The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem – The Momentum Wavefunction and the Wavefunction Entropies

Abstract

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R+M independently of all the probabilities of the input stochastic variable distribution in R, and subsequently, the output of the random phenomenon in R can be determined perfectly in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C=R+M.

1. Introduction

1.1 The momentum wavefunction and CPP

1.1.1 The momentum wavefunction probability distribution and CPP

The probability density for finding a particle with a given momentum is derived from the wavefunction as $f\left(p\right)={\left|\varphi \left(p\right)\right|}^2$. As with position, the wavefunction momentum probability density function (PDF) for finding the particle at a given momentum depends upon its state, and is given by [1, 2]:

$$f\left(p\right)={\left|\varphi \left(p\right)\right|}^2=\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]$$

Where $\mathrm{\hslash }=\frac{h}{2\pi }$ is the reduced Planck constant and $sinc\left(x\right)=\frac{sin\left(x\right)}{x}$ is the cardinal sine sinc function.

Therefore, the wavefunction momentum cumulative probability distribution function (CDF) which is equal to $P_r\left(P\right)$ in $\mathcal{R}$ is:

$$\begin{array}{lll}{P}_r\left(P\right)& =& F\left(p_j\right)=P_{\mathit{rob}}\left(P\le p_j\right)=\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\\ & =& \underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

And the real complementary probability to $P_r\left(P\right)$ in $\mathcal{R}$ which is $P_m\left(P\right)/i$ is:

$$\begin{array}{lll}{P}_m\left(P\right)/i& =& 1-P_r\left(P\right)=\begin{array}{l}1-F\left(p_j\right)=1-P_{\mathit{rob}}\left(P\le p_j\right)=P_{\mathit{rob}}\left(P>p_j\right)\\ =1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}=\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\end{array}\\ & & =1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ & & =\underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

Consequently, the imaginary complementary probability to $P_r\left(P\right)$ in $\mathcal{M}$ which is $P_m\left(P\right)$ is:

$$\begin{array}{l}{P}_m\left(P\right)=i\left[1-P_r\left(P\right)\right]=i\left[1-F\left(p_j\right)\right]=i\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]={\mathit{iP}}_{\mathit{rob}}\left(P>p_j\right)\\ =i\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]=i\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\\ =i\left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]\\ =i\underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

Furthermore, the complex random number or vector in $\mathcal{C}=\mathcal{R}+\mathcal{M}$ which is $Z\left(P\right)$ is:

$$\begin{array}{lll}Z\left(P\right)& =& P_r\left(P\right)+P_m\left(P\right)=P_r\left(P\right)+i\left[1-P_r\left(P\right)\right]=F\left(p_j\right)+i\left[1-F\left(p_j\right)\right]\\ & =& P_{\mathit{rob}}\left(P\le p_j\right)+i\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]=P_{\mathit{rob}}\left(P\le p_j\right)+{\mathit{iP}}_{\mathit{rob}}\left(P>p_j\right)\\ & =& \underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+i\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]=\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+i\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\\ & =& \underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ & & +i\left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]\\ & =& \underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ & & +i\underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

Additionally, the degree of our knowledge which is $\mathit{DOK}\left(P\right)$ is:

$$\begin{array}{l}\mathit{DOK}\left(P\right)={\left[P_r\left(P\right)\right]}^2+{\left[P_m\left(P\right)/i\right]}^2={\left[P_r\left(P\right)\right]}^2+{\left[1-P_r\left(P\right)\right]}^2={\left[F\left(p_j\right)\right]}^2+{\left[1-F\left(p_j\right)\right]}^2\\ ={\left[P_{\mathit{rob}}\left(P\le p_j\right)\right]}^2+{\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]}^2={\left[P_{\mathit{rob}}\left(P\le p_j\right)\right]}^2+{\left[P_{\mathit{rob}}\left(P>p_j\right)\right]}^2\end{array}$$

$$={\left[\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]}^2+{\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]}^2={\left[\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]}^2+{\left[\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]}^2$$

$$\begin{array}{l}={\left[\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]}^2\\ +{\left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]}^2\end{array}$$

$$\begin{array}{l}={\left[\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]}^2\\ +{\left[\underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]}^2\end{array}$$

Moreover, the chaotic factor which is $\mathit{Chf}\left(P\right)$ is:

$$\begin{array}{lll}\mathit{Chf}\left(P\right)& =& 2{\mathit{iP}}_r\left(P\right)P_m\left(P\right)\\ & =& 2{\mathit{iP}}_r\left(P\right)\times i\left[1-P_r\left(P\right)\right]=-2P_r\left(P\right)\left[1-P_r\left(P\right)\right]=-2F\left(p_j\right)\left[1-F\left(p_j\right)\right]\\ & =& -2P_{\mathit{rob}}\left(P\le p_j\right)\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]=-2P_{\mathit{rob}}\left(P\le p_j\right)P_{\mathit{rob}}\left(P>p_j\right)\\ & =& -2\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\times \left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]=-2\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\times \underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\\ & =& -2\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ & & \times \left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]\\ & =& -2\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ & & \times \underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

In addition, the magnitude of the chaotic factor which is $\mathit{\text{MChf}}\left(P\right)$ is:

$$\begin{array}{l}\mathit{\text{MChf}}\left(P\right)=\left|\mathit{Chf}\left(P\right)\right|=-2{\mathit{iP}}_r\left(P\right)P_m\left(P\right)=-2{\mathit{iP}}_r\left(P\right)\times i\left[1-P_r\left(P\right)\right]\\ =2P_r\left(P\right)\left[1-P_r\left(P\right)\right]=2F\left(p_j\right)\left[1-F\left(p_j\right)\right]\\ =2P_{\mathit{rob}}\left(P\le p_j\right)\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]=2P_{\mathit{rob}}\left(P\le p_j\right)P_{\mathit{rob}}\left(P>p_j\right)\\ =2\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\times \left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]=2\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\times \underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\end{array}$$

$$\begin{array}{l}=2\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ \times \left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]\end{array}$$

$$\begin{array}{l}=2\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ \times \underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}$$

