**5. The infinite potential well problem in quantum mechanics and the complex probability paradigm (***CPP***) parameters**

In this section, we will relate and link quantum mechanics to the complex probability paradigm with all its parameters by applying it to the infinite potential well problem and by using the four *CPP* concepts which are: the real probability *Pr* in the real probability set **R**, the imaginary probability *Pm* in the imaginary probability set**M**, the complex random vector or number *Z* in the complex probability set **C** ¼ **R** þ**M**, and the deterministic real probability *Pc* also in the probability set **C** [1–22, 66–99].

### **5.1 The position wave function and** *CPP***: The position wave function solution**

In quantum mechanics, the wave function gives the most fundamental description of the behavior of a particle; the measurable properties of the particle (such as its position, momentum, and energy) may all be derived from the wave function. The wave function *ψ*ð Þ *x*, *t* can be found by solving the Schrödinger equation for the system:

$$i\hbar\frac{\partial}{\partial t}\psi(\mathbf{x},t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial \mathbf{x}^2}\psi(\mathbf{x},t) + V(\mathbf{x})\psi(\mathbf{x},t)$$

where <sup>ℏ</sup> <sup>¼</sup> *<sup>h</sup>* <sup>2</sup>*<sup>π</sup>* is the reduced Planck constant, *m* is the mass of the particle, *i* is the imaginary unit, and *t* is time.

Inside the box, no forces act upon the particle, which means that the part of the wave function inside the box oscillates through space and time in the same form as a free particle:

$$\boldsymbol{\Psi}(\mathbf{x},t) = [\boldsymbol{A}\sin\left(k\mathbf{x}\right) + \boldsymbol{B}\cos\left(k\mathbf{x}\right)]e^{-i\alpha t}$$

where *A* and *B* are arbitrary complex numbers. The frequency of the oscillations through space and time is given by the wave number *k* and the angular frequency *ω*, respectively.

$$\Leftrightarrow \varphi\_n(\mathbf{x}, t) = \begin{cases} A \sin \left[ k\_n \left( \mathbf{x} - \mathbf{x}\_c + \frac{L}{2} \right) \right] e^{-i\alpha\_n t} & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x} < \mathbf{x}\_c + \frac{L}{2} \\\ 0 & \text{elsewhere} \end{cases}$$

where *kn* <sup>¼</sup> *<sup>n</sup><sup>π</sup> L* .

The unknown constant *A* may be found by normalizing the wave function, so that the total probability density of finding the particle in the system is 1. It follows that:

$$|\mathbf{A}| = \sqrt{\frac{2}{L}}$$

Thus, *A* may be any complex number with an absolute value ffiffiffiffiffiffiffiffi 2*=L* p ; these different values of *A* yield the same physical state, so *A* = ffiffiffiffiffiffiffiffi 2*=L* p can be selected to simplify.

### **5.2 The position wave function probability distribution and** *CPP*

In classical physics, the particle can be detected anywhere in the box with equal probability. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wave function as *f x*ð Þ¼ j j *ψ*ð Þ *x* <sup>2</sup> . For the particle in a box, the wave function position probability density function (*PDF*) for finding the particle at a given position depends upon its state and is given by:

$$\left|f(\mathbf{x}) = \left|\boldsymbol{\mu}(\mathbf{x})\right|^2 = \left\{\frac{2}{L}\sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] \quad \mathbf{x}\_c - \frac{L}{2} < \mathbf{x} < \mathbf{x}\_c + \frac{L}{2}\right\}$$

Thus, for any value of *n* greater than one, there are regions within the box for which *f x*ð Þ¼ 0, indicating that *spatial nodes* exist at which the particle cannot be found.

Therefore, the wave function position cumulative probability distribution function (*CDF*), which is equal to *Pr*ð Þ *X* in **R** is:

$$P\_r(X) = F(\mathbf{x}\_j) = P\_{rob}\left(X \le \mathbf{x}\_j\right) = \int\_{-\infty}^{\mathbf{x}\_j} \left|\varphi(\mathbf{x})\right|^2 d\mathbf{x}$$

$$= \begin{cases} \int\_{-L}^{\mathbf{x}\_j} \frac{2}{L} \sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] d\mathbf{x} & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x}\_j < \mathbf{x}\_c + \frac{L}{2} \\\ \frac{L}{2} & \text{otherwise} \end{cases}$$

