**3. The Wavefunction Entropies in R,M,and C**

Another measure of uncertainty in position is the information entropy of the probability distribution *Hx* which is the entropy in **R** and is equal to:

$$H\_{\mathbf{x}} = -\sum\_{\mathbf{x} = -\infty}^{\mathbf{x} = +\infty} |\boldsymbol{\varphi}(\mathbf{x})|^{2} L \boldsymbol{n} \left[ |\boldsymbol{\varphi}(\mathbf{x})|^{2} \mathbf{x}\_{0} \right] = -\sum\_{\mathbf{x} = \mathbf{x} - \frac{\mathbf{i}}{2}}^{\mathbf{x} = \mathbf{x} + \frac{\mathbf{i}}{2}} |\boldsymbol{\varphi}(\mathbf{x})|^{2} L \boldsymbol{n} \left[ |\boldsymbol{\varphi}(\mathbf{x})|^{2} \mathbf{x}\_{0} \right] = H\_{\mathbf{x}}^{R} = L \boldsymbol{n} \left( \frac{2L}{\varepsilon \mathbf{x}\_{0}} \right)$$

where *x*<sup>0</sup> is an arbitrary reference length [1, 2]. Take *x*<sup>0</sup> ¼ 1:

$$\begin{aligned} \Leftrightarrow & H\_x^R = -\sum\_{\substack{\mathbf{x} = \mathbf{x}\_c = \mathbf{c} \\ \mathbf{x} = \mathbf{x}\_c = \mathbf{2}}}^{\mathbf{x} = \mathbf{x}\_c + \mathbf{2}} \left[ \left| \mathbf{y}(\mathbf{x}) \right|^2 \text{Ln} \left[ \left| \mathbf{y}(\mathbf{x}) \right|^2 \right] \right] \\ &= \text{Ln} \left( \frac{2L}{e} \right) = \text{Ln} (2L) - \text{Ln} (e) = \text{Ln} (2L) - 1 = \text{Ln} (2 \times 200) - 1 = 4.991464547\dots \end{aligned}$$

<sup>⇔</sup>∀*<sup>x</sup>* : *xc* � *<sup>L</sup>* <sup>2</sup> <sup>≤</sup> *<sup>x</sup>*<sup>≤</sup> *xc* <sup>þ</sup> *<sup>L</sup>* <sup>2</sup> ,we have : *d H<sup>R</sup> x* � �≥0, that means that *H<sup>R</sup> <sup>x</sup>* is a nondecreasing series with *x* and converging to *Ln* <sup>2</sup>*<sup>L</sup> e* � � and that also in **R**, chaos and disorder are increasing with *x*.

The negative real entropy corresponding to *H<sup>R</sup> <sup>x</sup>* in **R** is *NegH<sup>R</sup> <sup>x</sup>* and is the following:

$$\begin{split} \text{Neg} & H\_{\text{x}}^{R} = -H\_{\text{x}}^{R} = \sum\_{\mathbf{x} = -\infty}^{\mathbf{x} = +\infty} |\boldsymbol{\psi}(\mathbf{x})|^{2} L \boldsymbol{n} \left[ |\boldsymbol{\psi}(\mathbf{x})|^{2} \right] = \sum\_{\mathbf{x} = \mathbf{x}\_{\varepsilon}}^{\mathbf{x} = \pm} \frac{L}{2} |\boldsymbol{\psi}(\mathbf{x})|^{2} L \boldsymbol{n} \left[ |\boldsymbol{\psi}(\mathbf{x})|^{2} \right] = -L \boldsymbol{n} \left( \frac{2L}{e} \right) \\ &= 1 - L \boldsymbol{n} (2L) = 1 - L \boldsymbol{n} (2 \times 200) = -4.991464547\dots \end{split}$$

<sup>⇔</sup>∀*<sup>x</sup>* : *xc* � *<sup>L</sup>* <sup>2</sup> <sup>≤</sup> *<sup>x</sup>*<sup>≤</sup> *xc* <sup>þ</sup> *<sup>L</sup>* <sup>2</sup> ,we have : *d NegH<sup>R</sup> x* � �≤0, which means that *NegH<sup>R</sup> <sup>x</sup>* is a nonincreasing series with *<sup>x</sup>* and converging to �*Ln* <sup>2</sup>*<sup>L</sup> e* � �. Therefore, if *H<sup>R</sup> <sup>x</sup>* measures in **R** the amount of disorder, of uncertainty, of chaos, of ignorance, of unpredictability, and of information gain in a random system then since *NegH<sup>R</sup> <sup>x</sup>* ¼ �*H<sup>R</sup> <sup>x</sup>* , that means the opposite of *H<sup>R</sup> <sup>x</sup>* , *NegH<sup>R</sup> <sup>x</sup>* measures in **R** the amount of order, of certainty, of predictability, and of information loss in a stochastic system.