Finally, the real probability in the complex probability universe $\mathcal{C}=\mathcal{R}+\mathcal{M}$ which is $\mathit{Pc}\left(P\right)$ is:

$$\begin{array}{lll}{\mathit{Pc}}^2\left(P\right)& =& {\left\{\left[P_r\left(P\right)\right]+\left[P_m\left(P\right)/i\right]\right\}}^2={\left\{\left[P_r\left(P\right)\right]+\left[1-P_r\left(P\right)\right]\right\}}^2\\ & =& {\left\{\left[F\left(p_j\right)\right]+\left[1-F\left(p_j\right)\right]\right\}}^2={\left\{P_{\mathit{rob}}\left(P\le p_j\right)+\left[1-P_{\mathit{rob}}\left(P\le p_j\right)\right]\right\}}^2\\ & =& {\left\{P_{\mathit{rob}}\left(P\le p_j\right)+P_{\mathit{rob}}\left(P>p_j\right)\right\}}^2\\ & =& {\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]\right\}}^2\\ & =& {\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2={\left\{\underset{-\infty }{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2\\ & =& {\left\{\begin{array}{l}\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ +\left[1-\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right]\end{array}\right\}}^2\\ & =& {\left\{\begin{array}{l}\underset{-\infty }{\overset{p_j}{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\\ +\underset{p_j}{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\end{array}\right\}}^2\\ & =& {\left\{\underset{-\infty }{\overset{+\infty }{\int }}\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}\right\}}^2=1^2=1=\mathit{Pc}\left(P\right)\end{array}$$

And, $\mathit{Pc}\left(P\right)$ can be computed using CPP as follows:

$$\begin{array}{lll}{\mathit{Pc}}^2\left(P\right)& =& \mathit{DOK}\left(P\right)-\mathit{Chf}\left(P\right)={\left[P_r\left(P\right)\right]}^2+{\left[P_m\left(P\right)/i\right]}^2-2{\mathit{iP}}_r\left(P\right)P_m\left(P\right)\\ & =& {\left[P_r\left(P\right)\right]}^2+{\left[1-P_r\left(P\right)\right]}^2+2P_r\left(P\right)\left[1-P_r\left(P\right)\right]={\left\{P_r\left(P\right)+\left[1-P_r\left(P\right)\right]\right\}}^2\end{array}$$

$$\begin{array}{l}={\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]\right\}}^2={\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2\\ ={\left\{\underset{-\infty }{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2\end{array}$$

$$=1^2=1=\mathit{Pc}\left(P\right)$$

And, $\mathit{Pc}\left(P\right)$ can be computed using always CPP as follows also:

$$\begin{array}{lll}{\mathit{Pc}}^2\left(P\right)& =& \mathit{DOK}\left(P\right)+\mathit{\text{MChf}}\left(P\right)={\left[P_r\left(P\right)\right]}^2+{\left[P_m\left(P\right)/i\right]}^2+\left[-2{\mathit{iP}}_r\left(P\right)P_m\left(P\right)\right]\\ & =& {\left[P_r\left(P\right)\right]}^2+{\left[1-P_r\left(P\right)\right]}^2+2P_r\left(P\right)\left[1-P_r\left(P\right)\right]={\left\{P_r\left(P\right)+\left[1-P_r\left(P\right)\right]\right\}}^2\\ & =& {\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\left[1-\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right]\right\}}^2\\ & =& {\left\{\underset{-\infty }{\overset{p_j}{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\underset{p_j}{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2={\left\{\underset{-\infty }{\overset{+\infty }{\int }}{\left|\varphi \left(p\right)\right|}^2\mathit{dp}\right\}}^2=1^2=1=\mathit{Pc}\left(P\right)\end{array}$$

Hence, the prediction of all the wavefunction momentum probabilities of the random infinite potential well problem in the universe $\mathcal{C}=\mathcal{R}+\mathcal{M}$ is permanently certain and perfectly deterministic.

1.1.2 The new model simulations

The following figures (Figures 1–37) illustrate all the calculations done above.

Figure 1.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 1.

Figure 2.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 1.

Figure 3.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 1.

Figure 4.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 1.

Figure 5.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 1.

Figure 6.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 2.

Figure 7.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 2.

Figure 8.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 2.

Figure 9.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 2.

Figure 10.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 2.

Figure 11.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 3.

Figure 12.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 3.

Figure 13.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 3.

Figure 14.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 3.

Figure 15.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 3.

Figure 16.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 4.

Figure 17.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 4.

Figure 18.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 4.

Figure 19.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 4.

Figure 20.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 4.

Figure 21.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 5.

Figure 22.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 5.

Figure 23.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 5.

Figure 24.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 5.

Figure 25.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 5.

Figure 26.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 6.

Figure 27.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 6.

Figure 28.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 6.

Figure 29.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 6.

Figure 30.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 6.

Figure 31.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 7.

Figure 32.

The graphs of all the CPP parameters as functions of the random variable P for the wavefunction momentum probability distribution for n = 7.

Figure 33.

The graphs of DOK and Chf and the deterministic probability Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 7.

Figure 34.

The graphs of P_r and P_m/i and Pc in terms of P and of each other for the wavefunction momentum probability distribution for n = 7.

