form, as given by

**174**

Since every subspace within our universe is created by an amount of energy ΔE and a section of time Δt, we see that; Boltzmann's entropy, Shannon's information, Heisenberg's uncertainty and Einstein's relativity has a profound association with a section of Δt and of ΔE since they are coexisted. In other words, all the laws, principles, and theories as well the paradoxes have to comply with the "coexistence" of ΔE and Δt, otherwise those laws and principles cannot guarantee to be existed

Nevertheless, increasing entropy is regarded as a "degradation" of energy by Kelvin [19], although entropy was originated by Clausius [19]. But he might have intended it to be used as a "negative" of entropy (i.e., neg-entropy) in which we see that as entropy or amount of information increases means that there is "energy degradation". This is also meant that entropy or amount of information "degrades with time". Let me stress again "energy degradation" within our universe is due to boundary expansion of our universe at the speed of light [17]. For which I see it entropy increases with time is "no longer" a myth, as most scientists believed it is. Since all the laws and principles are attached with a price-tag of (ΔE, Δt), but it is the Δt⟶0 that "cannot" be reached, even though we assumed having all the energy of ΔE to pay for! This is precisely the "physical" limit of a temporal (t > 0) subspace, by which the "instantaneous" moment of time (i.e., t = 0) can be approached but can "never" be able to attend, regardless how much of energy ΔE

Every substance or subspace has a piece of information which includes all the elementary particles, basic building blocks of the subspaces, atoms, papers, our planet, solar system, galaxy, and even our universe! In other words, the universe is flooded with information (i.e., spatial and temporal), or information fills up the whole universe. Strictly speaking, when one is dealing with the origin of the universe, the aspect of information has never been absence. Then, one would ask: What would be the amount of information, aside the needed energy ΔE, is required to create a specific substance? Or equivalently, what would be the "cost" of entropy to create it? To answer this question is to let me start with the law of uncertainty, in

where υ is the bandwidth, in which there exists a profound relationship of an "information cell" [20], as illustrated in **Figure 6**. In which we see that, the shape of (Δυ, Δt) or equivalently (ΔE, Δt) can be "mutually" exchanged. Since every bit of information can be efficiently transmitted, if and only if it is transmitting within the constraint of the uncertainty principle (i.e., Δυ�Δt ≥ 1). This relationship implies that the signal bandwidth should be either equal or smaller than the system band-

width (i.e., 1/Δt ≤ Δυ). In which we see that Δt and Δυ can be "traded".

It is however the unit region but not the shape of the information cell that determines the limit, as illustrated in **Figure 6**, we see that; within each unit cell, that is (Δυ, Δt) [or equivalently (ΔE, Δt)] can be mutually traded. But it is from Δυ

Δυ Δt ¼ 1 (29)

*Various (Δυ, Δt) information cells, where Δυ<sup>n</sup> and Δtn are the bandwidths and time-limited sections, and υ<sup>1</sup> > υ<sup>2</sup> >υ<sup>3</sup> > … >υ<sup>n</sup> are the frequencies.*

to Δt or from ΔE to Δt, since Δυ and ΔE are physical quantities. For which we see that; once a section of Δt is "used", we "cannot" get back the same moment of Δt, although we can create the same section of Δt, since time is a forward dependent variable.

Nevertheless, there are basically two types of information transmission; one is limited by uncertainty Principle and the other is constrained within the "certainty subspace". And the boundary between these two regimes is given by Δυ�Δt = 1 (or ΔE�Δt = h) as I called this limit a Quantum Unit [21]. In which we see that Δυ can be traded for Δt. But under uncertainty regime, information is carried by means of intensity (i.e., amplitude square) variation. Yet, information can also be transmitted within the certainty regime, such as applied to complexamplitude communication [22, 23]. As limited by the law of uncertainty, a quantum unit subspace QLS, for (ΔE, Δt) and (Δυ, Δt), are shown in **Figure 7** for reference.

Since every subspace within our universe is a temporal (t > 0) subspace, the radius of any subspace can be described by a time-dependent variable as given by

$$
\mathbf{r} = \mathbf{c} \cdot \Delta t \tag{30}
$$

where c is the speed of light, and Δt represents a section of time. In which we see that the size of the subspace enlarges rapidly as Δt increases as given by

$$\mathbf{V} = \left(\%\right) \pi \left(\mathbf{c} \,\Delta \mathbf{t}\right)^{\text{3}} \tag{31}$$

quantum limited subspace is for complex information transmission, for example, as applied to complex wave front construction (i.e., holographic recording) [23], complex-match filter synthesis [24], as well as synthetic aperture radar imaging [22]. But there is an apparent price paid for using a "wider" section of time Δt;

One of the important aspects of information transmission is that "reliable" information can be transmitted, such that information can be reached to the receiver with a "high degree of certainty". Let me take two key equations from information theory, "mutual information" transmission through a "passive additive

where H(A) is the information provided by the sender, H(A/B) is the information loss (or equivocation) through transmission due to noise, H(B) is information

However, there is a basic distinction between these two equations: one is for **"**reliable" information transmission and the other is for "retrievable" information. Although both equations represent the mutual information transmission between sender and receiver; but their objectives are rather different. Example; using Eq. (32) is purposely designed for "reliable information transmission" in which the transmitted information has a high degree of "certainty" to reach the receiver. While Eq. (33) is purposely designed to "retrieve information" from "unreliable" information" by the receiver. For which we see that; for "reliable" information transmission, one can simply increase the signal to noise ratio at the transmitting end. While for "unreliable" information transmission is to extract information from ambiguous information. In other words, one is to be sure information will be reached to the receiver "before" information is transmitted, and the other is to

In communication, basically there are two orientations: one by Norbert Wiener [25, 26] and the other by Claude Shannon [16]. But there is a major distinction between them; Wiener's communication strategy is that; if the information is corrupted through transmission, it may be recovered at the receiving end, but with a "cost" mostly at the receiving end. While Shannon's communication strategy carries a step further by encoding the information before it is transmitted such that, information can be "reliably" transmitted, also with a "cost" mostly at the transmitting end. In view of the Wiener and Shannon information transmission orientations; mutual information transfer of Eq. (32) is kind of Shannon type, while Eq. (33) is kind of Wiener type. In which we see that; "reliable" information transmission is basically controlled by the sender; It is to "minimize" the noise

One simple way to do it is by increasing the signal to noise ratio, with a "cost" of

received by the receiver, and H(B/A) is noise entropy of channel.

retrieve the information "after" information has been received.

entropy H(A/B) (or equivocation) of the channel, as shown by

higher signal energy (i.e., ΔE).

**177**

I A; B ð Þ¼ H Að Þ–H Að Þ *=*B (32)

I A; B ð Þ¼ H Bð Þ–H Bð Þ *=*A (33)

I A; B ð Þ≈ H Að Þ (34)

which "deviates" further away from real-time transmission.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

**8. Reliable communication**

noise channel" as given by [19]

and

Since the carrier bandwidth Δυ and time resolution Δt are exchangeable, we see that the size of the QLS enlarges as the carrier bandwidth Δυ decreases. In other words, narrower the carrier bandwidth Δυ has the advantage of having a larger quantum limited subspace for complex-amplitude communication as depicted in **Figure 8**.