$$\mathbf{0} \qquad \text{otherwise}$$

And the real complementary probability to *Pr*ð Þ *X* in **R**, which is *Pm*ð Þ *X =i* is:

$$\begin{aligned} P\_m(X)/i &= \mathbf{1} - P\_r(X) = \mathbf{1} - F\left(\mathbf{x}\_j\right) = \mathbf{1} - P\_{mb}\left(X \le \mathbf{x}\_j\right) = P\_{mb}\left(X > \mathbf{x}\_j\right) \\ &= \mathbf{1} - \int\_{-\infty}^{\infty} |y(\mathbf{x})|^2 d\mathbf{x} = \int\_{\mathbf{x}\_j}^{+\infty} |y(\mathbf{x})|^2 d\mathbf{x} \\\\ &= \begin{cases} 1 - \int\_{-L}^{\infty} \frac{2}{L} \sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] d\mathbf{x} & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x}\_j < \mathbf{x}\_c + \frac{L}{2} \\\\ \mathbf{1} & \text{otherwise} \end{cases} \\\\ &= \begin{cases} \frac{L}{2} \\\\ \int\_{-L}^{\infty} \frac{2}{L} \sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] d\mathbf{x} & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x}\_j < \mathbf{x}\_c + \frac{L}{2} \\\\ & \text{otherwise} \end{cases} \end{aligned}$$

Consequently, the imaginary complementary probability to *Pr*ð Þ *X* in **M**, which is *Pm*ð Þ *X* is:

$$P\_m(X) = i[1 - P\_r(X)] = i\left[1 - P\_r(\mathbf{x}\_j)\right] = i\left[1 - P\_{mb}\left(X \le \mathbf{x}\_j\right)\right] = iP\_{mb}\left(X > \mathbf{x}\_j\right)$$

$$= i\left[1 - \int\_{-\infty}^{\mathbf{x}\_j} |y(\mathbf{x})|^2 d\mathbf{x}\right] = i\int\_{\mathbf{x}\_j}^{+\infty} |y(\mathbf{x})|^2 d\mathbf{x}$$

$$= \begin{cases} i\left[1 - \int\_{-\infty}^{\mathbf{x}\_j} \frac{2}{L} \sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] d\mathbf{x}\right] & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x}\_j < \mathbf{x}\_c + \frac{L}{2} \\\\ \text{s.t.} & \mathbf{0} \\\\ & \mathbf{0} & \text{otherwise} \end{cases}$$

$$= \begin{cases} i\left[\int\_{\mathbf{x}\_j}^{\mathbf{x}\_c} \frac{L}{2} \sin^2\left[k\_n\left(\mathbf{x} - \mathbf{x}\_c + \frac{L}{2}\right)\right] d\mathbf{x}\right] & \mathbf{x}\_c - \frac{L}{2} < \mathbf{x}\_j < \mathbf{x}\_c + \frac{L}{2} \\\\ & \mathbf{0} & \text{otherwise} \end{cases}$$

$$\begin{split}M\text{safety}^{\circ}(X) = & [2P\_{\text{s}}(X)] = -2H\_{\text{s}}(X)P\_{\text{s}}(X) = -2H\_{\text{s}}(X) \times [1 - P\_{\text{c}}(X)] \\ &= 2P\_{\text{s}}(X \in \mathbf{x}\_{\text{s}})[1 - P\_{\text{c}}(X \in \mathbf{x}\_{\text{s}})] \\ &= 2P\_{\text{s}ol}\left(X \le \mathbf{x}\_{\text{s}}\right)[1 - P\_{\text{col}}\left(X \le \mathbf{x}\_{\text{s}}\right)] = 2P\_{\text{col}}\left(X \le \mathbf{x}\_{\text{s}}\right)P\_{\text{col}}\left(X > \mathbf{x}\_{\text{s}}\right) \\ &= 2\int\_{-\infty}^{\frac{\pi}{2}} |\mathbf{w}(\mathbf{x})|^{2} d\mathbf{x} \times \left[1 - \int\_{-\infty}^{\frac{\pi}{2}} |\mathbf{w}(\mathbf{x})|^{2} d\mathbf{x}\right] = 2\int\_{-\infty}^{\frac{\pi}{2}} |\mathbf{w}(\mathbf{x})|^{2} d\mathbf{x} \times \int\_{-\pi}^{+\pi} |\mathbf{w}(\mathbf{x})|^{2} d\mathbf{x} \\ &= \begin{cases} 2\int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{2}{L} \sin^{2}\left[k\_{\text{c}}\left(\mathbf{x} - \mathbf{x}\_{\text{c}} + \frac{L}{2}\right)\right] \text{d}\mathbf{x} \times \left[1 - \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{L}{L} \sin^{2}\left[k\_{\text{c}}\left(\mathbf{x} - \mathbf{x}\_{\text{c}} + \frac{L}{2}\right)\right] d\mathbf{x}\$$