Figure 35.

The graphs of the probabilities P_r and P_m and Z in terms of P for the wavefunction momentum probability distribution for n = 7.

Figure 36.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 12.

Figure 37.

The graph of the PDF of the wavefunction momentum probability distribution as a function of the random variable P for n = 100.

1.1.2.1 Simulations interpretation

In Figures 1,6,11,16,21,26,31,36, and37 we can see the graphs of the probability density functions (PDF) of the wavefunction momentum probability distribution for this problem as functions of the random variable P for n = 1, 2, 3, 4, 5, 6, 7, 12, 100.

In Figures 2,7,12,17,22,27, and32 we can see also the graphs and the simulations of all the CPP parameters (Chf, MChf, DOK, P_r, P_m/i, Pc) as functions of the random variable P for the wavefunction momentum probability distribution of the infinite potential well problem for n = 1, 2, 3, 4, 5, 6, 7. Hence, we can visualize all the new paradigm functions for this problem.

In the cubes (Figures 3,8,13,18,23,28, and 33), the simulation of DOK and Chf as functions of each other and the random variable P for the infinite potential well problem wavefunction momentum probability distribution can be seen. The thick line in cyan is the projection of the plane Pc²(P) = DOK(P) – Chf(P) = 1 = Pc(P) on the plane P = L_b = lower bound of P. This thick line starts at the point (DOK = 1, Chf = 0) when P = L_b, reaches the point (DOK = 0.5, Chf = −0.5) when P = 0, and returns at the end to (DOK = 1, Chf = 0) when P = U_b = upper bound of P. The other curves are the graphs of DOK(P) (red) and Chf(P) (green, blue, pink) in different simulation planes. Notice that they all have a minimum at the point (DOK = 0.5, Chf = −0.5, P = 0). The last simulation point corresponds to (DOK = 1, Chf = 0, P = U_b).

In the cubes (Figures 4,9,14,19,24,29, and 34), we can notice the simulation of the real probability P_r(P) in $\mathcal{R}$ and its complementary real probability P_m(P)/i in $\mathcal{R}$ also in terms of the random variable P for the infinite potential well problem wavefunction momentum probability distribution. The thick line in cyan is the projection of the plane Pc²(P) = P_r(P) + P_m(P)/i = 1 = Pc(P) on the plane P = L_b = lower bound of P. This thick line starts at the point (P_r = 0, P_m/i = 1) and ends at the point (P_r = 1, P_m/i = 0). The red curve represents P_r(P) in the plane P_r(P) = P_m(P)/i in light gray. This curve starts at the point (P_r = 0, P_m/i = 1, P = L_b = lower bound of P), reaches the point (P_r = 0.5, P_m/i = 0.5, P = 0), and gets at the end to (P_r = 1, P_m/i = 0, P = U_b = upper bound of P). The blue curve represents P_m(P)/i in the plane in cyan P_r(P) + P_m(P)/i = 1 = Pc(P). Notice the importance of the point which is the intersection of the red and blue curves at P = 0 and when P_r(P) = P_m(P)/i = 0.5.

In the cubes (Figures 5,10,15,20,25,30, and 35), we can notice the simulation of the complex probability Z(P) in $\mathcal{C}=\mathcal{R}+\mathcal{M}$ as a function of the real probability P_r(P) = Re(Z) in $\mathcal{R}$ and of its complementary imaginary probability P_m(P) = i × Im(Z) in $\mathcal{M}$, and this in terms of the random variable P for the infinite potential well problem wavefunction momentum probability distribution. The red curve represents P_r(P) in the plane P_m(P) = 0 and the blue curve represents P_m(P) in the plane P_r(P) = 0. The green curve represents the complex probability Z(P) = P_r(P) + P_m(P) = Re(Z) + i × Im(Z) in the plane P_r(P) = iP_m(P) + 1 or Z(P) plane in cyan. The curve of Z(P) starts at the point (P_r = 0, P_m = i, P = L_b = lower bound of P) and ends at the point (P_r = 1, P_m = 0, P = U_b = upper bound of P). The thick line in cyan is P_r(P = L_b) = iP_m(P = L_b) + 1 and it is the projection of the Z(P) curve on the complex probability plane whose equation is P = L_b. This projected thick line starts at the point (P_r = 0, P_m = i, P = L_b) and ends at the point (P_r = 1, P_m = 0, P = L_b). Notice the importance of the point corresponding to P = 0 and Z = 0.5 + 0.5i when P_r = 0.5 and P_m = 0.5i.

1.1.3 The characteristics of the momentum probability distribution

In quantum mechanics, the average, or expectation value of the momentum of a particle is given by: $⟨p⟩=\underset{-\infty }{\overset{+\infty }{\int }}p{\left|\varphi \left(p\right)\right|}^2\mathit{dp}=\underset{-\infty }{\overset{+\infty }{\int }}p\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\mathit{dp}$.

