Section 1 Solvable Systems

**Chapter 1**

**Abstract**

**1. Dipolar interactions**

where *μ*<sup>1</sup>

nucleus spin).

**3**

! and *μ*<sup>2</sup>

The dipolar interaction Hamiltonian is expressed as

*<sup>ϰ</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> !*:μ*<sup>2</sup> ! *r*<sup>3</sup> �

2 4

Dipolar Interactions: Hyperfine

The interaction between the nuclear spin and the electron spin creates a hyperfine structure. Hyperfine structure interaction occurs in paramagnetic structures with unpaired electrons. Therefore, hyperfine structure interaction is the most important of the fundamental parameters investigated by electron paramagnetic resonance (EPR) spectroscopy. For EPR spectroscopy the two effective Hamiltonian terms are the hyperfine structure interaction and the electronic Zeeman interaction. The hyperfine structure interaction has two types as isotropic and anisotropic hyperfine structure interactions. The zero-field splitting term (electronic quadrupole fine structure), the nuclear Zeeman term, and the nuclear quadrupole interaction term are among the Hamiltonian terms used in EPR. However, their effects are not as much as the term of the hyperfine structure interaction. The zero-field splitting term and the nuclear quadrupole interaction term are the fine structure terms. The interaction of two electron spins create a zero-field splitting, the interaction between the two nucleus spins form the nuclear quadrupole interaction. Hyperfine structure interaction, zero-field interaction, and nuclear quadrupole interaction are subclasses of dipolar interaction. Interaction tensors are available for all three interactions.

**Keywords:** dipolar interaction hyperfine structure, isotropic hyperfine structure, anisotropic hyperfine structure, the zero-field splitting, the nuclear quadrupole interaction, the electronic Zeeman interaction, the nuclear Zeeman term, EPR

Dipolar interaction occurs due to the interaction between the two spins. If one spin becomes an electron spin and the other spin becomes a nucleus spin, this interaction is called a hyperfine structure interaction. If two of the spins are electron spin or both are nucleus spin, this interaction is called fine structure interaction.

> 3 *μ*<sup>1</sup> !*:r* ! � � *<sup>μ</sup>*<sup>2</sup>

!*:r* ! � �

! are the magnetic dipole moments for each spin (electron spin or

3

5 (1)

*r*5

Structure Interaction and Fine

Structure Interactions

*Betül Çalişkan and Ali Cengiz Çalişkan*

#### **Chapter 1**

## Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions

*Betül Çalişkan and Ali Cengiz Çalişkan*

#### **Abstract**

The interaction between the nuclear spin and the electron spin creates a hyperfine structure. Hyperfine structure interaction occurs in paramagnetic structures with unpaired electrons. Therefore, hyperfine structure interaction is the most important of the fundamental parameters investigated by electron paramagnetic resonance (EPR) spectroscopy. For EPR spectroscopy the two effective Hamiltonian terms are the hyperfine structure interaction and the electronic Zeeman interaction. The hyperfine structure interaction has two types as isotropic and anisotropic hyperfine structure interactions. The zero-field splitting term (electronic quadrupole fine structure), the nuclear Zeeman term, and the nuclear quadrupole interaction term are among the Hamiltonian terms used in EPR. However, their effects are not as much as the term of the hyperfine structure interaction. The zero-field splitting term and the nuclear quadrupole interaction term are the fine structure terms. The interaction of two electron spins create a zero-field splitting, the interaction between the two nucleus spins form the nuclear quadrupole interaction. Hyperfine structure interaction, zero-field interaction, and nuclear quadrupole interaction are subclasses of dipolar interaction. Interaction tensors are available for all three interactions.

**Keywords:** dipolar interaction hyperfine structure, isotropic hyperfine structure, anisotropic hyperfine structure, the zero-field splitting, the nuclear quadrupole interaction, the electronic Zeeman interaction, the nuclear Zeeman term, EPR

#### **1. Dipolar interactions**

Dipolar interaction occurs due to the interaction between the two spins. If one spin becomes an electron spin and the other spin becomes a nucleus spin, this interaction is called a hyperfine structure interaction. If two of the spins are electron spin or both are nucleus spin, this interaction is called fine structure interaction. The dipolar interaction Hamiltonian is expressed as

$$\mathbf{x} = \left[ \frac{\overrightarrow{\mu\_1} \overrightarrow{\mu\_2}}{r^3} - \frac{\Im \left( \overrightarrow{\mu\_1} . \overrightarrow{r} \right) \left( \overrightarrow{\mu\_2} . \overrightarrow{r} \right)}{r^5} \right] \tag{1}$$

where *μ*<sup>1</sup> ! and *μ*<sup>2</sup> ! are the magnetic dipole moments for each spin (electron spin or nucleus spin).

#### **1.1 Hyperfine structure interaction**

The interaction between the magnetic dipole moment of the nucleus and the magnetic dipole moment of the electron gives the hyperfine structure interaction. There are two types of hyperfine structure interaction. These are isotropic hyperfine interaction and anisotropic hyperfine interaction.

#### *1.1.1 Isotropic hyperfine structure*

Isotropic superfine interaction is also known as Fermi contact interaction. The Hamiltonian term of isotropic hyperfine structure interaction is expressed as

$$\mathbf{x} = \mathbf{g}\_e \mathbf{g}\_N \beta\_e \beta\_N \left[ \frac{8\pi}{3} \overrightarrow{\mathbf{S}} . \overrightarrow{I} . \delta(r) \right] \tag{2}$$

In a shorter way, it is expressed as

*DOI: http://dx.doi.org/10.5772/intechopen.91791*

The diagonal elements of the tensor is expressed as

*A*0

*A*0

structure interaction Hamiltonian and is expressed as

*ϰ* ¼ *aS* ! *:I* ! þ *S* ! *: A*<sup>0</sup> !! *:I* ! ¼ *S* ! *:A* !! *:I* !

is the general hyperfine structure tensor.

where *A*<sup>0</sup> !!

where *A* !!

**Figure 1.**

**5**

*The formation of the hyperfine structure splittings.*

**1.2 Fine structure interaction**

*ϰ* ¼ *S* ! *: A*<sup>0</sup> !! *:I* !

> 3*i* <sup>2</sup> � *<sup>r</sup>*<sup>2</sup> *r*5

> > 3*ij r*5

The sum of the isotropic and anisotropic terms fully expresses the hyperfine

**Figure 1** shows the formation of the hyperfine structure splittings. **Figure 2** shows the formation of an EPR spectrum due to the hyperfine structure splittings.

The fine structure is seen in two ways. The first is the fine structure interaction between two electron spins. The second is the fine structure interaction between the two nucleus spins. The fine structure interaction between two electron spin is also

expressed in two ways as diagonal elements and non-diagonal elements.

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions*

*ii* ¼ *gegNβeβ<sup>N</sup>*

The non-diagonal elements of the tensor is expressed as

*ij* ¼ *gegNβeβ<sup>N</sup>*

is called the anisotropic hyperfine coupling tensor. The tensor is

, *<sup>i</sup>* <sup>¼</sup> *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* (8)

, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* (9)

(7)

(10)

where *ge*= *g*-value of the electron, *gN*= *g*-value of the nucleus, *β<sup>e</sup>* = Bohr

magneton, *βN*= nuclear magneton, *S* ! ¼ electron spin operator, *I* ! = nuclear spin operator, and *δ*ð Þ*r* = Dirac delta function for the distance between the electron and the nucleus.

In a shorter way, it is expressed as

$$\varkappa = \mathsf{a}\overrightarrow{\mathsf{S}}.\overrightarrow{I}\tag{3}$$

The isotropic hyperfine constant is written as

$$\mathfrak{a} = \frac{8\pi}{3} \mathbf{g}\_e \mathbf{g}\_N \beta\_e \beta\_N \delta(r) \tag{4}$$

Here *a* is called the isotropic hyperfine constant, *S* ! is the spin angular momentum of the electron, and *I* ! is the spin angular momentum of the nucleus.

#### *1.1.2 Anisotropic hyperfine structure*

Anisotropic hyperfine interaction is also called dipolar interaction or dipole– dipole interaction. The Hamiltonian term of anisotropic hyperfine structure interaction is expressed as

$$\mathbf{x} = \mathbf{g}\_o \mathbf{g}\_N \boldsymbol{\beta}\_e \boldsymbol{\beta}\_N \left[ \frac{\mathbf{3} \left( \overrightarrow{\mathbf{S}} . \overrightarrow{r} \right) \left( \overrightarrow{I} . \overrightarrow{r} \right)}{r^5} - \frac{\overrightarrow{\mathbf{S}} . \overrightarrow{I}}{r^3} \right] \tag{5}$$

More specifically, the expression of the anisotropic hyperfine interaction in the Cartesian coordinate is written as

$$\begin{split} \mathbf{x} &= \mathbf{g}\_{\sigma} \mathbf{g}\_{N} \boldsymbol{\rho}\_{r} \boldsymbol{\rho}\_{N} \left[ \frac{(3\mathbf{x}^{2} - r^{2})}{r^{5}} I\_{x} \mathbf{S}\_{x} + \frac{(3\mathbf{y}^{2} - r^{2})}{r^{5}} I\_{y} \mathbf{S}\_{y} + \frac{(3\mathbf{z}^{2} - r^{2})}{r^{5}} I\_{x} \mathbf{S}\_{x} + \frac{3\mathbf{x}\mathbf{y}}{r^{5}} \left( I\_{x} \mathbf{S}\_{y} + I\_{y} \mathbf{S}\_{x} \right) \right] \\ &+ \frac{3\mathbf{y}z}{r^{5}} \left( I\_{y} \mathbf{S}\_{x} + I\_{z} \mathbf{S}\_{y} \right) + \frac{3\mathbf{x}z}{r^{5}} \left( I\_{x} \mathbf{S}\_{x} + I\_{z} \mathbf{S}\_{x} \right) \end{split} \tag{6}$$

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions DOI: http://dx.doi.org/10.5772/intechopen.91791*

In a shorter way, it is expressed as

**1.1 Hyperfine structure interaction**

*Quantum Mechanics*

*1.1.1 Isotropic hyperfine structure*

magneton, *βN*= nuclear magneton, *S*

tum of the electron, and *I*

interaction is expressed as

*1.1.2 Anisotropic hyperfine structure*

Cartesian coordinate is written as

*<sup>r</sup>*<sup>5</sup> *IySz* <sup>þ</sup> *IzSy* � � <sup>þ</sup>

<sup>3</sup>*x*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> ð Þ

*<sup>r</sup>*<sup>5</sup> *IxSx* <sup>þ</sup>

3*xz*

*ϰ* ¼ *gegNβeβ<sup>N</sup>*

þ 3*yz*

**4**

In a shorter way, it is expressed as

The isotropic hyperfine constant is written as

Here *a* is called the isotropic hyperfine constant, *S*

*ϰ* ¼ *gegNβeβ<sup>N</sup>*

!

the nucleus.

The interaction between the magnetic dipole moment of the nucleus and the magnetic dipole moment of the electron gives the hyperfine structure interaction.

Isotropic superfine interaction is also known as Fermi contact interaction. The

8*π* 3 *S* ! *:I* ! *:δ*ð Þ*r* � �

¼ electron spin operator, *I*

!

<sup>3</sup> *gegNβeβNδ*ð Þ*<sup>r</sup>* (4)

is the spin angular momen-

5 (5)

3*xy*

*<sup>r</sup>*<sup>5</sup> *IxSy* <sup>þ</sup> *IySx*

(6)

!

is the spin angular momentum of the nucleus.

*I* ! *:r* ! � � *<sup>r</sup>*<sup>5</sup> � *<sup>S</sup>*

� � �

! *:I* ! *r*3

<sup>3</sup>*z*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> ð Þ

3

*<sup>r</sup>*<sup>5</sup> *IzSz* <sup>þ</sup>

= nuclear spin

(2)

(3)

Hamiltonian term of isotropic hyperfine structure interaction is expressed as

where *ge*= *g*-value of the electron, *gN*= *g*-value of the nucleus, *β<sup>e</sup>* = Bohr

operator, and *δ*ð Þ*r* = Dirac delta function for the distance between the electron and

*ϰ* ¼ *aS* ! *:I* !

Anisotropic hyperfine interaction is also called dipolar interaction or dipole– dipole interaction. The Hamiltonian term of anisotropic hyperfine structure

> 3 *S* ! *:r* ! � �

More specifically, the expression of the anisotropic hyperfine interaction in the

*<sup>r</sup>*<sup>5</sup> *IySy* <sup>þ</sup>

2 4

<sup>3</sup>*y*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> ð Þ

*<sup>r</sup>*<sup>5</sup> *IxSz* <sup>þ</sup> *IzSx* � � �

*ϰ* ¼ *gegNβeβ<sup>N</sup>*

!

*<sup>a</sup>* <sup>¼</sup> <sup>8</sup>*<sup>π</sup>*

There are two types of hyperfine structure interaction. These are isotropic

hyperfine interaction and anisotropic hyperfine interaction.

$$\varkappa = \overrightarrow{\mathcal{S}}.\overrightarrow{\overline{\mathcal{A}^0}}.\overrightarrow{I}\tag{7}$$

where *A*<sup>0</sup> !! is called the anisotropic hyperfine coupling tensor. The tensor is expressed in two ways as diagonal elements and non-diagonal elements. The diagonal elements of the tensor is expressed as

$$A\_{ii}^0 = \mathbf{g}\_{\alpha} \mathbf{g}\_N \beta\_{\epsilon} \beta\_N \left\langle \frac{3i^2 - r^2}{r^5} \right\rangle, i = \varkappa, \jmath, z \tag{8}$$

The non-diagonal elements of the tensor is expressed as

$$A^0\_{\vec{\imath}\vec{\jmath}} = \mathbf{g}\_e \mathbf{g}\_N \beta\_e \beta\_N \left< \frac{3i\vec{\jmath}}{r^5} \right>, i, j = \mathbf{x}, \mathbf{y}, z \tag{9}$$

The sum of the isotropic and anisotropic terms fully expresses the hyperfine structure interaction Hamiltonian and is expressed as

$$\boldsymbol{\varkappa} = \boldsymbol{\kappa} \overrightarrow{\boldsymbol{\mathcal{S}}} \overrightarrow{\boldsymbol{I}} + \overrightarrow{\boldsymbol{\mathcal{S}}} . \overrightarrow{\boldsymbol{A}^{0}} . \overrightarrow{\boldsymbol{I}} = \overrightarrow{\boldsymbol{\mathcal{S}}} . \overrightarrow{\boldsymbol{\mathcal{A}}} . \overrightarrow{\boldsymbol{I}} \tag{10}$$

!!

where *A* is the general hyperfine structure tensor.

**Figure 1** shows the formation of the hyperfine structure splittings. **Figure 2** shows the formation of an EPR spectrum due to the hyperfine structure splittings.

#### **1.2 Fine structure interaction**

The fine structure is seen in two ways. The first is the fine structure interaction between two electron spins. The second is the fine structure interaction between the two nucleus spins. The fine structure interaction between two electron spin is also

**Figure 1.** *The formation of the hyperfine structure splittings.*

*Dii* ¼ *ge* 2 *βe*

The non-diagonal elements of the tensor is expressed as.

*Dij* ¼ *ge* 2 *βe* <sup>2</sup> 3*ij r*5

*1.2.2 Nuclear quadrupole interaction*

*DOI: http://dx.doi.org/10.5772/intechopen.91791*

the nuclear quadrupole interaction is expressed as

6*I*ð Þ 2*I* � 1

integral was taken over the volume of the nucleus.

*The zero-field splittings for (a) s = 1/2, (b) s = 1, and (c) s = 3/2.*

**Figure 3.**

**7**

X *α*, *β*¼*x*, *y*, *z*

spherical symmetry. The nuclear quadrupole moment is expressed as

*eQ* ¼ ð

*<sup>ϰ</sup>* <sup>¼</sup> *eQ*

<sup>2</sup> *<sup>r</sup>*<sup>2</sup> � <sup>3</sup>*<sup>i</sup>*

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions*

The zero-field splittings for s = 1/2, s = 1, and s = 3/2 are shown in **Figure 3**.

The interaction between the nucleus spins is known as the nuclear quadrupole interaction. The effects of nuclear quadrupole interaction can be observed on the energy levels of the hyperfine structure for a nucleus with *I* ≥1. The Hamiltonian of

> *Vαβ* 3 2

where *Vαβ* is the component of the field gradient tensor and *eQ* is the nuclear quadrupole moment, and it is a measure of the deviation of charge distribution from

*<sup>ρ</sup><sup>N</sup>* <sup>3</sup>*z*<sup>2</sup> � *<sup>r</sup>*

where *e* is the proton charge, *ρ<sup>N</sup>* is the distribution function of the nuclear charge, *z* is the *z*-coordinate of the charge element a distance *r* from the origin. The

In general, the nuclear quadrupole interaction Hamiltonian is written as

*ϰ* ¼ *I* ! *: P* !! *:I* !

*IαI<sup>β</sup>* þ *IβI<sup>α</sup>* � � � *δαβ<sup>I</sup>*

*r*5

2

� �, *<sup>i</sup>* <sup>¼</sup> *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* (15)

� �, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* (16)

2 � � (17)

<sup>2</sup> � �*dV* (18)

(19)

**Figure 2.** *The formation of an EPR spectrum due to the hyperfine structure splittings.*

referred to as zero-field interaction or zero-field splitting. The interaction between two nuclear spin is called nuclear quadrupole interaction.

#### *1.2.1 Zero-field splitting (interaction)*

This interaction between two electron spins is the dipolar interaction. When writing Hamiltonian for zero-field interaction, the magnetic dipole moments in Eq. (1) are arranged for two electron spins. In this case, the Hamiltonian of the zero-field splitting is written as

$$\mathbf{x} = \mathbf{g}\_{\epsilon}^{-2} \boldsymbol{\beta}\_{\epsilon}^{-2} \left[ \frac{\overrightarrow{\mathbf{S}\_{1}} \cdot \overrightarrow{\mathbf{S}\_{2}}}{r^{3}} - \frac{\mathbf{3} \left( \overrightarrow{\mathbf{S}\_{1}} \cdot \overrightarrow{r} \right) \left( \overrightarrow{\mathbf{S}\_{2}} \cdot \overrightarrow{r} \right)}{r^{5}} \right] \tag{11}$$

More specifically, the expression of the anisotropic hyperfine interaction in the Cartesian coordinate is written as

$$\begin{split} \chi &= \mathbf{g}\_{\epsilon}^{2} \boldsymbol{\beta}\_{\epsilon}^{2} \left[ \frac{(r^{2} - 3\mathbf{x}^{2})}{r^{5}} \mathbf{S}\_{1x} \mathbf{S}\_{2x} + \frac{(r^{2} - 3\mathbf{y}^{2})}{r^{5}} \mathbf{S}\_{1y} \mathbf{S}\_{2y} + \frac{(r^{2} - 3\mathbf{z}^{2})}{r^{5}} \mathbf{S}\_{1x} \mathbf{S}\_{2x} \\ &- \frac{3\mathbf{x}\mathbf{y}}{r^{5}} \left( \mathbf{S}\_{1x} \mathbf{S}\_{2y} + \mathbf{S}\_{1y} \mathbf{S}\_{2x} \right) - \frac{3\mathbf{y}\mathbf{z}}{r^{5}} \left( \mathbf{S}\_{1y} \mathbf{S}\_{2x} + \mathbf{S}\_{1x} \mathbf{S}\_{2y} \right) - \frac{3\mathbf{x}\mathbf{z}}{r^{5}} \left( \mathbf{S}\_{1x} \mathbf{S}\_{2x} + \mathbf{S}\_{1x} \mathbf{S}\_{2x} \right) \end{split} \tag{12}$$

In a shorter way, it is expressed as

$$\varkappa = \overrightarrow{\mathbf{S}\_1}.\overrightarrow{\overrightarrow{D}}.\overrightarrow{\mathbf{S}\_2} \tag{13}$$

In general, the Hamiltonian of the zero-field splitting is written as

$$\varkappa = \overrightarrow{\mathcal{S}}.\overrightarrow{\bar{D}}.\overrightarrow{\mathcal{S}}\tag{14}$$

where *D* !! is called the zero-field splitting tensor or the spin–spin coupling tensor. The tensor is expressed in two ways as diagonal elements and non-diagonal elements. The diagonal elements of the tensor is expressed as

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions DOI: http://dx.doi.org/10.5772/intechopen.91791*

$$D\_{ii} = \mathbf{g}\_{\varepsilon}^{\ 2} \boldsymbol{\beta}\_{\varepsilon}^{\ 2} \left\langle \frac{r^2 - \mathfrak{H}^2}{r^5} \right\rangle, i = \varkappa, y, z \tag{15}$$

The non-diagonal elements of the tensor is expressed as.

$$D\_{\vec{\eta}} = \mathbf{g}\_{\epsilon}^{\ 2} \beta\_{\epsilon}^{\ 2} \left\langle \frac{\mathfrak{A}\vec{\eta}}{r^{5}} \right\rangle, i, j = \propto, y, z \tag{16}$$

The zero-field splittings for s = 1/2, s = 1, and s = 3/2 are shown in **Figure 3**.

#### *1.2.2 Nuclear quadrupole interaction*

The interaction between the nucleus spins is known as the nuclear quadrupole interaction. The effects of nuclear quadrupole interaction can be observed on the energy levels of the hyperfine structure for a nucleus with *I* ≥1. The Hamiltonian of the nuclear quadrupole interaction is expressed as

$$\chi = \frac{eQ}{6I(2I-1)} \sum\_{a,\beta=x,y,z} V\_{a\beta} \left\{ \frac{3}{2} \left( I\_a I\_\beta + I\_\beta I\_a \right) - \delta\_{a\beta} I^2 \right\} \tag{17}$$

where *Vαβ* is the component of the field gradient tensor and *eQ* is the nuclear quadrupole moment, and it is a measure of the deviation of charge distribution from spherical symmetry. The nuclear quadrupole moment is expressed as

$$eQ = \int \rho\_N (3x^2 - r^2)dV\tag{18}$$

where *e* is the proton charge, *ρ<sup>N</sup>* is the distribution function of the nuclear charge, *z* is the *z*-coordinate of the charge element a distance *r* from the origin. The integral was taken over the volume of the nucleus.

In general, the nuclear quadrupole interaction Hamiltonian is written as

$$\mathbf{x} = \overrightarrow{\dot{I}} . \overrightarrow{\dot{P}} . \overrightarrow{\dot{I}}\tag{19}$$

**Figure 3.** *The zero-field splittings for (a) s = 1/2, (b) s = 1, and (c) s = 3/2.*

referred to as zero-field interaction or zero-field splitting. The interaction between

This interaction between two electron spins is the dipolar interaction. When writing Hamiltonian for zero-field interaction, the magnetic dipole moments in Eq. (1) are arranged for two electron spins. In this case, the Hamiltonian of the

> 3 *S*<sup>1</sup> !*:r* ! � �

More specifically, the expression of the anisotropic hyperfine interaction in the

*<sup>r</sup>*<sup>5</sup> *<sup>S</sup>*1*yS*2*<sup>y</sup>* <sup>þ</sup>

� � � <sup>3</sup>*xz*

is called the zero-field splitting tensor or the spin–spin coupling tensor.

*<sup>r</sup>*<sup>5</sup> *<sup>S</sup>*1*yS*2*<sup>z</sup>* <sup>þ</sup> *<sup>S</sup>*1*zS*2*<sup>y</sup>*

*ϰ* ¼ *S*<sup>1</sup> !*:D* !! *:S*<sup>2</sup>

*ϰ* ¼ *S* ! *:D* !! *:S* !

The tensor is expressed in two ways as diagonal elements and non-diagonal

elements. The diagonal elements of the tensor is expressed as

In general, the Hamiltonian of the zero-field splitting is written as

*S*2 !*:r* ! � �

*<sup>r</sup>*<sup>2</sup> � <sup>3</sup>*z*<sup>2</sup> ð Þ

3

*<sup>r</sup>*<sup>5</sup> *<sup>S</sup>*1*zS*2*<sup>z</sup>*

*<sup>r</sup>*<sup>5</sup> ð Þ *<sup>S</sup>*1*zS*2*<sup>x</sup>* <sup>þ</sup> *<sup>S</sup>*1*xS*2*<sup>z</sup>*

! (13)

5 (11)

�

(12)

(14)

*r*5

two nuclear spin is called nuclear quadrupole interaction.

*The formation of an EPR spectrum due to the hyperfine structure splittings.*

*ϰ* ¼ *ge* 2 *βe* <sup>2</sup> *S*<sup>1</sup> !*: <sup>S</sup>*<sup>2</sup> �! *r*<sup>3</sup> �

*<sup>r</sup>*<sup>5</sup> *<sup>S</sup>*1*xS*2*<sup>x</sup>* <sup>þ</sup>

� � � <sup>3</sup>*yz*

2 4

*<sup>r</sup>*<sup>2</sup> � <sup>3</sup>*y*<sup>2</sup> ð Þ

*1.2.1 Zero-field splitting (interaction)*

zero-field splitting is written as

Cartesian coordinate is written as

<sup>2</sup> *<sup>r</sup>*<sup>2</sup> � <sup>3</sup>*x*<sup>2</sup> ð Þ

*<sup>r</sup>*<sup>5</sup> *<sup>S</sup>*1*xS*2*<sup>y</sup>* <sup>þ</sup> *<sup>S</sup>*1*yS*2*<sup>x</sup>*

In a shorter way, it is expressed as

�

� <sup>3</sup>*xy*

where *D* !!

**6**

*ϰ* ¼ *ge* 2 *βe*

**Figure 2.**

*Quantum Mechanics*

where *gs* is the spectroscopic splitting factor of the electron spin and is written as

*β<sup>e</sup> S* !

> *β<sup>e</sup> S* ! *:H* !

The nuclear Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the nucleus with the applied magnetic field:

> *ϰ* ¼ �*μ<sup>I</sup>* !*:H* !

*ϰ* ¼ � *γ<sup>I</sup> I* ! *:H* !

where *gI* is the spectroscopic splitting factor of the nucleus spin and is written as

*β<sup>N</sup> I* !

> *β<sup>N</sup> I* ! *:H* !

The effective spin Hamiltonian for EPR spectroscopy can be written as [1–9].

Dipolar interaction can be seen in three ways. These are the hyperfine structure interaction, the zero-field splitting interaction, and the nuclear quadrupole interaction. Each interaction involves the interaction of two spins. The interaction between a nucleus spin and an electron spin is mentioned in the hyperfine structure interaction. The interaction of two electron spins is mentioned in the zero-field splitting interaction. The interaction of two nuclear spins is mentioned in the nuclear quadrupole interaction. The last two interactions are also known as fine structure interactions. The hyperfine structure interaction is an important interaction for EPR spectroscopy. In EPR spectroscopy, the effect of the hyperfine structure interaction is taken into account together with the electron Zeeman interaction [10–24]. In addition, nuclear Zeeman interaction, the zero-field interaction, and the nuclear quadrupole interaction have an effect on EPR spectroscopy. However, their effects are

*:H* !

*ϰ* ¼ � *gI*

*ϰ* ¼ �*gI*

The general spin Hamiltonian for EPR spectroscopy can be written as

*β<sup>e</sup> S* ! *:H* ! þ *S* ! *:A* !! *:I* !

*ϰ* ¼ *gs*

where *γ<sup>I</sup>* is the nuclear gyromagnetic ratio and is written as.