Finally, the real probability in the complex probability universe **C** ¼ **R** þ**M** which is *Pc X*ð Þ is:

*Pc*<sup>2</sup> ð Þ¼ *<sup>X</sup>* f g ½ �þ *Pr*ð Þ *<sup>X</sup>* ½ � *Pm*ð Þ *<sup>X</sup> <sup>=</sup><sup>i</sup>* <sup>2</sup> <sup>¼</sup> f g ½ �þ *Pr*ð Þ *<sup>X</sup>* ½ � <sup>1</sup> � *Pr*ð Þ *<sup>X</sup>* <sup>2</sup> <sup>¼</sup> *F xj* � � � � <sup>þ</sup> <sup>1</sup> � *F xj* � � � � � � <sup>2</sup> ¼ *Prob X* ≤*xj* � � <sup>þ</sup> <sup>1</sup> � *Prob <sup>X</sup>* <sup>≤</sup>*xj* � � � � � � <sup>2</sup> <sup>¼</sup> *Prob <sup>X</sup>* <sup>≤</sup>*xj* � � <sup>þ</sup> *Prob <sup>X</sup>* <sup>&</sup>gt;*xj* � � � � <sup>2</sup> ¼ ð *xj* �∞ j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> *dx* þ 1 � ð *xj* �∞ j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> *dx* 2 4 3 5 8 < : 9 = ; 2 ¼ ð *xj* �∞ j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> *dx* þ þð∞ *xj* j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> *dx* 8 >< >: 9 >= >; 2 ¼ þð∞ �∞ j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> *dx* 8 < : 9 = ; 2 ¼ ð *xj xc*�*<sup>L</sup>* 2 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* <sup>þ</sup> <sup>1</sup> � ð *xj xc*�*<sup>L</sup>* 2 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* 2 6 4 3 7 5 8 >< >: 9 >= >; 2 *xc* � *<sup>L</sup>* <sup>2</sup> <sup>&</sup>lt;*xj* <sup>&</sup>lt;*xc* <sup>þ</sup> *L* 2 0 otherwise 8 >>>>< >>>>: ¼ ð *xj xc*�*<sup>L</sup>* 2 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* <sup>þ</sup> ð*xc*þ*L* 2 *xj* 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* 8 >< >: 9 >= >; 2 *xc* � *<sup>L</sup>* <sup>2</sup> <sup>&</sup>lt;*xj* <sup>&</sup>lt;*xc* <sup>þ</sup> *L* 2 0 otherwise 8 >>>>< >>>>: ¼ ð*xc*þ*L* 2 *xc*�*<sup>L</sup>* 2 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* 8 >< >: 9 >= >; 2 *xc* � *<sup>L</sup>* <sup>2</sup> <sup>&</sup>lt;*xj* <sup>&</sup>lt;*xc* <sup>þ</sup> *L* 2 0 otherwise 8 >>>>< >>>>: <sup>¼</sup> <sup>1</sup><sup>2</sup> *xc* � *<sup>L</sup>* <sup>2</sup> <sup>&</sup>lt;*xj* <sup>&</sup>lt;*xc* <sup>þ</sup> *L* 2 0 otherwise 8 < : <sup>¼</sup> <sup>1</sup> *xc* � *<sup>L</sup>* <sup>2</sup> <sup>&</sup>lt;*xj* <sup>&</sup>lt;*xc* <sup>þ</sup> *L* 2 0 otherwise 8 < : ¼ *Pc X*ð Þ