For the steady state particle in a box, it can be shown that the average momentum is always $⟨p⟩=0$ regardless of the state of the particle. In the probability set and universe $\mathcal{R}$, we have:

$${⟨p⟩}_R=⟨p⟩=0$$

The variance in the momentum is a measure of the uncertainty in momentum of the particle, so in the probability set and universe $\mathcal{R}$, we have:

$$\begin{array}{lll}{\mathrm{Var}}_{p,R}& =& \mathrm{Var}\left(p\right)={⟨p^2⟩}_R-{⟨p⟩}_R^2=\underset{-\infty }{\overset{+\infty }{\int }}{p}^2{\left|\varphi \left(p\right)\right|}^2\mathit{dp}-0\\ & =& \underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left\{\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\right\}\mathit{dp}={\left(\frac{\mathrm{\hslash }\mathit{n\pi }}{L}\right)}^2\end{array}$$

In the probability set and universe $\mathcal{M}$, we have:

$$\begin{array}{l}{⟨p⟩}_M=\underset{-\infty }{\overset{+\infty }{\int }}p\left\{i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\right\}\mathit{dp}=i\underset{-\infty }{\overset{+\infty }{\int }}p\left\{1-\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\right\}\mathit{dp}\\ =i\left\{\underset{-\infty }{\overset{+\infty }{\int }}\mathit{pdp}-\underset{-\infty }{\overset{+\infty }{\int }}p\left\{\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\right\}\mathit{dp}\right\}\\ =i\left\{{\left[\frac{p^2}{2}\right]}_{-\infty }^{+\infty }-{⟨p⟩}_R\right\}=i\left\{{\left[\frac{p^2}{2}\right]}_{-U_b}^{U_b}-{⟨p⟩}_R\right\}=i\left\{0-0\right\}=0\end{array}$$

$$\begin{array}{l}{\mathrm{Var}}_{p,M}={⟨p^2⟩}_M-{⟨p⟩}_M^2\\ =\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left\{i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\right\}\mathit{dp}-0\\ =i\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left\{1-\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\right\}\mathit{dp}\\ =i\left\{\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\mathit{dp}-\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left\{\frac{L}{\pi \mathrm{\hslash }}{\left(\frac{\mathit{n\pi }}{\mathit{n\pi }+\mathit{pL}/\mathrm{\hslash }}\right)}^2{sinc}^2\left[\frac{1}{2}\left(\mathit{n\pi }-\mathit{pL}/\mathrm{\hslash }\right)\right]\right\}\mathit{dp}\right\}\end{array}$$

$$=i\left\{\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\mathit{dp}-{\mathrm{Var}}_{p,R}\right\}=i\left\{{\left[\frac{p^3}{3}\right]}_{-\infty }^{+\infty }-{\mathrm{Var}}_{p,R}\right\}\to i\left\{+\infty -{\left(\frac{\mathrm{\hslash }\mathit{n\pi }}{L}\right)}^2\right\}\to +\infty$$

In the probability set and the universe $\mathcal{C}=\mathcal{R}+\mathcal{M}$, we have from CPP:

$$\begin{array}{l}{⟨p⟩}_C=\underset{-\infty }{\overset{+\infty }{\int }}p\left[z\left(p\right)\right]\mathit{dp}=\underset{-\infty }{\overset{+\infty }{\int }}p\left\{{\left|\varphi \left(p\right)\right|}^2+i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\right\}\mathit{dp}\\ =\underset{-\infty }{\overset{+\infty }{\int }}p{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\underset{-\infty }{\overset{+\infty }{\int }}\mathit{pi}\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\mathit{dp}\\ ={⟨p⟩}_R+{⟨p⟩}_M=0+i\left(0\right)=0\end{array}$$

$$\begin{array}{lll}{\mathrm{Var}}_{p,C}& =& {⟨p^2⟩}_C-{⟨p⟩}_C^2=\left[\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left[z\left(p\right)\right]\mathit{dp}\right]-{\left[{⟨p⟩}_R+{⟨p⟩}_M\right]}^2\\ & =& \left[\underset{-\infty }{\overset{+\infty }{\int }}{p}^2\left\{{\left|\varphi \left(p\right)\right|}^2+i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\right\}\mathit{dp}\right]-{\left[{⟨p⟩}_R+{⟨p⟩}_M\right]}^2\\ & =& \left[\underset{-\infty }{\overset{+\infty }{\int }}{p}^2{\left|\varphi \left(p\right)\right|}^2\mathit{dp}+\underset{-\infty }{\overset{+\infty }{\int }}{p}^{2}i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\mathit{dp}\right]-{\left[{⟨p⟩}_R+{⟨p⟩}_M\right]}^2\\ & =& \left[{⟨p^2⟩}_R+{⟨p^2⟩}_M\right]-{\left[{⟨p⟩}_R+{⟨p⟩}_M\right]}^2\\ & =& \left[{⟨p^2⟩}_R+{⟨p^2⟩}_M\right]-\left[{⟨p⟩}_R^2+{⟨p⟩}_M^2+2{⟨p⟩}_R{⟨p⟩}_M\right]\\ & =& \left[{⟨p^2⟩}_R-{⟨p⟩}_R^2\right]+\left[{⟨p^2⟩}_M-{⟨p⟩}_M^2\right]-2{⟨p⟩}_R{⟨p⟩}_M\\ & =& {\mathrm{Var}}_{p,R}+{\mathrm{Var}}_{p,M}-2{⟨p⟩}_R{⟨p⟩}_M\to {\left(\frac{\mathrm{\hslash }\mathit{n\pi }}{L}\right)}^2+\infty -2\left(0\right)\left(0\right)\to +\infty \end{array}$$

The following tables (Tables 1–4) compute the momentum distribution characteristics for $L=200$, $h=1$, and $n=1,2,8,10000$.

Momentum distribution characteristics	$L=200$, $h=1$, $n=1$
${⟨p⟩}_R$	0
${\mathrm{Var}}_{p,R}$	6.2500e−06
${⟨p⟩}_M$	0
${\mathrm{Var}}_{p,M}$	+∞
${⟨p⟩}_C={⟨p⟩}_R+{⟨p⟩}_M$	0 + i(0)
${\mathrm{Var}}_{p,C}={\mathrm{Var}}_{p,R}+{\mathrm{Var}}_{p,M}-2{⟨p⟩}_R{⟨p⟩}_M$	+∞

Table 1.