*<sup>γ</sup><sup>I</sup>* <sup>¼</sup> *gI*

*DOI: http://dx.doi.org/10.5772/intechopen.91791*

*ϰ* ¼ *gs*

**3. Conclusion**

negligible.

**9**

*β<sup>e</sup> S* ! *:H* ! þ *S* ! *:A* !! *:I* ! � *gI β<sup>N</sup> I* ! *: H* ! <sup>þ</sup> *<sup>I</sup>* ! *: P* !! *:I* ! þ *S* ! *:D* !! *:S* !

*βN* <sup>ℏ</sup> <sup>¼</sup> *gI* *:H* !

*ϰ* ¼� �*gs*

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions*

*ϰ* ¼ *gs*

*gs* ¼ 2 (23)

*<sup>β</sup><sup>N</sup>* in the atomic unit system, <sup>ℏ</sup> <sup>¼</sup> <sup>1</sup> (28)

*gI* ¼ 1 (29)

(24)

(25)

(26)

(27)

(30)

(31)

(32)

(33)

**Figure 4.** *The nuclear quadrupole splittings for (a) Hquadrupole* 6¼ *0, H = 0 and (b) for HZeeman* ≫ *Hquadrupole.*

where *P* !! is called the nuclear quadrupole coupling tensor. The nuclear quadrupole splittings are shown in **Figure 4**.

#### **2. Effective Hamiltonian terms in electron paramagnetic resonance spectroscopy**

The hyperfine structure Hamiltonian term, electron Zeeman Hamiltonian term, nuclear Zeeman Hamiltonian term, the term of the zero-field splitting, and the term of the nuclear quadrupole interaction are Hamiltonian terms in EPR Spectroscopy. However, in EPR spectroscopy, the electron Zeeman term and the hyperfine structure term are effective Hamiltonian terms. Therefore, the effect of the terms other than the electron Zeeman term and the hyperfine structure term is not taken into account, since the effect is minimal compared to these two terms. The electron Zeeman term and the nuclear Zeeman term have not been mentioned before. Therefore, it will be explained briefly below.

The electron Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the electron with the applied magnetic field:

$$\varkappa = -\overrightarrow{\mu\_s}\overrightarrow{H} \tag{20}$$

$$\mathbf{x} = -\left(\chi\_i \vec{\mathbf{S}}\right) \vec{H} \tag{21}$$

where *γ<sup>s</sup>* is the gyromagnetic ratio of electron spin and is written as.

$$\gamma\_{\varepsilon} = -\frac{\mathbf{g}\_{\nu}\boldsymbol{\beta}\_{\varepsilon}}{\hbar} = -\mathbf{g}\_{\nu}\boldsymbol{\beta}\_{\varepsilon} \text{ (in the atomic unit system, } \hbar = 1\text{)}\tag{22}$$

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions DOI: http://dx.doi.org/10.5772/intechopen.91791*

where *gs* is the spectroscopic splitting factor of the electron spin and is written as

$$\mathbf{g}\_s = \mathbf{2} \tag{23}$$

$$\mathbf{x} = -\left(-\mathbf{g}\_s \boldsymbol{\beta}\_e \vec{\mathbf{S}}\right) \vec{H} \tag{24}$$

$$
\varkappa = \lg \beta\_\epsilon \overline{\mathcal{S}} \,\, \overline{H} \tag{25}
$$

The nuclear Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the nucleus with the applied magnetic field:

$$\varkappa = -\overrightarrow{\mu\_{I}}\overrightarrow{H} \tag{26}$$

$$\mathbf{x} = -\left(\chi\_I \overrightarrow{I}\right) \overrightarrow{H} \tag{27}$$

where *γ<sup>I</sup>* is the nuclear gyromagnetic ratio and is written as.

$$\gamma\_I = \frac{\mathbf{g}\_I \boldsymbol{\theta}\_N}{\hbar} = \mathbf{g}\_I \boldsymbol{\theta}\_N \text{ (in the atomic unit system, } \hbar = 1\text{)}\tag{28}$$

where *gI* is the spectroscopic splitting factor of the nucleus spin and is written as

$$\mathbf{g}\_I = \mathbf{1} \tag{29}$$

$$\mathbf{x} = -\left(\mathbf{g}\_I \boldsymbol{\beta}\_N \vec{I}\right) \vec{H} \tag{30}$$

$$\mathbf{x} = -\mathbf{g}\_l \beta\_N \overrightarrow{I} \,\overrightarrow{H} \tag{31}$$

The general spin Hamiltonian for EPR spectroscopy can be written as

$$\varkappa = \lg \rho\_e \overrightarrow{\mathbf{S}} \overrightarrow{H} + \overrightarrow{\mathbf{S}} \overrightarrow{\mathbf{A}} \overrightarrow{I} - \lg \rho\_N \overrightarrow{I} \overrightarrow{\cdot H}^\circ + \overrightarrow{I} \overrightarrow{\cdot P} \overrightarrow{I} + \overrightarrow{\mathbf{S}} \overrightarrow{\cdot D} \overrightarrow{\cdot S} \tag{32}$$

The effective spin Hamiltonian for EPR spectroscopy can be written as [1–9].

$$\mathbf{x} = \mathbf{g}\_s \boldsymbol{\theta}\_e \overrightarrow{\mathbf{S}} \, \overrightarrow{H} + \overrightarrow{\mathbf{S}} . \overrightarrow{\vec{A}} \, \overrightarrow{I} \tag{33}$$

#### **3. Conclusion**

where *P* !!

*Quantum Mechanics*

**Figure 4.**

**spectroscopy**

netic field:

**8**

is called the nuclear quadrupole coupling tensor.

*The nuclear quadrupole splittings for (a) Hquadrupole* 6¼ *0, H = 0 and (b) for HZeeman* ≫ *Hquadrupole.*

**2. Effective Hamiltonian terms in electron paramagnetic resonance**

The hyperfine structure Hamiltonian term, electron Zeeman Hamiltonian term, nuclear Zeeman Hamiltonian term, the term of the zero-field splitting, and the term of the nuclear quadrupole interaction are Hamiltonian terms in EPR Spectroscopy. However, in EPR spectroscopy, the electron Zeeman term and the hyperfine structure term are effective Hamiltonian terms. Therefore, the effect of the terms other than the electron Zeeman term and the hyperfine structure term is not taken into account, since the effect is minimal compared to these two terms. The electron Zeeman term and the nuclear Zeeman term have not been mentioned before.

The electron Zeeman interaction occurs as a result of the interaction of the magnetic dipole moment caused by the spin of the electron with the applied mag-

> *ϰ* ¼ �*μ<sup>s</sup>* !*:H* !

*ϰ* ¼ � *γ<sup>s</sup> S* ! *:H* !

*<sup>β</sup><sup>e</sup>* in the atomic unit system, <sup>ℏ</sup> <sup>¼</sup> <sup>1</sup> (22)

where *γ<sup>s</sup>* is the gyromagnetic ratio of electron spin and is written as.

(20)

(21)

The nuclear quadrupole splittings are shown in **Figure 4**.

Therefore, it will be explained briefly below.

*<sup>γ</sup><sup>s</sup>* ¼ � *gs*

*βe* <sup>ℏ</sup> ¼ �*gs*

Dipolar interaction can be seen in three ways. These are the hyperfine structure interaction, the zero-field splitting interaction, and the nuclear quadrupole interaction. Each interaction involves the interaction of two spins. The interaction between a nucleus spin and an electron spin is mentioned in the hyperfine structure interaction. The interaction of two electron spins is mentioned in the zero-field splitting interaction. The interaction of two nuclear spins is mentioned in the nuclear quadrupole interaction. The last two interactions are also known as fine structure interactions.

The hyperfine structure interaction is an important interaction for EPR spectroscopy. In EPR spectroscopy, the effect of the hyperfine structure interaction is taken into account together with the electron Zeeman interaction [10–24]. In addition, nuclear Zeeman interaction, the zero-field interaction, and the nuclear quadrupole interaction have an effect on EPR spectroscopy. However, their effects are negligible.

*Quantum Mechanics*

### **Author details**

Betül Çalişkan<sup>1</sup> \* and Ali Cengiz Çalişkan<sup>2</sup>

1 Faculty of Arts and Science, Department of Physics, Pamukkale University, Kinikli, Denizli, Turkey

2 Faculty of Science, Department of Chemistry, Gazi University, Ankara, Turkey

**References**

[1] Atherton NM. Electron Spin Resonance Theory and Applications. New York: John Wiley & Sons Inc.; 1993

[2] Weil JA, Bolton JR, Wertz JE. Electron Paramagnetic Resonance Elementary Theory and Practical Applications. New York: John Wiley & Sons Inc.; 1994

*DOI: http://dx.doi.org/10.5772/intechopen.91791*

Acta Physica Polonica A. 2014;**125**(1): 135-138. DOI: 10.12693/APhysPolA.

[12] Caliskan B, Caliskan AC, Yerli R. Electron paramagnetic resonance study of radiation damage in isonipecotic acid single crystal. Journal of Molecular Structure. 2014;**1075**:12-16. DOI: 10.1016/j.mol.struc.2014.06.030

[13] Caliskan B, Caliskan AC. EPR study

of radiation damage in gamma irradiated 3-nitroacetophenone single crystal. Radiation Effects and Defects in Solids. 2017;**172**(5–6):398-410. DOI: 10.1080/10420150.2017.1320800

[14] Caliskan B, Caliskan AC, Er E. Electron paramagnetic resonance study of radiation-induced paramagnetic centers in succinic anhydride single crystal. Journal of Molecular Structure. 2017;**1144**:421-431. DOI: 10.1016/j.

[15] Caliskan B, Caliskan AC. EPR study of free radical in gamma-irradiated bis (cyclopentadienyl)zirconium dichloride single crystal. Radiation Effects and Defects in Solids. 2017;**172**(5–6): 507-516. DOI: 10.1080/10420150.

[16] Caliskan B, Aras E, Asik B, Buyum M, Birey M. EPR of gamma irradiated single crystals of cholesteryl benzoate. Radiation Effects and Defects

in Solids. 2004;**159**(1):1-5. DOI: 10.1080/10420150310001604101

[17] Caliskan B, Tokgoz H. Electron paramagnetic resonance study of gamma-irradiated phenidone single crystal. Radiation Effects and Defects in Solids. 2014;**169**(3):225-231. DOI: 10.1080/10420150.2013.834903

[18] Caliskan B, Civi M, Birey M. Electron paramagnetic resonance

molstruc.2017.05.039

2017.1346652

125.135

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions*

[3] Gordy W. Theory and Applications of Electron Spin Resonance. New York:

Biochemical Aspects of Electron-Spin Resonance Spectroscopy. New York: Van Nostrand Reinhold Company; 1978

Introduction to Electron Paramagnetic Resonance. New York: W.A. Benjamin,

[6] Assenheim HM. Introduction to Electron Spin Resonance. New York:

[7] Ingram DJE. Free Radicals as Studied by Electron Spin Resonance. Butterworth's

John Wiley & Sons Inc.; 1980

[4] Symons M. Chemical and

[5] Bersohn M, Baird JC. An

Inc.; 1966

Plenum Press; 1967

August 1968

**11**

Scientific Publications; 1958

[8] Ranby B, Rabek JF. ESR

Spectroscopy in Polymer Research. Berlin: Springer-Verlag; 1977

[10] Caliskan B, Caliskan AC, Er E. Electron paramagnetic resonance study

[11] Caliskan B. EPR study of gamma irradiated cholestanone single crystal.

of gamma-irradiated potassium hydroquinone monosulfonate single crystal. Radiation Effects and Defects in Solids. 2016;**171**(5–6):440-450. DOI: 10.1080/10420150.2016.1203924

[9] Roylance DK. An EPR investigation of polymer fracture, PhD Dissertation, Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah.

\*Address all correspondence to: bcaliska@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.

*Dipolar Interactions: Hyperfine Structure Interaction and Fine Structure Interactions DOI: http://dx.doi.org/10.5772/intechopen.91791*

#### **References**

[1] Atherton NM. Electron Spin Resonance Theory and Applications. New York: John Wiley & Sons Inc.; 1993

[2] Weil JA, Bolton JR, Wertz JE. Electron Paramagnetic Resonance Elementary Theory and Practical Applications. New York: John Wiley & Sons Inc.; 1994

[3] Gordy W. Theory and Applications of Electron Spin Resonance. New York: John Wiley & Sons Inc.; 1980

[4] Symons M. Chemical and Biochemical Aspects of Electron-Spin Resonance Spectroscopy. New York: Van Nostrand Reinhold Company; 1978

[5] Bersohn M, Baird JC. An Introduction to Electron Paramagnetic Resonance. New York: W.A. Benjamin, Inc.; 1966

[6] Assenheim HM. Introduction to Electron Spin Resonance. New York: Plenum Press; 1967

[7] Ingram DJE. Free Radicals as Studied by Electron Spin Resonance. Butterworth's Scientific Publications; 1958

[8] Ranby B, Rabek JF. ESR Spectroscopy in Polymer Research. Berlin: Springer-Verlag; 1977

[9] Roylance DK. An EPR investigation of polymer fracture, PhD Dissertation, Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah. August 1968

[10] Caliskan B, Caliskan AC, Er E. Electron paramagnetic resonance study of gamma-irradiated potassium hydroquinone monosulfonate single crystal. Radiation Effects and Defects in Solids. 2016;**171**(5–6):440-450. DOI: 10.1080/10420150.2016.1203924

[11] Caliskan B. EPR study of gamma irradiated cholestanone single crystal. Acta Physica Polonica A. 2014;**125**(1): 135-138. DOI: 10.12693/APhysPolA. 125.135

[12] Caliskan B, Caliskan AC, Yerli R. Electron paramagnetic resonance study of radiation damage in isonipecotic acid single crystal. Journal of Molecular Structure. 2014;**1075**:12-16. DOI: 10.1016/j.mol.struc.2014.06.030

[13] Caliskan B, Caliskan AC. EPR study of radiation damage in gamma irradiated 3-nitroacetophenone single crystal. Radiation Effects and Defects in Solids. 2017;**172**(5–6):398-410. DOI: 10.1080/10420150.2017.1320800

[14] Caliskan B, Caliskan AC, Er E. Electron paramagnetic resonance study of radiation-induced paramagnetic centers in succinic anhydride single crystal. Journal of Molecular Structure. 2017;**1144**:421-431. DOI: 10.1016/j. molstruc.2017.05.039

[15] Caliskan B, Caliskan AC. EPR study of free radical in gamma-irradiated bis (cyclopentadienyl)zirconium dichloride single crystal. Radiation Effects and Defects in Solids. 2017;**172**(5–6): 507-516. DOI: 10.1080/10420150. 2017.1346652

[16] Caliskan B, Aras E, Asik B, Buyum M, Birey M. EPR of gamma irradiated single crystals of cholesteryl benzoate. Radiation Effects and Defects in Solids. 2004;**159**(1):1-5. DOI: 10.1080/10420150310001604101

[17] Caliskan B, Tokgoz H. Electron paramagnetic resonance study of gamma-irradiated phenidone single crystal. Radiation Effects and Defects in Solids. 2014;**169**(3):225-231. DOI: 10.1080/10420150.2013.834903

[18] Caliskan B, Civi M, Birey M. Electron paramagnetic resonance

**Author details**

*Quantum Mechanics*

Betül Çalişkan<sup>1</sup>

**10**

Kinikli, Denizli, Turkey

\* and Ali Cengiz Çalişkan<sup>2</sup>

\*Address all correspondence to: bcaliska@gmail.com

provided the original work is properly cited.

1 Faculty of Arts and Science, Department of Physics, Pamukkale University,

2 Faculty of Science, Department of Chemistry, Gazi University, Ankara, Turkey

© 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,

analysis of gamma irradiated 4-nitropyridine N-oxide single crystal. Radiation Effects and Defects in Solids. 2006;**161**(5):313-317. DOI: 10.1080/ 10420150600576049

**Chapter 2**

**Abstract**

Exactly Solvable Problems in

*Kanagasabapathi Gnanasekar and Marimuthu Karunakaran*

Some of the problems in quantum mechanics can be exactly solved without any approximation. Some of the exactly solvable problems are discussed in this chapter. Broadly there are two main approaches to solve such problems. They are (i) based on the solution of the Schrödinger equation and (ii) based on operators. The normalized eigen function, eigen values, and the physical significance of some of the

**Keywords:** exactly solvable, Schrödinger equation, eigen function, eigen values

In general, the potential of the confined region is lower than the surroundings

The potential well is the region where the particle is confined in a small region.

<sup>V</sup> <sup>¼</sup> 0, �L<x<<sup>L</sup> <sup>∞</sup>, Otherwise

The one dimensional Schrödinger equation in Cartesian coordinate is given as

In the infinite potential well, the confined particle is present in the well region (Region-II) for an infinitely long time. So the solution of the Schrödinger equation in the Region-II and Region-III can be omitted for our discussion right now. The

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>

<sup>Ψ</sup> <sup>¼</sup> 0, where *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE*

Ψ ¼ *A*<sup>1</sup> sin *αx* þ *A*<sup>2</sup> cos *αx* (3)

2*m*

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> � <sup>V</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (1)

<sup>ℏ</sup><sup>2</sup> (2)

Ψ<sup>00</sup> þ *V*Ψ ¼ *E*Ψ¼)Ψ<sup>00</sup> þ

Ψ<sup>00</sup> þ

<sup>Ψ</sup><sup>00</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup>

2*m*

Quantum Mechanics

*Lourdhu Bruno Chandrasekar,*

selected problems are discussed.

The potential of the system is defined as

�ℏ<sup>2</sup> 2*m*

The solution of the Eq. (2) is

**13**

Schrödinger equation in the Region-II is written as

**1. Potential well**

(**Figure 1**) [1, 2].

[19] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the paramagnetic center in gammairradiated sulfanilic acid single crystal. Acta Physica Polonica A. 2019;**135**(3): 480-484. DOI: 10.12693/APhysPolA. 135.480

[20] Caliskan B, Civi M, Birey M. Electron paramagnetic resonance characterization of gamma irradiation damage centers in S-butyrylthiocholine iodide single crystal. Radiation Effects and Defects in Solids. 2007;**162**(2): 87-93. DOI: 10.1080/1042015 0600907632

[21] Aras E, Asik B, Caliskan B, Buyum M, Birey M. Electron paramagnetic resonance study of irradiated tetramethyl-4-piperidion. Radiation Effects and Defects in Solids. 2004;**159**(6):353-358. DOI: 10.1080/ 10420150410001731820

[22] Asik B, Aras E, Caliskan B, Eken M, Birey M. EPR study of irradiated 4 chloromethyl pyridinium chloride. Radiation Effects and Defects in Solids. 2004;**159**(1):55-60. DOI: 10.1080/ 10420150310001639770

[23] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the radiation damage in phosphoryethanolamine single crystal. Journal of Molecular Structure. 2018; **1173**:781-791. DOI: 10.1016/j.molstruc. 2018.07.045

[24] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the radiation damage in trans-chalcone single crystal. Acta Physica Polonica A. 2019;**136**(1):92-100. DOI: 10.12693/ APhysPolA.136.92

#### **Chapter 2**

analysis of gamma irradiated

10420150600576049

*Quantum Mechanics*

135.480

0600907632

4-nitropyridine N-oxide single crystal. Radiation Effects and Defects in Solids. 2006;**161**(5):313-317. DOI: 10.1080/

[19] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the paramagnetic center in gammairradiated sulfanilic acid single crystal. Acta Physica Polonica A. 2019;**135**(3): 480-484. DOI: 10.12693/APhysPolA.

[20] Caliskan B, Civi M, Birey M. Electron paramagnetic resonance characterization of gamma irradiation damage centers in S-butyrylthiocholine iodide single crystal. Radiation Effects and Defects in Solids. 2007;**162**(2): 87-93. DOI: 10.1080/1042015

[21] Aras E, Asik B, Caliskan B, Buyum M, Birey M. Electron paramagnetic resonance study of irradiated tetramethyl-4-piperidion. Radiation Effects and Defects in Solids. 2004;**159**(6):353-358. DOI: 10.1080/

10420150410001731820

10420150310001639770

2018.07.045

APhysPolA.136.92

**12**

[23] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the radiation damage in phosphoryethanolamine single crystal. Journal of Molecular Structure. 2018; **1173**:781-791. DOI: 10.1016/j.molstruc.

[24] Caliskan B, Caliskan AC. Electron paramagnetic resonance study of the radiation damage in trans-chalcone single crystal. Acta Physica Polonica A. 2019;**136**(1):92-100. DOI: 10.12693/

[22] Asik B, Aras E, Caliskan B, Eken M, Birey M. EPR study of irradiated 4 chloromethyl pyridinium chloride. Radiation Effects and Defects in Solids. 2004;**159**(1):55-60. DOI: 10.1080/

## Exactly Solvable Problems in Quantum Mechanics

*Lourdhu Bruno Chandrasekar, Kanagasabapathi Gnanasekar and Marimuthu Karunakaran*

#### **Abstract**

Some of the problems in quantum mechanics can be exactly solved without any approximation. Some of the exactly solvable problems are discussed in this chapter. Broadly there are two main approaches to solve such problems. They are (i) based on the solution of the Schrödinger equation and (ii) based on operators. The normalized eigen function, eigen values, and the physical significance of some of the selected problems are discussed.

**Keywords:** exactly solvable, Schrödinger equation, eigen function, eigen values

#### **1. Potential well**

The potential well is the region where the particle is confined in a small region. In general, the potential of the confined region is lower than the surroundings (**Figure 1**) [1, 2].

The potential of the system is defined as

$$\mathbf{V} = \begin{cases} \mathbf{0}, & -\mathbf{L} < \mathbf{x} < \mathbf{L} \\ \mathbf{\stackrel{\bullet}{\mathbf{0}}}, & \text{Otherwise} \end{cases}$$

The one dimensional Schrödinger equation in Cartesian coordinate is given as

$$\frac{-\hbar^2}{2m}\Psi'' + V\Psi = E\Psi \Longrightarrow \Psi'' + \frac{2m}{\hbar^2}(\mathbf{E} - \mathbf{V})\Psi = \mathbf{0} \tag{1}$$

In the infinite potential well, the confined particle is present in the well region (Region-II) for an infinitely long time. So the solution of the Schrödinger equation in the Region-II and Region-III can be omitted for our discussion right now. The Schrödinger equation in the Region-II is written as

$$
\Psi'' + \frac{2m}{\hbar^2} (\mathbf{E}) \Psi = \mathbf{0}
$$

$$
\Psi'' + a^2 \Psi = \mathbf{0},
\text{where } a^2 = \frac{2mE}{\hbar^2} \tag{2}
$$

The solution of the Eq. (2) is

$$
\Psi = A\_1 \sin ax + A\_2 \cos ax \tag{3}
$$

**Figure 1.** *Infinite potential well.*

At *x* ¼ �*L*, and at *x* ¼ *L*, the wave function vanishes since the potential is infinite. Hence, At *x* ¼ �*L*,

$$-A\_1 \sin aL + A\_2 \cos aL = 0\tag{4}$$

The eigen function is Ψ ¼ *A*<sup>2</sup> cos *αx* and the normalized eigen function is

<sup>Ψ</sup> <sup>¼</sup> *<sup>L</sup>*�1*=*<sup>2</sup> cos ð Þ *<sup>n</sup>πx=*2*<sup>L</sup>* , *<sup>n</sup>* <sup>¼</sup> 1, 3, 5, …

• The minimum energy state can be calculated by setting *n* ¼ 1, which corresponds to the ground state. The ground state energy is

*<sup>E</sup>*<sup>1</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup>

*=*8*mL*<sup>2</sup>

between any two successive states is not the same.

energy eigen value remains the same.

*=*2*mL*<sup>2</sup> . *<sup>L</sup>*�1*=*<sup>2</sup> sin ð Þ *<sup>n</sup>πx=*2*<sup>L</sup>* , *<sup>n</sup>* <sup>¼</sup> 2, 4, 6, …

The integer "n" is the quantum number and it denotes the discrete energy states in the quantum well. We can extract some physical information from the eigen solutions.

ℏ2

This is known as zero-point energy in the case of the potential well. The excited

*=*8*mL*<sup>2</sup>

, *<sup>E</sup>*<sup>3</sup> <sup>¼</sup> <sup>9</sup>*π*<sup>2</sup>ℏ<sup>2</sup>

• The energy difference between the successive states is simply the difference between the energy eigen value of the corresponding state. For example, *ΔE*<sup>12</sup> ¼ *E*<sup>1</sup> � *E*<sup>2</sup> ¼ 3*E*<sup>1</sup> and *ΔE*<sup>23</sup> ¼ *E*<sup>2</sup> � *E*<sup>3</sup> ¼ 5*E*1. Hence the energy difference

• Though the eigen functions for odd and even values of "n" are different, the

• If the potential well is chosen in the limit 0 <*x*<2*L* (width of the well is 2*L*), the energy eigen value is the same as given in Eqs.(6) and (8). But if the limit is chosen as 0<*x*<*L* (width of the well is *L*), the for all positive integers of "n," the eigen function is <sup>Ψ</sup> <sup>¼</sup> ð Þ <sup>2</sup>*=<sup>L</sup>* <sup>1</sup>*=*<sup>2</sup> sin ð Þ *<sup>n</sup>πx=<sup>L</sup>* and the energy eigen function is

Step potential is a problem that has two different finite potentials [3]. Classically, the tunneling probability is 1 when the energy of the particle is greater than the height

of the barrier. But the result is not true based on wave mechanics (**Figure 2**).

In Summary, the eigen value is *<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*2*π*2ℏ<sup>2</sup>

(

of "n." The normalized eigen functions are

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

state energies are *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>*π*<sup>2</sup>ℏ<sup>2</sup>

on. In general, *En* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> � *<sup>E</sup>*1.