In this we see that it is possible to create a temporal (t > 0) subspace within a temporal (t > 0) space (i.e., our universe) for communication. We stress that; it is "not" possible to create any time independent or timeless (t = 0) subspace within our temporal universe, since timeless (t = 0) or time independent "cannot" be existed within temporal universe. And this timeless (t = 0) or the "instantaneous limit" (or the causal condition) is the fact of physical limit (i.e., Δt⟶0) within our universe. This limit can only be approached with huge amount of energy ΔE, but we can "never" be able to reach it?

Furthermore, let me note that; timeless (t = 0) or time independent subspace is "not" an "inaccessible" space as some scientists claimed, since inaccessible implies it existed within our universe. Nevertheless, one of the apparent aspects of using large

**Figure 8.**

*A "very large" quantum limited subspace as depicted in (a) can be realized in practice within our temporal (t > 0) space, for example, such as applied to synthetic aperture radar imaging shown in (b).*

quantum limited subspace is for complex information transmission, for example, as applied to complex wave front construction (i.e., holographic recording) [23], complex-match filter synthesis [24], as well as synthetic aperture radar imaging [22]. But there is an apparent price paid for using a "wider" section of time Δt; which "deviates" further away from real-time transmission.

#### **8. Reliable communication**

One of the important aspects of information transmission is that "reliable" information can be transmitted, such that information can be reached to the receiver with a "high degree of certainty". Let me take two key equations from information theory, "mutual information" transmission through a "passive additive noise channel" as given by [19]

$$\mathbf{H(A;B)} = \mathbf{H(A)} \mathbf{-H(A/B)}\tag{32}$$

and

where c is the speed of light, and Δt represents a section of time. In which we see

*A set of quantum limited subspaces (QLS). (a) Shows a ΔE limited subspace; (b) Shows a Δv limited subspace.*

Since the carrier bandwidth Δυ and time resolution Δt are exchangeable, we see that the size of the QLS enlarges as the carrier bandwidth Δυ decreases. In other words, narrower the carrier bandwidth Δυ has the advantage of having a larger quantum limited subspace for complex-amplitude communication as depicted in

In this we see that it is possible to create a temporal (t > 0) subspace within a temporal (t > 0) space (i.e., our universe) for communication. We stress that; it is "not" possible to create any time independent or timeless (t = 0) subspace within our temporal universe, since timeless (t = 0) or time independent "cannot" be existed within temporal universe. And this timeless (t = 0) or the "instantaneous limit" (or the causal condition) is the fact of physical limit (i.e., Δt⟶0) within our universe. This limit can only be approached with huge amount of energy ΔE,

Furthermore, let me note that; timeless (t = 0) or time independent subspace is "not" an "inaccessible" space as some scientists claimed, since inaccessible implies it existed within our universe. Nevertheless, one of the apparent aspects of using large

*A "very large" quantum limited subspace as depicted in (a) can be realized in practice within our temporal*

*(t > 0) space, for example, such as applied to synthetic aperture radar imaging shown in (b).*

<sup>V</sup> <sup>¼</sup> ð Þ <sup>¾</sup> <sup>π</sup> ð Þ <sup>c</sup> <sup>Δ</sup><sup>t</sup> <sup>3</sup> (31)

that the size of the subspace enlarges rapidly as Δt increases as given by

**Figure 8**.

**Figure 8.**

**176**

**Figure 7.**

*Quantum Mechanics*

but we can "never" be able to reach it?

$$\mathbf{H(A;B)} = \mathbf{H(B)} \mathbf{-H(B/A)}\tag{33}$$

where H(A) is the information provided by the sender, H(A/B) is the information loss (or equivocation) through transmission due to noise, H(B) is information received by the receiver, and H(B/A) is noise entropy of channel.

However, there is a basic distinction between these two equations: one is for **"**reliable" information transmission and the other is for "retrievable" information. Although both equations represent the mutual information transmission between sender and receiver; but their objectives are rather different. Example; using Eq. (32) is purposely designed for "reliable information transmission" in which the transmitted information has a high degree of "certainty" to reach the receiver. While Eq. (33) is purposely designed to "retrieve information" from "unreliable" information" by the receiver. For which we see that; for "reliable" information transmission, one can simply increase the signal to noise ratio at the transmitting end. While for "unreliable" information transmission is to extract information from ambiguous information. In other words, one is to be sure information will be reached to the receiver "before" information is transmitted, and the other is to retrieve the information "after" information has been received.

In communication, basically there are two orientations: one by Norbert Wiener [25, 26] and the other by Claude Shannon [16]. But there is a major distinction between them; Wiener's communication strategy is that; if the information is corrupted through transmission, it may be recovered at the receiving end, but with a "cost" mostly at the receiving end. While Shannon's communication strategy carries a step further by encoding the information before it is transmitted such that, information can be "reliably" transmitted, also with a "cost" mostly at the transmitting end. In view of the Wiener and Shannon information transmission orientations; mutual information transfer of Eq. (32) is kind of Shannon type, while Eq. (33) is kind of Wiener type. In which we see that; "reliable" information transmission is basically controlled by the sender; It is to "minimize" the noise entropy H(A/B) (or equivocation) of the channel, as shown by

$$\mathbf{I(A;B)} \approx \mathbf{H(A)}\tag{34}$$

One simple way to do it is by increasing the signal to noise ratio, with a "cost" of higher signal energy (i.e., ΔE).

On the other hand, to recovering the transmitted information is to "maximize" H(B/A) (the channel noise). Since the entropy H(B) at the receiving end is "larger" than the entropy at the sending end; that is H(B) > H(A), we have,

$$\mathbf{H(A;B)} = \mathbf{H(B)} \mathbf{-H(B/A)} \approx \mathbf{H(A)}\tag{35}$$

"narrower" section of relativistic time (Δt') with respect to an standstill subspace, since relativistic dilation time window is wider Δt' > Δt. In which we see that; "relativistic" uncertainty within the moving subspace, as with respect to a standstill

2

2

In which we see ΔE energy is "conserved". Thus a "narrower" time-window Δt can be squeeze as with respect to standstill subspace. This is precisely physically possible to exploit for "time-domain" digital communication, as from ground

One the other hand, as from satellite to ground station digital-transmission, we

� <sup>½</sup> <sup>¼</sup> <sup>h</sup> (38)

� <sup>½</sup> <sup>¼</sup> <sup>1</sup> (39)

� �<sup>2</sup> <sup>q</sup> <sup>¼</sup> *<sup>h</sup>* (40)

� �<sup>2</sup> <sup>q</sup> <sup>¼</sup> <sup>1</sup> (41)

ΔE Δt'½1–ðν*=*cÞ

Δυ Δt'½1–ðν*=*cÞ

might want to use digital-bandwidth (i.e., Δν). This is a "frequency-domain" information transmission strategy, as in contrast with time-domain, which has "not" fully exploited yet. In which the "relativistic" uncertainty relationship within the standstill subspace as with respect to the moving subspace can be written as

> Δ*E Δt* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>v</sup> c*

Δ*v Δt* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>v</sup> c*

In this we see that a narrower bandwidth Δ*v* can be used for "frequency

taneously" (i.e., t = 0) within our temporal (t > 0) universe.