The momentum distribution characteristics for $L=200$, $h=1$, and $n=1$.

Momentum distribution characteristics	$L=200$, $h=1$, $n=2$
${⟨p⟩}_R$	0
${\mathrm{Var}}_{p,R}$	2.500e−05
${⟨p⟩}_M$	0
${\mathrm{Var}}_{p,M}$	+∞
${⟨p⟩}_C={⟨p⟩}_R+{⟨p⟩}_M$	0 + i(0)
${\mathrm{Var}}_{p,C}={\mathrm{Var}}_{p,R}+{\mathrm{Var}}_{p,M}-2{⟨p⟩}_R{⟨p⟩}_M$	+∞

Table 2.

The momentum distribution characteristics for $L=200$, $h=1$, and $n=2$.

Momentum distribution characteristics	$L=200$, $h=1$, $n=8$
${⟨p⟩}_R$	0
${\mathrm{Var}}_{p,R}$	4.0000e−04
${⟨p⟩}_M$	0
${\mathrm{Var}}_{p,M}$	+∞
${⟨p⟩}_C={⟨p⟩}_R+{⟨p⟩}_M$	0 + i(0)
${\mathrm{Var}}_{p,C}={\mathrm{Var}}_{p,R}+{\mathrm{Var}}_{p,M}-2{⟨p⟩}_R{⟨p⟩}_M$	+∞

Table 3.

The momentum distribution characteristics for $L=200$, $h=1$, and $n=8$.

Momentum distribution characteristics	$L=200$, $h=1$, $n=10,000$
${⟨p⟩}_R$	0
${\mathrm{Var}}_{p,R}$	625
${⟨p⟩}_M$	0
${\mathrm{Var}}_{p,M}$	+∞
${⟨p⟩}_C={⟨p⟩}_R+{⟨p⟩}_M$	0 + i(0)
${\mathrm{Var}}_{p,C}={\mathrm{Var}}_{p,R}+{\mathrm{Var}}_{p,M}-2{⟨p⟩}_R{⟨p⟩}_M$	+∞

Table 4.

The momentum distribution characteristics for $L=200$, $h=1$, and $n=10,000$.

For $n\gg 1$ (large n) we get: ${\mathrm{Var}}_{p,R}={\left(\frac{\mathrm{\hslash }\mathit{n\pi }}{L}\right)}^2\to +\infty$.

2. Heisenberg uncertainty principle in $\mathcal{R}$, $\mathcal{M}$, and $\mathcal{C}$

The uncertainties in the probability set and universe $\mathcal{R}$ in position and momentum ($\mathrm{\Delta }{x}_R$ and $\mathrm{\Delta }{p}_R$) are defined as being equal to the square root of their respective variances in $\mathcal{R}$, so that:

$$\mathrm{\Delta }{x}_R\times \mathrm{\Delta }{p}_R=\sqrt{{\mathrm{Var}}_{x,R}}\times \sqrt{{\mathrm{Var}}_{p,R}}=\sqrt{\frac{L^2}{12}\left(1-\frac{6}{n^2{\pi }^2}\right)}\times \sqrt{\frac{{\mathrm{\hslash }}^2{n}^2{\pi }^2}{L^2}}=\frac{\mathrm{\hslash }}{2}\sqrt{\frac{n^2{\pi }^2}{3}-2}$$

This product increases with increasing n, having a minimum value for $n=1$. The value of this product for $n=1$ is about equal to 0.568 $\mathrm{\hslash }$ which obeys the Heisenberg uncertainty principle, which states that:

$$\mathrm{\Delta }x\times \mathrm{\Delta }p\ge \frac{\mathrm{\hslash }}{2}\iff \forall n\ge 1:\mathrm{\Delta }{x}_R\times \mathrm{\Delta }{p}_R\ge \frac{\mathrm{\hslash }}{2}$$

The uncertainties in the probability set and universe $\mathcal{M}$ in position and momentum ($\mathrm{\Delta }{x}_M$ and $\mathrm{\Delta }{p}_M$) are defined as being equal to the square root of their respective variances in $\mathcal{M}$, so that:

$$\mathrm{\Delta }{x}_M\times \mathrm{\Delta }{p}_M=\sqrt{{\mathrm{Var}}_{x,M}}\times \sqrt{{\mathrm{Var}}_{p,M}}\to \sqrt{i\left\{\frac{L^2}{12}\left[L-\left(1-\frac{6}{n^2{\pi }^2}\right)\right]\right\}}\times \sqrt{+\infty }\to +\infty$$

$\iff \forall n\ge 1:\mathrm{\Delta }{x}_M\times \mathrm{\Delta }{p}_M\ge \frac{\mathrm{\hslash }}{2}$, in accordance with the Heisenberg uncertainty principle.

The uncertainties in the probability set and universe $\mathcal{C}$ = $\mathcal{R}+\mathcal{M}$ in position and momentum ($\mathrm{\Delta }{x}_C$ and $\mathrm{\Delta }{p}_C$) are defined as being equal to the square root of their respective variances in $\mathcal{C}$, so that:

$$\begin{array}{l}\mathrm{\Delta }{x}_C\times \mathrm{\Delta }{p}_C=\sqrt{{\mathrm{Var}}_{x,C}}\times \sqrt{{\mathrm{Var}}_{p,C}}\\ \to \sqrt{\frac{L^2}{12}\left(1-\frac{6}{n^2{\pi }^2}\right)+i\left\{\frac{L^2}{12}\left[L-\left(1-\frac{6}{n^2{\pi }^2}\right)\right]\right\}}\times \sqrt{+\infty }\to +\infty \end{array}$$

$\iff \forall n\ge 1:\mathrm{\Delta }{x}_C\times \mathrm{\Delta }{p}_C\ge \frac{\mathrm{\hslash }}{2}$, in accordance with the Heisenberg uncertainty principle.