*<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*π*<sup>2</sup>ℏ<sup>2</sup>

**2. Step potential**

**Figure 2.** *Step potential.*

**15**

<sup>Ψ</sup> <sup>¼</sup> *<sup>L</sup>*�1*=*<sup>2</sup> sin ð Þ *<sup>n</sup>πx=*2*<sup>L</sup>* (9)

*=*8*mL*<sup>2</sup> for all positive integer values

*=*8*mL*<sup>2</sup> (11)

, *<sup>E</sup>*<sup>4</sup> <sup>¼</sup> <sup>16</sup>*π*<sup>2</sup>ℏ<sup>2</sup>

*=*8*mL*<sup>2</sup>

, and so

(10)

Similarly, at *x* ¼ *L*

$$A\_1 \sin aL + A\_2 \cos aL = 0\tag{5}$$

The addition and subtraction of these equations give two different solutions.

i. <sup>2</sup>*A*<sup>2</sup> cos *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼) cos *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼)*α<sup>L</sup>* <sup>¼</sup> *<sup>n</sup>π=*2¼)*α*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*π*<sup>2</sup>*=*4*L*<sup>2</sup> ; *n* ¼ 1, 3, 5, … … . Since *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE* <sup>ℏ</sup><sup>2</sup> , <sup>2</sup>*mE* <sup>ℏ</sup><sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*π*<sup>2</sup>*=*4*L*<sup>2</sup> , the energy eigen value is found as

$$E = n^2 \pi^2 \hbar^2 / 8mL^2 \tag{6}$$

The eigen function is Ψ ¼ *A*<sup>1</sup> cos *αx*

$$=A\_1 \cos\ (n\pi\alpha/2L)$$

According to the normalization condition,

$$\int\_{-L}^{L} \Psi^\* \Psi \mathbf{dx} = \mathbf{0} \Longrightarrow A\_1 = L^{-1/2}$$

Hence the normalized eigen function for *n* ¼ 1, 3, 5, … … is

$$\Psi = L^{-1/2}\cos\left(n\pi x/2L\right)\tag{7}$$

ii. <sup>2</sup>*A*<sup>1</sup> sin *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼) sin *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼)*α<sup>L</sup>* <sup>¼</sup> *<sup>n</sup>π=*2¼)*α*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*2*π*2*=*4*L*<sup>2</sup> ; *n* ¼ 2, 4, 6, … … . For this case, *n* ¼ 2, 4, 6, … … , the corresponding energy eigen value is

$$E = n^2 \pi^2 \hbar^2 / 8mL^2 \tag{8}$$

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

The eigen function is Ψ ¼ *A*<sup>2</sup> cos *αx* and the normalized eigen function is

$$
\Psi = L^{-1/2} \sin^{\circ}(n\pi\infty/2L) \tag{9}
$$

In Summary, the eigen value is *<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*2*π*2ℏ<sup>2</sup> *=*8*mL*<sup>2</sup> for all positive integer values of "n." The normalized eigen functions are

$$\Psi = \begin{cases} L^{-1/2}\cos\ (n\pi\infty/2L), n = \mathbf{1}, \mathbf{3}, \mathbf{5}, \dots \\ L^{-1/2}\sin\ (n\pi\infty/2L), n = \mathbf{2}, \mathbf{4}, \mathbf{6}, \dots \end{cases} \tag{10}$$

The integer "n" is the quantum number and it denotes the discrete energy states in the quantum well. We can extract some physical information from the eigen solutions.

• The minimum energy state can be calculated by setting *n* ¼ 1, which corresponds to the ground state. The ground state energy is

$$E\_1 = \pi^2 \hbar^2 / 8mL^2 \tag{11}$$

This is known as zero-point energy in the case of the potential well. The excited state energies are *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>*π*<sup>2</sup>ℏ<sup>2</sup> *=*8*mL*<sup>2</sup> , *<sup>E</sup>*<sup>3</sup> <sup>¼</sup> <sup>9</sup>*π*<sup>2</sup>ℏ<sup>2</sup> *=*8*mL*<sup>2</sup> , *<sup>E</sup>*<sup>4</sup> <sup>¼</sup> <sup>16</sup>*π*<sup>2</sup>ℏ<sup>2</sup> *=*8*mL*<sup>2</sup> , and so on. In general, *En* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> � *<sup>E</sup>*1.


#### **2. Step potential**

At *x* ¼ �*L*, and at *x* ¼ *L*, the wave function vanishes since the potential is

The addition and subtraction of these equations give two different solutions.

¼ *A*<sup>1</sup> cos ð Þ *nπx=*2*L*

<sup>Ψ</sup><sup>∗</sup> <sup>Ψ</sup>d*<sup>x</sup>* <sup>¼</sup> <sup>0</sup>¼)*A*<sup>1</sup> <sup>¼</sup> *<sup>L</sup>*�1*=*<sup>2</sup>

For this case, *n* ¼ 2, 4, 6, … … , the corresponding energy eigen value is

i. <sup>2</sup>*A*<sup>2</sup> cos *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼) cos *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼)*α<sup>L</sup>* <sup>¼</sup> *<sup>n</sup>π=*2¼)*α*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*π*<sup>2</sup>*=*4*L*<sup>2</sup>

*<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> *π*2 ℏ2

<sup>ℏ</sup><sup>2</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*π*<sup>2</sup>*=*4*L*<sup>2</sup>

�*A*<sup>1</sup> sin *αL* þ *A*<sup>2</sup> cos *αL* ¼ 0 (4)

*A*<sup>1</sup> sin *αL* þ *A*<sup>2</sup> cos *αL* ¼ 0 (5)

, the energy eigen value is found as

<sup>Ψ</sup> <sup>¼</sup> *<sup>L</sup>*�1*=*<sup>2</sup> cos ð Þ *<sup>n</sup>πx=*2*<sup>L</sup>* (7)

*=*8*mL*<sup>2</sup> (8)

*=*8*mL*<sup>2</sup> (6)

; *n* ¼ 1, 3, 5, … … .

; *n* ¼ 2, 4, 6, … … .

infinite. Hence, At *x* ¼ �*L*,

**Figure 1.**

*Infinite potential well.*

*Quantum Mechanics*

Similarly, at *x* ¼ *L*

Since *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE*

**14**

<sup>ℏ</sup><sup>2</sup> , <sup>2</sup>*mE*

The eigen function is Ψ ¼ *A*<sup>1</sup> cos *αx*

According to the normalization condition,

ð *L*

�*L*

Hence the normalized eigen function for *n* ¼ 1, 3, 5, … … is

ii. <sup>2</sup>*A*<sup>1</sup> sin *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼) sin *<sup>α</sup><sup>L</sup>* <sup>¼</sup> <sup>0</sup>¼)*α<sup>L</sup>* <sup>¼</sup> *<sup>n</sup>π=*2¼)*α*<sup>2</sup> <sup>¼</sup> *<sup>n</sup>*2*π*2*=*4*L*<sup>2</sup>

*<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup> *π*2 ℏ2

Step potential is a problem that has two different finite potentials [3]. Classically, the tunneling probability is 1 when the energy of the particle is greater than the height of the barrier. But the result is not true based on wave mechanics (**Figure 2**).

**Figure 2.** *Step potential.* The potential of the system

$$\mathbf{V} = \begin{cases} \mathbf{0}, & -\infty < \mathbf{x} < \mathbf{0} \\ \mathbf{V}\_0, & \mathbf{0} \le \mathbf{x} < \infty \end{cases}$$

The Schrödinger equation in the Region-I and Region-II is given, respectively as,

$$
\Psi'' + \frac{2m}{\hbar^2} (\mathbf{E}) \Psi = \mathbf{0} \tag{12}
$$

where *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*m*ð Þ <sup>E</sup>�*V*<sup>0</sup>

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

From these equations,

given as

<sup>ℏ</sup><sup>2</sup> . As *x* ! ∞, the wave function Ψ<sup>2</sup> must be finite. Hence

*A*<sup>1</sup> þ *B*<sup>1</sup> ¼ *A*<sup>2</sup> (19)

*<sup>A</sup>*<sup>2</sup> (20)

ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* <sup>2</sup> (22)

¼ 1 (23)

(21)

Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þ *iβx* by setting *B*<sup>2</sup> ¼ 0. According to the boundary conditions at *x* ¼ 0,

*α*

*<sup>A</sup>*<sup>1</sup> � *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*

A2 <sup>¼</sup> <sup>2</sup><sup>α</sup>

B1 <sup>¼</sup> <sup>α</sup> � <sup>β</sup> α þ β A1

2

2

*<sup>R</sup>* <sup>¼</sup> j j *<sup>B</sup>*<sup>1</sup>

*<sup>T</sup>* <sup>¼</sup> j j *<sup>A</sup>*<sup>2</sup>

*<sup>T</sup>* <sup>þ</sup> *<sup>R</sup>* <sup>¼</sup> <sup>4</sup>*αβ*

From these easily one can show that

potential of the system is given as (**Figure 3**)

the step potential is conserved.

**3. Potential barrier**

**Figure 3.** *Potential barrier.*

**17**

j j *A*<sup>1</sup>

j j *A*<sup>1</sup>

α þ β A1

The reflection coefficient R and the transmission coefficient T, respectively, are

<sup>2</sup> <sup>¼</sup> *<sup>α</sup>* � *<sup>β</sup> α* þ *β* <sup>2</sup>

<sup>2</sup> <sup>¼</sup> <sup>4</sup>*αβ*

ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* <sup>2</sup> <sup>þ</sup> *<sup>α</sup>* � *<sup>β</sup>*

The results again indicate that the total number of particles which encounters

This problem clearly explains the wave-mechanical tunneling [3, 4]. The

<sup>V</sup> <sup>¼</sup> V0, 0 <sup>&</sup>lt;x<<sup>L</sup> 0, Otherwise

*α* þ *β* <sup>2</sup>

$$
\Psi'' + \frac{2m}{\hbar^2} (\mathbf{E} - \mathbf{V}) \Psi = \mathbf{0} \tag{13}
$$

Case (i): when *E* <*V*0, the solutions of the Schrödinger equations in the Region-I and Region-II, respectively, are given as

$$\Psi\_1 = A\_1 \exp\left(i\alpha\mathbf{x}\right) + B\_1 \exp\left(-i\alpha\mathbf{x}\right) \tag{14}$$

$$\Psi\_2 = A\_2 \exp\left(-\beta\mathbf{x}\right) + B\_2 \exp\left(\beta\mathbf{x}\right)$$

where *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE* <sup>ℏ</sup><sup>2</sup> and *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*m*ð Þ <sup>E</sup>�*V*<sup>0</sup> <sup>ℏ</sup><sup>2</sup> . Here, *B*<sup>2</sup> exp ð Þ *βx* represents the exponentially increasing wave along the x-direction. The wave function Ψ<sup>2</sup> must be finite as *x* ! ∞. This is possible only by setting *B*<sup>2</sup> ¼ 0. Hence the eigen function in the Region-II is

$$
\Psi\_2 = A\_2 \exp\left(-\beta \mathbf{x}\right) \tag{15}
$$

According to admissibility conditions of wave functions, at *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ<sup>2</sup> and Ψ0 <sup>1</sup> ¼ Ψ<sup>0</sup> 2. It gives us

$$A\_1 + B\_1 = A\_2 \tag{16}$$

$$A\_1 - B\_1 = i \left(\frac{\beta}{a}\right) A\_2 \tag{17}$$

From these two equations,

$$\mathbf{A}\_{2} = \left(\frac{2\alpha}{\alpha + \mathbf{i}\beta}\right) \mathbf{A}\_{1}$$

$$\mathbf{B}\_{1} = \left(\frac{\alpha - \mathbf{i}\beta}{\alpha + \mathbf{i}\beta}\right) \mathbf{A}\_{1}$$

The reflection coefficient R is given as

$$R = \frac{\left|B\_1\right|^2}{\left|A\_1\right|^2} = \left|\frac{\alpha - i\beta}{\alpha + i\beta}\right|^2 = 1\tag{18}$$

It is interesting to note that all the particles that encounter the step potential are reflected back. This is due to the fact that the width of the step potential is infinite. The number of particles in the process is conserved, which leads that *T* ¼ 0, since *T* þ *R* ¼ 1.

Case (ii): when *E*>*V*0, the solutions are given as

$$\begin{aligned} \Psi\_1 &= A\_1 \exp\left(i\alpha\pi\right) + B\_1 \exp\left(-i\alpha\pi\right), \\ \Psi\_2 &= A\_2 \exp\left(i\beta\pi\right) + B\_2 \exp\left(-i\beta\pi\right). \end{aligned}$$

where *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*m*ð Þ <sup>E</sup>�*V*<sup>0</sup> <sup>ℏ</sup><sup>2</sup> . As *x* ! ∞, the wave function Ψ<sup>2</sup> must be finite. Hence Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þ *iβx* by setting *B*<sup>2</sup> ¼ 0. According to the boundary conditions at *x* ¼ 0,

$$A\_1 + B\_1 = A\_2 \tag{19}$$

$$A\_1 - B\_1 = \left(\frac{\beta}{a}\right) A\_2 \tag{20}$$

From these equations,

The potential of the system

*Quantum Mechanics*

and Region-II, respectively, are given as

<sup>ℏ</sup><sup>2</sup> and *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*m*ð Þ <sup>E</sup>�*V*<sup>0</sup>

where *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE*

2. It gives us

From these two equations,

The reflection coefficient R is given as

Ψ0 <sup>1</sup> ¼ Ψ<sup>0</sup>

*T* þ *R* ¼ 1.

**16**

<sup>V</sup> <sup>¼</sup> 0, �<sup>∞</sup> <sup>&</sup>lt;x<<sup>0</sup> V0, 0≤x< ∞

2*m*

The Schrödinger equation in the Region-I and Region-II is given, respectively as,

Case (i): when *E* <*V*0, the solutions of the Schrödinger equations in the Region-I

Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þþ �*βx B*<sup>2</sup> exp ð Þ *βx*

increasing wave along the x-direction. The wave function Ψ<sup>2</sup> must be finite as *x* ! ∞. This is possible only by setting *B*<sup>2</sup> ¼ 0. Hence the eigen function in the Region-II is

According to admissibility conditions of wave functions, at *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ<sup>2</sup> and

*α* 

A1

A1

 

2

*<sup>A</sup>*<sup>1</sup> � *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>i</sup> <sup>β</sup>*

A2 <sup>¼</sup> <sup>2</sup><sup>α</sup>

B1 <sup>¼</sup> <sup>α</sup> � <sup>i</sup><sup>β</sup> α þ iβ 

2

*<sup>R</sup>* <sup>¼</sup> j j *<sup>B</sup>*<sup>1</sup>

Case (ii): when *E*>*V*0, the solutions are given as

j j *A*<sup>1</sup>

α þ iβ 

<sup>2</sup> <sup>¼</sup> *<sup>α</sup>* � *<sup>i</sup><sup>β</sup> α* þ *iβ*

It is interesting to note that all the particles that encounter the step potential are reflected back. This is due to the fact that the width of the step potential is infinite. The number of particles in the process is conserved, which leads that *T* ¼ 0, since

> Ψ<sup>1</sup> ¼ *A*<sup>1</sup> exp ð Þþ *iαx B*<sup>1</sup> exp ð Þ �*iαx* Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þþ *iβx B*<sup>2</sup> exp ð Þ �*iβx*

 

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (12)

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> � <sup>V</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (13)

Ψ<sup>1</sup> ¼ *A*<sup>1</sup> exp ð Þþ *iαx B*<sup>1</sup> exp ð Þ �*iαx* (14)

<sup>ℏ</sup><sup>2</sup> . Here, *B*<sup>2</sup> exp ð Þ *βx* represents the exponentially

Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þ �*βx* (15)

*A*<sup>1</sup> þ *B*<sup>1</sup> ¼ *A*<sup>2</sup> (16)

*A*<sup>2</sup> (17)

¼ 1 (18)

Ψ<sup>00</sup> þ

2*m*

Ψ<sup>00</sup> þ

$$\mathbf{A}\_2 = \left(\frac{2\alpha}{\alpha + \beta}\right) \mathbf{A}\_1$$

$$\mathbf{B}\_1 = \left(\frac{\alpha - \beta}{\alpha + \beta}\right) \mathbf{A}\_1$$

The reflection coefficient R and the transmission coefficient T, respectively, are given as

$$R = \frac{\left|B\_1\right|^2}{\left|A\_1\right|^2} = \left(\frac{\alpha - \beta}{\alpha + \beta}\right)^2\tag{21}$$

$$T = \frac{\left|A\_2\right|^2}{\left|A\_1\right|^2} = \frac{4a\beta}{\left(a+\beta\right)^2} \tag{22}$$

From these easily one can show that

$$T + R = \frac{4a\beta}{\left(a + \beta\right)^2} + \left(\frac{a - \beta}{a + \beta}\right)^2 = 1\tag{23}$$

The results again indicate that the total number of particles which encounters the step potential is conserved.

#### **3. Potential barrier**

This problem clearly explains the wave-mechanical tunneling [3, 4]. The potential of the system is given as (**Figure 3**)

$$\mathbf{V} = \begin{cases} \mathbf{V}\_0, & \mathbf{0} < \mathbf{x} < \mathbf{L} \\ \mathbf{0}, & \text{Otherwise} \end{cases}$$

**Figure 3.** *Potential barrier.*

In the Region-I, the Schrödinger equation is <sup>Ψ</sup><sup>00</sup> <sup>þ</sup> *<sup>α</sup>*2<sup>Ψ</sup> <sup>¼</sup> 0. The wave function in this region is given as

$$A\Psi\_1 = A\_1 \exp\left(i a \mathbf{x}\right) + B\_1 \exp\left(-i a \mathbf{x}\right) \text{ where } a^2 = \frac{2mE}{\hbar^2} \tag{24}$$

In Region-II, if *<sup>E</sup>*<*V*0, the Schrödinger equation is <sup>Ψ</sup><sup>00</sup> � *<sup>β</sup>*<sup>2</sup> Ψ ¼ 0. The solution of the equation is given as

$$\Psi\_2 = A\_2 \exp\left(\beta \mathbf{x}\right) + B\_2 \exp\left(-\beta \mathbf{x}\right) \text{ where } \beta^2 = \frac{2m(\mathbf{E} - V\_0)}{\hbar^2} \tag{25}$$

The Schrödinger equation in the Region-III is <sup>Ψ</sup><sup>00</sup> <sup>þ</sup> *<sup>α</sup>*2<sup>Ψ</sup> <sup>¼</sup> 0. The corresponding solution is Ψ<sup>3</sup> ¼ *A*<sup>3</sup> exp ð Þþ *iαx B*<sup>3</sup> exp ð Þ �*iαx* . But in the Region-III, the waves can travel only along positive x-direction and there is no particle coming from the right, *B*<sup>3</sup> ¼ 0. Hence

$$
\Psi\_3 = A\_3 \exp\left(i\alpha\mathbf{x}\right) \tag{26}
$$

• When the length of the barrier is an integral multiple of *π=β*, there is no reflection from the barrier. This is termed as resonance scattering.

• The tunneling probability depends on the height and width of the barrier.

lattices.

**4. Delta potential**

The Schrödinger equation is

Ψ<sup>00</sup> þ

where *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> �2*mE*

This gives us

related by the following equation

Ψ0 <sup>1</sup> 6¼ Ψ<sup>0</sup>

**Figure 4.**

**19**

*Dirac delta potential.*

2*m*

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

• Later, Kronig and Penney extended this idea to explain the motion of a charge carrier in a periodic potential which is nothing but the one-dimensional

The Dirac delta potential is infinitesimally narrow potential only at some point

Here *λ* is the positive constant, which is the strength of the delta potential. Here,

2*m*

Region � I : Ψ<sup>1</sup> ¼ *A*<sup>1</sup> exp ð Þ *βx* (34)

Region � II : Ψ<sup>2</sup> ¼ *A*<sup>2</sup> exp ð Þ �*βx* (35)

<sup>ℏ</sup><sup>2</sup> . At *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ2. So the coefficients *A*<sup>1</sup> and *A*<sup>2</sup> are equal. But

2, since the first derivative causes the discontinuity. The first derivatives are

<sup>1</sup> ¼ � <sup>2</sup>*m<sup>λ</sup>*

*<sup>β</sup>* <sup>¼</sup> *<sup>m</sup><sup>λ</sup>*

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> <sup>þ</sup> *λδ*ð Þ *<sup>x</sup>* <sup>Ψ</sup> <sup>¼</sup> 0 (33)

<sup>ℏ</sup><sup>2</sup> (36)

<sup>ℏ</sup><sup>2</sup> (37)

<sup>V</sup> <sup>¼</sup> �λδð Þ <sup>x</sup> , x <sup>¼</sup> <sup>0</sup> 0, Otherwise

(generally at the origin, for convenience) [3]. The potential of the system

we confine ourselves only to the bound states, hence *E* <0 (**Figure 4**).

<sup>ℏ</sup><sup>2</sup> ð Þ <sup>E</sup> � <sup>V</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>¼)Ψ<sup>00</sup> <sup>þ</sup>

Ψ0 <sup>2</sup> � Ψ<sup>0</sup>

The solution of the Schrödinger equation is given as

At *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ<sup>2</sup> and Ψ<sup>0</sup> <sup>1</sup> ¼ Ψ<sup>0</sup> 2. These give us two equations

$$A\_1 + B\_1 = A\_2 + B\_2 \tag{27}$$

$$A\_1 - B\_1 = \left(\frac{\beta}{ia}\right)(A\_2 - B\_2) \tag{28}$$

At *x* ¼ *L*, Ψ<sup>2</sup> ¼ Ψ<sup>3</sup> and Ψ<sup>0</sup> <sup>2</sup> ¼ Ψ<sup>0</sup> 3. These conditions give us another two equations

$$A\_2 \exp\left(\beta L\right) + B\_2 \exp\left(-\beta L\right) = A\_3 \exp\left(iaL\right) \tag{29}$$

$$A\_2 \exp\left(\beta L\right) - B\_2 \exp\left(-\beta L\right) = A\_3 \left(\frac{ia}{\beta}\right) \exp\left(iaL\right) \tag{30}$$

Solving the equations from (27) to (30), one can find the coefficients in the equations. The reflection coefficient is R is found as

$$R = \frac{|B\_1|^2}{|A\_1|^2} = \left[\frac{V\_0^2}{4E(V\_0 - E)}\sinh^2(\beta L)\right] \left[1 + \frac{V\_0^2}{4E(V\_0 - E)}\sinh^2(\beta L)\right]^{-1} \tag{31}$$

The transmission coefficient T is found as

$$T = \frac{|A\_2|^2}{|A\_1|^2} = \left[1 + \frac{V\_0^2}{4E(V\_0 - E)} \sinh^2(\beta L)\right]^{-1} \tag{32}$$

From Eqs. (31) and (32), one can show that *T* þ *R* ¼ 1. The following are the conclusions obtained from the above mathematical analysis.


#### **4. Delta potential**

In the Region-I, the Schrödinger equation is <sup>Ψ</sup><sup>00</sup> <sup>þ</sup> *<sup>α</sup>*2<sup>Ψ</sup> <sup>¼</sup> 0. The wave function in

<sup>Ψ</sup><sup>1</sup> <sup>¼</sup> *<sup>A</sup>*<sup>1</sup> exp ð Þþ *<sup>i</sup>α<sup>x</sup> <sup>B</sup>*<sup>1</sup> exp ð Þ �*iα<sup>x</sup>* where *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*mE*

<sup>Ψ</sup><sup>2</sup> <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> exp ð Þþ *<sup>β</sup><sup>x</sup> <sup>B</sup>*<sup>2</sup> exp ð Þ �*β<sup>x</sup>* where *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*m*ð Þ <sup>E</sup> � *<sup>V</sup>*<sup>0</sup>

The Schrödinger equation in the Region-III is <sup>Ψ</sup><sup>00</sup> <sup>þ</sup> *<sup>α</sup>*2<sup>Ψ</sup> <sup>¼</sup> 0. The corresponding solution is Ψ<sup>3</sup> ¼ *A*<sup>3</sup> exp ð Þþ *iαx B*<sup>3</sup> exp ð Þ �*iαx* . But in the Region-III, the waves can travel only along positive x-direction and there is no particle coming from the

Ψ<sup>3</sup> ¼ *A*<sup>3</sup> exp ð Þ *iαx* (26)

*A*<sup>1</sup> þ *B*<sup>1</sup> ¼ *A*<sup>2</sup> þ *B*<sup>2</sup> (27)

3. These conditions give us another two equations

ð Þ *A*<sup>2</sup> � *B*<sup>2</sup> (28)

exp ð Þ *iαL* (30)

ð Þ *βL*

(31)

(32)

2. These give us two equations

*A*<sup>2</sup> exp ð Þþ *βL B*<sup>2</sup> exp ð Þ¼ �*βL A*<sup>3</sup> exp ð Þ *iαL* (29)

*iα β* 

*V*2 0 <sup>4</sup>*E V*ð Þ <sup>0</sup> � *<sup>E</sup> sinh* <sup>2</sup>

�<sup>1</sup>

ð Þ *βL*

In Region-II, if *<sup>E</sup>*<*V*0, the Schrödinger equation is <sup>Ψ</sup><sup>00</sup> � *<sup>β</sup>*<sup>2</sup>

<sup>1</sup> ¼ Ψ<sup>0</sup>

<sup>2</sup> ¼ Ψ<sup>0</sup>

equations. The reflection coefficient is R is found as

0 <sup>4</sup>*E V*ð Þ <sup>0</sup> � *<sup>E</sup> sinh* <sup>2</sup>

The transmission coefficient T is found as

*<sup>T</sup>* <sup>¼</sup> j j *<sup>A</sup>*<sup>2</sup>

j j *A*<sup>1</sup>

2

conclusions obtained from the above mathematical analysis.

<sup>2</sup> ¼ 1 þ

*A*<sup>2</sup> exp ð Þ� *βL B*<sup>2</sup> exp ð Þ¼ �*βL A*<sup>3</sup>

*<sup>A</sup>*<sup>1</sup> � *<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*

*iα* 

Solving the equations from (27) to (30), one can find the coefficients in the

1 þ

*V*2 0 <sup>4</sup>*E V*ð Þ <sup>0</sup> � *<sup>E</sup> sinh* <sup>2</sup>

From Eqs. (31) and (32), one can show that *T* þ *R* ¼ 1. The following are the

• When *E*<*V*0, though the energy of the incident particles is less than the height of the barrier, the particle can tunnel into the barrier region. This is in contrast

• As *V*<sup>0</sup> ! ∞, the transmission coefficient is zero. Hence the tunneling is not

to the laws of classical physics. This is known as the tunnel effect.

�<sup>1</sup>

ð Þ *βL*

<sup>ℏ</sup><sup>2</sup> (24)

Ψ ¼ 0. The solution

<sup>ℏ</sup><sup>2</sup> (25)

this region is given as

*Quantum Mechanics*

of the equation is given as

right, *B*<sup>3</sup> ¼ 0. Hence

*<sup>R</sup>* <sup>¼</sup> j j *<sup>B</sup>*<sup>1</sup>

**18**

j j *A*<sup>1</sup>

2

<sup>2</sup> <sup>¼</sup> *<sup>V</sup>*<sup>2</sup>

possible only when *V*<sup>0</sup> ! ∞.

At *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ<sup>2</sup> and Ψ<sup>0</sup>

At *x* ¼ *L*, Ψ<sup>2</sup> ¼ Ψ<sup>3</sup> and Ψ<sup>0</sup>

The Dirac delta potential is infinitesimally narrow potential only at some point (generally at the origin, for convenience) [3]. The potential of the system

$$\mathbf{V} = \begin{cases} -\lambda \delta(\mathbf{x}), & \mathbf{x} = \mathbf{0} \\ \mathbf{0}, & \text{Otherwise} \end{cases}$$

Here *λ* is the positive constant, which is the strength of the delta potential. Here, we confine ourselves only to the bound states, hence *E* <0 (**Figure 4**).