And, *Pc X*ð Þ can be computed using *CPP* as follows:

$$\begin{split}P\mathbf{P}^{2}(X)&=\text{DOK}(X)-\text{Qf}(X)=[P\_{r}(X)]^{2}+[P\_{m}(X)/t]^{2}-2\text{P}\_{r}(X)\mathbf{P}\_{m}(X) \\ &=[P\_{r}(X)]^{2}+[1-P\_{r}(X)]^{2}+2P\_{r}(X)[1-\mathbf{P}\_{r}(X)] = \left\{P\_{r}(X)+[1-P\_{r}(X)]\right\}^{2} \\ &=\left\{\left[\int\_{-\infty}^{\frac{\pi}{2}}|\mathbf{y}(\mathbf{x})|^{2}d\mathbf{x}+\left[1-\int\_{-\infty}^{\frac{\pi}{2}}|\mathbf{y}(\mathbf{x})|^{2}d\mathbf{x}\right]\right\}^{2} = \left\{\left[\int\_{-\infty}^{\frac{\pi}{2}}|\mathbf{y}(\mathbf{x})|^{2}d\mathbf{x}+\int\_{\frac{\pi}{2}}^{+\infty}|\mathbf{y}(\mathbf{x})|^{2}d\mathbf{x}\right\}^{2} = \left\{\left[\int\_{-\infty}^{+\infty}|\mathbf{y}(\mathbf{x})|^{2}d\mathbf{x}\right]^{2}\right\}^{2} \\ &=\begin{cases} 1^{2} & \text{x}\_{\varepsilon}-\frac{L}{2} <\mathbf{x}\_{\mathbb{X}} <\mathbf{x}\_{\varepsilon}+\frac{L}{2} \\ 0 & \text{otherwise} \end{cases} = \begin{cases} 1 & \text{x}\_{\varepsilon}-\frac{L}{2} <\mathbf{x}\_{\mathbb{X}} <\mathbf{x}\_{\varepsilon}+\frac{L}{2} \\ 0 & \text{otherwise} \end{cases} \\ \end{split}$$

And, *Pc X*ð Þ can be computed using always *CPP* as follows also:

$$\begin{split}Pc^2(X) &= DOK(X) + MCdf(X) = [P\_r(X)]^2 + [P\_m(X)/i]^2 + [-2iP\_r(X)P\_m(X)] \\ &= [P\_r(X)]^2 + [1 - P\_r(X)]^2 + 2P\_r(X)[1 - P\_r(X)] = [P\_r(X) + [1 - P\_r(X)]]^2 \\ &= \left\{ \int\_{-\infty}^{x\_j} |\psi(x)|^2 dx + \left[1 - \int\_{-\infty}^{x\_j} |\psi(x)|^2 dx\right] \right\}^2 \\ &= \left\{ \int\_{-\infty}^{x\_j} |\psi(x)|^2 dx + \int\_{x\_j}^{+\infty} |\psi(x)|^2 dx \right\}^2 = \left\{ \int\_{-\infty}^{+\infty} |\psi(x)|^2 dx \right\}^2 \end{split}$$

$$\begin{aligned} &= \begin{cases} 1^2 & \text{x}\_c - \frac{L}{2} < \text{x}\_j < \text{x}\_c + \frac{L}{2} \\ 0 & \text{otherwise} \end{cases} = \begin{cases} 1 & \text{x}\_c - \frac{L}{2} < \text{x}\_j < \text{x}\_c + \frac{L}{2} \\ 0 & \text{otherwise} \end{cases} \\ &= \text{Pc}(X) \end{aligned}$$

$$\begin{aligned} \left\{ \int\_{L}^{U\_{b}} \text{sinc}^{2} \left[ k\_{n} \left( \mathbf{x} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right] d\mathbf{x} &= \frac{2}{L} \int\_{L}^{U\_{b}} \left\{ \frac{1 - \cos \left[ 2k\_{n} \left( \mathbf{x} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right]}{2} \right\} d\mathbf{x} \\ &= \frac{1}{L} \int\_{L}^{U\_{b}} \left\{ 1 - \cos \left[ 2k\_{n} \left( \mathbf{x} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right] \right\} d\mathbf{x} \\ &= \frac{1}{L} \left\{ \mathbf{x} - \frac{\sin \left[ 2k\_{n} \left( \mathbf{x} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right]}{2k\_{n}} \right\}\_{L}^{U\_{b}} \\ &= \frac{1}{2k\_{n}L} \left\{ \left[ 2k\_{n}U\_{b} - \sin \left[ 2k\_{n} \left( U\_{b} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right] \right] \right\} \end{aligned}$$