The complementary real entropy to *H<sup>R</sup> <sup>x</sup>* in **<sup>R</sup>** is *<sup>H</sup><sup>R</sup> <sup>x</sup>* and is the following:

$$\overline{H}\_{\mathbf{x}}^{\mathbb{R}} = -\sum\_{\mathbf{x}=-\infty}^{\mathbf{x}=+\infty} \left[\mathbf{1} - \left|\boldsymbol{\nu}(\mathbf{x})\right|^{2}\right] \text{Ln}\left[\mathbf{1} - \left|\boldsymbol{\nu}(\mathbf{x})\right|^{2}\right] = -\sum\_{\mathbf{x}=\mathbf{x}\_{\mathbf{c}}+\frac{\mathbf{L}}{2}}^{\mathbf{x}=\mathbf{x}\_{\mathbf{c}}+\frac{\mathbf{L}}{2}} \left[\mathbf{1} - \left|\boldsymbol{\nu}(\mathbf{x})\right|^{2}\right] \text{Ln}\left[\mathbf{1} - \left|\boldsymbol{\nu}(\mathbf{x})\right|^{2}\right] = \mathbf{1}$$

In the complementary real probability set to **R**, we denote the corresponding real entropy by *H<sup>R</sup> x* .

The meaning of *H<sup>R</sup> <sup>x</sup>* is the following: it is the real entropy in the real set **R** and which is related to the complementary real probability *Pm=i* ¼ 1 � *Pr*.

$$\begin{aligned} \text{Let } & \boldsymbol{\mathfrak{x}} : \boldsymbol{\mathfrak{x}}\_{\boldsymbol{\mathfrak{t}}} - \frac{\boldsymbol{L}}{2} \leq \boldsymbol{\mathfrak{x}} \leq \boldsymbol{\mathfrak{x}}\_{\boldsymbol{\mathfrak{t}}} + \frac{\boldsymbol{L}}{2} \text{, we have } : \quad d\left[ \overline{\boldsymbol{H}}\_{\boldsymbol{\mathfrak{x}}}^{\mathbb{R}} \right] \geq \boldsymbol{0}, \text{ that means that } \overline{\boldsymbol{H}}\_{\boldsymbol{\mathfrak{x}}}^{\mathbb{R}} \text{ is a.e.}\\ \text{If } & \boldsymbol{\mathfrak{x}} : \boldsymbol{\mathfrak{x}} \text{ is a.e., then } \quad \boldsymbol{\mathfrak{x}} : \boldsymbol{\mathfrak{x}} \text{ is a.e.} \end{aligned}$$

nondecreasing series with *x* and converging to 1 and that also means that in the complementary real probability set to **R**, chaos and disorder are increasing with *x*.

In the complementary imaginary probability set **M** to the set **R**, we denote the corresponding imaginary entropy by *H<sup>M</sup> <sup>x</sup>* . The meaning of *H<sup>M</sup> <sup>x</sup>* is the following: it is the

imaginary entropy in the imaginary set **M** and which is related to the complementary imaginary probability *Pm* <sup>¼</sup> *<sup>i</sup>*ð Þ <sup>1</sup> � *Pr* . The complementary entropy to *<sup>H</sup><sup>R</sup> <sup>x</sup>* in **M** is *H<sup>M</sup> x* and is computed as follows:

*H<sup>M</sup> <sup>x</sup>* ¼ � *<sup>x</sup>*X¼þ<sup>∞</sup> *x*¼�∞ *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln i* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln i* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i *Lni* <sup>þ</sup> *Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *i Lni* <sup>þ</sup> *Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Lni* � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *iLni* <sup>þ</sup> *iLn* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i � *<sup>i</sup>* j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Lni* � *<sup>i</sup>* j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *iLni* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *iLni* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i � *<sup>i</sup>* <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i*Ln* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i ¼ � <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 *iLni* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i <sup>þ</sup> *iH<sup>R</sup> <sup>x</sup>* ¼ �*iLni* <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> h i <sup>þ</sup> *iH<sup>R</sup> x* ¼ �*iLni x*¼ X*xc*þ*<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*�*<sup>L</sup>* 2 1 � *x*¼ X*xc*þ*<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*�*<sup>L</sup>* 2 j j *ψ*ð Þ *x* 2 8 < : 9 = ; <sup>þ</sup> *iH<sup>R</sup> x* ¼ �*iLni xc* þ *L* 2 � � � *xc* � *<sup>L</sup>* 2 � � þ 1 � � � 1 � � <sup>þ</sup> *iH<sup>R</sup> <sup>x</sup>* since <sup>X</sup> *<sup>x</sup>*¼*xc*<sup>þ</sup> *<sup>L</sup>* 2 *<sup>x</sup>*¼*xc*� *<sup>L</sup>* 2 j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> <sup>¼</sup> <sup>1</sup> ¼ �ð Þ *iLni <sup>L</sup>* <sup>þ</sup> *iH<sup>R</sup> x*