The Schrödinger equation is

$$
\Psi'' + \frac{2m}{\hbar^2} (\mathbf{E} - \mathbf{V}) \Psi = \mathbf{0} \\
\Longrightarrow \Psi'' + \frac{2m}{\hbar^2} (\mathbf{E} + \lambda \delta(\mathbf{x})) \Psi = \mathbf{0} \tag{33}
$$

The solution of the Schrödinger equation is given as

$$\text{Region} - \text{I} : \Psi\_1 = A\_1 \exp\left(\beta \mathbf{x}\right) \tag{34}$$

$$\text{Region} - \text{II} : \Psi\_{\text{2}} = A\_{\text{2}} \exp \left( -\beta \mathbf{x} \right) \tag{35}$$

where *<sup>β</sup>*<sup>2</sup> <sup>¼</sup> �2*mE* <sup>ℏ</sup><sup>2</sup> . At *x* ¼ 0, Ψ<sup>1</sup> ¼ Ψ2. So the coefficients *A*<sup>1</sup> and *A*<sup>2</sup> are equal. But Ψ0 <sup>1</sup> 6¼ Ψ<sup>0</sup> 2, since the first derivative causes the discontinuity. The first derivatives are related by the following equation

$$
\Psi\_2' - \Psi\_1' = -\frac{2m\lambda}{\hbar^2} \tag{36}
$$

This gives us

$$
\beta = \frac{m\lambda}{\hbar^2} \tag{37}
$$

$$\begin{array}{c|c} \multicolumn{1}{c}{\bullet \quad x = \emptyset \quad \begin{array}{c} x = \emptyset \quad \begin{array}{c} x = \emptyset \end{array} \\ \cline{2-3} \text{Region - } I & \begin{array}{c} \text{Region - } II \end{array} \end{array} \end{array}$$

**Figure 4.** *Dirac delta potential.*

Equating the value of *β* gives the energy eigen value as

$$E = -\frac{m\lambda^2}{2\hbar^2} \tag{38}$$

The time-independent Schrödinger equation is given as

2*m*

*<sup>α</sup>* <sup>¼</sup> *<sup>m</sup><sup>ω</sup>* ℏ <sup>1</sup>*=*<sup>2</sup>

> *d*2 Ψ

The asymptotic Schrödinger equation ð Þ *α* ! ∞ is given as

*d*2 Ψ *<sup>d</sup>α*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup>

<sup>ℏ</sup><sup>2</sup> *<sup>E</sup>* � *<sup>m</sup>ω*2*x*<sup>2</sup>

The potential is not constant since it is a function of "x"; Eq. (40) cannot solve

Using the new constant *β* and the variable *α*, the Schrödinger equation has the

The general solution of the equation is exp �*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> . As *<sup>α</sup>* ! <sup>∞</sup>, exp <sup>þ</sup>*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> becomes infinite, hence it cannot be a solution. So the only possible solution is exp �*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> . Based on the asymptotic solution, the general solution of Eq. (42) is

<sup>Ψ</sup> <sup>¼</sup> <sup>H</sup>*n*ð Þ <sup>α</sup> exp �*a*<sup>2</sup>

<sup>2</sup>*<sup>n</sup>* � *<sup>n</sup>*!

*E* ¼ *n* þ

• The integer *n* ¼ 0 represents the ground state, *n* ¼ 1 represents the first excited state, and so on. The ground state energy of the linear harmonic oscillator is *E* ¼ ℏ*ω=*2. This minimum energy is known as ground state energy.

> *mω* ℏ*π* <sup>1</sup>*=*<sup>4</sup>

1 2

exp � *<sup>m</sup>ωx*<sup>2</sup> 2ℏ

*=*2

<sup>H</sup>*n*ð Þ <sup>α</sup> exp �*a*<sup>2</sup>

<sup>ℏ</sup>*<sup>ω</sup>* (44)

(45)

2

*<sup>x</sup>* and *<sup>β</sup>* <sup>¼</sup> <sup>2</sup>*<sup>E</sup>*

ℏ*ω:*

*<sup>d</sup>α*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>α</sup>*<sup>2</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (41)

Ψ ¼ 0 (42)

*=*2 (43)

<sup>ℏ</sup>*<sup>ω</sup>* ¼ 0 holds.

<sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (40)

Ψ<sup>00</sup> þ

directly as the previous problems. Let

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

The normalized eigen function is

This gives the energy eigen value as

<sup>Ψ</sup> <sup>¼</sup> *<sup>m</sup><sup>ω</sup>* ℏ*π*

The important results are given as follows:

• The ground state normalized eigen function is

Ψ0ð Þ¼ *x*

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

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

The solution given in Eq. (43) is valid if the condition 2*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � <sup>2</sup>*<sup>E</sup>*

form

given as

**21**

The energy eigen value expression does not have any integer like in the case of the potential well. Hence there is only one bound state which is available for a particular value of "m."

The eigen function can be evaluated as follows: The eigen function is always continuous. At *x* ¼ 0 gives us *A*<sup>1</sup> ¼ *A*<sup>2</sup> ¼ *A*. Hence the eigen function is

$$\Psi = A \exp\left(\beta |\mathbf{x}|\right)$$

To normalize Ψ,

$$\int\_{-\infty}^{\infty} |\Psi|^2 d\mathbf{x} = \mathbf{1} \Rightarrow 2 \int\_{0}^{\infty} |\Psi|^2 d\mathbf{x} = \mathbf{1}$$

This gives us *<sup>A</sup>* <sup>¼</sup> ffiffiffi *<sup>β</sup>* <sup>p</sup> <sup>¼</sup> ffiffiffiffi *<sup>m</sup><sup>λ</sup>* <sup>p</sup> ℏ .

#### **5. Linear harmonic oscillator**

Simple harmonic oscillator, damped harmonic oscillator, and force harmonic oscillator are the few famous problems in classical physics. But if one looks into the atomic world, the atoms are vibrating even at 0 K. Such atomic oscillations need the tool of quantum physics to understand its nature. In all the previous examples, the potential is constant in any particular region. But in this case, the potential is a function of the position coordinate "x."

#### **5.1 Schrodinger method**

The potential of the linear harmonic oscillator as a function of "x" is given as (**Figure 5**) [4–6]:

**Figure 5.** *Potential energy of the linear harmonic oscillator.*

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

Equating the value of *β* gives the energy eigen value as

∞ð

j j <sup>Ψ</sup> <sup>2</sup>

�∞

*<sup>β</sup>* <sup>p</sup> <sup>¼</sup> ffiffiffiffi *<sup>m</sup><sup>λ</sup>* <sup>p</sup> ℏ .

particular value of "m."

*Quantum Mechanics*

To normalize Ψ,

This gives us *<sup>A</sup>* <sup>¼</sup> ffiffiffi

**5.1 Schrodinger method**

(**Figure 5**) [4–6]:

**Figure 5.**

**20**

**5. Linear harmonic oscillator**

function of the position coordinate "x."

*Potential energy of the linear harmonic oscillator.*

*<sup>E</sup>* ¼ � *<sup>m</sup>λ*<sup>2</sup>

The energy eigen value expression does not have any integer like in the case of the potential well. Hence there is only one bound state which is available for a

The eigen function can be evaluated as follows: The eigen function is always

Ψ ¼ *A* exp ð Þ *β*j j *x*

Simple harmonic oscillator, damped harmonic oscillator, and force harmonic oscillator are the few famous problems in classical physics. But if one looks into the atomic world, the atoms are vibrating even at 0 K. Such atomic oscillations need the tool of quantum physics to understand its nature. In all the previous examples, the potential is constant in any particular region. But in this case, the potential is a

The potential of the linear harmonic oscillator as a function of "x" is given as

*<sup>V</sup>* <sup>¼</sup> *<sup>m</sup>ω*<sup>2</sup>*x*<sup>2</sup>

∞ð

j j <sup>Ψ</sup> <sup>2</sup>

*dx* ¼ 1

0

*dx* ¼ 1 ) 2

continuous. At *x* ¼ 0 gives us *A*<sup>1</sup> ¼ *A*<sup>2</sup> ¼ *A*. Hence the eigen function is

<sup>2</sup>ℏ<sup>2</sup> (38)

<sup>2</sup> (39)

The time-independent Schrödinger equation is given as

$$
\Psi'' + \frac{2m}{\hbar^2} \left( E - \frac{m\alpha^2 \mathbf{x}^2}{2} \right) \Psi = \mathbf{0} \tag{40}
$$

The potential is not constant since it is a function of "x"; Eq. (40) cannot solve directly as the previous problems. Let

$$a = \left(\frac{ma}{\hbar}\right)^{1/2} \ge \text{and } \beta = \frac{2E}{\hbar a \lambda}$$

Using the new constant *β* and the variable *α*, the Schrödinger equation has the form

$$\frac{d^2\Psi}{da^2} + (\beta - a^2)\Psi = 0\tag{41}$$

The asymptotic Schrödinger equation ð Þ *α* ! ∞ is given as

$$a\frac{d^2\Psi}{da^2} - a^2\Psi = 0\tag{42}$$

The general solution of the equation is exp �*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> . As *<sup>α</sup>* ! <sup>∞</sup>, exp <sup>þ</sup>*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> becomes infinite, hence it cannot be a solution. So the only possible solution is exp �*a*<sup>2</sup> ð Þ *<sup>=</sup>*<sup>2</sup> . Based on the asymptotic solution, the general solution of Eq. (42) is given as

$$\Psi = \mathcal{H}\_n(\mathfrak{a}) \exp\left(-a^2/2\right).$$

The normalized eigen function is

$$\Psi = \left[ \left( \frac{m\alpha}{\hbar \pi} \right)^{1/2} \left( \frac{1}{2^n \times n!} \right) \right]^{1/2} \mathbf{H}\_n(\alpha) \exp \left( -a^2/2 \right) \tag{43}$$

The solution given in Eq. (43) is valid if the condition 2*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � <sup>2</sup>*<sup>E</sup>* <sup>ℏ</sup>*<sup>ω</sup>* ¼ 0 holds. This gives the energy eigen value as

$$E = \left(n + \frac{1}{2}\right)\hbar\omega\tag{44}$$

The important results are given as follows:


$$
\Psi\_0(\mathbf{x}) = \left(\frac{ma}{\hbar\pi}\right)^{1/4} \exp\left(-\frac{ma\mathbf{x}^2}{2\hbar}\right) \tag{45}
$$

• The energy difference between any two successive levels is ℏ*ω*. Hence the energy difference between any two successive levels is constant. But this is not true in the case of real oscillators.

#### **5.2 Operator method**

The operator method is also one of the convenient methods to solve the exactly solvable problem as well as approximation methods in quantum mechanics [5]. The Hamiltonian of the linear harmonic oscillator is given as,

$$H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 \mathbf{x}^2 \tag{46}$$

The expectation value of *a*þ*a* is

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

Let us consider the ground state ∣0⟩*:*

Since *a* ∣0⟩ ¼ 0, ⟨0j*a*þ*a*j0⟩ ¼ 0. Thus,

the result as

found as

**23**

a<sup>þ</sup> h i� a ⟨njaþajn⟩ ¼ ⟨nj

⟨njHjn⟩ � ⟨nj

<sup>ℏ</sup>*ωEn*⟨*n*j*n*⟩ � <sup>1</sup>

⟨0jaþaj0⟩ <sup>¼</sup> E0

¼ 1 ℏω

¼ 1

*E*0 <sup>ℏ</sup>*<sup>ω</sup>* � <sup>1</sup>

Similarly, the energy of the first excited state is found as follows:

ffiffi 1

ffiffi 1 <sup>p</sup> *:* ffiffi 1 <sup>p</sup> ⟨1j1⟩ <sup>¼</sup> E1

<sup>1</sup> <sup>¼</sup> *<sup>E</sup>*<sup>1</sup> <sup>ℏ</sup>*<sup>ω</sup>* � <sup>1</sup>

⟨1jaþaj1⟩ <sup>¼</sup> E1

<sup>p</sup> ⟨1jaþj0⟩ <sup>¼</sup> E1

H <sup>ℏ</sup><sup>ω</sup> � <sup>1</sup> 2 jn⟩

> 1 2 jn⟩

(54)

<sup>2</sup> (55)

ℏ*ω* (56)

(58)

(59)

<sup>2</sup> have to be

� �ℏ*<sup>ω</sup>* (57)

, *<sup>p</sup>*<sup>2</sup> � �, and h i *<sup>p</sup>*

*a* þ *a*<sup>þ</sup> ð Þ (60)

� � (61)

<sup>2</sup> <sup>¼</sup> *En* <sup>ℏ</sup>*<sup>ω</sup>* � <sup>1</sup> 2

<sup>ℏ</sup><sup>ω</sup> � <sup>1</sup> 2

<sup>2</sup> <sup>¼</sup> <sup>0</sup> ) *<sup>E</sup>*<sup>0</sup> <sup>¼</sup> <sup>ℏ</sup>*<sup>ω</sup>*

<sup>ℏ</sup><sup>ω</sup> � <sup>1</sup> 2

> <sup>ℏ</sup><sup>ω</sup> � <sup>1</sup> 2

<sup>ℏ</sup><sup>ω</sup> � <sup>1</sup> 2

2

<sup>2</sup> ) *<sup>E</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>

1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *x*<sup>2</sup> h i � h i *x*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *p*<sup>2</sup> h i � h i *p*

2

2

2

*a* � *a*<sup>þ</sup> *i*

In the same way, *E*<sup>2</sup> ¼ 5ℏ*ω=*2, *E*<sup>3</sup> ¼ 7ℏ*ω=*2, and so on. Hence, one can generalize

*En* ¼ *n* þ

The uncertainties in position and momentum, respectively, are given as

q

q

evaluated. From Eqs. (47) and (48) the position and momentum operators are

*Δx* ¼

*Δp* ¼

*<sup>x</sup>* <sup>¼</sup> <sup>ℏ</sup> 2*mω* � �<sup>1</sup>*=*<sup>2</sup>

*<sup>p</sup>* <sup>¼</sup> *<sup>m</sup>ω*<sup>ℏ</sup> 2 � �<sup>1</sup>*=*<sup>2</sup>

In order to evaluate the uncertainties *<sup>x</sup>*<sup>2</sup> � �, h i *<sup>x</sup>*

Let us define the operator "a," lowering operator, in such a way that

$$\mathfrak{a} = (2m\mathfrak{o}\hbar)^{-1/2}(ma\mathfrak{x} + ip\ )\tag{47}$$

and the corresponding Hermitian adjoint, raising operator, is

$$a^{+} = \left(2ma\hbar\right)^{-1/2} (ma\infty - ip\ )\tag{48}$$

$$\begin{aligned} \mathbf{a}^+\mathbf{a} &= (2\mathbf{m}o\hbar)^{-1}(\mathbf{m}\mathbf{ox} - \mathbf{ip})(\mathbf{m}\mathbf{ox} + \mathbf{ip}) \\\\ &= (2\mathbf{m}o\hbar)^{-1}(\mathbf{m}^2o^2\mathbf{x}^2 + \mathbf{p}^2 + \mathbf{im}\mathbf{om}\mathbf{p} - \mathbf{imopx}) \\\\ &= (2\mathbf{m}o\hbar)^{-1}(m^2o^2\mathbf{x}^2 + p^2 + \mathbf{imo}[\mathbf{x}, p]) \end{aligned} \tag{49}$$

Here, ½ � *x*, *p* represents the commutation between the operators *x* and *p*. ½ �¼ *x*, *p i* and Eq. (49) becomes

$$\mathbf{a}^+\mathbf{a} = \left(2\text{mo}\hbar\right)^{-1}\left(\text{m}^2\text{o}^2\text{x}^2 + \text{p}^2 - \text{mo}\hbar\right)$$

$$= \frac{1}{\text{ao}\hbar}\left(\frac{1}{2}\text{mo}^2\text{x}^2 + \frac{\text{p}^2}{2\text{m}}\right) - \frac{1}{2}$$

$$= \frac{H}{\hbar\omega} - \frac{1}{2}\tag{50}$$

In the same way, one can find the *aa*<sup>þ</sup> and it is given as

$$
tau^{+} = \frac{H}{\hbar o} + \frac{1}{2} \tag{51}$$

Adding Eqs. (50) and (51) gives us the Hamiltonian in terms of the operators.

$$H = \frac{\hbar a}{2} \left( a a^+ + a^+ a \right) \tag{52}$$

Subtracting Eq. (50) from (51) gives, *aa*<sup>þ</sup> � *a*þ*a* ¼ 1. This can be simplified as

$$[a, a^{+}] = \mathbf{1} \tag{53}$$

The Hamiltonian H acts on any state ∣*n*⟩ that gives the eigen value *En* times the same state ∣*n*⟩, that is, *H* ∣*n*⟩ ¼ *En* ∣*n*⟩.

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

The expectation value of *a*þ*a* is

• The energy difference between any two successive levels is ℏ*ω*. Hence the energy difference between any two successive levels is constant. But this is not

The operator method is also one of the convenient methods to solve the exactly solvable problem as well as approximation methods in quantum mechanics [5]. The

ð Þ mωx � ip ð Þ mωx þ ip

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *imω*½ � *<sup>x</sup>*, *<sup>p</sup>*

Here, ½ � *x*, *p* represents the commutation between the operators *x* and *p*. ½ �¼ *x*, *p i*

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> 2m

m<sup>2</sup> ω2

<sup>¼</sup> *<sup>H</sup>* <sup>ℏ</sup>*<sup>ω</sup>* � <sup>1</sup>

*aa*<sup>þ</sup> <sup>¼</sup> *<sup>H</sup>*

*<sup>H</sup>* <sup>¼</sup> <sup>ℏ</sup>*<sup>ω</sup>*

ℏ*ω* þ 1

Adding Eqs. (50) and (51) gives us the Hamiltonian in terms of the operators.

Subtracting Eq. (50) from (51) gives, *aa*<sup>þ</sup> � *a*þ*a* ¼ 1. This can be simplified as

The Hamiltonian H acts on any state ∣*n*⟩ that gives the eigen value *En* times the

x2 <sup>þ</sup> p2 <sup>þ</sup> imωxp � imωpx

x2 <sup>þ</sup> p2 � <sup>m</sup>ω<sup>ℏ</sup>

� 1 2

*x*<sup>2</sup> (46)

ð Þ *mωx* þ *ip* (47)

ð Þ *mωx* � *ip* (48)

<sup>2</sup> (50)

<sup>2</sup> (51)

<sup>2</sup> *aa*<sup>þ</sup> <sup>þ</sup> *<sup>a</sup>*<sup>þ</sup> ð Þ *<sup>a</sup>* (52)

*a*, *a*<sup>þ</sup> ½ �¼ 1 (53)

(49)

true in the case of real oscillators.

Hamiltonian of the linear harmonic oscillator is given as,

<sup>a</sup>þ<sup>a</sup> <sup>¼</sup> ð Þ 2mω<sup>ℏ</sup> �<sup>1</sup>

<sup>¼</sup> ð Þ 2mω<sup>ℏ</sup> �<sup>1</sup>

<sup>¼</sup> ð Þ <sup>2</sup>*mω*<sup>ℏ</sup> �<sup>1</sup>

<sup>a</sup>þ<sup>a</sup> <sup>¼</sup> ð Þ 2mω<sup>ℏ</sup> �<sup>1</sup>

¼ 1 ωℏ

In the same way, one can find the *aa*<sup>þ</sup> and it is given as

same state ∣*n*⟩, that is, *H* ∣*n*⟩ ¼ *En* ∣*n*⟩.

**22**

*<sup>H</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> 2*m* þ 1 2 *mω*<sup>2</sup>

*<sup>a</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*mω*<sup>ℏ</sup> �1*=*<sup>2</sup>

*<sup>a</sup>*<sup>þ</sup> <sup>¼</sup> ð Þ <sup>2</sup>*mω*<sup>ℏ</sup> �1*=*<sup>2</sup>

m<sup>2</sup> ω2

*m*2 *ω*2

> 1 2 mω<sup>2</sup>

and the corresponding Hermitian adjoint, raising operator, is

Let us define the operator "a," lowering operator, in such a way that

**5.2 Operator method**

*Quantum Mechanics*

and Eq. (49) becomes

$$
\langle \mathbf{a}^+ \mathbf{a} \rangle \equiv \langle \mathbf{n} | \mathbf{a}^+ \mathbf{a} | \mathbf{n} \rangle = \langle \mathbf{n} | \frac{\mathbf{H}}{\hbar \alpha} - \frac{1}{2} | \mathbf{n} \rangle
$$

$$
= \frac{1}{\hbar \alpha} \langle \mathbf{n} | \mathbf{H} | \mathbf{n} \rangle - \langle \mathbf{n} | \frac{1}{2} | \mathbf{n} \rangle
\tag{54}
$$

$$
= \frac{1}{\hbar \alpha} E\_n \langle n | n \rangle - \frac{1}{2} = \frac{E\_n}{\hbar \alpha} - \frac{1}{2}
$$

Let us consider the ground state ∣0⟩*:*

$$\langle \mathbf{0} | \mathbf{a}^+ \mathbf{a} | \mathbf{0} \rangle = \frac{\mathbf{E}\_0}{\hbar \alpha} - \frac{\mathbf{1}}{2}$$

Since *a* ∣0⟩ ¼ 0, ⟨0j*a*þ*a*j0⟩ ¼ 0. Thus,

$$\frac{E\_0}{\hbar o} - \frac{1}{2} = 0 \Rightarrow E\_0 = \frac{\hbar o}{2} \tag{55}$$

Similarly, the energy of the first excited state is found as follows:

$$
\langle \mathbf{1} | \mathbf{a}^+ \mathbf{a} | \mathbf{1} \rangle = \frac{\mathbf{E}\_1}{\hbar \alpha} - \frac{\mathbf{1}}{2}
$$

$$
\sqrt{1} \langle \mathbf{1} | \mathbf{a}^+ | \mathbf{0} \rangle = \frac{\mathbf{E}\_1}{\hbar \alpha} - \frac{1}{2}
$$

$$
\sqrt{1} . \sqrt{1} \ \langle \mathbf{1} | \mathbf{1} \rangle = \frac{\mathbf{E}\_1}{\hbar \alpha} - \frac{1}{2}
$$

$$
\mathbf{1} = \frac{E\_1}{\hbar \alpha} - \frac{1}{2} \Rightarrow E\_1 = \frac{3}{2} \hbar \alpha \tag{56}
$$

In the same way, *E*<sup>2</sup> ¼ 5ℏ*ω=*2, *E*<sup>3</sup> ¼ 7ℏ*ω=*2, and so on. Hence, one can generalize the result as

$$E\_n = \left(n + \frac{1}{2}\right)\hbar\nu\tag{57}$$

The uncertainties in position and momentum, respectively, are given as

$$
\Delta \mathbf{x} = \sqrt{\left< \mathbf{x}^2 \right> - \left< \mathbf{x} \right>^2} \tag{58}
$$

$$
\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2} \tag{59}
$$

In order to evaluate the uncertainties *<sup>x</sup>*<sup>2</sup> � �, h i *<sup>x</sup>* 2 , *<sup>p</sup>*<sup>2</sup> � �, and h i *<sup>p</sup>* <sup>2</sup> have to be evaluated. From Eqs. (47) and (48) the position and momentum operators are found as

$$
\omega = \left(\frac{\hbar}{2ma}\right)^{1/2} (a + a^+) \tag{60}
$$

$$p = \left(\frac{mo\hbar}{2}\right)^{1/2} \left(\frac{a - a^{+}}{i}\right) \tag{61}$$

a. The expectation value of 'x' is given as,

$$
\langle \mathbf{x} \rangle \equiv \langle \mathbf{n} | \mathbf{x} | \mathbf{n} \rangle = \left(\frac{\hbar}{2\mathbf{m}\mathbf{o}}\right)^{1/2} \langle \mathbf{n} | (\mathbf{a} + \mathbf{a}^+) | \mathbf{n} \rangle
$$

$$
= \left(\frac{\hbar}{2\mathbf{m}\mathbf{o}}\right)^{1/2} \left(\langle \mathbf{n} | (\mathbf{a}) | \mathbf{n} \rangle + \langle \mathbf{n} | (\mathbf{a}^+) | \mathbf{n} \rangle\right)
$$

$$
= \left(\frac{\hbar}{2\mathbf{m}\mathbf{o}}\right)^{1/2} \left(\sqrt{\mathbf{n}} \left\langle \mathbf{n} | \mathbf{n} - \mathbf{1} \right\rangle + \sqrt{\mathbf{n} + \mathbf{1}} \left\langle \mathbf{n} | \mathbf{n} + \mathbf{1} \right\rangle\right)
$$

*<sup>Δ</sup><sup>p</sup>* <sup>¼</sup> *<sup>m</sup>ω*<sup>ℏ</sup>

• Hence the minimum uncertainty product is *<sup>Δ</sup>x:Δ<sup>p</sup>* <sup>¼</sup> <sup>ℏ</sup>

ground state eigen function. This can be done as follows:

*x* þ *i*

*d* Ψ0ð Þ *x*

ln Ψ0ð Þ¼� *x*

!

*<sup>∂</sup>* <sup>Ψ</sup>0ð Þ *<sup>x</sup>*

<sup>Ψ</sup>0ð Þ *<sup>x</sup>* ¼ � *<sup>m</sup>ω<sup>x</sup>*

*mω* ℏ

<sup>Ψ</sup>0ð Þ¼ *<sup>x</sup> A exp* � *<sup>m</sup>ωx*<sup>2</sup>

*mω* ℏ*π* � �<sup>1</sup>*=*<sup>4</sup>

ð Þ <sup>2</sup>*mω*<sup>ℏ</sup> �1*=*<sup>2</sup>

ℏ *mω*

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

Integrating the above equation gives,

The normalized eigen function is given as

*=* ffiffiffiffi *<sup>n</sup>*! � � <sup>p</sup> <sup>Ψ</sup>0ð Þ *<sup>x</sup>* .

<sup>Ψ</sup>*n*ð Þ¼ *<sup>x</sup> <sup>a</sup>*<sup>þ</sup> ð Þ*<sup>n</sup>*

**25**

Ψ0ð Þ¼ *x*

One can see that this result is identical to Eq. (45).

• The other eigen states can be evaluated using the equation,

**6. Conclusions**

and *<sup>Δ</sup><sup>p</sup>* <sup>¼</sup> *<sup>m</sup><sup>ω</sup>*

*Δx:Δp* ≥ <sup>ℏ</sup>

relation.

2 � �1*=*<sup>2</sup> .