$$\left[ -\left[ 2k\_{n}L\_{b} - \sin \left[ 2k\_{n} \left( L\_{b} - \mathbf{x}\_{c} + \frac{L}{2} \right) \right] \right] \right]$$

$$\begin{aligned} \int\_{-\infty}^{+\infty} f(x)dx &= \int\_{-\infty}^{\infty} f(x)dx + \int\_{-\frac{L}{2}}^{+\frac{L}{2}} f(x)dx + \int\_{x\_{\epsilon} + \frac{L}{2}}^{+\infty} f(x)dx\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{ $L\_n \gets \frac{L}{2}$ }\\ &= 0 + \int\_{-\frac{L}{2}}^{+\frac{L}{2}} |\Psi(x)|^2 dx + 0 = \quad \quad \int \frac{2}{L} \sin^2 \left[k\_n \left(x - x\_c + \frac{L}{2}\right)\right] dx\\ &= \frac{1}{2k\_n L} \left\{ \left[2k\_n \left(x\_c + \frac{L}{2}\right) - \sin \left[2k\_n \left(x\_c + \frac{L}{2} - x\_c + \frac{L}{2}\right)\right] \right] \right.\\ &\qquad \qquad \qquad \qquad - \left[2k\_n \left(x\_c - \frac{L}{2}\right) - \sin \left[2k\_n \left(x\_c - \frac{L}{2} - x\_c + \frac{L}{2}\right)\right] \right] \right\} \\ &= \frac{1}{2k\_n L} \left\{ \left[2k\_n x\_c + k\_n L - \sin \left[2k\_n L\right] \right] - \left[2k\_n x\_c - k\_n L - \sin \left[2k\_n (0)\right] \right] \right\} \\ &= \frac{1}{2k\_n L} \left\{ 2k\_n L - \sin \left[2k\_n L \right] \right\} \end{aligned}$$

$$\begin{aligned} \frac{1}{2k\_n L} \left\{ 2k\_n L - \sin \left[ \frac{2n\pi L}{L} \right] \right\} &= \frac{1}{2k\_n L} \left\{ 2k\_n L - \sin \left[ 2n\pi \right] \right\} = \frac{1}{2k\_n L} \left\{ 2k\_n L - 0 \right\}, \\\\ \text{where } n &= 1, 2, 3, \dots \\\\ \frac{2k\_n L}{2k\_n L} &= 1 \end{aligned}$$

$$\begin{aligned} \left|1.\,\forall \mathbf{x}: \, 0 \le \left|\nu(\mathbf{x})\right|^2 \le 1, \,\text{as } \forall \mathbf{x}: \, -1 \le \sin\left(\mathbf{x}\right) \le 1 \,\Leftrightarrow \forall \mathbf{x}: \, 0 \le \sin^2\left(\mathbf{x}\right) \le 1 \\\\ \text{2. } \int\_{-\infty}^{+\infty} \left|\nu(\mathbf{x})\right|^2 d\mathbf{x} = \mathbf{1} \end{aligned}$$

$$\begin{split} \int\_{\mathbf{x}\_{c} - \frac{L}{2}}^{L} & \ln\left(\left[2k\_{n}\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right]\right) d\mathbf{x} \\ &= \frac{1}{2k\_{n}L} \Bigg\{ \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right)\right]\right] - \left[2k\_{n}\left(\mathbf{x}\_{c} - \frac{L}{2}\right) - \sin\left[2k\_{n}\left(\mathbf{x}\_{c} - \frac{L}{2} - \mathbf{x}\_{c} + \frac{L}{2}\right)\right]\right] \Bigg\} \\ &= \frac{1}{2k\_{n}L} \Bigg\{ \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right)\right]\right] - \left[2k\_{n}\mathbf{x}\_{c} - k\_{n}L - \sin\left[2k\_{n}(\mathbf{0})\right]\right] \Bigg\} \\ &= \frac{1}{2k\_{n}L} \Bigg\{ \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right)\right]\right] - \left[2k\_{n}\mathbf{x}\_{c} - k\_{n}L\right] \Bigg\} \\ &= \frac{1}{2k\_{n}L} \Bigg\{ 2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right) - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{c} + \frac{L}{2}\right)\right] \Bigg\} \end{split}$$