Nevertheless, the essence of ΔE Δt = h (or Δυ Δt = 1) shows that ΔE and Δt or Δυ and Δt can be mutually traded. Again, trading from ΔE for Δt or equivalently from Δυ for Δt is physically viable, since ΔE and Δυ are physical quantities and Δt is "not". Since Δt is coexisted with ΔE (or equivalently with frequency Δυ), we can change Δt, but we "cannot" change the speed of time. In other words, it is time dictates the science but "not" science changes or "curves" the speed of time. In which we have shown that in principle, we can "squeeze" Δt as small as we wish with a huge price of ΔE, but we can "never" able to squeeze Δt to zero (i.e., Δt = 0). In which we see that; it is "not" possible to transmit a "bit" of information "instan-

Since digital communication requires a "narrower" Δt for rapid transmission and complex amplitude communication needs a "wider" Δt for transmission, this is what communication between satellites and ground stations can do with the "relativistic" uncertainty principle. For example, using digital transmission from ground station to satellite stations has the advantage to squeeze the relativistic Δt somewhat at receiving satellite station. On the other hand, from a satellite station to ground stations, one might use wider relativistic Δt for digital frequency signal transmission. Wider Δt also offers a lager "certainty" communication space for complex

Let me assume a "relativistic" communication scenario as depicted in **Figure 9**,

in which we assume Q1 and Q2 satellite stations situated within two distinct

subspace, can be shown as given by

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

Or equivalently we have,

Or equivalently we have,

domain" digital communication.

wave front transmission [22].

**179**

station to satellite information transmission.

Eq. (35) essentially shows us that; information can be "recovered" after being received, again with a price; ΔE and Δt. In view of these strategies; we see that the cost paid for using Weiner type for information transmission is "much higher" than the Shannon type; aside the cost of higher energy of ΔE it needs extra amount of time Δt for "post processing". Thus, we see that Wiener communication strategy is effective for a "none cooperating" sender, for example, applied to radar detection, and others. One the other hand, Shannon type provides a more reliable information transmission, by simply increasing the signal to noise so that every bit of information can be "reliably transmitted" to the receiver.

Therefore, we see that quantum entanglement communication [13] is basically using Wiener communication strategy. The price will be "much higher and very inefficient", such as post processing is one thing. And it is "illogical" to require the received signal be "more equivocal" (i.e., uncertain); the better the information recovery it can be received at the receiving end. In which quantum entanglement communication is designed for extracting information as Weiner type communication. However, it is "not" the purpose for reliable information-transmission of Shannon.

#### **9. Relativistic transmission**

One of most esoteric aspects in time must be Einstein's special theory of relativity [1] as stated approximately as follows: when a subspace moves faster than the other, there is a "relative" time speed between them, although time speed within the subspaces is the "same". In this we see that the "relativistic time" within a vast cosmological space may not be the same. Let me start with the relativistic time dilation as given by

$$
\Delta t' = \frac{\Delta t}{\sqrt{1 - \nu^2/c^2}}\tag{36}
$$

where Δt <sup>0</sup> is the relativistic time window, as compared with the time window Δt of a standstill subspace; v is the velocity of a moving subspace; and c is the velocity of light. In which we see that time dilation Δt <sup>0</sup> within a moving subspace, "relative" to the time duration of the standstill subspace Δt, appears to be wider as velocity increases.

In view of law of uncertainty limit as given by

$$
\Delta \mathbf{E} \,\Delta \mathbf{t} = \mathbf{h} \tag{37}
$$

we see that every subspace is limited by ΔE and Δt. In other words, it is the h region, but not the shape of that determines the boundary of (ΔE, Δt). For example, the shape can be either elongated or compressed, as long as it is equaled to h region, as can be seen depicted in **Figure 6**.

Incidentally, the uncertainty limit of Eq. (37) is also the limit of "reliable" bit information transmission [16]. Nonetheless the connection with the special theory of relativity is that; subspaces near the edge of our universe will receive a

"narrower" section of relativistic time (Δt') with respect to an standstill subspace, since relativistic dilation time window is wider Δt' > Δt. In which we see that; "relativistic" uncertainty within the moving subspace, as with respect to a standstill subspace, can be shown as given by

$$
\Delta \mathbf{E} \,\Delta \mathbf{t}" [\mathbf{1} - (\nu/\mathbf{c})^2] \,\,^{\prime \prime} = \mathbf{h} \tag{38}
$$

Or equivalently we have,

On the other hand, to recovering the transmitted information is to "maximize" H(B/A) (the channel noise). Since the entropy H(B) at the receiving end is "larger"

Eq. (35) essentially shows us that; information can be "recovered" after being received, again with a price; ΔE and Δt. In view of these strategies; we see that the cost paid for using Weiner type for information transmission is "much higher" than the Shannon type; aside the cost of higher energy of ΔE it needs extra amount of time Δt for "post processing". Thus, we see that Wiener communication strategy is effective for a "none cooperating" sender, for example, applied to radar detection, and others. One the other hand, Shannon type provides a more reliable information transmission, by simply increasing the signal to noise so that every bit of informa-

Therefore, we see that quantum entanglement communication [13] is basically using Wiener communication strategy. The price will be "much higher and very inefficient", such as post processing is one thing. And it is "illogical" to require the received signal be "more equivocal" (i.e., uncertain); the better the information recovery it can be received at the receiving end. In which quantum entanglement communication is designed for extracting information as Weiner type communication. However, it is "not" the purpose for reliable information-transmission of

One of most esoteric aspects in time must be Einstein's special theory of relativity [1] as stated approximately as follows: when a subspace moves faster than the other, there is a "relative" time speed between them, although time speed within the subspaces is the "same". In this we see that the "relativistic time" within a vast cosmological space may not be the same. Let me start with the relativistic time

<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup><sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

of a standstill subspace; v is the velocity of a moving subspace; and c is the velocity

we see that every subspace is limited by ΔE and Δt. In other words, it is the h region, but not the shape of that determines the boundary of (ΔE, Δt). For example, the shape can be either elongated or compressed, as long as it is equaled to h region,

Incidentally, the uncertainty limit of Eq. (37) is also the limit of "reliable" bit information transmission [16]. Nonetheless the connection with the special theory

of relativity is that; subspaces near the edge of our universe will receive a

to the time duration of the standstill subspace Δt, appears to be wider as velocity

<sup>0</sup> is the relativistic time window, as compared with the time window Δt

**<sup>1</sup>** � *<sup>v</sup>***<sup>2</sup>***=c***<sup>2</sup>** <sup>p</sup> (36)

ΔE Δt ¼ h (37)

<sup>0</sup> within a moving subspace, "relative"

*Δt*

of light. In which we see that time dilation Δt

as can be seen depicted in **Figure 6**.