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

> 2 � �

• The minimum uncertainty state is the ground state. In this state, *<sup>Δ</sup><sup>x</sup>* <sup>¼</sup> <sup>ℏ</sup>

have higher uncertainty than the ground state, the general uncertainty is

• Since Ψ0ð Þ *x* corresponds to the low energy state, a Ψ0ð Þ¼ *x* 0. This gives us the

a Ψ0ð Þ¼ *x* 0

ð Þ �*i*ℏ∂*=∂<sup>x</sup>* ð Þ <sup>2</sup>*mω*<sup>ℏ</sup> <sup>1</sup>*=*<sup>2</sup>

*<sup>∂</sup><sup>x</sup>* ¼ �*<sup>x</sup>* <sup>Ψ</sup>0ð Þ *<sup>x</sup>*

<sup>ℏ</sup> *dx*

*x*2 2 � �

þ ln *A*

2ℏ � �

*exp* � *<sup>m</sup>ωx*<sup>2</sup> 2ℏ � �

<sup>2</sup>. This is the mathematical representation of Heisenberg's uncertainty

ð Þ *mωx* þ *ip* Ψ0ð Þ¼ *x* 0

Ψ0ð Þ¼ *x* 0

*<sup>Δ</sup>x:Δ<sup>p</sup>* <sup>¼</sup> <sup>ℏ</sup>

ð Þ 2*n* þ 1 � �1*=*<sup>2</sup>

<sup>2</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> (64)

(63)

2*mω* � �1*=*<sup>2</sup>

<sup>2</sup>. Since the other states

Since the states *n* � 1, *n*, *n* þ 1 are orthogonal to each other, ⟨*n*j*n* � 1⟩ ¼ 0 and ⟨*n*j*n* þ 1⟩ ¼ 0. So h i *x* ¼ 0. The expectation value of the position in any state is zero.

b. The expectation value of momentum is

$$\langle p \rangle \equiv \langle n|p|n \rangle = \left(\frac{m\alpha\hbar}{2}\right)^{1/2} \left(\frac{1}{i}\right) \langle n|a - a^+|n\rangle \Longrightarrow \langle p \rangle = \mathbf{0}.$$

Not only position, the expectation value of momentum in any state is also zero. c.

$$\begin{aligned} \langle \mathbf{x}^2 \rangle &\equiv \langle n|\mathbf{x}^2|n\rangle = \frac{\hbar}{2m\alpha} \langle n|(a+a^+)(a+a^+)|n\rangle \\ &= \frac{\hbar}{2\text{mo}} \langle n|\left(\mathbf{a}^2 + \mathbf{a}^{+2} + \mathbf{a}\mathbf{a}^+ + \mathbf{a}^+\mathbf{a}\right)|n\rangle \\ &= \frac{\hbar}{2\text{mo}} \left( \langle n|\mathbf{a}^2|\mathbf{n}\rangle + \langle n|\mathbf{a}^{+2}|\mathbf{n}\rangle + \langle n|\mathbf{a}\mathbf{a}^+|\mathbf{n}\rangle + \langle n|\mathbf{a}^+\mathbf{a}|\mathbf{n}\rangle \right) \\ &= \frac{\hbar}{2\text{mo}} \left( \sqrt{\mathbf{n}}\sqrt{\mathbf{n}-1} \left( \mathbf{n}|\mathbf{n}-2\rangle + \sqrt{\mathbf{n}+1}\sqrt{\mathbf{n}+2} \langle \mathbf{n}|\mathbf{n}+2\rangle + (\mathbf{n}+1)\langle \mathbf{n}|\mathbf{n}\rangle + \mathbf{n}\langle \mathbf{n}|\mathbf{n}\rangle \right) \right) \\ &= \frac{\hbar}{2\text{mo}} (2\mathbf{n}+1) \end{aligned}$$

d.

$$\begin{aligned} \langle p^2 \rangle &= \langle n|p^2|n\rangle = -\left(\frac{\text{mo}\hbar}{2}\right) \langle n|(a-a^+)(a-a^+)|n\rangle \\ &= -\left(\frac{\text{mo}\hbar}{2}\right) \left(\langle \mathbf{n}|\mathbf{a}^2|\mathbf{n}\rangle + \langle \mathbf{n}|\mathbf{a}^+|\mathbf{n}\rangle - \langle \mathbf{n}|\mathbf{a}\mathbf{a}^+|\mathbf{n}\rangle - \langle \mathbf{n}|\mathbf{a}^+\mathbf{a}|\mathbf{n}\rangle\right) \\ &= -\left(\frac{\text{mo}\hbar}{2}\right) \left(\sqrt{\mathbf{n}}\sqrt{\mathbf{n}-1}\left\langle \mathbf{n}|\mathbf{n}-2\right\rangle + \sqrt{\mathbf{n}+1}\sqrt{\mathbf{n}+2}\langle \mathbf{n}|\mathbf{n}+2\rangle - (\mathbf{n}+1)\langle \mathbf{n}|\mathbf{n}\rangle - \mathbf{n}\left\langle \mathbf{n}|\mathbf{n}\right\rangle\right) \\ &= \left(\frac{\text{mo}\hbar}{2}\right) (2\mathbf{n}+\mathbf{1}) \end{aligned}$$

From Eq. (58) and (59), the uncertainty in position and momentum, respectively are given as,

$$
\Delta \mathbf{x} = \left(\frac{\hbar}{2m\alpha} (2n+1)\right)^{1/2} \tag{62}
$$

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

$$
\Delta p = \left( \left( \frac{mo\hbar}{2} \right) (2n+1) \right)^{1/2} \tag{63}
$$

$$
\Delta \mathfrak{x}. \Delta p = \frac{\hbar}{2} (2n + 1) \tag{64}
$$

#### **6. Conclusions**

a. The expectation value of 'x' is given as,

<sup>¼</sup> <sup>ℏ</sup> 2mω � �1*=*<sup>2</sup>

<sup>¼</sup> <sup>ℏ</sup> 2mω � �1*=*<sup>2</sup> ffiffiffi

b. The expectation value of momentum is

h i *<sup>p</sup>* � ⟨*n*j*p*j*n*⟩ <sup>¼</sup> *<sup>m</sup>ω*<sup>ℏ</sup>

2 � �<sup>1</sup>*=*<sup>2</sup> 1

<sup>2</sup>*mω*⟨*n*<sup>j</sup> *<sup>a</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>þ</sup> ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>þ</sup> ð Þj*n*⟩

<sup>n</sup> � <sup>1</sup> <sup>p</sup> ⟨nj<sup>n</sup> � <sup>2</sup>⟩ <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffi

<sup>j</sup>n⟩ <sup>þ</sup> ⟨njaþ<sup>2</sup>

zero.

*Quantum Mechanics*

c.

*<sup>x</sup>*<sup>2</sup> � � � ⟨*n*j*x*<sup>2</sup>

<sup>¼</sup> <sup>ℏ</sup>

<sup>¼</sup> <sup>ℏ</sup>

<sup>¼</sup> <sup>ℏ</sup> 2mω

<sup>¼</sup> <sup>ℏ</sup>

d.

*<sup>p</sup>*<sup>2</sup> � � � ⟨*n*j*p*<sup>2</sup>

¼ � <sup>m</sup>ω<sup>ℏ</sup> 2 � �

¼ � <sup>m</sup>ω<sup>ℏ</sup> 2 � � ffiffiffi

<sup>¼</sup> <sup>m</sup>ω<sup>ℏ</sup> 2 � �

tively are given as,

**24**

2m<sup>ω</sup> ⟨nja<sup>2</sup>

2m<sup>ω</sup> ð Þ 2n <sup>þ</sup> <sup>1</sup>

ffiffiffi <sup>n</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffi

<sup>j</sup>*n*⟩ ¼ � *<sup>m</sup>ω*<sup>ℏ</sup> 2 � �

⟨nja<sup>2</sup>

ð Þ 2n þ 1

<sup>n</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffi

<sup>j</sup>*n*⟩ <sup>¼</sup> <sup>ℏ</sup>

2m<sup>ω</sup> ⟨n<sup>j</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>a</sup>þ<sup>2</sup> <sup>þ</sup> aa<sup>þ</sup> <sup>þ</sup> <sup>a</sup>þ<sup>a</sup>

<sup>j</sup>n⟩ <sup>þ</sup> ⟨njaþ<sup>2</sup>

� �

h i <sup>x</sup> � ⟨njxjn⟩ <sup>¼</sup> <sup>ℏ</sup>

2mω � �1*=*<sup>2</sup>

⟨nj a þ a<sup>þ</sup> ð Þjn⟩

<sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> ⟨nj<sup>n</sup> <sup>þ</sup> <sup>1</sup>⟩ � �

⟨*n*j*a* � *a*þj*n*⟩¼)h i *p* ¼ 0*:*

<sup>n</sup> <sup>þ</sup> <sup>2</sup> <sup>p</sup> ⟨nj<sup>n</sup> <sup>þ</sup> <sup>2</sup>⟩ <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> ⟨njn⟩ <sup>þ</sup> <sup>n</sup> ⟨njn⟩

<sup>n</sup> <sup>þ</sup> <sup>2</sup> <sup>p</sup> ⟨nj<sup>n</sup> <sup>þ</sup> <sup>2</sup>⟩ � ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> ⟨njn⟩ � <sup>n</sup> ⟨njn⟩

(62)

⟨njð Þj a n⟩ þ ⟨nj a<sup>þ</sup> ð Þ ð Þjn⟩

<sup>n</sup> <sup>p</sup> ⟨nj<sup>n</sup> � <sup>1</sup>⟩ <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffi

Since the states *n* � 1, *n*, *n* þ 1 are orthogonal to each other, ⟨*n*j*n* � 1⟩ ¼ 0 and ⟨*n*j*n* þ 1⟩ ¼ 0. So h i *x* ¼ 0. The expectation value of the position in any state is

> *i* � �

Not only position, the expectation value of momentum in any state is also zero.

jn⟩

� �

⟨*n*j *a* � *a*<sup>þ</sup> ð Þ *a* � *a*<sup>þ</sup> ð Þj*n*⟩

<sup>n</sup> � <sup>1</sup> <sup>p</sup> ⟨nj<sup>n</sup> � <sup>2</sup>⟩ <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffi

*<sup>Δ</sup><sup>x</sup>* <sup>¼</sup> <sup>ℏ</sup>

� �

jn⟩ þ ⟨njaaþjn⟩ þ ⟨njaþajn⟩

<sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffi

jn⟩ � ⟨njaaþjn⟩ � ⟨njaþajn⟩

<sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffi

From Eq. (58) and (59), the uncertainty in position and momentum, respec-

<sup>2</sup>*mω*ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> � �<sup>1</sup>*=*<sup>2</sup>

� �

� �


$$\begin{aligned} \mathbf{a} \, \Psi\_0(\mathbf{x}) &= \mathbf{0} \end{aligned}$$

$$(2m\boldsymbol{\alpha}\hbar)^{-1/2}(m\boldsymbol{\alpha}\mathbf{x} + \boldsymbol{i}\boldsymbol{p}\, )\, \Psi\_0(\mathbf{x}) = \mathbf{0}$$

$$\left(\left(\frac{m\boldsymbol{\alpha}}{2\hbar}\right)^{1/2}\mathbf{x} + \mathbf{i}\frac{(-i\hbar\partial\!\!/\partial\mathbf{x})}{(2m\boldsymbol{\alpha}\hbar)^{1/2}}\right)\Psi\_0(\mathbf{x}) = \mathbf{0}$$

$$\frac{\hbar}{m\boldsymbol{\alpha}}\frac{\partial\,\Psi\_0(\mathbf{x})}{\partial\mathbf{x}} = -\mathbf{x}\,\Psi\_0(\mathbf{x})$$

$$\frac{d\,\Psi\_0(\mathbf{x})}{\Psi\_0(\mathbf{x})} = -\frac{m\boldsymbol{\alpha}\mathbf{x}}{\hbar}d\mathbf{x}$$

Integrating the above equation gives,

$$
\ln \Psi\_0(\mathbf{x}) = -\frac{m\alpha}{\hbar} \left(\frac{\mathbf{x}^2}{2}\right) + \ln A
$$

$$
\Psi\_0(\mathbf{x}) = A \exp\left(-\frac{m\alpha \mathbf{x}^2}{2\hbar}\right)
$$

The normalized eigen function is given as

$$
\Psi\_0(\mathbf{x}) = \left(\frac{ma}{\hbar\pi}\right)^{1/4} \exp\left(-\frac{mo\mathbf{x}^2}{2\hbar}\right),
$$

One can see that this result is identical to Eq. (45).

• The other eigen states can be evaluated using the equation, <sup>Ψ</sup>*n*ð Þ¼ *<sup>x</sup> <sup>a</sup>*<sup>þ</sup> ð Þ*<sup>n</sup> =* ffiffiffiffi *<sup>n</sup>*! � � <sup>p</sup> <sup>Ψ</sup>0ð Þ *<sup>x</sup>* .

#### **7. Particle in a 3D box**

The confinement of a particle in a three-dimensional potential is discussed in this section [4, 6]. The potential is defined as (**Figure 6**)

$$\mathbf{V} = \begin{cases} \mathbf{0}, & \mathbf{0} \le \mathbf{x} < \mathbf{a}; \; \mathbf{0} \le \mathbf{y} < \mathbf{b}; \; \mathbf{0} \le \mathbf{z} < \mathbf{c} \\\; \mathbf{a}, & \text{Otherwise} \end{cases}$$

The three dimensional time-independent Schrödinger equation is given as

$$
\nabla^2 \Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) - \frac{2m}{\hbar^2} V \Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = -E \Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) \tag{65}
$$

Now the equation can be separated as follows:

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

> *d*2 Ψ*x*ð Þ *x dx*<sup>2</sup> <sup>þ</sup>

*d*2 Ψ*y*ð Þ*y dy*<sup>2</sup> <sup>þ</sup>

*d*2 Ψ*z*ð Þ*z dz*<sup>2</sup> <sup>þ</sup>

The normalized eigen function Ψ*x*ð Þ *x* is given as

In the same way, Ψ*y*ð Þ*y* and Ψ*z*ð Þ*z* are given as

Hence, the eigen function Ψ x, y, z � � is given as

<sup>Ψ</sup> x, y, z � � <sup>¼</sup> <sup>Ψ</sup>*x*ð Þ *<sup>x</sup>* <sup>Ψ</sup>*y*ð Þ*<sup>y</sup>* <sup>Ψ</sup>*z*ð Þ¼ *<sup>z</sup>*

The total energy E is

**27**

The energy given values are given as

Ψ*x*ð Þ¼ *x*

Ψ*y*ð Þ¼ *y*

Ψ*z*ð Þ¼ *z*

2*m*

2*m*

2*m*

2 *a* � �<sup>1</sup>*=*<sup>2</sup>

2 *b* � �<sup>1</sup>*=*<sup>2</sup>

2 *c* � �<sup>1</sup>*=*<sup>2</sup>

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

Exð Þ¼ <sup>x</sup> <sup>n</sup><sup>2</sup>

Ey y � � <sup>¼</sup> n2

*<sup>E</sup>* <sup>¼</sup> *Ex*ð Þþ *<sup>x</sup> Ey*ð Þþ *<sup>y</sup> Ez*ð Þ¼ *<sup>z</sup> <sup>π</sup>*<sup>2</sup>ℏ<sup>2</sup>

*<sup>E</sup>* <sup>¼</sup> *<sup>π</sup>*<sup>2</sup>ℏ<sup>2</sup> <sup>2</sup>*ma*<sup>2</sup> *<sup>n</sup>*<sup>2</sup>

Some of the results are summarized here:

Ezð Þ¼ z

<sup>ℏ</sup><sup>2</sup> *Ex*ð Þ *<sup>x</sup>* <sup>Ψ</sup>*x*ð Þ¼ *<sup>x</sup>* <sup>0</sup>

<sup>ℏ</sup><sup>2</sup> *Ey*ð Þ*<sup>y</sup>* <sup>Ψ</sup>*y*ð Þ¼ *<sup>y</sup>* <sup>0</sup>

<sup>ℏ</sup><sup>2</sup> *Ez*ð Þ*<sup>z</sup>* <sup>Ψ</sup>*z*ð Þ¼ *<sup>z</sup>* <sup>0</sup>

sin *nxπ<sup>x</sup> a* � �

> *nyπy b* � �

sin *nzπ<sup>z</sup> c* � �

sin *nxπ<sup>x</sup> a* � � sin

xπ<sup>2</sup>ℏ<sup>2</sup> 2ma2

yπ<sup>2</sup>ℏ<sup>2</sup> 2mb2

2*m*

*n*2 *x a*<sup>2</sup> þ *n*2 *y <sup>b</sup>*<sup>2</sup> <sup>þ</sup>

!

*n*2 *z c*2

n2 zπ<sup>2</sup>ℏ<sup>2</sup> 2mc2

• In a cubical potential box, *a* ¼ *b* ¼ *c*, then the energy eigen value becomes,

*<sup>x</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup>

� �*:*

*<sup>y</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> *z* *nyπy b*

� � sin *nzπ<sup>z</sup>*

*c* � � (68)

(69)

sin

Let the eigen function <sup>Ψ</sup> x, y, z is taken as the product of <sup>Ψ</sup>*x*ð Þ *<sup>x</sup>* , <sup>Ψ</sup>*y*ð Þ*<sup>y</sup>* and <sup>Ψ</sup>*z*ð Þ*<sup>z</sup>* according to the technique of separation of variables. i.e., <sup>Ψ</sup> x, y, z <sup>¼</sup> Ψ*x*ð Þ *x* Ψ*y*ð Þ*y* Ψ*z*ð Þ*z* .

$$\begin{split} & \Psi\_{\mathbf{y}}(\mathbf{y}) \Psi\_{x}(\mathbf{z}) \frac{d^{2} \Psi\_{x}(\mathbf{x})}{d\mathbf{x}^{2}} + \Psi\_{x}(\mathbf{x}) \Psi\_{x}(\mathbf{z}) \frac{d^{2} \Psi\_{y}(\mathbf{y})}{d\mathbf{y}^{2}} + \Psi\_{x}(\mathbf{x}) \Psi\_{y}(\mathbf{y}) \frac{d^{2} \Psi\_{x}(\mathbf{z})}{d\mathbf{z}^{2}} - \frac{2m}{\hbar^{2}} V \Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) \\ & = -\frac{2m}{\hbar^{2}} \mathbf{E} \Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) \end{split}$$

Divide the above equation by Ψ x, y, z gives us

$$\frac{1}{\Psi\_x(\mathbf{x})} \frac{d^2 \Psi\_x(\mathbf{x})}{d\mathbf{x}^2} + \frac{1}{\Psi\_y(\mathbf{y})} \frac{d^2 \Psi\_y(\mathbf{y})}{d\mathbf{y}^2} + \frac{1}{\Psi\_z(\mathbf{z})} \frac{d^2 \Psi\_z(\mathbf{z})}{d\mathbf{z}^2} = -\frac{2m}{\hbar^2} E \tag{66}$$

Now we can boldly write E as *Ex*ð Þþ *x Ey*ð Þþ *y Ez*ð Þ*z*

$$\frac{1}{\Psi\_x(\mathbf{x})}\frac{d^2\Psi\_x(\mathbf{x})}{d\mathbf{x}^2} + \frac{1}{\Psi\_y(\mathbf{y})}\frac{d^2\Psi\_y(\mathbf{y})}{d\mathbf{y}^2} + \frac{1}{\Psi\_z(\mathbf{z})}\frac{d^2\Psi\_z(\mathbf{z})}{d\mathbf{z}^2} = -\frac{2m}{\hbar^2} \left( E\_x(\mathbf{x}) + E\_y(\mathbf{y}) + E\_z(\mathbf{z}) \right) \tag{67}$$

**Figure 6.** *Three-dimensional potential box.*

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

**7. Particle in a 3D box**

*Quantum Mechanics*

The confinement of a particle in a three-dimensional potential is discussed in

<sup>V</sup> <sup>¼</sup> 0, 0<sup>≤</sup> <sup>x</sup><sup>&</sup>lt; a; 0≤y<b; 0≤z<sup>&</sup>lt; <sup>c</sup>

The three dimensional time-independent Schrödinger equation is given as

<sup>ℏ</sup><sup>2</sup> *<sup>V</sup>*<sup>Ψ</sup> x, y, z

¼ �E<sup>Ψ</sup> x, y, z

*dy*<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup>*x*ð Þ *<sup>x</sup>* <sup>Ψ</sup>*y*ð Þ*<sup>y</sup>*

1 Ψ*z*ð Þ*z* *d*2 Ψ*z*ð Þ*z dz*<sup>2</sup> ¼ � <sup>2</sup>*<sup>m</sup>*

gives us

is taken as the product of <sup>Ψ</sup>*x*ð Þ *<sup>x</sup>* , <sup>Ψ</sup>*y*ð Þ*<sup>y</sup>* and <sup>Ψ</sup>*z*ð Þ*<sup>z</sup>*

*d*2 Ψ*z*ð Þ*z dz*<sup>2</sup> � <sup>2</sup>*<sup>m</sup>*

(65)

<sup>ℏ</sup><sup>2</sup> *<sup>V</sup>*<sup>Ψ</sup> x, y, z 

<sup>ℏ</sup><sup>2</sup> *<sup>E</sup>* (66)

(67)

<sup>ℏ</sup><sup>2</sup> *Ex*ð Þþ *<sup>x</sup> Ey*ð Þþ *<sup>y</sup> Ez*ð Þ*<sup>z</sup>*

<sup>¼</sup>

∞, Otherwise

according to the technique of separation of variables. i.e., Ψ x, y, z

1 Ψ*y*ð Þ*y*

Now we can boldly write E as *Ex*ð Þþ *x Ey*ð Þþ *y Ez*ð Þ*z*

*d*2 Ψ*y*ð Þ*y dy*<sup>2</sup> <sup>þ</sup> *d*2 Ψ*y*ð Þ*y dy*<sup>2</sup> <sup>þ</sup>

1 Ψ*z*ð Þ*z* *d*2 Ψ*z*ð Þ*z dz*<sup>2</sup> ¼ � <sup>2</sup>*<sup>m</sup>*

*d*2 Ψ*y*ð Þ*y*

this section [4, 6]. The potential is defined as (**Figure 6**)

Ψ x, y, z � <sup>2</sup>*<sup>m</sup>*

*dx*<sup>2</sup> <sup>þ</sup> <sup>Ψ</sup>*x*ð Þ *<sup>x</sup>* <sup>Ψ</sup>*z*ð Þ*<sup>z</sup>*

Divide the above equation by Ψ x, y, z

*d*2 Ψ*x*ð Þ *x dx*<sup>2</sup> <sup>þ</sup>

1 Ψ*y*ð Þ*y*

∇2

Let the eigen function Ψ x, y, z

Ψ*x*ð Þ *x* Ψ*y*ð Þ*y* Ψ*z*ð Þ*z* .

*d*2 Ψ*x*ð Þ *x*

<sup>ℏ</sup><sup>2</sup> <sup>E</sup><sup>Ψ</sup> x, y, z 

> 1 Ψ*x*ð Þ *x*

Ψ*y*ð Þ*y* Ψ*z*ð Þ*z*

¼ � <sup>2</sup>*<sup>m</sup>*

1 Ψ*x*ð Þ *x*

**Figure 6.**

**26**

*Three-dimensional potential box.*

*d*2 Ψ*x*ð Þ *x dx*<sup>2</sup> <sup>þ</sup> Now the equation can be separated as follows:

$$\begin{aligned} \frac{d^2 \Psi\_x(\mathbf{x})}{d\mathbf{x}^2} + \frac{2m}{\hbar^2} E\_x(\mathbf{x}) \Psi\_x(\mathbf{x}) &= \mathbf{0} \\\\ \frac{d^2 \Psi\_y(\mathbf{y})}{d\mathbf{y}^2} + \frac{2m}{\hbar^2} E\_y(\mathbf{y}) \Psi\_y(\mathbf{y}) &= \mathbf{0} \\\\ \frac{d^2 \Psi\_z(\mathbf{z})}{d\mathbf{z}^2} + \frac{2m}{\hbar^2} E\_z(\mathbf{z}) \Psi\_z(\mathbf{z}) &= \mathbf{0} \end{aligned}$$

The normalized eigen function Ψ*x*ð Þ *x* is given as

$$
\Psi\_{\mathbf{x}}(\mathbf{x}) = \left(\frac{2}{a}\right)^{1/2} \sin\left(\frac{n\_{\mathbf{x}}\pi\mathbf{x}}{a}\right),
$$

In the same way, Ψ*y*ð Þ*y* and Ψ*z*ð Þ*z* are given as

$$
\Psi\_{\mathcal{V}}(\mathcal{Y}) = \left(\frac{2}{b}\right)^{1/2} \sin\left(\frac{n\_{\mathcal{V}}\pi\mathcal{Y}}{b}\right)
$$

$$
\Psi\_{\mathcal{z}}(\mathcal{z}) = \left(\frac{2}{c}\right)^{1/2} \sin\left(\frac{n\_{\mathcal{z}}\pi\mathcal{z}}{c}\right)
$$

Hence, the eigen function Ψ x, y, z � � is given as

$$\Psi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \Psi\_{\mathbf{x}}(\mathbf{x})\Psi\_{\mathbf{y}}(\mathbf{y})\Psi\_{\mathbf{z}}(\mathbf{z}) = \left(\frac{8}{abc}\right)^{1/2} \sin\left(\frac{n\_x \pi \chi}{a}\right) \sin\left(\frac{n\_y \pi \chi}{b}\right) \sin\left(\frac{n\_z \pi \mathbf{z}}{c}\right) \tag{68}$$

The energy given values are given as

$$\mathbf{E}\_{\mathbf{x}}(\mathbf{x}) = \frac{\mathbf{n}\_{\mathbf{x}}^2 \pi^2 \hbar^2}{2\mathbf{m}\mathbf{a}^2}$$

$$\mathbf{E}\_{\mathbf{y}}(\mathbf{y}) = \frac{\mathbf{n}\_{\mathbf{y}}^2 \pi^2 \hbar^2}{2\mathbf{m}\mathbf{b}^2}$$

$$\mathbf{E}\_{\mathbf{z}}(\mathbf{z}) = \frac{\mathbf{n}\_{\mathbf{z}}^2 \pi^2 \hbar^2}{2\mathbf{m}\mathbf{c}^2}$$

The total energy E is

$$E = E\_x(\varkappa) + E\_y(\wp) + E\_z(z) = \frac{\pi^2 \hbar^2}{2m} \left( \frac{n\_x^2}{a^2} + \frac{n\_y^2}{b^2} + \frac{n\_z^2}{c^2} \right) \tag{69}$$

Some of the results are summarized here:

• In a cubical potential box, *a* ¼ *b* ¼ *c*, then the energy eigen value becomes,

$$E = \frac{\pi^2 \hbar^2}{2ma^2} \left( n\_x^2 + n\_y^2 + n\_x^2 \right).$$

• The minimum energy that corresponds to the ground state is *<sup>E</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*π*2ℏ<sup>2</sup> <sup>2</sup>*ma*<sup>2</sup> . Here *nx* ¼ *ny* ¼ *nz* ¼ 1.

**References**

Pearson

Editions

**29**

[1] Griffiths DJ. Introduction to Quantum Mechanics. 2nd ed. India:

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

[2] Singh K, Singh SP. Elements of Quantum Mechanics. 1st ed. India: S.

[3] Gasiorowicz S. Quantum Mechanics.

[4] Schiff LI. Quantum Mechanics. 4th ed. India: McGraw Hill International

[5] Peleg Y, Pnini R, Zaarur E, Hecht E. Quantum Mechanics. 2nd ed. India:

[6] Aruldhas G. Quantum Mechanics.

Chand & Company Ltd

3rd ed. India: Wiley

McGraw Hill Editions

2nd ed. India: Prentice-Hall


### **Author details**

Lourdhu Bruno Chandrasekar<sup>1</sup> \*, Kanagasabapathi Gnanasekar<sup>2</sup> and Marimuthu Karunakaran<sup>3</sup>

1 Department of Physics, Periyar Maniammai Institute of Science and Technology, Vallam, India

2 Department of Physics, The American College, Madurai, India

3 Department of Physics, Alagappa Government Arts College, Karaikudi, India

\*Address all correspondence to: brunochandrasekar@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.

*Exactly Solvable Problems in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.93317*

### **References**

• The minimum energy that corresponds to the ground state is *<sup>E</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*π*2ℏ<sup>2</sup>

• The states (111), (222), (333), (444), … . has no degeneracy.

• Different states with different quantum numbers may have the same energy. This phenomenon is known as degeneracy. For example, the states (i) *nx* ¼ 2; *ny* ¼ *nz* ¼ 1, (ii) *ny* ¼ 2; *nx* ¼ *nz* ¼ 1; and (iii) *nz* ¼ 2; *nx* ¼ *ny* ¼ 1 have the

*ma*<sup>2</sup> . So we can say that the energy <sup>6</sup>*π*2ℏ<sup>2</sup>

• In this problem, the state may have zero-fold degeneracy, 3-fold degeneracy or

\*, Kanagasabapathi Gnanasekar<sup>2</sup> and

1 Department of Physics, Periyar Maniammai Institute of Science and Technology,

3 Department of Physics, Alagappa Government Arts College, Karaikudi, India

© 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,

2 Department of Physics, The American College, Madurai, India

\*Address all correspondence to: brunochandrasekar@gmail.com

*nx* ¼ *ny* ¼ *nz* ¼ 1.