From the properties of logarithms, we have: *<sup>θ</sup>Lnx* <sup>¼</sup> *Ln x<sup>θ</sup>* � � then *iLni* <sup>¼</sup> *Lni<sup>i</sup>* . Moreover, Leonhard Euler's formula for complex numbers gives: *<sup>e</sup>i<sup>θ</sup>* <sup>¼</sup> cos *<sup>θ</sup>* <sup>þ</sup> *<sup>i</sup>*sin *<sup>θ</sup>*. Take *<sup>θ</sup>* <sup>¼</sup> *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*kπ*⇔*ei*ð Þ *<sup>π</sup>=*2þ2*k<sup>π</sup>* <sup>¼</sup> cosð Þþ *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*k<sup>π</sup> <sup>i</sup>*sin ð Þ¼ *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*k<sup>π</sup>*

0 þ *i*ð Þ¼ 1 *i*, then:

*i <sup>i</sup>* <sup>¼</sup> *ei*ð Þ *<sup>π</sup>=*2þ2*k<sup>π</sup>* � �*<sup>i</sup>* <sup>¼</sup> *ei* <sup>2</sup>ð Þ *<sup>π</sup>=*2þ2*k<sup>π</sup>* <sup>¼</sup> *<sup>e</sup>*�ð Þ *<sup>π</sup>=*2þ2*k<sup>π</sup>* since *<sup>i</sup>* <sup>2</sup> ¼ �1, therefore:

�*iLni* ¼ �*Lni<sup>i</sup>* ¼ �*Ln e*�ð Þ *<sup>π</sup>=*2þ2*k<sup>π</sup>* � � <sup>¼</sup> *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*k<sup>π</sup>* since *Ln e*½�¼ 1 and where *<sup>k</sup>* belongs to the set of integer numbers Z.

Consequently,

$$H\_{\mathbf{x}}^{M} = -(iL\boldsymbol{n}\mathbf{i})L + i\overline{H}\_{\mathbf{x}}^{\mathbb{R}} = (\pi/2 + 2k\pi)L + i\overline{H}\_{\mathbf{x}}^{\mathbb{R}}$$

That means that *H<sup>M</sup> <sup>x</sup>* is a complex number where:

the real part is: Re *H<sup>M</sup> x* � � <sup>¼</sup> ð Þ *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*k<sup>π</sup> <sup>L</sup>*, and the imaginary part is: Im *<sup>H</sup><sup>M</sup> x* � � <sup>¼</sup> *<sup>H</sup><sup>R</sup> x* . For *k* ¼ �1 then

Re *H<sup>M</sup> x* � � ¼ �ð Þ <sup>3</sup>*π=*<sup>2</sup> *<sup>L</sup>* ¼ �4*:*71238898*<sup>L</sup>* ¼ �942*:*<sup>4777961</sup> … for *<sup>L</sup>* <sup>¼</sup> 200.

For *<sup>k</sup>* <sup>¼</sup> 0 then Re *<sup>H</sup><sup>M</sup> x* � � <sup>¼</sup> ð Þ *<sup>π</sup>=*<sup>2</sup> *<sup>L</sup>* <sup>¼</sup> <sup>1</sup>*:*570796327*<sup>L</sup>* <sup>¼</sup> <sup>314</sup>*:*<sup>1592654</sup> … for *<sup>L</sup>* <sup>¼</sup> 200. For *k* ¼ 1 then

Re *H<sup>M</sup> x* � � <sup>¼</sup> ð Þ <sup>5</sup>*π=*<sup>2</sup> *<sup>L</sup>* <sup>¼</sup> <sup>7</sup>*:*853981634*<sup>L</sup>* <sup>¼</sup> <sup>1570</sup>*:*<sup>796327</sup> … for *<sup>L</sup>* <sup>¼</sup> 200, etc.

Finally, the entropy *H<sup>C</sup> <sup>x</sup>* in **C** = **R** þ**M** is the following:

$$\begin{aligned} H\_{\mathbf{x}}^{C} &= -\sum\_{\mathbf{x}=\mathbf{x}\_{\cdot}}^{L} \operatorname{Pc}(\mathbf{x}) L n [\operatorname{Pc}(\mathbf{x})] \\ &\quad \mathop{\rm x=\mathbf{x}\_{\cdot}-\frac{L}{2}} \\ &= -\sum\_{\mathbf{x}=\mathbf{x}\_{\cdot}+\frac{L}{2}}^{L} \mathbf{1} \times L n [\mathbf{1}] = -\sum\_{\mathbf{x}=\mathbf{x}\_{\cdot}-\frac{L}{2}}^{L} (\mathbf{1} \times \mathbf{0}) = \mathbf{0} \\ &= H\_{\mathbf{x}}^{R} + N \text{eg} H\_{\mathbf{x}}^{R} \end{aligned}$$