Consequently, the Heisenberg uncertainty principle is verified in the universe $\mathcal{R}$, in the universe $\mathcal{M}$, and the complex universe $\mathcal{C}$.

3. The Wavefunction Entropies in $\mathcal{R},\mathcal{M},$and $\mathcal{C}$

Another measure of uncertainty in position is the information entropy of the probability distribution $H_x$ which is the entropy in $\mathcal{R}$ and is equal to:

$$H_x=-\sum _{x=-\infty }^{x=+\infty }{\left|\psi \left(x\right)\right|}^2\mathit{Ln}\left[{\left|\psi \left(x\right)\right|}^2{x}_0\right]=-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}{\left|\psi \left(x\right)\right|}^2\mathit{Ln}\left[{\left|\psi \left(x\right)\right|}^2{x}_0\right]=H_x^R=\mathit{Ln}\left(\frac{2L}{{\mathit{ex}}_0}\right)$$

where $x_0$ is an arbitrary reference length [1, 2]. Take $x_0=1$:

$$\begin{array}{l}\iff H_x^R=-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}{\left|\psi \left(x\right)\right|}^2\mathit{Ln}\left[{\left|\psi \left(x\right)\right|}^2\right]\\ =\mathit{Ln}\left(\frac{2L}{e}\right)=\mathit{Ln}\left(2L\right)-\mathit{Ln}\left(e\right)=\mathit{Ln}\left(2L\right)-1=\mathit{Ln}\left(2\times 200\right)-1=4.991464547\dots \end{array}$$

$\iff \forall x:x_c-\frac{L}{2}\le x\le x_c+\frac{L}{2},\text{we have}:d\left[H_x^R\right]\ge 0$, that means that $H_x^R$ is a nondecreasing series with x and converging to $\mathit{Ln}\left(\frac{2L}{e}\right)$ and that also in $\mathcal{R}$, chaos and disorder are increasing with x.

The negative real entropy corresponding to $H_x^R$ in $\mathcal{R}$ is ${\mathit{\text{NegH}}}_x^R$ and is the following:

$$\begin{array}{l}{\mathit{\text{NegH}}}_x^R=-H_x^R=\sum _{x=-\infty }^{x=+\infty }{\left|\psi \left(x\right)\right|}^2\mathit{Ln}\left[{\left|\psi \left(x\right)\right|}^2\right]=\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}{\left|\psi \left(x\right)\right|}^2\mathit{Ln}\left[{\left|\psi \left(x\right)\right|}^2\right]=-\mathit{Ln}\left(\frac{2L}{e}\right)\\ =1-\mathit{Ln}\left(2L\right)=1-\mathit{Ln}\left(2\times 200\right)=-4.991464547\dots \end{array}$$

$\iff \forall x:x_c-\frac{L}{2}\le x\le x_c+\frac{L}{2},\text{we have}:d\left[{\mathit{\text{NegH}}}_x^R\right]\le 0$, which means that ${\mathit{\text{NegH}}}_x^R$ is a nonincreasing series with x and converging to $-\mathit{Ln}\left(\frac{2L}{e}\right)$. Therefore, if $H_x^R$ measures in $\mathcal{R}$ the amount of disorder, of uncertainty, of chaos, of ignorance, of unpredictability, and of information gain in a random system then since ${\mathit{\text{NegH}}}_x^R=-H_x^R$, that means the opposite of $H_x^R$, ${\mathit{\text{NegH}}}_x^R$ measures in $\mathcal{R}$ the amount of order, of certainty, of predictability, and of information loss in a stochastic system.

The complementary real entropy to $H_x^R$ in $\mathcal{R}$ is ${\overline{H}}_x^R$ and is the following:

$${\overline{H}}_x^R=-\sum _{x=-\infty }^{x=+\infty }\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]=-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]=1$$

In the complementary real probability set to $\mathcal{R}$, we denote the corresponding real entropy by ${\overline{H}}_x^R$.

The meaning of ${\overline{H}}_x^R$ is the following: it is the real entropy in the real set $\mathcal{R}$ and which is related to the complementary real probability $P_m/i=1-P_r$.

$\iff \forall x:x_c-\frac{L}{2}\le x\le x_c+\frac{L}{2},\text{we have}:d\left[{\overline{H}}_x^R\right]\ge 0$, that means that ${\overline{H}}_x^R$ is a nondecreasing series with x and converging to 1 and that also means that in the complementary real probability set to $\mathcal{R}$, chaos and disorder are increasing with x.

In the complementary imaginary probability set $\mathcal{M}$ to the set $\mathcal{R}$, we denote the corresponding imaginary entropy by $H_x^M$. The meaning of $H_x^M$ is the following: it is the imaginary entropy in the imaginary set $\mathcal{M}$ and which is related to the complementary imaginary probability $P_m=i\left(1-P_r\right)$. The complementary entropy to $H_x^R$ in $\mathcal{M}$ is $H_x^M$ and is computed as follows:

$$\begin{array}{lll}{H}_x^M& =& -\sum _{x=-\infty }^{x=+\infty }i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left\{i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\right\}\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left\{i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\right\}\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\left\{\mathit{Lni}+\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\right\}\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}i\left\{\mathit{Lni}+\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]-\left[{\left|\psi \left(x\right)\right|}^2\right]\mathit{Lni}-\left[{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\right\}\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\mathit{\text{iLni}}+\mathit{iLn}\left[1-{\left|\psi \left(x\right)\right|}^2\right]-i\left[{\left|\psi \left(x\right)\right|}^2\right]\mathit{Lni}-i\left[{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\mathit{\text{iLni}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]+i\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\mathit{\text{iLni}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]-i\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\psi \left(x\right)\right|}^2\right]\\ & =& -\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\mathit{\text{iLni}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]+i{\overline{H}}_x^R=-\mathit{\text{iLni}}\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\left[1-{\left|\psi \left(x\right)\right|}^2\right]+i{\overline{H}}_x^R\end{array}$$

$$=-\mathit{\text{iLni}}\left\{\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}1-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}{\left|\psi \left(x\right)\right|}^2\right\}+i{\overline{H}}_x^R$$

$$\begin{array}{l}=-\mathit{\text{iLni}}\left\{\left[\left(x_c+\frac{L}{2}\right)-\left(x_c-\frac{L}{2}\right)+1\right]-1\right\}+i{\overline{H}}_x^R\text{since}\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}{\left|\psi \left(x\right)\right|}^2=1\\ =-\left(\mathit{\text{iLni}}\right)L+i{\overline{H}}_x^R\end{array}$$

From the properties of logarithms, we have: $\mathit{\text{θLnx}}=\mathit{Ln}\left(x^{\theta }\right)$ then $\mathit{\text{iLni}}={\mathit{Lni}}^i$.

Moreover, Leonhard Euler’s formula for complex numbers gives: $e^{\mathit{i\theta }}=cos\theta +isin\theta$.

Take $\theta =\pi /2+2\mathit{k\pi }\iff e^{i\left(\pi /2+2\mathit{k\pi }\right)}=cos\left(\pi /2+2\mathit{k\pi }\right)+isin\left(\pi /2+2\mathit{k\pi }\right)=0+i\left(1\right)=i$, then:

$i^i={\left(e^{i\left(\pi /2+2\mathit{k\pi }\right)}\right)}^i=e^{i^2\left(\pi /2+2\mathit{k\pi }\right)}=e^{-\left(\pi /2+2\mathit{k\pi }\right)}$ since $i^2=-1$, therefore:

$-\mathit{\text{iLni}}=-{\mathit{Lni}}^i=-\mathit{Ln}\left[e^{-\left(\pi /2+2\mathit{k\pi }\right)}\right]=\pi /2+2\mathit{k\pi }$ since $\mathit{Ln}\left[e\right]=1$ and where k belongs to the set of integer numbers Z.

Consequently, $$H_x^M=-\left(\mathit{\text{iLni}}\right)L+i{\overline{H}}_x^R=\left(\pi /2+2\mathit{k\pi }\right)L+i{\overline{H}}_x^R$$

That means that $H_x^M$ is a complex number where:

the real part is: $Re\left(H_x^M\right)=\left(\pi /2+2\mathit{k\pi }\right)L$, and the imaginary part is: $Im\left(H_x^M\right)={\overline{H}}_x^R$.

For $k=-1$ then $Re\left(H_x^M\right)=\left(-3\pi /2\right)L=-4.71238898L=-942.4777961\dots \text{for}L=200$.

For $k=0$ then $Re\left(H_x^M\right)=\left(\pi /2\right)L=1.570796327L=314.1592654\dots \text{for}L=200$.

For $k=1$ then $Re\left(H_x^M\right)=\left(5\pi /2\right)L=7.853981634L=1570.796327\dots \text{for}L=200$,

etc.

Finally, the entropy $H_x^C$ in $\mathcal{C}$ = $\mathcal{R}+\mathcal{M}$ is the following:

$$\begin{array}{l}{H}_x^C=-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\mathit{Pc}\left(x\right)\mathit{Ln}\left[\mathit{Pc}\left(x\right)\right]\\ =-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}1\times \mathit{Ln}\left[1\right]=-\sum _{x=x_c-\frac{L}{2}}^{x=x_c+\frac{L}{2}}\left(1\times 0\right)=0\\ =H_x^R+{\mathit{\text{NegH}}}_x^R\end{array}$$

$\iff \forall x:x_c-\frac{L}{2}\le x\le x_c+\frac{L}{2},\text{we have}:d\left[H_x^C\right]=0$, that means that $H_x^C$ 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 $\mathcal{C}=\mathcal{R}+\mathcal{M}$, 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 ($\mathit{Pc}\left(x\right)=1$ and $H_x^C=0$).

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

$$H_p=-\sum _{p=-\infty }^{p=+\infty }{\left|\varphi \left(p\right)\right|}^2\mathit{Ln}\left[{\left|\varphi \left(p\right)\right|}^2{p}_0\right]=\mathit{Ln}\left(\frac{4\pi \mathrm{\hslash }{e}^{2\left(1-\gamma \right)}}{{\mathit{Lp}}_0}\right)=\underset{n\to +\infty }{lim}{H}_p\left(n\right)$$