$$\begin{split} &\mathbb{E}\int\_{\mathbf{x}\_{\ell}}^{\mathbf{x}\_{\ell}+\mathbf{L}} \frac{1}{2} \sin^{2}\left[k\_{n}\left(\mathbf{x}-\mathbf{x}\_{\ell}+\frac{L}{2}\right)\right] d\mathbf{x} \\ &= \frac{1}{2k\_{n}L} \left\{ \left[2k\_{n}\left(\mathbf{x}\_{\ell}+\frac{L}{2}\right) - \sin\left[2k\_{n}\left(\mathbf{x}\_{\ell}+\frac{L}{2}-\mathbf{x}\_{\ell}+\frac{L}{2}\right)\right] \right] - \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j}-\mathbf{x}\_{\ell}+\frac{L}{2}\right)\right]\right] \right\} \\ &= \frac{1}{2k\_{n}L} \left\{ \left[2k\_{n}\mathbf{x}\_{\ell}+k\_{n}L-\sin\left[2k\_{n}L\right]\right] - \left[2k\_{n}\mathbf{x}\_{j}-\sin\left[2k\_{n}\left(\mathbf{x}\_{j}-\mathbf{x}\_{\ell}+\frac{L}{2}\right)\right]\right] \right\} \end{split}$$

$$\frac{1}{2k\_{n}L}\left\{\left[2k\_{n}\mathbf{x}\_{\varepsilon} + k\_{n}L - \sin\left[\frac{2n\pi L}{L}\right]\right] - \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{\varepsilon} + \frac{L}{2}\right)\right]\right]\right\}$$

$$=\frac{1}{2k\_{n}L}\left\{\left[2k\_{n}\mathbf{x}\_{\varepsilon} + k\_{n}L - \sin\left[2n\pi\right]\right] - \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{\varepsilon} + \frac{L}{2}\right)\right]\right]\right\},$$

$$\text{where } n = 1, 2, 3, \dots \\ = \frac{1}{2k\_{n}L}\left\{\left[2k\_{n}\mathbf{x}\_{\varepsilon} + k\_{n}L - 0\right] - \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{\varepsilon} + \frac{L}{2}\right)\right]\right]\right\}$$

$$=\frac{1}{2k\_{n}L}\left\{\left[2k\_{n}\mathbf{x}\_{\varepsilon} + k\_{n}L\right] - \left[2k\_{n}\mathbf{x}\_{j} - \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{\varepsilon} + \frac{L}{2}\right)\right]\right]\right\}$$

$$=\frac{1}{2k\_{n}L}\left\{2k\_{n}\left(\mathbf{x}\_{\varepsilon} - \mathbf{x}\_{j} + \frac{L}{2}\right) + \sin\left[2k\_{n}\left(\mathbf{x}\_{j} - \mathbf{x}\_{\varepsilon} + \frac{L}{2}\right)\right]\right\}$$

#### **Figure 5.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 1.*

#### **Figure 6.**

*The graphs of* DOK *and* Chf,*, and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 1.*

#### **Figure 7.**

*The graphs of* Pr *and* Pm*/*i*, and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 1.*

#### **Figure 8.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 1.*

#### **Figure 9.**

*The graph of the* PDF *of the wave function position probability distribution as a function of the random variable* X *for* n *= 2.*

#### **Figure 10.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 2.*

#### **Figure 11.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 2.*

#### **Figure 12.**

*The graphs of* Pr *and* Pm*/*i*, and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 2.*

#### **Figure 13.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 2.*

#### **Figure 15.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 3.*

#### **Figure 16.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 3.*

#### **Figure 17.**

*The graphs of* Pr *and* Pm*/*i*, and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 3.*

#### **Figure 18.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 3.*

**Figure 19.**

*The graph of the* PDF *of the wave function position probability distribution as a function of the random variable* X *for* n *= 4.*

**Figure 20.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 4.*

#### **Figure 21.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 4.*

#### **Figure 22.**

*The graphs of* Pr *and* Pm*/*i, *and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 4.*

#### **Figure 23.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 4.*

#### **Figure 25.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 5.*

#### **Figure 26.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 5.*

#### **Figure 27.**

*The graphs of* Pr *and* Pm*/*i, *and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 5.*

#### **Figure 28.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 5.*

**Figure 29.**

*The graph of the* PDF *of the wave function position probability distribution as a function of the random variable* X *for* n *= 20.*

#### **Figure 30.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 20.*

#### **Figure 31.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 20.*

#### **Figure 32.**

*The graphs of* Pr *and* Pm*/*i, *and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 20.*