<sup>⇔</sup>∀*<sup>x</sup>* : *xc* � *<sup>L</sup>* <sup>2</sup> <sup>≤</sup> *<sup>x</sup>*<sup>≤</sup> *xc* <sup>þ</sup> *<sup>L</sup>* <sup>2</sup> , we have: *d H<sup>C</sup> x* � � <sup>¼</sup> 0, that means that *<sup>H</sup><sup>C</sup> <sup>x</sup>* is a constant series with *x* and is always equal to 0. That means also and most importantly, for the wavefunction position distribution and in the probability set and universe **C** ¼ **R** þ**M**, we have complete order, no chaos, no ignorance, no uncertainty, no disorder, no randomness, no information loss or gain but a conservation of information, and no unpredictability since all measurements are completely and perfectly deterministic (*Pc x*ð Þ¼ 1 and *<sup>H</sup><sup>C</sup> <sup>x</sup>* ¼ 0).

Similarly, we can determine another measure of uncertainty in momentum which is the information entropy of the probability distribution *Hp* and which is [1, 2]:

$$H\_p = -\sum\_{p=-\infty}^{p=+\infty} |\phi(p)|^2 L n\left[|\phi(p)|^2 p\_0\right] = L n \left(\frac{4\pi\hbar e^{2(1-\gamma)}}{L p\_0}\right) = \lim\_{n \to +\infty} H\_p(n)$$

Where *γ* is Euler's constant and is equal to: 0.577215664901532 …

For *p*<sup>0</sup> ¼ 1 we can compute all the defined entropies in **R**, **M**, and **C** and which are [1–30]:

$$H\_p^R = -\sum\_{p=-\infty}^{p=+\infty} |\phi(p)|^2 L n\left[|\phi(p)|^2\right] = L n\left(\frac{4\pi\hbar e^{2(1-r)}}{L}\right) = \lim\_{n \to +\infty} H\_p(n)$$

$$N \text{eg} H\_p^R = \sum\_{p=-\infty}^{p=+\infty} |\phi(p)|^2 L n\left[|\phi(p)|^2\right] = -L n\left(\frac{4\pi\hbar e^{2(1-r)}}{L}\right) = -\lim\_{n \to +\infty} H\_p(n)$$

$$H\_p^R = -\sum\_{p=-\infty}^{p=+\infty} \left[1 - |\phi(p)|^2\right] L n\left[1 - |\phi(p)|^2\right]$$

$$H\_p^M = -\sum\_{p=-\infty}^{p=+\infty} i\left[1 - |\phi(p)|^2\right] L n\left\{i\left[1 - |\phi(p)|^2\right]\right\}$$

$$H\_p^C = -\sum\_{p=-\infty}^{p=+\infty} P c(p) L n \left[P c(p)\right] = -\sum\_{p=-\infty}^{p=+\infty} \mathbf{1} \times L n(1) = -\sum\_{p=-\infty}^{p=+\infty} (\mathbf{1} \times 0) = \mathbf{0} = H\_p^R + N \text{eg} H\_p^R$$

That means also and most importantly, for the wavefunction momentum distribution and in the probability set and universe **C** ¼ **R** þ**M**, we have complete order, no chaos, no ignorance, no uncertainty, no disorder, no randomness, no information loss or gain but a conservation of information, and no unpredictability since all measurements are completely and perfectly deterministic (*Pc p*ð Þ¼ 1 and *<sup>H</sup><sup>C</sup> <sup>p</sup>* ¼ 0).

*p*

The quantum mechanical entropic uncertainty principle states that for *x*0*p*<sup>0</sup> ¼ ℏ then:

*H<sup>R</sup> <sup>x</sup>* <sup>þ</sup> *<sup>H</sup><sup>R</sup> <sup>p</sup>* ð Þ *n* ≥*Ln e*ð Þffi *π* 2*:*144729886 … nats, (base *e* in *Ln* gives the "natural units" nat).

For *x*0*p*<sup>0</sup> ¼ ℏ, the sum of the position and momentum entropies yields:

*H<sup>R</sup> <sup>x</sup>* <sup>þ</sup> *<sup>H</sup><sup>R</sup> <sup>p</sup>* ð Þ¼ <sup>∞</sup> *Ln* <sup>8</sup>*πe*<sup>1</sup>�2*<sup>γ</sup>* ð Þffi <sup>3</sup>*:*<sup>069740098</sup> … nats, (base *<sup>e</sup>* in *Ln* gives the "natural units" nat).

which satisfies the quantum entropic uncertainty principle.

The following figures (**Figures 38**–**51**) illustrate all the computations done above.