*Quantum Mechanics*

degenerate.

**Author details**

Vallam, India

**28**

Lourdhu Bruno Chandrasekar<sup>1</sup>

provided the original work is properly cited.

Marimuthu Karunakaran<sup>3</sup>

same energy of *<sup>E</sup>* <sup>¼</sup> <sup>6</sup>*π*2ℏ<sup>2</sup>

6-fold degeneracy.

<sup>2</sup>*ma*<sup>2</sup> . Here

*ma*<sup>2</sup> has a 3-fold

[1] Griffiths DJ. Introduction to Quantum Mechanics. 2nd ed. India: Pearson

[2] Singh K, Singh SP. Elements of Quantum Mechanics. 1st ed. India: S. Chand & Company Ltd

[3] Gasiorowicz S. Quantum Mechanics. 3rd ed. India: Wiley

[4] Schiff LI. Quantum Mechanics. 4th ed. India: McGraw Hill International Editions

[5] Peleg Y, Pnini R, Zaarur E, Hecht E. Quantum Mechanics. 2nd ed. India: McGraw Hill Editions

[6] Aruldhas G. Quantum Mechanics. 2nd ed. India: Prentice-Hall

**Chapter 3**

**Abstract**

Problem

*María Esther Burgos*

interpretations of quantum theory

**1. Introduction and outlook**

projection postulate ([3], pp. 5–6).

Niels Bohr.

sense [6–8].

**31**

Transitions between Stationary

Accounting for projections during measurements is the traditional measurement problem. Transitions between stationary states require measurements, posing a different measurement problem. Both are compared. Several interpretations of quantum mechanics attempting to solve the traditional measurement problem are summarized. A highly desirable aim is to account for both problems. Not every

**Keywords:** quantum measurement problem, transitions between stationary states,

In 1930 Paul Dirac published *The Principles of Quantum Mechanics* [1]. Two years later John von Neumann published *Mathematische Grundlagen der Quantenmechanik* [2]. These initial versions of quantum theory share two characteristics, (i) the state vector ∣*ψ*i (wave function *ψ*) describes the state of an individual system, and (ii) they involve two laws of change of the state of the system: spontaneous processes, governed by the Schrödinger equation, and measurement processes, ruled by the

Many other versions of quantum theory followed. Those where *ψ* describes the state of an individual system and where the projection postulate is included among its axioms are generally called standard, ordinary, or orthodox quantum mechanics (OQM), sometimes referred to as the Copenhagen interpretation, associated to

The most relevant differences between spontaneous processes (SP) and measurement processes (MP) are as follows [4]: in SP the observer plays no role, in MP the observer (or the measuring device) plays a paramount role; in SP the state vector ∣*ψ*ð Þi *t* is continuous, in MP ∣*ψ*ð Þi *t* collapses (jumps, is projected, is reduced); in SP the superposition principle applies, in MP the superposition principle breaks down; SP are ruled by a deterministic law, MP are ruled by probability laws; in SP every action is localized, in MP there is a kind of action-at-a-distance [5]; and in SP conservation laws are strictly valid, in MP conservation laws have only a statistical

Since the projection postulate contradicts the fundamental Schrödinger equation of motion, some authors rushed to the conclusion that it was defective.

States and the Measurement

interpretation of quantum mechanics achieves this goal.

#### **Chapter 3**

## Transitions between Stationary States and the Measurement Problem

*María Esther Burgos*

#### **Abstract**

Accounting for projections during measurements is the traditional measurement problem. Transitions between stationary states require measurements, posing a different measurement problem. Both are compared. Several interpretations of quantum mechanics attempting to solve the traditional measurement problem are summarized. A highly desirable aim is to account for both problems. Not every interpretation of quantum mechanics achieves this goal.

**Keywords:** quantum measurement problem, transitions between stationary states, interpretations of quantum theory

#### **1. Introduction and outlook**

In 1930 Paul Dirac published *The Principles of Quantum Mechanics* [1]. Two years later John von Neumann published *Mathematische Grundlagen der Quantenmechanik* [2]. These initial versions of quantum theory share two characteristics, (i) the state vector ∣*ψ*i (wave function *ψ*) describes the state of an individual system, and (ii) they involve two laws of change of the state of the system: spontaneous processes, governed by the Schrödinger equation, and measurement processes, ruled by the projection postulate ([3], pp. 5–6).

Many other versions of quantum theory followed. Those where *ψ* describes the state of an individual system and where the projection postulate is included among its axioms are generally called standard, ordinary, or orthodox quantum mechanics (OQM), sometimes referred to as the Copenhagen interpretation, associated to Niels Bohr.

The most relevant differences between spontaneous processes (SP) and measurement processes (MP) are as follows [4]: in SP the observer plays no role, in MP the observer (or the measuring device) plays a paramount role; in SP the state vector ∣*ψ*ð Þi *t* is continuous, in MP ∣*ψ*ð Þi *t* collapses (jumps, is projected, is reduced); in SP the superposition principle applies, in MP the superposition principle breaks down; SP are ruled by a deterministic law, MP are ruled by probability laws; in SP every action is localized, in MP there is a kind of action-at-a-distance [5]; and in SP conservation laws are strictly valid, in MP conservation laws have only a statistical sense [6–8].

Since the projection postulate contradicts the fundamental Schrödinger equation of motion, some authors rushed to the conclusion that it was defective. Henry Margenau suggested in a manuscript sent to Albert Einstein on November 13, 1935, that this postulate should be abandoned. Einstein replied that the formalism of quantum mechanics inevitably requires the following postulate: "If a measurement performed upon a system yields a value *m*, then the same measurement performed immediately afterwards yields again the value *m* with certainty" ([3], p. 228). The projection postulate guarantees compliance with this principle.

The eigenvalue equations of E are

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

*t*0, the system is in the state ∣*ψ*ð Þi *t*<sup>0</sup> = ∣φ *<sup>j</sup>*

the state to change. At time *t* the state of the system will be

*Transitions between Stationary States and the Measurement Problem*

and depending only on H, *t*, and *t*<sup>0</sup> ([1], p. 109).

authors may have been changed for homogeneity.

**3. TBSS require measurements**

∣*ψ*ð Þi ¼ *t*<sup>0</sup> ∣φ *<sup>j</sup>*

initial state ∣φ*<sup>j</sup>*

**33**

E ∣φ*n*i ¼ *En* ∣φ*n*i (2)

i, the eigenvector of the non-perturbed

i∣

i (3)

<sup>2</sup> (4)

i to

where *En* (*n* = 1, 2, … , *N*) are the eigenvalues of E and ∣φ*n*i the corresponding eigenvectors. For simplicity we shall consider the spectrum of E to be entirely discrete and non-degenerate. All the *En* and ∣φ*n*i are supposed to be known.

Let ∣*ψ*ð Þi *t* be the state of the system at time *t*. We assume that at the initial time

Hamiltonian E corresponding to the eigenvalue *Ej*. If there is no perturbation, i.e., if the Hamiltonian were E, this state would be stationary. But the perturbation causes

∣*ψ*ð Þi ¼ *t UH*ð Þ *t*, *t*<sup>0</sup> j j *ψ*ð Þi ¼ *t*<sup>0</sup> *UH*ð Þ *t*, *t*<sup>0</sup> φ *<sup>j</sup>*

where *UH*ð Þ *t*, *t*<sup>0</sup> is the evolution operator, a linear operator independent on ∣*ψ*i

The probability of a transition taking place from the initial stationary state ∣φ *<sup>j</sup>*

the final stationary state ∣φ*k*i (respectively corresponding to the eigenvalues *Ej* and *Ek* of E) induced by the perturbation W(*t*) during the time interval (*t*0, *t*) is then

*Pjk*ð Þ¼ *t*0, *t* ∣⟨φ*k*∣*UH*ð Þ *t*, *t*<sup>0</sup> ∣φ*<sup>j</sup>*

See, for instance, [1], Chapter VII; [9], Chapter 9; [10], Chapter XIII; [11], Chapter IV; [12], Chapter 19; and [13], Chapter XVII. Note: symbols used by these

TDPT includes two clearly different stages. The first governed by the Schrödinger equation and the second ruled by probability laws [14]. Concerning this issue Dirac points out: "When one makes an observation on the dynamical system, the state of the system gets changed in an unpredictable way, but in between observations causality applies, in quantum mechanics as in classical mechanics, and the system is governed by equations of motion which make the state at one time determine the state at a later time. These equations of motion … will apply so long as the dynamical system is left undisturbed by any observation or similar process … Let us consider a particular state of motion through the time during which the system is left undisturbed. We shall have the state at any time *t* corresponding to a certain ket which depends on *t* and which may be written ∣*ψ*ð Þi *t* … The requirement that the state at one time [*t*0] determines the state at another time [*t*] means that ∣*ψ*ð Þi *t*<sup>0</sup> determines ∣*ψ*ð Þi *t* … " ([1], p. 108).

During the first stage of TDPT the process is ruled by the Schrödinger equation:

where Hð Þ*t* is the total Hamiltonian of the system and ℏ is Planck's constant divided by 2π. The solution of Eq. (5) corresponding to the initial condition

and H(*t*), which includes the perturbation W(*t*). Since ∣*ψ*ð Þi *t* depends only on the

i is unique; ∣*ψ*ð Þi *t* is completely determined by the initial state ∣*ψ*ð Þi *t*<sup>0</sup>

i and on H(*t*), or if preferred on the perturbation W(*t*), then

dt∣*ψ*ð Þi ¼ *<sup>t</sup>* <sup>H</sup>ð Þ*<sup>t</sup>* <sup>∣</sup>*ψ*ð Þi *<sup>t</sup>* (5)

<sup>i</sup><sup>ℏ</sup> <sup>d</sup>

The traditional measurement problem in quantum mechanics is how (or whether) wave function collapse occurs when a measurement is performed. Although *a similar measurement problem* is implied in transitions between stationary states (TBSS) induced by a time-dependent perturbation, *it is conspicuously absent* from the specialized literature on the subject.

The contents of this paper are as follows: time-dependent perturbation theory (TDPT) is summarized in Section 2. Section 3 shows that according to TDPT, measurements are required for TBSS to occur. Section 4 highlights the similarities and differences between the traditional measurement problem and that implied in TBSS. Section 5 includes several interpretations of quantum mechanics which attempt to solve the traditional measurement problem: Bohmian mechanics, decoherence, spontaneous localization, and spontaneous projection approach (SPA). Section 6 shows that SPA accounts for TBSS, and in cooperation with decoherence, it also accounts for the traditional measurement problem. Section 7 compiles conclusions.

#### **2. The formulation of TDPT**

TDPT was formulated by Dirac in 1930 ([1], Chapter VII). In his words: "In [TDPT] we do not consider any modification to be made in the states of the unperturbed system, but we *suppose* that the perturbed system, instead of remaining permanently in one of these states, is continually changing from one to another, or making transitions, under the influence of the perturbation" ([1], p. 167; emphasis added). The aim of TDPT is, then, to calculate the probability of TBSS which can be induced by the perturbation during a given time interval.

Dirac points out that "this method must … be used for solving *all* problems involving a consideration of time, such as those about the transient phenomena that occur when the perturbation is suddenly applied, or more generally problems in which the perturbation varies with the time in any way (i.e. in which the perturbing energy involves the time explicitly). [It must also] be used in collision problems, even though the perturbing energy does not here involve the time explicitly, if one wishes to calculate absorption and emission probabilities, since these probabilities, unlike a scattering probability, cannot be defined without reference to a state of affairs that varies with the time" ([1], p. 168; emphasis added).

TDPT is a key ingredient of OQM. It has many applications and is at the basis of quantum electrodynamics, the extension of OQM accounting for the interactions between matter and radiation ([1], Chapter X; [9], Chapter 9). Without TDPT, OQM would hardly be such a powerful and successful theory.

To develop TDPT one starts by splitting in two the total Hamiltonian H(*t*) acting on the system:

$$\mathbf{H}(t) = \mathbf{E} + \mathbf{W}(t) \tag{1}$$

E is the Hamiltonian of an unperturbed system, which can be dealt with exactly. Every dependence on time is included in W(*t*). Dirac asserts that "the perturbing energy W(*t*) can be an arbitrary function of the time" ([1], p. 172).

*Transitions between Stationary States and the Measurement Problem DOI: http://dx.doi.org/10.5772/intechopen.91801*

The eigenvalue equations of E are

Henry Margenau suggested in a manuscript sent to Albert Einstein on November 13, 1935, that this postulate should be abandoned. Einstein replied that the formalism of quantum mechanics inevitably requires the following postulate: "If a measurement performed upon a system yields a value *m*, then the same measurement performed immediately afterwards yields again the value *m* with certainty" ([3], p. 228). The projection postulate guarantees compliance with this principle. The traditional measurement problem in quantum mechanics is how (or whether) wave function collapse occurs when a measurement is performed.

Although *a similar measurement problem* is implied in transitions between stationary states (TBSS) induced by a time-dependent perturbation, *it is conspicuously absent*

The contents of this paper are as follows: time-dependent perturbation theory (TDPT) is summarized in Section 2. Section 3 shows that according to TDPT, measurements are required for TBSS to occur. Section 4 highlights the similarities and differences between the traditional measurement problem and that implied in TBSS. Section 5 includes several interpretations of quantum mechanics which attempt to solve the traditional measurement problem: Bohmian mechanics, decoherence, spontaneous localization, and spontaneous projection approach (SPA). Section 6 shows that SPA accounts for TBSS, and in cooperation with decoherence, it also accounts for the traditional measurement problem. Section 7

TDPT was formulated by Dirac in 1930 ([1], Chapter VII). In his words: "In [TDPT] we do not consider any modification to be made in the states of the unperturbed system, but we *suppose* that the perturbed system, instead of

remaining permanently in one of these states, is continually changing from one to another, or making transitions, under the influence of the perturbation" ([1], p. 167; emphasis added). The aim of TDPT is, then, to calculate the probability of TBSS which can be induced by the perturbation during a given time interval. Dirac points out that "this method must … be used for solving *all* problems involving a consideration of time, such as those about the transient phenomena that occur when the perturbation is suddenly applied, or more generally problems in which the perturbation varies with the time in any way (i.e. in which the perturbing energy involves the time explicitly). [It must also] be used in collision problems, even though the perturbing energy does not here involve the time explicitly, if one wishes to calculate absorption and emission probabilities, since these probabilities, unlike a scattering probability, cannot be defined without reference to a state of

TDPT is a key ingredient of OQM. It has many applications and is at the basis of quantum electrodynamics, the extension of OQM accounting for the interactions between matter and radiation ([1], Chapter X; [9], Chapter 9). Without TDPT,

To develop TDPT one starts by splitting in two the total Hamiltonian H(*t*) acting

E is the Hamiltonian of an unperturbed system, which can be dealt with exactly. Every dependence on time is included in W(*t*). Dirac asserts that "the perturbing

Hð*t*Þ ¼ E þ Wð*t*Þ (1)

affairs that varies with the time" ([1], p. 168; emphasis added).

OQM would hardly be such a powerful and successful theory.

energy W(*t*) can be an arbitrary function of the time" ([1], p. 172).

from the specialized literature on the subject.

compiles conclusions.

*Quantum Mechanics*

on the system:

**32**

**2. The formulation of TDPT**

$$E \mid \!\!\!\!\!\!\!p\!\_n\rangle = E\_n \mid \!\!\!\!p\!\_n\rangle \tag{2}$$

where *En* (*n* = 1, 2, … , *N*) are the eigenvalues of E and ∣φ*n*i the corresponding eigenvectors. For simplicity we shall consider the spectrum of E to be entirely discrete and non-degenerate. All the *En* and ∣φ*n*i are supposed to be known.

Let ∣*ψ*ð Þi *t* be the state of the system at time *t*. We assume that at the initial time *t*0, the system is in the state ∣*ψ*ð Þi *t*<sup>0</sup> = ∣φ *<sup>j</sup>* i, the eigenvector of the non-perturbed Hamiltonian E corresponding to the eigenvalue *Ej*. If there is no perturbation, i.e., if the Hamiltonian were E, this state would be stationary. But the perturbation causes the state to change. At time *t* the state of the system will be

$$|\psi(t)\rangle = U\_H(t, t\_0)|\psi(t\_0)\rangle = U\_H(t, t\_0)|\Phi\_j\rangle \tag{3}$$

where *UH*ð Þ *t*, *t*<sup>0</sup> is the evolution operator, a linear operator independent on ∣*ψ*i and depending only on H, *t*, and *t*<sup>0</sup> ([1], p. 109).

The probability of a transition taking place from the initial stationary state ∣φ *<sup>j</sup>* i to the final stationary state ∣φ*k*i (respectively corresponding to the eigenvalues *Ej* and *Ek* of E) induced by the perturbation W(*t*) during the time interval (*t*0, *t*) is then

$$P\_{jk}(t\_0, t) = |\langle \!\langle \!\!\phi\_k \vert U\_H(t, t\_0) \vert \!\!\!\phi\_j \rangle|^2 \tag{4}$$

See, for instance, [1], Chapter VII; [9], Chapter 9; [10], Chapter XIII; [11], Chapter IV; [12], Chapter 19; and [13], Chapter XVII. Note: symbols used by these authors may have been changed for homogeneity.

#### **3. TBSS require measurements**

TDPT includes two clearly different stages. The first governed by the Schrödinger equation and the second ruled by probability laws [14]. Concerning this issue Dirac points out: "When one makes an observation on the dynamical system, the state of the system gets changed in an unpredictable way, but in between observations causality applies, in quantum mechanics as in classical mechanics, and the system is governed by equations of motion which make the state at one time determine the state at a later time. These equations of motion … will apply so long as the dynamical system is left undisturbed by any observation or similar process … Let us consider a particular state of motion through the time during which the system is left undisturbed. We shall have the state at any time *t* corresponding to a certain ket which depends on *t* and which may be written ∣*ψ*ð Þi *t* … The requirement that the state at one time [*t*0] determines the state at another time [*t*] means that ∣*ψ*ð Þi *t*<sup>0</sup> determines ∣*ψ*ð Þi *t* … " ([1], p. 108).

During the first stage of TDPT the process is ruled by the Schrödinger equation:

$$\dot{\mathbf{u}}\hbar\frac{\mathbf{d}}{\mathbf{d}t}|\psi(t)\rangle = \mathbf{H}(t)\,|\psi(t)\rangle\tag{5}$$

where Hð Þ*t* is the total Hamiltonian of the system and ℏ is Planck's constant divided by 2π. The solution of Eq. (5) corresponding to the initial condition ∣*ψ*ð Þi ¼ *t*<sup>0</sup> ∣φ *<sup>j</sup>* i is unique; ∣*ψ*ð Þi *t* is completely determined by the initial state ∣*ψ*ð Þi *t*<sup>0</sup> and H(*t*), which includes the perturbation W(*t*). Since ∣*ψ*ð Þi *t* depends only on the initial state ∣φ*<sup>j</sup>* i and on H(*t*), or if preferred on the perturbation W(*t*), then

$$|\psi(t)\rangle \equiv |\psi\_{j,H}(t)\rangle = U\_H(t, t\_0)|\psi(t\_0)\rangle = U\_H(t, t\_0)|\Phi\_j\rangle \tag{6}$$

of ε is performed, OQM states that the system continues to evolve in

*Transitions between Stationary States and the Measurement Problem*

**4. Two kinds of measurement problems: similarities and differences**

It is often overlooked that TDPT requires a measurement of ε in order to

There are, then, two kinds of measurement problems: (i) the traditional measurement problem and (ii) the measurement problem related to TBSS. Both of them are measurement problems for in both the Schrödinger evolution is interrupted and the state of the system instantaneously collapses as established by the projection

i. In the traditional measurement problem, the experimenter chooses the physical quantity to be measured. This quantity can be, in principle, any physical quantity such as the position, a component of the angular momentum, the energy, etc. Measurements of these quantities have been performed many times, with different methods, by different people, and in

ii. In TBSS the system jumps to an eigenstate of E, the operator representing ε. The experimenter has no choice; the only physical quantity susceptible to be "measured" is the non-perturbed energy ε. We say "measured" because it seems difficult to admit that TBSS involve measurements of any physical quantity. It seems even more difficult to admit that ε is measured every time a photon is either emitted or absorbed by an atom, as TDPT requires. TBSS could be considered "measurements" without observers or measuring devices.

"In most cases, the wave function evolves gently, in a perfectly predictable and continuous way, according to the Schrödinger equation; in some cases only (as soon as a measurement is performed), unpredictable changes take place, according to the postulate of wave packet reduction" [15]. TBSS, which are happening everywhere all the time, must also be included in *some of the cases* where unpredictable changes

In previous papers we have pointed out the following contradiction: On the one hand, according to OQM *there is no room for the projection postulate* as long as we are dealing with spontaneous processes. On the other hand, to account for spontaneous processes involving a consideration of time OQM requires, through TDPT, *the application of the projection postulate*. This is a flagrant incoherence absent from the

Quantum weirdness has been associated with the traditional measurement problem. To solve it, several interpretations of quantum mechanics have been proposed. In the following section, we shall address a few of them. For a critical review of the most popular interpretations of quantum theory, see the interesting

study of Franck Laloë *Do we really understand quantum mechanics?* [15].

perturbations [14]. A perturbation is something completely different from a measurement. When the perturbation W(*t*) is applied, the Hamiltonian changes from E to E + W(*t*), but the Schrödinger evolution is not suspended. By contrast, a measurement interrupts the Schrödinger evolution. According to TDPT the perturbation W(*t*) applied during the interval (*t*0, *tf* ) as well as the measurement of

i ! <sup>∣</sup>φ*k*i, suggesting that TBSS are simply the result of

i ! ∣φ*k*i to occur.

compliance with Schrödinger's equation.

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

ε at *tf* are necessary for the transition ∣φ*<sup>j</sup>*

different circumstances.

take place according to the projection postulate.

literature [14, 16].

**35**

obtain the collapse ∣*ψj*,*<sup>H</sup> tf*

postulate.

The evolution from ∣φ *<sup>j</sup>* i to ∣*ψj*,*H*ð Þi *t* given by Eq. (6) is *automatic*. No transition from the initial state ∣φ*<sup>j</sup>* i to a stationary state ∣φ*k*i results until time *t*.

In the second stage of TDPT, *it is assumed* that at a time *tf* , a measurement is performed. As a consequence, a projection from ∣*ψj*,*<sup>H</sup> tf* <sup>i</sup> to <sup>∣</sup>φ*k*<sup>i</sup> takes place. In the words of Albert Messiah: "We suppose that at the initial time *t*<sup>0</sup> the system is in an eigenstate of E, the state ∣φ*<sup>j</sup>* i say. We wish to calculate the probability that *if a measurement is made* at a later time *tf* , the system *will be found* to be in a different eigenstate of E, the state ∣φ*k*i say. This quantity, by definition the probability of transition from ∣φ *<sup>j</sup>* i to ∣φ*k*i, will be denoted by *Pjk t*0, *tf* " ([13], p. 725; emphases added). Clearly

$$P\_{jk}(\mathfrak{t}\_0, \mathfrak{t}\_f) = |\langle \mathfrak{q}\_k | U\_H(\mathfrak{t}\_f, \mathfrak{t}\_0) | \mathfrak{q}\_j \rangle|^2 \tag{7}$$

Dirac does not explicitly mention measurements. He supposes that at the initial time *t*0, the system is in a state for which E has the value *E <sup>j</sup>* with certainty. The ket corresponding to this state is ∣φ*<sup>j</sup>* i. At time *tf* the corresponding ket will be *UH tf* , *t*<sup>0</sup> <sup>∣</sup>φ*<sup>j</sup>* i ([1], p. 172). The probability of E then *having* the value *Ek* is given by Eq. (7). For *Ek* ¼6 *Ej*, *Pjk t*0, *tf* is the probability of a transition taking place from ∣φ *<sup>j</sup>* i to ∣φ*k*i during the time interval (*t*0, *tf* ), while *Pjj t*0, *tf* is the probability of no transition taking place at all. The sum of *Pjk t*0, *tf* for all *k* is unity ([1], pp. 172–173).

Note that where Messiah says "the probability that *if a measurement* [of E] *is made* … the system *will be found* to be in … the state ∣φ*k*i *…* " Dirac says "the probability of E then *having* the value *Ek* … " Dirac's assertion, however, has exactly the same meaning as Messiah's, as shown in the following quote from Dirac's book *The Principles of Quantum Mechanics*: "The expression that an observable 'has a particular value' for a particular state is permissible in quantum mechanics in the special case when a measurement of the observable is certain to lead to the particular value, so that the state is an eigenstate of the observable … In the general case we cannot speak of an observable having a value for a particular state … [but] we can go further and speak of the probability of its having any specified value for the state, meaning *the probability of this specified value being obtained when one makes a measurement of the observable*" ([1], pp. 46–47; emphases added). Hence Dirac's statement "the probability of E then *having* the value *Ek* is given by Eq. (7)" should be understood as "the probability of *Ek* being obtained when one makes a measurement of E is given by Eq. (7)." *Both Dirac (the author of TDPT) and Messiah place measurements at the very heart of TDPT*.

The following diagram illustrates the complete process leading the system from the initial state ∣φ*<sup>j</sup>* i to the final state ∣φ*k*i:

$$|\psi(\mathfrak{t}\_0)\rangle = |\mathfrak{o}\_j\rangle \xrightarrow{\quad} \xrightarrow{\quad} |\psi\_{j,H}(\mathfrak{t}\_f)\rangle = U\_H\{\mathfrak{t}\_f, \mathfrak{t}\_0\} \xrightarrow{\quad} |\mathfrak{o}\_j\rangle$$


Let ε be the non-perturbed energy represented by the operator E. *Everything happens as if at time tf a measurement of* ε *is performed* [14]. If no measurement

∣*ψ*ð Þi � *t* ∣*ψj*,*H*ð Þi ¼ *t UH*ð Þ *t*, *t*<sup>0</sup> j j *ψ*ð Þi ¼ *t*<sup>0</sup> *UH*ð Þ *t*, *t*<sup>0</sup> φ *<sup>j</sup>*

In the second stage of TDPT, *it is assumed* that at a time *tf* , a measurement is

words of Albert Messiah: "We suppose that at the initial time *t*<sup>0</sup> the system is in an

*measurement is made* at a later time *tf* , the system *will be found* to be in a different eigenstate of E, the state ∣φ*k*i say. This quantity, by definition the probability of

<sup>¼</sup> <sup>∣</sup>⟨φ*k*∣*UH tf* , *<sup>t</sup>*<sup>0</sup>

Dirac does not explicitly mention measurements. He supposes that at the initial time *t*0, the system is in a state for which E has the value *E <sup>j</sup>* with certainty. The ket

Note that where Messiah says "the probability that *if a measurement* [of E] *is*

The following diagram illustrates the complete process leading the system from

Let ε be the non-perturbed energy represented by the operator E. *Everything happens as if at time tf a measurement of* ε *is performed* [14]. If no measurement

*made* … the system *will be found* to be in … the state ∣φ*k*i *…* " Dirac says "the probability of E then *having* the value *Ek* … " Dirac's assertion, however, has exactly the same meaning as Messiah's, as shown in the following quote from Dirac's book *The Principles of Quantum Mechanics*: "The expression that an observable 'has a particular value' for a particular state is permissible in quantum mechanics in the special case when a measurement of the observable is certain to lead to the particular value, so that the state is an eigenstate of the observable … In the general case we cannot speak of an observable having a value for a particular state … [but] we can go further and speak of the probability of its having any specified value for the state, meaning *the probability of this specified value being obtained when one makes a measurement of the observable*" ([1], pp. 46–47; emphases added). Hence Dirac's statement "the probability of E then *having* the value *Ek* is given by Eq. (7)" should be understood as "the probability of *Ek* being obtained when one makes a measurement of E is given by Eq. (7)." *Both Dirac (the author of TDPT) and Messiah place*

i to ∣φ*k*i, will be denoted by *Pjk t*0, *tf*

performed. As a consequence, a projection from ∣*ψj*,*<sup>H</sup> tf*

*Pjk t*0, *tf*

to ∣φ*k*i during the time interval (*t*0, *tf* ), while *Pjj t*0, *tf*

transition taking place at all. The sum of *Pjk t*0, *tf*

*measurements at the very heart of TDPT*.

state is ruled by the Schrödinger equation

i to the final state ∣φ*k*i:

**First stage**: during the interval (*t*0, *t <sup>f</sup>* ) the evolution of the

the initial state ∣φ*<sup>j</sup>*

**34**

i to a stationary state ∣φ*k*i results until time *t*.

i to ∣*ψj*,*H*ð Þi *t* given by Eq. (6) is *automatic*. No transition

i say. We wish to calculate the probability that *if a*

∣φ*<sup>j</sup>*

i. At time *tf* the corresponding ket will be

is the probability of a transition taking place from ∣φ *<sup>j</sup>*

i ([1], p. 172). The probability of E then *having* the value *Ek* is given by

i∣

The evolution from ∣φ *<sup>j</sup>*

eigenstate of E, the state ∣φ*<sup>j</sup>*

corresponding to this state is ∣φ*<sup>j</sup>*

Eq. (7). For *Ek* ¼6 *Ej*, *Pjk t*0, *tf*

transition from ∣φ *<sup>j</sup>*

added). Clearly

*UH tf* , *t*<sup>0</sup> <sup>∣</sup>φ*<sup>j</sup>*

from the initial state ∣φ*<sup>j</sup>*

*Quantum Mechanics*

i (6)

<sup>i</sup> to <sup>∣</sup>φ*k*<sup>i</sup> takes place. In the

" ([13], p. 725; emphases

is the probability of no

for all *k* is unity ([1], pp. 172–173).