#### **Figure 33.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 20.*

#### **Figure 35.**

*The graphs of all the* CPP *parameters as functions of the random variable* X *for the wave function position probability distribution for* n *= 100.*

#### **Figure 36.**

*The graphs of* DOK *and* Chf, *and the deterministic probability* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 100.*

#### **Figure 37.**

*The graphs of* Pr *and* Pm*/*i, *and* Pc *in terms of* X *and of each other for the wave function position probability distribution for* n *= 100.*

#### **Figure 38.**

*The graphs of the probabilities* Pr *and* Pm *and* Z *in terms of* X *for the wave function position probability distribution for* n *= 100.*

### *5.3.1 Simulations interpretation*

In **Figures 4, 9, 14, 19, 24, 29**, and **34**, we can see the graphs of the probability density functions (*PDF*) of the wave function position probability distribution for this problem as functions of the random variable *X* : �100 ≤*X* ≤100 for *n* = 1, 2, 3, 4, 5, 20, and 100.

In **Figures 5, 10, 15, 20, 25, 30**, and **35**, we can see also the graphs and the simulations of all the *CPP* parameters (*Chf*, *MChf*, *DOK*, *Pr*, *Pm*/*i*, and *Pc*) as functions of the random variable *X* for the wave function position probability distribution of the infinite potential well problem for *n* = 1, 2, 3, 4, 5, 20, and 100. Hence, we can visualize all the new paradigm functions for this problem.

In the cubes (**Figures 6, 11, 16, 21, 26, 31**, and **36**), the simulation of *DOK* and *Chf* as functions of each other and of the random variable *X* for the infinite potential well problem wave function position probability distribution can be seen. The thick line in cyan is the projection of the plane *Pc*<sup>2</sup> (*X*) = *DOK*(*X*) – *Chf*(*X*)=1= *Pc*(*X*) on the plane *X* = *Lb* = lower bound of *X* = �100. This thick line starts at the point (*DOK* = 1, *Chf* = 0) when *X* = *Lb* = �100, reaches the point (*DOK* = 0.5, *Chf* = �0.5) when *X* = 0, and returns at the end to (*DOK* = 1, *Chf* = 0) when *X=Ub* = upper bound of *X* = 100. The other curves are the graphs of *DOK*(*X*) (red) and *Chf*(*X*) (green, blue, and pink) in different simulation planes. Notice that they all have a minimum at the point (*DOK* = 0.5, *Chf* = �0.5, and *X =* 0). The last simulation point corresponds to (*DOK* = 1, *Chf* = 0, and *X=Ub* = 100).

In the cubes (**Figures 7, 12, 17, 22, 27, 32**, and **37**), we can notice the simulation of the real probability *Pr*(*X*) in **R** and its complementary real probability *Pm*(*X*)/*i* in **R** also in terms of the random variable *X* for the infinite potential well problem wave function position probability distribution. The thick line in cyan is the projection of the plane *Pc*<sup>2</sup> (*X*) = *Pr*(*X*) + *Pm*(*X*)/*i* =1= *Pc*(*X*) on the plane *X* = *Lb* = lower bound of *X* = �100. This thick line starts at the point (*Pr* = 0, *Pm*/*i* = 1) and ends at the point (*Pr* = 1, *Pm*/*i* = 0). The red curve represents *Pr*(*X*) in the plane *Pr*(*X*) = *Pm*(*X*)/*i* in light gray. This curve starts at the point (*Pr* = 0, *Pm*/*i* = 1, and *X* = *Lb* = lower bound of *X* = �100), reaches the point (*Pr* = 0.5, *Pm*/*i* = 0.5, and *X* = 0), and gets at the end to (*Pr* = 1, *Pm*/*i* = 0, and *X=Ub* = upper bound of *X* = 100). The blue curve represents *Pm*(*X*)/*i* in the plane in cyan *Pr*(*X*) + *Pm*(*X*)/*i* =1= *Pc*(*X*). Notice the importance of the point, which is the intersection of the red and blue curves at *X* = 0, and when *Pr*(*X*) = *Pm*(*X*)/*i* = 0.5.