In view of law of uncertainty limit as given by

I A; B ð Þ¼ H Bð Þ–H Bð Þ *=*A ≈ H Að Þ (35)

than the entropy at the sending end; that is H(B) > H(A), we have,

tion can be "reliably transmitted" to the receiver.

Shannon.

*Quantum Mechanics*

**9. Relativistic transmission**

dilation as given by

where Δt

increases.

**178**

$$
\Delta \mathbf{u} \,\, \Delta \mathbf{t}' [\mathbf{1} - (\nu/\mathbf{c})^2] \,\, ^\vee = \mathbf{1} \tag{39}
$$

In which we see ΔE energy is "conserved". Thus a "narrower" time-window Δt can be squeeze as with respect to standstill subspace. This is precisely physically possible to exploit for "time-domain" digital communication, as from ground station to satellite information transmission.

One the other hand, as from satellite to ground station digital-transmission, we might want to use digital-bandwidth (i.e., Δν). This is a "frequency-domain" information transmission strategy, as in contrast with time-domain, which has "not" fully exploited yet. In which the "relativistic" uncertainty relationship within the standstill subspace as with respect to the moving subspace can be written as

$$\frac{\Delta E \,\Delta t}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} = h \tag{40}$$

Or equivalently we have,

$$\frac{\Delta v \,\Delta t}{\sqrt{1 - \left(\frac{p}{c}\right)^2}} = 1 \tag{41}$$

In this we see that a narrower bandwidth Δ*v* can be used for "frequency domain" digital communication.

Nevertheless, the essence of ΔE Δt = h (or Δυ Δt = 1) shows that ΔE and Δt or Δυ and Δt can be mutually traded. Again, trading from ΔE for Δt or equivalently from Δυ for Δt is physically viable, since ΔE and Δυ are physical quantities and Δt is "not". Since Δt is coexisted with ΔE (or equivalently with frequency Δυ), we can change Δt, but we "cannot" change the speed of time. In other words, it is time dictates the science but "not" science changes or "curves" the speed of time. In which we have shown that in principle, we can "squeeze" Δt as small as we wish with a huge price of ΔE, but we can "never" able to squeeze Δt to zero (i.e., Δt = 0). In which we see that; it is "not" possible to transmit a "bit" of information "instantaneously" (i.e., t = 0) within our temporal (t > 0) universe.

Since digital communication requires a "narrower" Δt for rapid transmission and complex amplitude communication needs a "wider" Δt for transmission, this is what communication between satellites and ground stations can do with the "relativistic" uncertainty principle. For example, using digital transmission from ground station to satellite stations has the advantage to squeeze the relativistic Δt somewhat at receiving satellite station. On the other hand, from a satellite station to ground stations, one might use wider relativistic Δt for digital frequency signal transmission. Wider Δt also offers a lager "certainty" communication space for complex wave front transmission [22].

Let me assume a "relativistic" communication scenario as depicted in **Figure 9**, in which we assume Q1 and Q2 satellite stations situated within two distinct

As I see it; it is our universe governs the science and it is not the science dictates our universe. Within our universe every subspace is created by an amount of energy ΔE and a section of time Δt. Once a section of Δt has been used, it cannot bring it back, although we can create the same Δt at a different time. Although ΔE can be traded for Δt, but it is "impossible" to squeeze Δt equals to zero (i.e., t = 0), and this is the "temporal limit" of our universe. In this we see that there is "no" substance that can travel instantly (i.e., t = 0) within our universe. Even someday we may discover substance that travels beyond the speed of light, this is by "no" means that

Nevertheless, the nature of a section of time Δt is all about our temporal (t > 0) universe, in which time is space and space is time. I have shown that within our universe every subspace takes an amount of energy ΔE and a section of time Δt to "tangle"; by which ΔE and Δt cannot be separated. Although ΔE and Δt can be mutually traded, it is trading ΔE for Δt, or Δυ for Δt, but not trading for ΔE or Δt for Δυ since Δt is a real variable but "not" a physical quantity. But we cannot trade Δt for ΔE; once a section of Δt has been used, it cannot bring it back since time is a forward dependent variable. It is however, in principle, possible to trade ΔE (or Δυ) for a smallest Δt, but it is "not" possible to squeeze Δt to zero, no matter how much energy ΔE that one is willing to pay. Since Δt = 0 is the "instantaneous" response that "cannot" be reached within a temporal (t > 0) subspace, in which we see that Δt is lower bounded by Δt =0. But Δt = 0 exists only within a timeless (t = 0) space

In view of the laws of entropy, information, uncertainty, relativity, and universe

<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup><sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

One of the most interesting topics in science must be time traveling for which I

Now we let the photonic traveler start his voyage with a "narrow pulse" width Δt' from subspace 1 (i.e., very closed to our planet earth) to a distant subspace 3,

assume a photonic traveler (i.e., a photon) is situated within subspace 1 at the center of our temporal universe in **Figure 11**. In view of this figure, the outward speed of subspaces 3 moves somewhat faster than subspace 2 (i.e., v3 > v2) toward

the boundary of our universe since subspace 3 is closer to the boundary.

Notice that law of universe in Eq. (46) has a set of equations; one is for an isolated mass m and the other is for isolated photonic-particle, since photon is a "virtual" particle has no mass. Nevertheless, these laws and principles are profoundly associated with (ΔE, Δt), where unit (ΔE, Δt) is the "necessary" cost within our universe. We have shown that it is possible to "squeeze" Δt by widening ΔE. This corresponds to a higher energy of shorter wavelength λ. But it is "impossible" to trade for infinitesimal small section of Δt (i.e., t ≈ 0), which is physical

*Δt*

<sup>U</sup> : *<sup>Δ</sup>*<sup>E</sup> *<sup>Δ</sup>*<sup>t</sup> <sup>≥</sup>ðΔmc2

limited as imposed by our temporal (t > 0) universe.

ΔI � ΔE Δt ¼ h, per bit of information (42) ΔS � E Δt*=*T ¼ h*=*T, per bit of information (43)

ΔE Δt≥h (44)

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup>*=c*<sup>2</sup> <sup>p</sup> (45)

Þ*Δ*t, *Δ*E *Δ*t≥h (46)

the substance can travel instantly (i.e., t = 0) within our universe.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

but not within our universe.

**10. Time traveling?**

**181**

as given by

**Figure 9.** *Relativistic digital transmission within temporal subspace.*

subspaces, one travels at a velocity *v* and the other is stand still. In view of this figure, we see that the hypothetical scenario is a "physical realizable" paradigm, since these two subspaces are embedded within a temporal (t > 0) space.