**Second stage**: ∣*ψ<sup>j</sup>*,*<sup>H</sup> t <sup>f</sup>*

with probability *Pjk t*0, *t <sup>f</sup>*

<sup>i</sup> jumps to <sup>∣</sup>φ*k*<sup>i</sup>

<sup>2</sup> (7)

i

of ε is performed, OQM states that the system continues to evolve in compliance with Schrödinger's equation.

#### **4. Two kinds of measurement problems: similarities and differences**

It is often overlooked that TDPT requires a measurement of ε in order to obtain the collapse ∣*ψj*,*<sup>H</sup> tf* i ! <sup>∣</sup>φ*k*i, suggesting that TBSS are simply the result of perturbations [14]. A perturbation is something completely different from a measurement. When the perturbation W(*t*) is applied, the Hamiltonian changes from E to E + W(*t*), but the Schrödinger evolution is not suspended. By contrast, a measurement interrupts the Schrödinger evolution. According to TDPT the perturbation W(*t*) applied during the interval (*t*0, *tf* ) as well as the measurement of ε at *tf* are necessary for the transition ∣φ*<sup>j</sup>* i ! ∣φ*k*i to occur.

There are, then, two kinds of measurement problems: (i) the traditional measurement problem and (ii) the measurement problem related to TBSS. Both of them are measurement problems for in both the Schrödinger evolution is interrupted and the state of the system instantaneously collapses as established by the projection postulate.


"In most cases, the wave function evolves gently, in a perfectly predictable and continuous way, according to the Schrödinger equation; in some cases only (as soon as a measurement is performed), unpredictable changes take place, according to the postulate of wave packet reduction" [15]. TBSS, which are happening everywhere all the time, must also be included in *some of the cases* where unpredictable changes take place according to the projection postulate.

In previous papers we have pointed out the following contradiction: On the one hand, according to OQM *there is no room for the projection postulate* as long as we are dealing with spontaneous processes. On the other hand, to account for spontaneous processes involving a consideration of time OQM requires, through TDPT, *the application of the projection postulate*. This is a flagrant incoherence absent from the literature [14, 16].

Quantum weirdness has been associated with the traditional measurement problem. To solve it, several interpretations of quantum mechanics have been proposed. In the following section, we shall address a few of them. For a critical review of the most popular interpretations of quantum theory, see the interesting study of Franck Laloë *Do we really understand quantum mechanics?* [15].

#### **5. Some alternative interpretations to OQM**

#### **5.1 Bohmian mechanics (BM)**

It is also called the causal interpretation of quantum mechanics and the pilotwave model. Its first version was proposed by Louis de Broglie in 1927, rapidly abandoned and forgotten, and reformulated by David Bohm in 1952 [17].

In BM it is assumed that particles are point-like. They have well-defined positions at each instant and thus describe trajectories. A system of *N* particles with masses *mk* and *actual* positions Q*k*(*t*) (*k* = 1, … , *N*) can be described by the couple (Q(*t*), *ψ*(*t*)), where Q(*t*) = (Q1(*t*), … , QN(*t*)) is the *actual* configuration of the system. The wave function of the system is *ψ* = *ψ* (*q*, *t*) = *ψ* (*q*1, … , *qN*; *t*), a function on the space of *possible* configurations *q* of the system. The wave function evolves according to the Schrödinger equation:

$$\mathrm{i}\hbar\,\frac{\partial}{\partial t}\Psi = \mathrm{H}\,\psi\,\tag{8}$$

*Φ<sup>i</sup>* j ⟩ *η*<sup>0</sup> j ⟩ ! *Φ<sup>i</sup>* j ⟩ *η<sup>i</sup>* j ⟩ (10)

<sup>2</sup> <sup>j</sup>*Φ*1⟩ ⟨*Φ*2<sup>∣</sup> *<sup>η</sup>*<sup>1</sup> <sup>j</sup> ⟩ ⟨*η*2<sup>∣</sup>

<sup>2</sup> <sup>j</sup>*Φ*2⟩ ⟨*Φ*2<sup>∣</sup> *<sup>η</sup>*<sup>2</sup> <sup>j</sup> ⟩ ⟨*η*2<sup>∣</sup>

� � ð Þ <sup>j</sup>*Φ*1⟩ � <sup>j</sup>*Φ*2⟩ , the linearity of the

(12)

j*Φ*1⟩ *η*<sup>1</sup> j ⟩ � j*Φ*2⟩ *η*<sup>2</sup> ð Þ j ⟩ (11)

<sup>2</sup> <sup>j</sup>*Φ*2⟩ ⟨*Φ*2<sup>∣</sup> (13)

2 p

1

Assuming that the environment states are almost orthogonal to each other, i.e.,

"Eq. (13) does not imply that the system is in a mixture of states j*Φ*1⟩ and j*Φ*2⟩. Since these two states are simultaneously present in Eqs. (11) and (12), the composite system + environment displays superposition and associated interferences. However, Eq. (13) says that such quantum manifestations will not appear as long as

It has been proven that for large classical objects, decoherence would be virtually instantaneous because of the high probability of interaction of such systems with some environmental quantum. Several models illustrate the gradual cancelation of the off-diagonal elements with decoherence over time. Experiments also show that,

unobservable ([9], p. 251). "These experiments provide impressive direct evidence for how the interaction with the environment gradually delocalizes the quantum coherence required for the interference effects to be observed … We find our observations to be in excellent agreement with theoretical predictions" ([21],

The key assumption is that each elementary constituent of any physical system is subject, at random times, to spontaneous localization processes (called hittings) around random positions. The best known mathematical model stating which modifications of the wave function are induced by localizations, where and when they occur, is usually referred to as the Ghirardi-Rimini-Weber (GRW) theory [22, 23].

� � be the wave function of a system of *N* particles. "If a hitting occurs for the *i*th particle at point *x*, the wave function is instantaneously multiplied

2*d*<sup>2</sup>

� � *qi* � *<sup>x</sup>* � �<sup>2</sup>

� � (14)

1

1

If the initial state of the system is <sup>j</sup>*Φ*�⟩ <sup>=</sup> <sup>1</sup>ffiffi

<sup>j</sup>*Φ*�⟩ *<sup>η</sup>*<sup>0</sup> <sup>j</sup> ⟩ ! <sup>1</sup>

*Transitions between Stationary States and the Measurement Problem*

<sup>2</sup> <sup>j</sup>*Φ*1⟩ ⟨*Φ*1<sup>∣</sup> *<sup>η</sup>*<sup>1</sup> <sup>j</sup> ⟩ ⟨*η*1<sup>∣</sup> �

⟨*η*<sup>1</sup> *η*<sup>2</sup> j ⟩ ≈ 0 ([9], p. 248), the reduced density matrix becomes

experiments are performed only on the system" ([9], p. 248).

by a Gaussian function (appropriately normalized)" [24]:

, *<sup>x</sup>* � � <sup>¼</sup> K exp � <sup>1</sup>

*G qi*

*ρ*0 ≈ 1

<sup>2</sup> <sup>j</sup>*Φ*2⟩ ⟨*Φ*1<sup>∣</sup> *<sup>η</sup>*<sup>2</sup> <sup>j</sup> ⟩ ⟨*η*1<sup>∣</sup> <sup>þ</sup>

<sup>2</sup> <sup>j</sup>*Φ*1⟩ ⟨*Φ*1<sup>∣</sup> <sup>þ</sup>

due to the interaction with the environment, superposition states become

The corresponding pure state density matrix is

ffiffi 2 p � �

Schrödinger equation yields entangled states:

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

*<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup>

p. 265).

**37**

**5.3 Spontaneous localization**

It holds as follows [24]: Let *ψ q*1, … , *qN*

� 1

where H is the nonrelativistic Hamiltonian. The actual positions of the particles evolve according to the guiding equation:

$$\frac{\mathbf{d}}{\mathbf{d}t} \mathbf{Q}\_k(t) = \frac{\hbar}{m\_k} \text{Im} \left[ \frac{\boldsymbol{\Psi}^\* \ \partial\_k \boldsymbol{\Psi}}{\boldsymbol{\Psi}^\* \ \boldsymbol{\Psi}} \right] \tag{9}$$

where Im [] is the imaginary part of [] and *∂<sup>k</sup>* = (∂/∂x*k*, ∂/∂y*k*, ∂/∂z*k*) is the gradient with respect to the generic coordinates *qk* = (x*k*, y*k*, z*k*) of the *k*th particle. For a system of *N* particles, Eqs. (8) and (9) completely define BM [18]. It is worth stressing that (i) BM is a nonlocal theory and (ii) BM is a deterministic theory: the initial couple (Q(*t*0), *ψ*(*t*0)) determines the couple at any time *t* > *t*0.

BM accounts for all of the phenomena governed by nonrelativistic quantum mechanics, from spectral lines and scattering theory to superconductivity, the quantum Hall effect and quantum computing [18]. A proposed extension of BM describes creation and annihilation events: the world lines for the particles can begin and end [19]. For any experiment the deterministic Bohmian model yields the usual quantum predictions [18].

In BM the usual measurement postulates of quantum theory emerge from an analysis of the Eqs. (8) and (9). In the collapse of the wave function, the interaction of the quantum system with the environment (air molecules, cosmic rays, internal microscopic degrees of freedom, etc.) plays a significant role. Even if the Schrödinger evolution is not interrupted, replacing the original wave function for its "collapsed" derivative is justified as a pragmatic affair [18]. In this regard BM appeals for processes of decoherence.

#### **5.2 Decoherence**

Decoherence is an interesting physical phenomenon entirely contained in the linear Schrödinger equation and does not imply any particular conceptual problem [15]. It is a consequence of the unavoidable coupling of the quantum system with the surrounding medium which "looks and smells as a collapse" [20].

Decoherence is currently the subject of a great deal of research. To grasp how it works, let us consider the following case, taken from Daniel Bes' *Quantum Mechanics* ([9], pp. 247–248).

A quantum system in the state *Φ<sup>i</sup>* j ⟩ (i = 1, 2) interacts with the environment, initially in the state *η*<sup>0</sup> j ⟩, resulting in

*Transitions between Stationary States and the Measurement Problem DOI: http://dx.doi.org/10.5772/intechopen.91801*

**5. Some alternative interpretations to OQM**

It is also called the causal interpretation of quantum mechanics and the pilotwave model. Its first version was proposed by Louis de Broglie in 1927, rapidly abandoned and forgotten, and reformulated by David Bohm in 1952 [17].

In BM it is assumed that particles are point-like. They have well-defined positions at each instant and thus describe trajectories. A system of *N* particles with masses *mk* and *actual* positions Q*k*(*t*) (*k* = 1, … , *N*) can be described by the couple (Q(*t*), *ψ*(*t*)), where Q(*t*) = (Q1(*t*), … , QN(*t*)) is the *actual* configuration of the system. The wave function of the system is *ψ* = *ψ* (*q*, *t*) = *ψ* (*q*1, … , *qN*; *t*), a function on the space of *possible* configurations *q* of the system. The wave function evolves

where H is the nonrelativistic Hamiltonian. The actual positions of the particles

Im *<sup>ψ</sup>* <sup>∗</sup> *<sup>∂</sup>k<sup>ψ</sup> ψ* <sup>∗</sup> *ψ* 

ℏ *mk*

where Im [] is the imaginary part of [] and *∂<sup>k</sup>* = (∂/∂x*k*, ∂/∂y*k*, ∂/∂z*k*) is the gradient with respect to the generic coordinates *qk* = (x*k*, y*k*, z*k*) of the *k*th particle. For a system of *N* particles, Eqs. (8) and (9) completely define BM [18]. It is worth stressing that (i) BM is a nonlocal theory and (ii) BM is a deterministic theory: the

BM accounts for all of the phenomena governed by nonrelativistic quantum mechanics, from spectral lines and scattering theory to superconductivity, the quantum Hall effect and quantum computing [18]. A proposed extension of BM describes creation and annihilation events: the world lines for the particles can begin and end [19]. For any experiment the deterministic Bohmian model yields the

In BM the usual measurement postulates of quantum theory emerge from an analysis of the Eqs. (8) and (9). In the collapse of the wave function, the interaction of the quantum system with the environment (air molecules, cosmic rays, internal

Schrödinger evolution is not interrupted, replacing the original wave function for its "collapsed" derivative is justified as a pragmatic affair [18]. In this regard BM

Decoherence is an interesting physical phenomenon entirely contained in the linear Schrödinger equation and does not imply any particular conceptual problem [15]. It is a consequence of the unavoidable coupling of the quantum system with

Decoherence is currently the subject of a great deal of research. To grasp how it works, let us consider the following case, taken from Daniel Bes' *Quantum Mechanics*

A quantum system in the state *Φ<sup>i</sup>* j ⟩ (i = 1, 2) interacts with the environment,

microscopic degrees of freedom, etc.) plays a significant role. Even if the

the surrounding medium which "looks and smells as a collapse" [20].

*ψ* ¼ H *ψ* (8)

(9)

iℏ *∂ ∂t*

**5.1 Bohmian mechanics (BM)**

*Quantum Mechanics*

according to the Schrödinger equation:

evolve according to the guiding equation:

usual quantum predictions [18].

appeals for processes of decoherence.

initially in the state *η*<sup>0</sup> j ⟩, resulting in

**5.2 Decoherence**

([9], pp. 247–248).

**36**

d d*t*

Q*k*ðÞ¼ *t*

initial couple (Q(*t*0), *ψ*(*t*0)) determines the couple at any time *t* > *t*0.

$$\left|\Phi\_i\right>\left|\eta\_0\right> \to \left|\Phi\_i\right>\left|\eta\_i\right>\tag{10}$$

If the initial state of the system is <sup>j</sup>*Φ*�⟩ <sup>=</sup> <sup>1</sup>ffiffi 2 p � � ð Þ <sup>j</sup>*Φ*1⟩ � <sup>j</sup>*Φ*2⟩ , the linearity of the Schrödinger equation yields entangled states:

$$\left| \left| \Phi\_{\pm} \right> \left| \eta\_{0} \right> \to \left( \frac{1}{\sqrt{2}} \right) \left( \left| \Phi\_{1} \right> \left| \eta\_{1} \right> \pm \left| \Phi\_{2} \right> \left| \eta\_{2} \right> \right) \tag{11}$$

The corresponding pure state density matrix is

$$\begin{aligned} \rho &= \frac{1}{2} \left| \left| \Phi\_1 \right\rangle \left\langle \Phi\_1 \right| \left| \eta\_1 \right\rangle \left\langle \eta\_1 \right| \pm \frac{1}{2} \left| \left| \Phi\_1 \right\rangle \left\langle \Phi\_2 \right| \left| \eta\_1 \right\rangle \left\langle \eta\_2 \right| \\\\ &\pm \frac{1}{2} \left| \left| \Phi\_2 \right\rangle \left\langle \Phi\_1 \right| \left| \eta\_2 \right\rangle \left\langle \eta\_1 \right| + \frac{1}{2} \left| \left| \Phi\_2 \right\rangle \left\langle \Phi\_2 \right| \left| \eta\_2 \right\rangle \left\langle \eta\_2 \right| \end{aligned} \tag{12}$$

Assuming that the environment states are almost orthogonal to each other, i.e., ⟨*η*<sup>1</sup> *η*<sup>2</sup> j ⟩ ≈ 0 ([9], p. 248), the reduced density matrix becomes

$$
\rho' \approx \frac{1}{2} \left| \Phi\_1 \right\rangle \left\langle \Phi\_1 \right| + \frac{1}{2} \left| \Phi\_2 \right\rangle \left\langle \Phi\_2 \right| \tag{13}
$$

"Eq. (13) does not imply that the system is in a mixture of states j*Φ*1⟩ and j*Φ*2⟩. Since these two states are simultaneously present in Eqs. (11) and (12), the composite system + environment displays superposition and associated interferences. However, Eq. (13) says that such quantum manifestations will not appear as long as experiments are performed only on the system" ([9], p. 248).

It has been proven that for large classical objects, decoherence would be virtually instantaneous because of the high probability of interaction of such systems with some environmental quantum. Several models illustrate the gradual cancelation of the off-diagonal elements with decoherence over time. Experiments also show that, due to the interaction with the environment, superposition states become unobservable ([9], p. 251). "These experiments provide impressive direct evidence for how the interaction with the environment gradually delocalizes the quantum coherence required for the interference effects to be observed … We find our observations to be in excellent agreement with theoretical predictions" ([21], p. 265).

#### **5.3 Spontaneous localization**

The key assumption is that each elementary constituent of any physical system is subject, at random times, to spontaneous localization processes (called hittings) around random positions. The best known mathematical model stating which modifications of the wave function are induced by localizations, where and when they occur, is usually referred to as the Ghirardi-Rimini-Weber (GRW) theory [22, 23]. It holds as follows [24]:

Let *ψ q*1, … , *qN* � � be the wave function of a system of *N* particles. "If a hitting occurs for the *i*th particle at point *x*, the wave function is instantaneously multiplied by a Gaussian function (appropriately normalized)" [24]:

$$G(q\_i, \boldsymbol{\omega}) = \mathbf{K} \exp\left[ -\left(\frac{\mathbf{1}}{2d^2}\right) \left(q\_i - \boldsymbol{\omega}\right)^2 \right] \tag{14}$$

where *d* and K are constants. Let

$$\Phi\_i(q\_1, \ldots, q\_N; \mathbf{x}) = \boldsymbol{\psi}(q\_1, \ldots, q\_N) \; \mathbf{G}(q\_i, \mathbf{x}) \tag{15}$$

v. The relation

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

the Schrödinger channel is

Schrödinger evolution [25–27].

has its preferential set.

evolutions.

**39**

tion of α [25].

j*φk*i is

⟨*ψ*ð Þj *<sup>t</sup> <sup>A</sup>*j*ψ*ð Þ*<sup>t</sup>* ⟩ <sup>¼</sup> <sup>X</sup>

*Transitions between Stationary States and the Measurement Problem*

evolution follows. By contrast, if it has the preferential set *N<sup>φ</sup>*

*dPk*ðÞ¼ *<sup>t</sup> <sup>γ</sup><sup>k</sup>* j j ð Þ*<sup>t</sup>* <sup>2</sup> *dt*

where <sup>τ</sup>ð Þ*<sup>t</sup>* <sup>Δ</sup>EðÞ¼ *<sup>t</sup>* <sup>ℏ</sup>*=*2 and ½ � <sup>Δ</sup>Eð Þ*<sup>t</sup>* <sup>2</sup> <sup>¼</sup> ⟨*ψ*ð Þj *<sup>t</sup>* E2

*j*

must be fulfilled for every operator *A* representing a conserved quantity α when W(*t*) = 0. The validity of this relation ensures the statistical sense of the conserva-

If the system in the state ∣*ψ*ð Þi *t* does not have a preferential set, the Schrödinger

interval ð Þ *t*, *t* þ *dt* , the system can either remain in the Schrödinger channel or jump to one of its preferential states. The probability that it jumps to the preferential state

It is easily shown that in the interval ð Þ *t*, *t* þ *dt* , the probability that the system abandons the Schrödinger channel is *dt=*τð Þ*t* and the probability that it remains in

*dPS*ðÞ¼ *<sup>t</sup>* <sup>1</sup> � *dt*

In cases where the system remains in the Schrödinger channel, the transformation of the state yielded by SPA exactly coincides with that yielded by OQM. It could be wrongly assumed that there is a complete correspondence (i) between OQM spontaneous processes and SPA processes where the preferential set is absent; and (ii) between OQM measurement processes and SPA processes where the system

Certainly SPA processes where the preferential set is absent as well as OQM spontaneous processes are forcible Schrödinger evolutions. And unless the system is an eigenstate of the operator representing the quantity to be measured, OQM measurements entail projections. But if the system has its preferential set, according to SPA it can either be projected to a preferential state or remain in the Schrödinger channel [26, 27]. Differing from OQM, in SPA there is always room for Schrödinger

In sum, SPA states that in general the wave function evolves gently, in a perfectly predictable and continuous way, in agreement with the Schrödinger equation;

Measurement is a complicated and theory-laden business ([29], p. 208). When one talks about the measurement problem in quantum mechanics, one is not referring to a real and theory-laden process but just to the problem of *accounting in*

in some cases only, when the system jumps to one of its preferential states, unpredictable changes take place, according to the projection postulate. Assuming that projections are a law of nature, SPA succeeds in proposing a real physical mechanism for the emergence of a single result in a single experiment.

**6. Facing both measurement problems**

So the dominant process in a small time interval ð Þ *t*, *t* þ *dt* is always the

∣*γ <sup>j</sup>*ð Þ*t* ∣

<sup>2</sup> ⟨*φ <sup>j</sup>*

<sup>τ</sup>ð Þ*<sup>t</sup>* <sup>¼</sup> j j ⟨*φk*j*ψ*ð Þ*<sup>t</sup>* ⟩ <sup>2</sup> *dt*

∣*A*∣*φ <sup>j</sup>*

i (17)

� �, in the small time

<sup>τ</sup>ð Þ*<sup>t</sup>* (18)

<sup>j</sup>*ψ*ð Þ*<sup>t</sup>* ⟩ � ½ � ⟨*ψ*ð Þj *<sup>t</sup>* <sup>E</sup>j*ψ*ð Þ*<sup>t</sup>* ⟩ <sup>2</sup> [26, 27].

*<sup>τ</sup>*ð Þ*<sup>t</sup>* (19)

be the unnormalized wave function immediately after the localization and *P x*ð Þ the density probability of the hitting taking place at *x*. Assuming that *P x*ð Þ equals the integral of *<sup>Φ</sup><sup>i</sup>* j j<sup>2</sup> over the 3*N*-dimensional space implies that hittings occur with higher probability at those places where, in the standard quantum description, there is a higher probability of finding the particle. The constant K appearing in Eq. (14) is chosen in such a way that the integral of *P x*ð Þ over the whole space equals unity. Finally, it is assumed that the hittings occur at randomly distributed times, according to a Poisson distribution, with mean frequency f. The parameters chosen in the GRW-model are f = 10�<sup>16</sup> s �<sup>1</sup> and *d* = 10�<sup>5</sup> cm [24].

GRW aims to a unification of all kinds of physical evolution, including wave function reduction. On the one hand, the theory succeeds in proposing a real physical mechanism for the emergence of a single result in a single experiment, which is attractive from a physical point of view, and solves the "preferred basis problem," since the basis is that of localized states. The occurrence of superposition of far-away states is destroyed by the additional process of localization [15]. On the other hand, it fails to account for TBSS referred to in TDPT. Similar theories to GRW, like the continuous spontaneous localization, confront the same problem. The reason is simple: localizations localize (see Eqs. (14) and (15)). They do not yield the system to a stationary state.

#### **5.4 Spontaneous projection approach (SPA)**

Two kinds of processes irreducible to one another occur in nature: those strictly continuous and causal, governed by a deterministic law, and those implying discontinuities, ruled by probability laws. This is the main hypothesis of SPA [25]. Continuous and causal processes are Schrödinger's evolutions. Processes implying discontinuities are jumps to the preferential states ∣*φ <sup>j</sup>* i ð Þ *j* ¼ 1, … , *N* belonging to the preferential set *N<sup>φ</sup>* � � (= <sup>∣</sup>*φ*1i, … , <sup>∣</sup>*φN*iÞ of the system in a given state [26, 27].

In SPA conservation laws play a paramount role. The system has the tendency to jump to the eigenstates of every constant of the motion, while the jumps must respect the statistical sense of every conservation law [25].

The preferential set may or may not exist. If the system in the state ∣*ψ*ð Þi t has the preferential set *N<sup>φ</sup>* � �, we can write

$$\left|\left\langle\psi(t)\right\rangle\right\rangle = \sum\_{j} \left\langle r\_{j}(t) \left|\left\langle\rho\_{j}\right\rangle\right.\tag{16}$$

where *γ <sup>j</sup>* ð Þ*t* = ⟨*φj*j*ψ*ð Þ*t* ⟩ 6¼ 0 for every *j* = 1, … , *N* and *N* ≥2. Let us stress the following characteristics of the preferential set [26, 27]:


*Transitions between Stationary States and the Measurement Problem DOI: http://dx.doi.org/10.5772/intechopen.91801*

v. The relation

where *d* and K are constants. Let

*Quantum Mechanics*

in the GRW-model are f = 10�<sup>16</sup> s

the preferential set *N<sup>φ</sup>*

preferential set *N<sup>φ</sup>*

where *γ <sup>j</sup>*

**38**

**5.4 Spontaneous projection approach (SPA)**

discontinuities are jumps to the preferential states ∣*φ <sup>j</sup>*

� �, we can write

i. It depends on the state ∣*ψ*ð Þi *t* .

iv. At least ð Þ *N* � 1 members of *N<sup>φ</sup>*

referred to elsewhere [28].

have more than one preferential set.

respect the statistical sense of every conservation law [25].