Where $\gamma$ is Euler’s constant and is equal to: 0.577215664901532…

For $p_0=1$ we can compute all the defined entropies in $\mathcal{R}$, $\mathcal{M}$, and $\mathcal{C}$ and which are [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]:

$$H_p^R=-\sum _{p=-\infty }^{p=+\infty }{\left|\varphi \left(p\right)\right|}^2\mathit{Ln}\left[{\left|\varphi \left(p\right)\right|}^2\right]=\mathit{Ln}\left(\frac{4\pi \mathrm{\hslash }{e}^{2\left(1-\gamma \right)}}{L}\right)=\underset{n\to +\infty }{lim}{H}_p\left(n\right)$$

$${\mathit{\text{NegH}}}_p^R=\sum _{p=-\infty }^{p=+\infty }{\left|\varphi \left(p\right)\right|}^2\mathit{Ln}\left[{\left|\varphi \left(p\right)\right|}^2\right]=-\mathit{Ln}\left(\frac{4\pi \mathrm{\hslash }{e}^{2\left(1-\gamma \right)}}{L}\right)=-\underset{n\to +\infty }{lim}{H}_p\left(n\right)$$

$${\overline{H}}_p^R=-\sum _{p=-\infty }^{p=+\infty }\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\mathit{Ln}\left[1-{\left|\varphi \left(p\right)\right|}^2\right]$$

$$H_p^M=-\sum _{p=-\infty }^{p=+\infty }i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\mathit{Ln}\left\{i\left[1-{\left|\varphi \left(p\right)\right|}^2\right]\right\}$$

$$H_p^C=-\sum _{p=-\infty }^{p=+\infty }\mathit{Pc}\left(p\right)\mathit{Ln}\left[\mathit{Pc}\left(p\right)\right]=-\sum _{p=-\infty }^{p=+\infty }1\times \mathit{Ln}\left[1\right]=-\sum _{p=-\infty }^{p=+\infty }\left(1\times 0\right)=0=H_p^R+{\mathit{\text{NegH}}}_p^R$$

That means also and most importantly, for the wavefunction momentum distribution and in the probability set and universe $\mathcal{C}=\mathcal{R}+\mathcal{M}$, 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 ($\mathit{Pc}\left(p\right)=1$ and $H_p^C=0$).

The quantum mechanical entropic uncertainty principle states that for $x_0{p}_0=\mathrm{\hslash }$ then:

$H_x^R+H_p^R\left(n\right)\ge \mathit{Ln}\left(\mathit{e\pi }\right)\cong 2.144729886\dots \text{nats}$, (base e in Ln gives the “natural units” nat).

For $x_0{p}_0=\mathrm{\hslash }$, the sum of the position and momentum entropies yields:

$H_x^R+H_p^R\left(\infty \right)=\mathit{Ln}\left(8\pi e^{1-2\gamma }\right)\cong 3.069740098\dots \text{nats}$, (base e 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.

Figure 38.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=1$.

Figure 39.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=1$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 40.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=2$.

Figure 41.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=2$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 42.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=3$.

Figure 43.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=3$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 44.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=4$.

Figure 45.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=4$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 46.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=5$.

Figure 47.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=5$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 48.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=20$.

Figure 49.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=20$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

Figure 50.

The graphs of $H_x^R,{\overline{H}}_x^R,H_x^C,{\mathit{\text{NegH}}}_x^R$ as functions of X for $n=100$.

Figure 51.

The graph of $H_x^M=Re\left(H_x^M\right)+iIm\left(H_x^M\right)$ in red as functions of X for $n=100$ and for $k=-1,0,1$ in the planes in yellow, in cyan, and in light gray, respectively.

4. Conclusion and perspectives

In the current research work, the original extended model of eight axioms (EKA) of A. N. Kolmogorov was connected and applied to the infinite potential well problem in quantum mechanics theory. Thus, a tight link between quantum mechanics and the novel paradigm (CPP) was achieved. Consequently, the model of “Complex Probability” was more developed beyond the scope of my 19 previous research works on this topic.

Additionally, as it was proved and verified in the novel model, before the beginning of the random phenomenon simulation and at its end we have the chaotic factor (Chf and MChf) is zero and the degree of our knowledge (DOK) is one since the stochastic fluctuations and effects have either not started yet or they have terminated and finished their task on the probabilistic phenomenon. During the execution of the nondeterministic phenomenon and experiment we also have: 0.5 ≤ DOK < 1, −0.5 ≤ Chf < 0, and 0 < MChf ≤ 0.5. We can see that during this entire process we have incessantly and continually Pc² = DOK – Chf = DOK + MChf = 1 = Pc, that means that the simulation which behaved randomly and stochastically in the real set and universe $\mathcal{R}$ is now certain and deterministic in the complex probability set and universe $\mathcal{C}=\mathcal{R}+\mathcal{M}$, and this after adding to the random experiment executed in the real universe $\mathcal{R}$ the contributions of the imaginary set and universe $\mathcal{M}$ and hence after eliminating and subtracting the chaotic factor from the degree of our knowledge. Furthermore, the real, imaginary, complex, and deterministic probabilities and that correspond to each value of the momentum random variable P have been determined in the three probabilities sets and universes which are $\mathcal{R}$, $\mathcal{M}$, and $\mathcal{C}$ by $P_r$, $P_m$, Z and Pc respectively. Consequently, at each value of P, the novel quantum mechanics and CPP parameters P_r, P_m, $P_m/i$, DOK, Chf, MChf, Pc, and Z are surely and perfectly predicted in the complex probabilities set and universe $\mathcal{C}$ with Pc maintained equal to one permanently and repeatedly.

In addition, referring to all these obtained graphs and executed simulations throughout the whole research work, we are able to quantify and visualize both the system chaos and stochastic effects and influences (expressed and materialized by Chf and MChf) and the certain knowledge (expressed and materialized by DOK and Pc) of the new paradigm. This is without any doubt very fruitful, wonderful, and fascinating and proves and reveals once again the advantages of extending A. N. Kolmogorov’s five axioms of probability and hence the novelty and benefits of my inventive and original model in the fields of prognostics, applied mathematics, and quantum mechanics that can be called verily: “The Complex Probability Paradigm”.

As prospective research, we aim to develop the novel prognostic paradigm conceived and implement it in a large set of nondeterministic phenomena in quantum mechanics.