Now, if we let Q1 station transmits a pulse signal with a duration Δt to Q2 station. Assuming without any significant time delay, the digital pulse as received by Q2 station appeal "wider" due to relativistic dilation as can be seen from Eq. (41). For instance, if we assume the time-dilation from Q1 station relatively with respect to Q2 station is two time wider (i.e., Δt' = 2Δt), then Δt' is two times wider as received by Q2; to complete for a bit" of information transmitted from Q1, as depicted in **Figure 10**, where we see that the transmitted ΔE is "conservation". Needless to say that if the received pulse of Δt is transmitted back to Q1 in motion; the receiving pulse width will be 2 time broader, as can be seen in the figure. In which we see that; one can exploits faster "time-digital" transmission from a static station Q2 to a moving station Q1. From Q1 to Q2 static station, one can take advantage for larger communication subspace, such as synthetic aperture radar imaging [22].

**Figure 10.** *A relativistic digital information transmission, Δt' = 2Δt.*

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

As I see it; it is our universe governs the science and it is not the science dictates our universe. Within our universe every subspace is created by an amount of energy ΔE and a section of time Δt. Once a section of Δt has been used, it cannot bring it back, although we can create the same Δt at a different time. Although ΔE can be traded for Δt, but it is "impossible" to squeeze Δt equals to zero (i.e., t = 0), and this is the "temporal limit" of our universe. In this we see that there is "no" substance that can travel instantly (i.e., t = 0) within our universe. Even someday we may discover substance that travels beyond the speed of light, this is by "no" means that the substance can travel instantly (i.e., t = 0) within our universe.

Nevertheless, the nature of a section of time Δt is all about our temporal (t > 0) universe, in which time is space and space is time. I have shown that within our universe every subspace takes an amount of energy ΔE and a section of time Δt to "tangle"; by which ΔE and Δt cannot be separated. Although ΔE and Δt can be mutually traded, it is trading ΔE for Δt, or Δυ for Δt, but not trading for ΔE or Δt for Δυ since Δt is a real variable but "not" a physical quantity. But we cannot trade Δt for ΔE; once a section of Δt has been used, it cannot bring it back since time is a forward dependent variable. It is however, in principle, possible to trade ΔE (or Δυ) for a smallest Δt, but it is "not" possible to squeeze Δt to zero, no matter how much energy ΔE that one is willing to pay. Since Δt = 0 is the "instantaneous" response that "cannot" be reached within a temporal (t > 0) subspace, in which we see that Δt is lower bounded by Δt =0. But Δt = 0 exists only within a timeless (t = 0) space but not within our universe.

In view of the laws of entropy, information, uncertainty, relativity, and universe as given by

$$
\Delta \mathbf{I} \sim \Delta \mathbf{E} \,\Delta \mathbf{t} = \mathbf{h} \text{, per bit of information} \tag{42}
$$

$$
\Delta \mathbf{S} \sim \mathbf{E} \,\Delta \mathbf{t}/\mathbf{T} = \mathbf{h}/\mathbf{T}, \text{per bit of information} \tag{43}
$$

$$
\Delta \mathbf{E} \,\,\Delta \mathbf{t} \ge \mathbf{h} \,\tag{44}
$$

$$
\Delta t' = \frac{\Delta t}{\sqrt{1 - v^2/c^2}}\tag{45}
$$

$$\mathbf{U}: \Delta \mathbf{E} \, \Delta \mathbf{t} \ge (\Delta \mathbf{m} \mathbf{c}^2) \Delta \mathbf{t}, \,\Delta \mathbf{E} \, \Delta \mathbf{t} \ge \mathbf{h} \tag{46}$$

Notice that law of universe in Eq. (46) has a set of equations; one is for an isolated mass m and the other is for isolated photonic-particle, since photon is a "virtual" particle has no mass. Nevertheless, these laws and principles are profoundly associated with (ΔE, Δt), where unit (ΔE, Δt) is the "necessary" cost within our universe. We have shown that it is possible to "squeeze" Δt by widening ΔE. This corresponds to a higher energy of shorter wavelength λ. But it is "impossible" to trade for infinitesimal small section of Δt (i.e., t ≈ 0), which is physical limited as imposed by our temporal (t > 0) universe.

#### **10. Time traveling?**

One of the most interesting topics in science must be time traveling for which I assume a photonic traveler (i.e., a photon) is situated within subspace 1 at the center of our temporal universe in **Figure 11**. In view of this figure, the outward speed of subspaces 3 moves somewhat faster than subspace 2 (i.e., v3 > v2) toward the boundary of our universe since subspace 3 is closer to the boundary.

Now we let the photonic traveler start his voyage with a "narrow pulse" width Δt' from subspace 1 (i.e., very closed to our planet earth) to a distant subspace 3,

subspaces, one travels at a velocity *v* and the other is stand still. In view of this figure, we see that the hypothetical scenario is a "physical realizable" paradigm, since these two subspaces are embedded within a temporal (t > 0) space.

*Relativistic digital transmission within temporal subspace.*

**Figure 9.**

*Quantum Mechanics*

**Figure 10.**

**180**

*A relativistic digital information transmission, Δt' = 2Δt.*

communication subspace, such as synthetic aperture radar imaging [22].

Now, if we let Q1 station transmits a pulse signal with a duration Δt to Q2 station. Assuming without any significant time delay, the digital pulse as received by Q2 station appeal "wider" due to relativistic dilation as can be seen from Eq. (41). For instance, if we assume the time-dilation from Q1 station relatively with respect to Q2 station is two time wider (i.e., Δt' = 2Δt), then Δt' is two times wider as received by Q2; to complete for a bit" of information transmitted from Q1, as depicted in **Figure 10**, where we see that the transmitted ΔE is "conservation". Needless to say that if the received pulse of Δt is transmitted back to Q1 in motion; the receiving pulse width will be 2 time broader, as can be seen in the figure. In which we see that; one can exploits faster "time-digital" transmission from a static station Q2 to a moving station Q1. From Q1 to Q2 static station, one can take advantage for larger

**Figure 11.**

*A schematic diagram of our expanding universe. It shows our universe is a temporal (t > 0) dynamic stochastic universe; time and space are "coexisted." (μo, εo) are the permeability and permittivity of space.*

which has an outward velocity of v3. If the "relativistic" time dilation Δt' between these two subspaces is "two" times wider than the static sunspace 1 (i.e., Δt=2 Δt'). Then velocity of subspace 3 can be calculated by means Einstein's special theory of relativity as given by

$$
\Delta t' = \frac{\Delta t}{\sqrt{1 - v^2/c^2}}\tag{47}
$$

After the long journey arrived at subspace 3, the traveler is contemplating when he should return back. The "dilemma" is that if he waited too long, he may not be able to return home soon enough to enjoy some of his time-gained, since subspace 3 is moving even faster closer to the speed of light. For which he has decided to return right away, since is a longer journey of "more" than 26 billion light-years to cover,