*<sup>Φ</sup><sup>i</sup> <sup>q</sup>*1, … , *qN*; *<sup>x</sup>* � � <sup>¼</sup> *<sup>ψ</sup> <sup>q</sup>*1, … , *qN*

be the unnormalized wave function immediately after the localization and *P x*ð Þ the density probability of the hitting taking place at *x*. Assuming that *P x*ð Þ equals the integral of *<sup>Φ</sup><sup>i</sup>* j j<sup>2</sup> over the 3*N*-dimensional space implies that hittings occur with higher probability at those places where, in the standard quantum description, there is a higher probability of finding the particle. The constant K appearing in Eq. (14) is chosen in such a way that the integral of *P x*ð Þ over the whole space equals unity. Finally, it is assumed that the hittings occur at randomly distributed times,

according to a Poisson distribution, with mean frequency f. The parameters chosen

Two kinds of processes irreducible to one another occur in nature: those strictly continuous and causal, governed by a deterministic law, and those implying discontinuities, ruled by probability laws. This is the main hypothesis of SPA [25]. Continuous and causal processes are Schrödinger's evolutions. Processes implying

In SPA conservation laws play a paramount role. The system has the tendency to

The preferential set may or may not exist. If the system in the state ∣*ψ*ð Þi t has the

*j γ j* ð Þ*t* ∣*φ<sup>j</sup>*

ii. If it exists, the preferential set is unique. A system in the state ∣*ψ*ð Þi *t* cannot

written H(*t*) = E + W(*t*), the preferential set does not depend on W(*t*).

the case where a preferential state is not a stationary state, has been

jump to the eigenstates of every constant of the motion, while the jumps must

<sup>∣</sup>*ψ*ð Þi ¼ *<sup>t</sup>* <sup>X</sup>

ð Þ*t* = ⟨*φj*j*ψ*ð Þ*t* ⟩ 6¼ 0 for every *j* = 1, … , *N* and *N* ≥2. Let us stress the following characteristics of the preferential set [26, 27]:

iii. Even if in the general case the Hamiltonian of the system can be

� � (= <sup>∣</sup>*φ*1i, … , <sup>∣</sup>*φN*iÞ of the system in a given state [26, 27].

GRW aims to a unification of all kinds of physical evolution, including wave function reduction. On the one hand, the theory succeeds in proposing a real physical mechanism for the emergence of a single result in a single experiment, which is attractive from a physical point of view, and solves the "preferred basis problem," since the basis is that of localized states. The occurrence of superposition of far-away states is destroyed by the additional process of localization [15]. On the other hand, it fails to account for TBSS referred to in TDPT. Similar theories to GRW, like the continuous spontaneous localization, confront the same problem. The reason is simple: localizations localize (see Eqs. (14) and (15)). They do not yield the system to a stationary state.

�<sup>1</sup> and *d* = 10�<sup>5</sup> cm [24].

� � *G qi*

, *x* � � (15)

i ð Þ *j* ¼ 1, … , *N* belonging to

i (16)

� � are eigenstates of E. The exception, i.e.,

$$
\langle \psi(t) | A | \psi(t) \rangle = \sum\_{j} | \gamma\_{j}(t) |^{2} \left\langle \rho\_{j} | A | \rho\_{j} \right\rangle \tag{17}
$$

must be fulfilled for every operator *A* representing a conserved quantity α when W(*t*) = 0. The validity of this relation ensures the statistical sense of the conservation of α [25].

If the system in the state ∣*ψ*ð Þi *t* does not have a preferential set, the Schrödinger evolution follows. By contrast, if it has the preferential set *N<sup>φ</sup>* � �, in the small time interval ð Þ *t*, *t* þ *dt* , the system can either remain in the Schrödinger channel or jump to one of its preferential states. The probability that it jumps to the preferential state j*φk*i is

$$dP\_k(t) = \left|\gamma\_k(t)\right|^2 \frac{dt}{\pi(t)} = \left|\langle\wp\_k|\wp(t)\rangle\right|^2 \frac{dt}{\pi(t)}\tag{18}$$

where <sup>τ</sup>ð Þ*<sup>t</sup>* <sup>Δ</sup>EðÞ¼ *<sup>t</sup>* <sup>ℏ</sup>*=*2 and ½ � <sup>Δ</sup>Eð Þ*<sup>t</sup>* <sup>2</sup> <sup>¼</sup> ⟨*ψ*ð Þj *<sup>t</sup>* E2 <sup>j</sup>*ψ*ð Þ*<sup>t</sup>* ⟩ � ½ � ⟨*ψ*ð Þj *<sup>t</sup>* <sup>E</sup>j*ψ*ð Þ*<sup>t</sup>* ⟩ <sup>2</sup> [26, 27].

It is easily shown that in the interval ð Þ *t*, *t* þ *dt* , the probability that the system abandons the Schrödinger channel is *dt=*τð Þ*t* and the probability that it remains in the Schrödinger channel is

$$dP\_S(t) = \mathbf{1} - \frac{dt}{\tau(t)}\tag{19}$$

So the dominant process in a small time interval ð Þ *t*, *t* þ *dt* is always the Schrödinger evolution [25–27].

In cases where the system remains in the Schrödinger channel, the transformation of the state yielded by SPA exactly coincides with that yielded by OQM. It could be wrongly assumed that there is a complete correspondence (i) between OQM spontaneous processes and SPA processes where the preferential set is absent; and (ii) between OQM measurement processes and SPA processes where the system has its preferential set.

Certainly SPA processes where the preferential set is absent as well as OQM spontaneous processes are forcible Schrödinger evolutions. And unless the system is an eigenstate of the operator representing the quantity to be measured, OQM measurements entail projections. But if the system has its preferential set, according to SPA it can either be projected to a preferential state or remain in the Schrödinger channel [26, 27]. Differing from OQM, in SPA there is always room for Schrödinger evolutions.

In sum, SPA states that in general the wave function evolves gently, in a perfectly predictable and continuous way, in agreement with the Schrödinger equation; in some cases only, when the system jumps to one of its preferential states, unpredictable changes take place, according to the projection postulate. Assuming that projections are a law of nature, SPA succeeds in proposing a real physical mechanism for the emergence of a single result in a single experiment.

#### **6. Facing both measurement problems**

Measurement is a complicated and theory-laden business ([29], p. 208). When one talks about the measurement problem in quantum mechanics, one is not referring to a real and theory-laden process but just to the problem of *accounting in*

*principle for projections resulting from measurements*, i.e., to the fact that the Schrödinger evolution is suspended when a measurement is performed.

SPA justifies Dirac's assertion: "in [TDPT] we do not consider any modification to be made in the states of the unperturbed system, but we suppose that the perturbed system, instead of remaining permanently in one of these states, is continually changing from one to another, or making transitions, under the influence of the perturbation" ([1], p. 167).

Laloë points out that "decoherence is not to be confused with the measurement

decoherence, the off-diagonal elements of the density matrix vanish … " [15]. In his view "the crux of most of our difficulties with quantum mechanics is the question: what is exactly the process that forces Nature … to make its choice among the various possibilities for the results of experiments?" [15]. SPA answers: spontaneous

SPA and decoherence are not opposed theories competing for "an explanation" to the measurement problem but cooperating theories. Projections break down the Schrödinger evolution, but they are not frequent. If the system has its preferential set, projections can take place at the very beginning of the process or not (in SPA there is always room for Schrödinger evolutions). As long as projections do not take place, decoherence can make its work entangling the system with the environment. But nothing prevents the total, entangled system, to have its preferential set. This may be why a spontaneous projection finally breaks down the superposition of states of the total system. Nature makes its choice, and it is only then that

Carlton Caves declares: "Mention collapse of the wave function, and you are likely to encounter vague uneasiness or, in extreme cases, real discomfort. This uneasiness can usually be traced to a feeling that wave-function collapse lies 'outside' quantum mechanics: The real quantum mechanics is said to be the unitary Schrödinger evolution; wave-function collapse is regarded as an ugly duckling of questionable status, dragged in to interrupt the beautiful flow of Schrödinger

If collapses implied in traditional measurement are regarded as an ugly duckling of questionable status, collapses implied in TBSS could result definitively unbearable. Neither observers nor measuring devices could be invoked to excuse their occurrence, but they are there, happening all the time, more or less everywhere,

The search for a solution to the traditional measurement problem is at the basis of most interpretations of quantum mechanics. In this paper we have summed up four of these interpretations which succeed in avoiding the quantum superposition of macroscopically distinct states, an important element of the traditional measurement problem. Every particular interpretation provides a particular point of view on the traditional measurement problem: (1) in Bohmian mechanics Schrödinger's evolution is not interrupted; replacing the original wave function for its "collapsed" derivative is just a pragmatic affair; (2) in decoherence the linear Schrödinger equation yields an unavoidable coupling of the quantum system with the surrounding medium, which is not a collapse but looks and smells as if it were; (3) in GRW collapses result from localizations; and (4) in SPA collapses result from jumps to

By contrast, no different interpretations of quantum mechanics are invoked to account for TBSS, as if the corresponding measurement problem were immune to the different interpretations of the theory. We have shown, however, that at least

Every proposed solution to the measurement problem should apply to both measurement problems: the traditional and that implied in TBSS. A solution to just

one interpretation of quantum mechanics does not account for TBSS.

e.g., every time a photon is either emitted or absorbed by an atom.

process itself; it is just the process which takes place just before: during

*Transitions between Stationary States and the Measurement Problem*

projections to the preferential states.

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

decoherence is completed.

**7. Conclusions**

evolution" [31].

preferential states.

**41**

one of them is not good enough.

On the one hand, in general the preferential states of the system are the eigenstates of E, which do not depend on the perturbation W(*t*). Hence no modification of these states should be considered. On the other hand, if the initial state of the system is ∣*ψ*ð Þi ¼ *t*<sup>0</sup> ∣*φ <sup>j</sup>* i, an eigenstate of E, the effect of the perturbation is to gently remove the state ∣*ψ*ð Þi *t*<sup>0</sup> from ∣*φ <sup>j</sup>*i, and yield it to the linear superposition ∣*ψ*ð Þi *t* given by Eq. (16). Once the system is in this linear superposition, it can either suddenly jump to a stationary state or remain in the Schrödinger channel. If it jumps, it can either go to a state ∣*φk*i (where *k* 6¼ *j*) or come back to its initial state ∣*φj*i. The result can be described as a system continually changing from one to another stationary state or making transitions, as Dirac asserts.

In principle SPA accounts for TBSS. By contrast, decoherence has little to contribute concerning this matter.

Assuming as valid the ideal measurement scheme, in previous papers we have addressed the traditional measurement problem as follows [4, 25].

Let *A* be the operator representing the physical quantity *α* referred to the system S. We shall denote by ∣*a <sup>j</sup>*i the eigenvector of *A* corresponding to the eigenvalue *a <sup>j</sup>* (*j* ¼ 1, 2, … Þ; for simplicity we shall refer to the discrete non-degenerate case. If the initial state of S is ∣*a <sup>j</sup>*i and the initial state of the measuring device M is ∣*m*0i, the initial state of the total system S + M (before the measurement takes place) will be denoted by ∣*a <sup>j</sup>*i ∣*m*0i. The final state of the total system (when the measurement is over) will be denoted by ∣Φi.

According to the ideal measurement scheme the Schrödinger evolution results

$$|a\_j\rangle \, |m\_0\rangle \to |\Phi\rangle = |\Phi\_j\rangle \,\tag{20}$$

This scheme is supposed to be valid in cases where the measured physical quantity is compatible with every conserved quantity referred to S + M [30].

If the initial state of S is P *j γ j* ∣*a <sup>j</sup>*i (where *γ <sup>j</sup>* 6¼ 0 for every *j* = 1, … , *N*), the linearity of the Schrödinger equation yields entangled states:

$$\left(\sum\_{j} \gamma\_{j} \left| a\_{j} \right>\right) \left| m\_{0} \right> \to \left| \Phi \right> = \sum\_{j} \gamma\_{j} \left| \Phi\_{j} \right>\tag{21}$$

The set f g *N*<sup>Φ</sup> = {∣Φ1i, … , ∣Φ*N*i} can be considered the preferential set of S + M in the state ∣Φi (as a matter of fact, f g *N*<sup>Φ</sup> clearly fulfills several of the requirements imposed to such a set). Hence, projections like ∣Φi ! ∣Φ1i, … . or ∣Φi ! ∣Φ*N*i may result. This is SPA proposed solution to the traditional measurement problem.

Decoherence invokes an alternative solution to the traditional measurement problem. Once the expansion (21) is obtained, the density matrix corresponding to the state ∣Φi is replaced by the reduced density matrix as previously done in Section 5.2 (see Eqs. (12) and (13)). It is claim that "there has been a leakage of coherence from the system to the composite entity (system + environment). Since we are not able to control this entity, *the decoherence has been completed to all practical purposes*" ([9], p. 248; emphases added).

#### *Transitions between Stationary States and the Measurement Problem DOI: http://dx.doi.org/10.5772/intechopen.91801*

Laloë points out that "decoherence is not to be confused with the measurement process itself; it is just the process which takes place just before: during decoherence, the off-diagonal elements of the density matrix vanish … " [15]. In his view "the crux of most of our difficulties with quantum mechanics is the question: what is exactly the process that forces Nature … to make its choice among the various possibilities for the results of experiments?" [15]. SPA answers: spontaneous projections to the preferential states.

SPA and decoherence are not opposed theories competing for "an explanation" to the measurement problem but cooperating theories. Projections break down the Schrödinger evolution, but they are not frequent. If the system has its preferential set, projections can take place at the very beginning of the process or not (in SPA there is always room for Schrödinger evolutions). As long as projections do not take place, decoherence can make its work entangling the system with the environment. But nothing prevents the total, entangled system, to have its preferential set. This may be why a spontaneous projection finally breaks down the superposition of states of the total system. Nature makes its choice, and it is only then that decoherence is completed.

#### **7. Conclusions**

*principle for projections resulting from measurements*, i.e., to the fact that the Schrödinger evolution is suspended when a measurement is performed.

to be made in the states of the unperturbed system, but we suppose that the perturbed system, instead of remaining permanently in one of these states, is continually changing from one to another, or making transitions, under the

On the one hand, in general the preferential states of the system are the eigenstates of E, which do not depend on the perturbation W(*t*). Hence no modification of these states should be considered. On the other hand, if the initial state of

gently remove the state ∣*ψ*ð Þi *t*<sup>0</sup> from ∣*φ <sup>j</sup>*i, and yield it to the linear superposition ∣*ψ*ð Þi *t* given by Eq. (16). Once the system is in this linear superposition, it can either suddenly jump to a stationary state or remain in the Schrödinger channel. If it jumps, it can either go to a state ∣*φk*i (where *k* 6¼ *j*) or come back to its initial state

i. The result can be described as a system continually changing from one to

In principle SPA accounts for TBSS. By contrast, decoherence has little to con-

Assuming as valid the ideal measurement scheme, in previous papers we have

Let *A* be the operator representing the physical quantity *α* referred to the system

According to the ideal measurement scheme the Schrödinger evolution results

<sup>∣</sup>*m*0i ! <sup>∣</sup>Φi ¼ <sup>X</sup>

The set f g *N*<sup>Φ</sup> = {∣Φ1i, … , ∣Φ*N*i} can be considered the preferential set of S + M in the state ∣Φi (as a matter of fact, f g *N*<sup>Φ</sup> clearly fulfills several of the requirements imposed to such a set). Hence, projections like ∣Φi ! ∣Φ1i, … . or ∣Φi ! ∣Φ*N*i may result. This is SPA proposed solution to the traditional measurement problem. Decoherence invokes an alternative solution to the traditional measurement problem. Once the expansion (21) is obtained, the density matrix corresponding to the state ∣Φi is replaced by the reduced density matrix as previously done in Section 5.2 (see Eqs. (12) and (13)). It is claim that "there has been a leakage of coherence from the system to the composite entity (system + environment). Since we are not able to control this entity, *the decoherence has been completed to all practical purposes*"

This scheme is supposed to be valid in cases where the measured physical quantity is compatible with every conserved quantity referred to S + M [30].

S. We shall denote by ∣*a <sup>j</sup>*i the eigenvector of *A* corresponding to the eigenvalue *a <sup>j</sup>* (*j* ¼ 1, 2, … Þ; for simplicity we shall refer to the discrete non-degenerate case. If the initial state of S is ∣*a <sup>j</sup>*i and the initial state of the measuring device M is ∣*m*0i, the initial state of the total system S + M (before the measurement takes place) will be denoted by ∣*a <sup>j</sup>*i ∣*m*0i. The final state of the total system (when the measurement is

another stationary state or making transitions, as Dirac asserts.

addressed the traditional measurement problem as follows [4, 25].

*j γ j*

linearity of the Schrödinger equation yields entangled states:

*γ <sup>j</sup>* j*a <sup>j</sup>*i !

X *j*

influence of the perturbation" ([1], p. 167).

the system is ∣*ψ*ð Þi ¼ *t*<sup>0</sup> ∣*φ <sup>j</sup>*

*Quantum Mechanics*

tribute concerning this matter.

over) will be denoted by ∣Φi.

If the initial state of S is P

([9], p. 248; emphases added).

**40**

∣*φj*

SPA justifies Dirac's assertion: "in [TDPT] we do not consider any modification

i, an eigenstate of E, the effect of the perturbation is to

∣*a <sup>j</sup>*i ∣*m*0i ! ∣Φi ¼ ∣Φ*j*i (20)

∣*a <sup>j</sup>*i (where *γ <sup>j</sup>* 6¼ 0 for every *j* = 1, … , *N*), the

*γ <sup>j</sup>* ∣Φ *<sup>j</sup>*i (21)

*j*

Carlton Caves declares: "Mention collapse of the wave function, and you are likely to encounter vague uneasiness or, in extreme cases, real discomfort. This uneasiness can usually be traced to a feeling that wave-function collapse lies 'outside' quantum mechanics: The real quantum mechanics is said to be the unitary Schrödinger evolution; wave-function collapse is regarded as an ugly duckling of questionable status, dragged in to interrupt the beautiful flow of Schrödinger evolution" [31].

If collapses implied in traditional measurement are regarded as an ugly duckling of questionable status, collapses implied in TBSS could result definitively unbearable. Neither observers nor measuring devices could be invoked to excuse their occurrence, but they are there, happening all the time, more or less everywhere, e.g., every time a photon is either emitted or absorbed by an atom.

The search for a solution to the traditional measurement problem is at the basis of most interpretations of quantum mechanics. In this paper we have summed up four of these interpretations which succeed in avoiding the quantum superposition of macroscopically distinct states, an important element of the traditional measurement problem. Every particular interpretation provides a particular point of view on the traditional measurement problem: (1) in Bohmian mechanics Schrödinger's evolution is not interrupted; replacing the original wave function for its "collapsed" derivative is just a pragmatic affair; (2) in decoherence the linear Schrödinger equation yields an unavoidable coupling of the quantum system with the surrounding medium, which is not a collapse but looks and smells as if it were; (3) in GRW collapses result from localizations; and (4) in SPA collapses result from jumps to preferential states.

By contrast, no different interpretations of quantum mechanics are invoked to account for TBSS, as if the corresponding measurement problem were immune to the different interpretations of the theory. We have shown, however, that at least one interpretation of quantum mechanics does not account for TBSS.

Every proposed solution to the measurement problem should apply to both measurement problems: the traditional and that implied in TBSS. A solution to just one of them is not good enough.

### **Acknowledgements**

We are indebted to Professor J.C. Centeno for many fruitful discussions. We thank Carlos Valero for the transcription of formulas into Math Type.

**References**

[1] Dirac PAM. The Principles of Quantum Mechanics. Oxford: Clarendon Press; 1930

*DOI: http://dx.doi.org/10.5772/intechopen.91801*

*Transitions between Stationary States and the Measurement Problem*

[11] Heitler W. The Quantum Theory of Radiation. 3rd ed. New York: Dover

[12] Merzbacher E. Quantum Mechanics. New York: John Wiley & Sons; 1961

[14] Burgos ME. Success and incoherence of orthodox quantum mechanics. Journal of Modern Physics. 2016;**7**:1449-1454. DOI: 10.4236/jmp.2016.712132

[15] Laloë F. Do we really understand quantum mechanics? American Journal of Physics, American Association of Physics Teachers. 2001;**69**:655-701

[16] Burgos ME. Zeno of elea shines a new light on quantum weirdness. Journal of Modern Physics. 2017;**8**:1382-1397. DOI: 10.4236/jmp.2017.88087

[17] Bohm D. A suggested interpretation of the quantum theory in terms of hidden variables. Physical Review. 1952;

[18] Bohmian GS. Mechanics. Stanford Encyclopedia of Philosophy; 2001.

[19] Dürr D, Goldstein S, Tumulka R, Zanghì N. Bohmian mechanics and quantum field theory. Physical Review Letters. 2004;**93**. Available from: https://arxiv.org/abs/quant-ph/0303156

[20] Tegmar M, Wheeler JA. 100 years of quantum mysteries. Scientific American. 2001;**284**(2):68-75

[21] Schlosshauer M. Decoherence and the Quantum-to-Classical Transition.

[22] Ghirardi GC, Rimini A, Weber T. Unified dynamics for microscopic and

Berlin: Springer-Verlag; 2007

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Revised 2017

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Publications Inc.; 1984

Company; 1965

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[3] Jammer M. The Philosophy of Quantum Mechanics. New York:

[4] Burgos ME. The measurement problem in quantum mechanics revisited. In: Pahlavani M, editor. Selected Topics in Applications of Quantum Mechanics. Croatia: IntechOpen; 2015. pp. 137-173. DOI:

[5] Burgos ME. Evidence of action-at-adistance in experiments with individual particles. Journal of Modern Physics. 2015:6, 1663-1670. DOI: 10.4236/

[6] Burgos ME, Criscuolo FG, Etter TL. Conservation laws, machines of the first type and superluminal communication. Speculations in Science and Technology.

[8] Burgos ME. Contradiction between conservation laws and orthodox

quantum mechanics. Journal of Modern Physics. 2010;**1**:137-142. DOI: 10.4236/

[9] Bes DR. Quantum Mechanics. 3rd ed. Berlin: Springer-Verlag; 2012. DOI:

[10] Cohen-Tannoudji C, Diu B, Laloë F. Quantum Mechanics. New York: John

10.1007/978-3-642-20556-9

Wiley & Sons; 1977

**43**

Berlin: Springer; 1932

John Wiley & Sons; 1974

10.5772/59209

jmp.2015.6111

1999;**21**(4):227-233

jmp.2010.12019

80-84

[7] Criscuolo FG, Burgos ME. Conservation laws in spontaneous and measurement-like individual processes. Physics Essays. 2000;**13**(1):

### **Author details**

María Esther Burgos Independent Scientist, Ciudad Autónoma de Buenos Aires, Argentina

\*Address all correspondence to: mburgos25@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.

*Transitions between Stationary States and the Measurement Problem DOI: http://dx.doi.org/10.5772/intechopen.91801*

#### **References**

**Acknowledgements**

*Quantum Mechanics*

**Author details**

**42**

María Esther Burgos

Independent Scientist, Ciudad Autónoma de Buenos Aires, Argentina

© 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: mburgos25@gmail.com

provided the original work is properly cited.

We are indebted to Professor J.C. Centeno for many fruitful discussions. We thank Carlos Valero for the transcription of formulas into Math Type.

[1] Dirac PAM. The Principles of Quantum Mechanics. Oxford: Clarendon Press; 1930

[2] von Neumann J. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer; 1932

[3] Jammer M. The Philosophy of Quantum Mechanics. New York: John Wiley & Sons; 1974

[4] Burgos ME. The measurement problem in quantum mechanics revisited. In: Pahlavani M, editor. Selected Topics in Applications of Quantum Mechanics. Croatia: IntechOpen; 2015. pp. 137-173. DOI: 10.5772/59209

[5] Burgos ME. Evidence of action-at-adistance in experiments with individual particles. Journal of Modern Physics. 2015:6, 1663-1670. DOI: 10.4236/ jmp.2015.6111

[6] Burgos ME, Criscuolo FG, Etter TL. Conservation laws, machines of the first type and superluminal communication. Speculations in Science and Technology. 1999;**21**(4):227-233

[7] Criscuolo FG, Burgos ME. Conservation laws in spontaneous and measurement-like individual processes. Physics Essays. 2000;**13**(1): 80-84

[8] Burgos ME. Contradiction between conservation laws and orthodox quantum mechanics. Journal of Modern Physics. 2010;**1**:137-142. DOI: 10.4236/ jmp.2010.12019

[9] Bes DR. Quantum Mechanics. 3rd ed. Berlin: Springer-Verlag; 2012. DOI: 10.1007/978-3-642-20556-9

[10] Cohen-Tannoudji C, Diu B, Laloë F. Quantum Mechanics. New York: John Wiley & Sons; 1977

[11] Heitler W. The Quantum Theory of Radiation. 3rd ed. New York: Dover Publications Inc.; 1984

[12] Merzbacher E. Quantum Mechanics. New York: John Wiley & Sons; 1961

[13] Messiah A. Quantum Mechanics. Amsterdam: North Holland Publishing Company; 1965

[14] Burgos ME. Success and incoherence of orthodox quantum mechanics. Journal of Modern Physics. 2016;**7**:1449-1454. DOI: 10.4236/jmp.2016.712132

[15] Laloë F. Do we really understand quantum mechanics? American Journal of Physics, American Association of Physics Teachers. 2001;**69**:655-701

[16] Burgos ME. Zeno of elea shines a new light on quantum weirdness. Journal of Modern Physics. 2017;**8**:1382-1397. DOI: 10.4236/jmp.2017.88087

[17] Bohm D. A suggested interpretation of the quantum theory in terms of hidden variables. Physical Review. 1952; **85**:166-179 180-193

[18] Bohmian GS. Mechanics. Stanford Encyclopedia of Philosophy; 2001. Revised 2017

[19] Dürr D, Goldstein S, Tumulka R, Zanghì N. Bohmian mechanics and quantum field theory. Physical Review Letters. 2004;**93**. Available from: https://arxiv.org/abs/quant-ph/0303156

[20] Tegmar M, Wheeler JA. 100 years of quantum mysteries. Scientific American. 2001;**284**(2):68-75

[21] Schlosshauer M. Decoherence and the Quantum-to-Classical Transition. Berlin: Springer-Verlag; 2007

[22] Ghirardi GC, Rimini A, Weber T. Unified dynamics for microscopic and macroscopic systems. Physical Review D. 1986;**34**:470-491

[23] Ghirardi GC, Rimini A, Weber T. Disentanglement of quantum wave functions. Physical Review D. 1987;**36**: 3287-3289

[24] Ghirardi GC. Collapse theories. Stanford Encyclopedia of Philosophy. 2002. Revised 2016

[25] Burgos ME. Which Natural Processes Have the Special Status of Measurements? Foundations of Physics. 1998;**28**(8):1323-1346

[26] Burgos ME. Unravelling the quantum maze. Journal of Modern Physics. 2018;**9**:1697-1711. DOI: 10.4236/jmp.2018.98106

[27] Burgos ME. The contradiction between two versions of quantum theory could be decided by experiment. Journal of Modern Physics. 2019;**10**: 1190-1208. DOI: 10.4236/ jmp.2019.1010079

Section 2

Foundational Problems

in Quantum Mechanics

**45**

[28] Burgos ME. Transitions to the continuum: Three different approaches. Foundations of Physics. 2008;**38**(10): 883-907

[29] Bell M, Gottfried K, Veltman M. John S. Bell on the Foundations of Quantum Mechanics. Word Scientific: Singapore; 2001

[30] Araki H, Yanase MM. Measurement of quantum mechanical operators. Physical Review. 1960;**120**(2):622-626

[31] Caves C. Quantum mechanics of measurements distributed in time. A path-integral formulation. Physical Review D. 1986;**33**:1643-1665

Section 2