*The "relative" time gain as the traveler reached subspace 3. BLY represents billion light-years.*

But as I see it; all the "relative" time-gained will be used up on his journey back home; it turns out the traveler will be home at precisely the same time of subspace 3 "without" any time gain. This part I will let you to figure out, since you have all the mathematics to play with. Yet the worst scenario is that; the traveler "cannot" find his home, since his home had been gone a few billion light-years ago after he had

On the other hand, if the traveler is "not" a cruising photonic particle, then the

which is a price that "nobody" can afford, even just for one-way trip to subspace 3, where m is the mass of the traveler, in which we see that "time traveling" to the

Nevertheless, every subspace within our universe is always attached a price; a section of time Δt and an amount of energyΔE, although the unit (ΔE, Δt) is a "necessary" cost. For example the "cost" to create a golf ball; it need a huge amount of energy ΔE and a section of time Δt, but without an amount of information ΔI (or

Another scenario is that traveling within "empty" space as depicted in **Figure 13**, as normally assumed. in spite it is a nonphysical paradigm; we see that traveler can reach subspace 3 instantly and return back as he wishes, since within a timeless (t = 0) space it has "no" distance and no time, although the diagram shows it has. And this is precisely a virtual mathematical paradigm do to science, even though the subspace has no time and yet appears it has. For which I have found; practically all the laws, principles and theories of science were developed from the same empty

Since science is a "principle" of logic, in which we see that a simple logic worth more than tons of mathematics. For example, as illustrated in **Figure 14**, if a timetraveler able to remove himself from current moment of 2020 and searching for last year of the same moment of 2019. The question is can he find it? The apparent answer is that; last year of our universe has been departed. Similarly, the traveler is wishing to visit next year 2021, but next year of our universe has not arrived yet. In short, I remark that it is physical realizable science that "directs" the mathematics, but not the virtual mathematics that leads science, although science needs

kinetic energy to reach a velocity of V3 = 161,820 miles/s can be calculated as

.

future is "unlikely", even assume we can travel at the speed of light.

equivalent an amount of ΔS) it will not make it happen.

space, which is "not" a physical realizable subspace.

in view of an outward velocity of subspace 3 to overcome.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

departed from subspace 1 to subspace 3.

**Figure 12.**

**183**

K.E. = ½ m v2 = ½ m (161,820)<sup>2</sup>

For which the outward velocity V3 is given by

V3 ¼ 0*:*87 c ¼ 0*:*87 � 186, 000 ¼ 161, 820 miles*=*s

With reference to Hubble space telescopic observation [27], the boundary of our universe is about 15 billion light years away from subspace 1; for which Subspace 3 is estimated about 13 billion light years away from the center of our universe. Which will take the photonic traveler a 13 billion light-years and possible added another 13 billion light-years to catch-up to subspace 3, since subspace 3 has moved away as traveler's voyage started. For which the traveler will take about 26 billion light-years to reach subspace 3, at speed of light.

Nevertheless, as arrived at subspace 3, the traveler's pulse pulse-width reduces to about 1/4 the size. Which has a 3/4 "gain" in relative time-duration with respect to the static subspace 1 and the gain can be translated into "duration" of time that has been taken during the voyage. Since it took about a total 26 billion light-years journey to reach subspace 3, there is a "net gain"**of** about 19.5 billion light-years ahead "relatively" to the time duration that has gone by at the subspace 1. In other words; there is a total 19.5 billion light-years "relatively ahead" of subspace 1, after a total 26 billion light-years journey to subspace 3, as illustrated in **Figure 12**.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

**Figure 12.**

which has an outward velocity of v3. If the "relativistic" time dilation Δt' between these two subspaces is "two" times wider than the static sunspace 1 (i.e., Δt=2 Δt'). Then velocity of subspace 3 can be calculated by means Einstein's special theory of

*A schematic diagram of our expanding universe. It shows our universe is a temporal (t > 0) dynamic stochastic*

*universe; time and space are "coexisted." (μo, εo) are the permeability and permittivity of space.*

<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup><sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V3 ¼ 0*:*87 c ¼ 0*:*87 � 186, 000 ¼ 161, 820 miles*=*s

With reference to Hubble space telescopic observation [27], the boundary of our universe is about 15 billion light years away from subspace 1; for which Subspace 3 is estimated about 13 billion light years away from the center of our universe. Which will take the photonic traveler a 13 billion light-years and possible added another 13 billion light-years to catch-up to subspace 3, since subspace 3 has moved away as traveler's voyage started. For which the traveler will take about 26 billion

Nevertheless, as arrived at subspace 3, the traveler's pulse pulse-width reduces to about 1/4 the size. Which has a 3/4 "gain" in relative time-duration with respect to the static subspace 1 and the gain can be translated into "duration" of time that has been taken during the voyage. Since it took about a total 26 billion light-years journey to reach subspace 3, there is a "net gain"**of** about 19.5 billion light-years ahead "relatively" to the time duration that has gone by at the subspace 1. In other words; there is a total 19.5 billion light-years "relatively ahead" of subspace 1, after a

total 26 billion light-years journey to subspace 3, as illustrated in **Figure 12**.

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup>*=c*<sup>2</sup> <sup>p</sup> (47)

*Δt*

For which the outward velocity V3 is given by

light-years to reach subspace 3, at speed of light.

relativity as given by

**Figure 11.**

*Quantum Mechanics*

**182**

*The "relative" time gain as the traveler reached subspace 3. BLY represents billion light-years.*

After the long journey arrived at subspace 3, the traveler is contemplating when he should return back. The "dilemma" is that if he waited too long, he may not be able to return home soon enough to enjoy some of his time-gained, since subspace 3 is moving even faster closer to the speed of light. For which he has decided to return right away, since is a longer journey of "more" than 26 billion light-years to cover, in view of an outward velocity of subspace 3 to overcome.

But as I see it; all the "relative" time-gained will be used up on his journey back home; it turns out the traveler will be home at precisely the same time of subspace 3 "without" any time gain. This part I will let you to figure out, since you have all the mathematics to play with. Yet the worst scenario is that; the traveler "cannot" find his home, since his home had been gone a few billion light-years ago after he had departed from subspace 1 to subspace 3.

On the other hand, if the traveler is "not" a cruising photonic particle, then the kinetic energy to reach a velocity of V3 = 161,820 miles/s can be calculated as K.E. = ½ m v2 = ½ m (161,820)<sup>2</sup> .

which is a price that "nobody" can afford, even just for one-way trip to subspace 3, where m is the mass of the traveler, in which we see that "time traveling" to the future is "unlikely", even assume we can travel at the speed of light.

Nevertheless, every subspace within our universe is always attached a price; a section of time Δt and an amount of energyΔE, although the unit (ΔE, Δt) is a "necessary" cost. For example the "cost" to create a golf ball; it need a huge amount of energy ΔE and a section of time Δt, but without an amount of information ΔI (or equivalent an amount of ΔS) it will not make it happen.

Another scenario is that traveling within "empty" space as depicted in **Figure 13**, as normally assumed. in spite it is a nonphysical paradigm; we see that traveler can reach subspace 3 instantly and return back as he wishes, since within a timeless (t = 0) space it has "no" distance and no time, although the diagram shows it has. And this is precisely a virtual mathematical paradigm do to science, even though the subspace has no time and yet appears it has. For which I have found; practically all the laws, principles and theories of science were developed from the same empty space, which is "not" a physical realizable subspace.

Since science is a "principle" of logic, in which we see that a simple logic worth more than tons of mathematics. For example, as illustrated in **Figure 14**, if a timetraveler able to remove himself from current moment of 2020 and searching for last year of the same moment of 2019. The question is can he find it? The apparent answer is that; last year of our universe has been departed. Similarly, the traveler is wishing to visit next year 2021, but next year of our universe has not arrived yet.

In short, I remark that it is physical realizable science that "directs" the mathematics, but not the virtual mathematics that leads science, although science needs

the "instantaneous and simultaneous" superimposing principle for quantum computing did "not" actually exist within our universe. One of the important aspects within our universe is that one cannot get something from nothing there is always a price to pay an amount of energy ΔE and a section of time Δt. The important is that

Any science that existed within our universe has time or temporal (t > 0), in which we see that any scientific law, principle, theory, and paradox has to comply with temporal (t > 0) aspect within our universe, otherwise it may not be a physically realizable science, as we know that science is mathematics but mathematics is not equal to science. In this we have shown that any analytic solution has to be temporal (t > 0), otherwise it cannot be implemented within our universe,

Since Schrödinger's quantum mechanics is a legacy of Hamiltonian classical mechanics, we have shown that Schrödinger's mechanics is a timeless (t = 0) machine since Hamiltonian mechanics is timeless (t = 0). This includes

Schrödinger's fundamental principle of superposition which is "not" a physically realizable principle. Since Schrödinger's cat is one of the most controversial para-

Schrödinger's cat is "not "a physically realizable paradox, which should not have

The most esoteric nature of our universe must be time, for which every fundamental law, principle, and theory is associated with a section of time Δt. We have shown that it is the section of Δt that we have used cannot bring it back. And this is the section of Δt that a set of most elegant laws and principles are associated with. In this we have shown that we can squeeze Δt approaches to zero, but it is "not" possible to reach zero even though we have all the energy ΔE to pay for it, in which we see that we can change the section of Δt, but we cannot change the speed of

Information is a very important aspect in science, since everything is a piece of

Nevertheless, time traveling is a very interesting topic for all scientists, in which

Overall, this chapter is to show that it is not how rigorous the mathematics is, it is the physically realizable paradigm that produces viable solution. If one used a nonphysical realizable model, it is very "likely" one will get a nonphysical realizable

Finally, I would stress that the nature of temporal (t > 0) quantum mechanics is all about the temporal (t > 0) universe, in which we have seen that it is our universe

information. Nevertheless, without the connection with entropy, information would be very difficult to apply in science. Since entropy is in energy form, but this is by "no" means that entropy is conserved implies that information is conserved since entropy is equivalent to an amount of information. We have shown; information has two major orientations; Shannon transmission is for "reliable" information while Weiner communication is for information "retrieval", for which we see that every bit of information takes an amount of energy ΔE and a section of time Δt to

I have shown it is physically "not" realizable; it is simply we cannot "curve" a temporal (t > 0) space, since time in a "dependent" forward variable with space. It is science can change a section of time Δt but "not" change the speed of time. In other words, we walk on the street and it is not the street that walks on us. However, time traveling is possible if our universe is embedded within an empty space. But emptiness is a timeless (t = 0) space which is "not" exited within our temporal universe. And this is precisely most of the scientists uses this empty space for over a few centuries since the dawn of science. And this is precisely why all the

laws, principles and theories are timeless (t = 0) or time-independent.

solution, virtual and fictitious as mathematics is.

doxes in modern history of science, we have shown that the paradox of

which includes all the laws, principles, and theories.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

they are not free!

been postulated!

transmit, and it is not free.

time.

**185**

#### **Figure 13.**

*Our universe model embedded within an empty space. This is a subspace that normally used since the dawn of science.*

#### **Figure 14.**

*A composited temporal universe as function of time. Notice that these temporal universes cannot be "simultaneous" existed as superposition principle of quantum mechanics.*

mathematics. In which I note that; it is "not" how rigorous the mathematics is, it is the physical realizable science we embrace. Otherwise more and more virtual sciences will continuingly emerge. In view of relativity, we can "relatively" slow down the time somewhat, but we can "never" change the speed of time. It is you walk with time, and it is "not" time walks with you.

#### **11. Conclusion**

In conclusion, I would point out that quantum scientists used amazing mathematical analyses added to their fantastic computer simulations that provide very convincing results. But mathematical analyses and computer animations are virtual and fictitious, and many of their animations are "not" physically real, for example,

#### *Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

the "instantaneous and simultaneous" superimposing principle for quantum computing did "not" actually exist within our universe. One of the important aspects within our universe is that one cannot get something from nothing there is always a price to pay an amount of energy ΔE and a section of time Δt. The important is that they are not free!

Any science that existed within our universe has time or temporal (t > 0), in which we see that any scientific law, principle, theory, and paradox has to comply with temporal (t > 0) aspect within our universe, otherwise it may not be a physically realizable science, as we know that science is mathematics but mathematics is not equal to science. In this we have shown that any analytic solution has to be temporal (t > 0), otherwise it cannot be implemented within our universe, which includes all the laws, principles, and theories.

Since Schrödinger's quantum mechanics is a legacy of Hamiltonian classical mechanics, we have shown that Schrödinger's mechanics is a timeless (t = 0) machine since Hamiltonian mechanics is timeless (t = 0). This includes Schrödinger's fundamental principle of superposition which is "not" a physically realizable principle. Since Schrödinger's cat is one of the most controversial paradoxes in modern history of science, we have shown that the paradox of Schrödinger's cat is "not "a physically realizable paradox, which should not have been postulated!

The most esoteric nature of our universe must be time, for which every fundamental law, principle, and theory is associated with a section of time Δt. We have shown that it is the section of Δt that we have used cannot bring it back. And this is the section of Δt that a set of most elegant laws and principles are associated with. In this we have shown that we can squeeze Δt approaches to zero, but it is "not" possible to reach zero even though we have all the energy ΔE to pay for it, in which we see that we can change the section of Δt, but we cannot change the speed of time.

Information is a very important aspect in science, since everything is a piece of information. Nevertheless, without the connection with entropy, information would be very difficult to apply in science. Since entropy is in energy form, but this is by "no" means that entropy is conserved implies that information is conserved since entropy is equivalent to an amount of information. We have shown; information has two major orientations; Shannon transmission is for "reliable" information while Weiner communication is for information "retrieval", for which we see that every bit of information takes an amount of energy ΔE and a section of time Δt to transmit, and it is not free.

Nevertheless, time traveling is a very interesting topic for all scientists, in which I have shown it is physically "not" realizable; it is simply we cannot "curve" a temporal (t > 0) space, since time in a "dependent" forward variable with space. It is science can change a section of time Δt but "not" change the speed of time. In other words, we walk on the street and it is not the street that walks on us. However, time traveling is possible if our universe is embedded within an empty space. But emptiness is a timeless (t = 0) space which is "not" exited within our temporal universe. And this is precisely most of the scientists uses this empty space for over a few centuries since the dawn of science. And this is precisely why all the laws, principles and theories are timeless (t = 0) or time-independent.

Overall, this chapter is to show that it is not how rigorous the mathematics is, it is the physically realizable paradigm that produces viable solution. If one used a nonphysical realizable model, it is very "likely" one will get a nonphysical realizable solution, virtual and fictitious as mathematics is.

Finally, I would stress that the nature of temporal (t > 0) quantum mechanics is all about the temporal (t > 0) universe, in which we have seen that it is our universe

mathematics. In which I note that; it is "not" how rigorous the mathematics is, it is the physical realizable science we embrace. Otherwise more and more virtual sciences will continuingly emerge. In view of relativity, we can "relatively" slow down the time somewhat, but we can "never" change the speed of time. It is you walk

*A composited temporal universe as function of time. Notice that these temporal universes cannot be*

*Our universe model embedded within an empty space. This is a subspace that normally used since the dawn of*

In conclusion, I would point out that quantum scientists used amazing mathematical analyses added to their fantastic computer simulations that provide very convincing results. But mathematical analyses and computer animations are virtual and fictitious, and many of their animations are "not" physically real, for example,

with time, and it is "not" time walks with you.

*"simultaneous" existed as superposition principle of quantum mechanics.*

**11. Conclusion**

**184**

**Figure 13.**

*Quantum Mechanics*

**Figure 14.**

*science.*

that governs our science; it is not our science that "curves" our universe. Although we can change a section of time Δt, we cannot change the speed of time. In short, it is the physically realizable science we value, but not the fancy mathematical solution we adored.

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Publishers; 1961

1049

**43**(3–4):172

**53**:195-220

Publication; 1938

2012

63-70

**187**

1913;**26**(1):1-23

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*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

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[15] Yu FTS. Aspect of particle and wave dynamics. In: Origin of Temporal (t > 0) Universe: Correcting with Relativity, Entropy, Communication and Quantum Mechanics, Appendix. New York: CRC

[13] Życzkowski K, Horodecki P, Horodecki M, Horodecki R. Dynamics of quantum entanglement. Physical

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Communication. Urbana, IL: University

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Mechanics, Chapter 1. New York: CRC

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[19] Yu FTS. Optics and Information Theory. New York: Wiley-Interscience;

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[9] Bohr N. On the constitution of atoms and molecules. Philosophical Magazine.

[10] MacKinnon E. De Broglie's thesis: A critical retrospective. American Journal

of Physics. 1976;**44**:1047-1055

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of the mechanics of atoms and

[3] Heisenberg W. Über den anschaulichen Inhalt der

## **Author details**

Francis T.S. Yu Penn State University, University Park, PA, USA

\*Address all correspondence to: fty1@psu.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Nature of Temporal (t > 0) Quantum Theory: Part II DOI: http://dx.doi.org/10.5772/intechopen.93562*

#### **References**

that governs our science; it is not our science that "curves" our universe. Although we can change a section of time Δt, we cannot change the speed of time. In short, it is the physically realizable science we value, but not the fancy mathematical solu-

tion we adored.

*Quantum Mechanics*

**Author details**

Francis T.S. Yu

**186**

Penn State University, University Park, PA, USA

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: fty1@psu.edu

provided the original work is properly cited.

[1] Einstein A. Relativity, the Special and General Theory. New York: Crown Publishers; 1961

[2] Schrödinger E. An undulatory theory of the mechanics of atoms and molecules. Physics Review. 1926;**28**(6): 1049

[3] Heisenberg W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik. 1927; **43**(3–4):172

[4] Boltzmann L. Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte. 1866; **53**:195-220

[5] Knudsen JM, Hjorth P. Elements of Newtonian Mechanics. Heidelberg: Springer Science & Business Media; 2012

[6] Tolman RC. The Principles of Statistical Mechanics. London: Dover Publication; 1938

[7] Lawden DF. The Mathematical Principles of Quantum Mechanics. London: Methuen & Co Ltd.; 1967

[8] Yu FTS. The fate of Schrodinger's cat. Asian Journal of Physics. 2019;**28**(1): 63-70

[9] Bohr N. On the constitution of atoms and molecules. Philosophical Magazine. 1913;**26**(1):1-23

[10] MacKinnon E. De Broglie's thesis: A critical retrospective. American Journal of Physics. 1976;**44**:1047-1055

[11] Yu FTS. What is "wrong" with current theoretical physicists? In: Bulnes F, Stavrou VN, Morozov O, Bourdine AV, editors. Advances in Quantum Communication and Information, Chapter 9. London: IntechOpen; 2020. pp. 123-143

[12] Bennett CH. Quantum information and computation. Physics Today. 1995; **48**(10):24-30

[13] Życzkowski K, Horodecki P, Horodecki M, Horodecki R. Dynamics of quantum entanglement. Physical Review A. 2001;**65**:1-10

[14] Einstein Attacks Quantum Theory. Scientist and Two Colleagues Find It Is Not 'Complete' Even though 'Correct'. New York City: The New York Times; 1935

[15] Yu FTS. Aspect of particle and wave dynamics. In: Origin of Temporal (t > 0) Universe: Correcting with Relativity, Entropy, Communication and Quantum Mechanics, Appendix. New York: CRC Press; 2019. pp. 145-147

[16] Shannon CE, Weaver W. The Mathematical Theory of Communication. Urbana, IL: University of Illinois Press; 1949

[17] Yu FTS. From relativity to discovery of temporal (t > 0) universe. In: Origin of Temporal (t > 0) Universe: Correcting with Relativity, Entropy, Communication and Quantum Mechanics, Chapter 1. New York: CRC Press; 2019. pp. 1-26

[18] Parzen E. Stochastic Processes. San Francisco: Holden Day, Inc.; 1962

[19] Yu FTS. Optics and Information Theory. New York: Wiley-Interscience; 1976

[20] Gabor D. Communication theory and physics. Philosophical Magazine. 1950;**41**(7):1161

[21] Yu FTS. Information transmission with quantum limited subspace. Asian Journal of Physics. 2018;**27**(1):1-12

[22] Cultrona LJ, Leith EN, Porcello LJ, Vivian WE. On the application of

coherent optical processing techniques to synthetic-aperture radar. Proceedings of the IEEE. 1966;**54**:1026

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[27] Zimmerman R. The Universe in a Mirror: The Saga of the Hubble Space Telescope. Princeton, NJ: Princeton Press; 2016

Section 3

Applications

**189**