In the cubes (**Figures 8, 13, 18, 23, 28, 33**, and **38**), we can notice the simulation of the complex probability *Z*(*X*) in **C** ¼ **R** þ**M** as a function of the real probability *Pr*(*X*) = Re(*Z*) in **R** and of its complementary imaginary probability *Pm*(*X*) = *i* � Im(*Z*) in**M**, and this in terms of the random variable *X* for the infinite potential well problem wave function position probability distribution. The red curve represents *Pr*(*X*) in the plane *Pm*(*X*) = 0, and the blue curve represents *Pm*(*X*) in the plane *Pr*(*X*) = 0. The green curve represents the complex probability *Z*(*X*) = *Pr*(*X*) + *Pm*(*X*) = Re(*Z*) + *i* � Im(*Z*) in the plane *Pr*(*X*) = *iPm*(*X*) + 1 or *Z*(*X*) plane in cyan. The curve of *Z*(*X*) starts at the point (*Pr* = 0, *Pm* = *i*, and *X=Lb* = lower bound of *X* = �100) and ends at the point (*Pr* = 1, *Pm* = 0, and *X=Ub* = upper bound of *X* = 100). The thick line in cyan is *Pr*(*X=Lb* = �100) = *iPm*(*X=Lb* = �100) + 1, and it is the projection of the *Z*(*X*) curve on the complex probability plane whose equation is *X=Lb* = �100. This projected thick line starts at the point (*Pr* = 0, *Pm* = *i*, *X=Lb* = �100) and ends at the point (*Pr* = 1, *Pm* = 0, and *X=Lb* = �100). Notice the importance of the point corresponding to *X* = 0 and *Z* = 0.5 + 0.5*i,* when *Pr* = 0.5 and *Pm* = 0.5*i*.

## **5.4 The characteristics of the position probability distribution**

In quantum mechanics, the average, or expectation value of the position of a particle is given by [10]:

$$\langle \boldsymbol{\omega} \rangle = \int\_{-\infty}^{+\infty} \boldsymbol{\omega} |\boldsymbol{\varphi}(\boldsymbol{\omega})|^2 d\boldsymbol{\omega} = \int\_{\boldsymbol{\omega}\_c - \frac{L}{2}}^{\boldsymbol{\omega}\_c + \frac{L}{2}} \frac{2}{L} \boldsymbol{\omega} \sin^2 \left[ k\_n \left( \boldsymbol{\omega} - \boldsymbol{\omega}\_c + \frac{L}{2} \right) \right] d\boldsymbol{\omega}$$

For the steady state particle in a box, it can be shown that the average position is always h i *x* ¼ *xc*, regardless of the state of the particle. For a superposition of states, the expectation value of the position will change based on the cross term, which is proportional to cosð Þ *ωt* . In the probability set and universe **R**, we have:

$$
\langle \mathfrak{x} \rangle\_R = \langle \mathfrak{x} \rangle = \mathfrak{x}\_c
$$

The variance in the position is a measure of the uncertainty in the position of the particle, so in the probability set and universe **R**, we have:

$$\begin{split} \mathsf{Var}\_{\mathsf{x},R} &= \mathsf{Var}(\mathsf{x}) = \left< \mathsf{x}^{2} \right>\_{R} - \left< \mathsf{x} \right>\_{R}^{2} = \left\{ \int\_{-\infty}^{+\infty} \mathsf{x}^{2} |\mathsf{y}(\mathsf{x})|^{2} d\mathsf{x} \right\} - \left\{ \int\_{-\infty}^{+\infty} \mathsf{x} |\mathsf{y}(\mathsf{x})|^{2} d\mathsf{x} \right\}^{2} \\ &= \left\{ \int\_{-\frac{L}{2}}^{\mathsf{x}\_{c} + \frac{L}{2}} \mathsf{x}^{2} \sin^{2} \left[ k\_{n} \left( \mathsf{x} - \mathsf{x}\_{c} + \frac{L}{2} \right) \right] d\mathsf{x} \right\} - \mathsf{x}\_{c}^{2} = \frac{L^{2}}{12} \left( 1 - \frac{6}{n^{2} \pi^{2}} \right) \end{split}$$

In the probability set and universe **M**, we have:

$$\begin{split} \langle \mathbf{x} \rangle\_{M} &= \int\_{-\infty}^{+\infty} \mathbf{x} \left\{ i \left[ 1 - \left| \boldsymbol{y}(\mathbf{x}) \right|^{2} \right] \right\} d\mathbf{x} = i \left. \int\_{\mathbf{x} - \frac{L}{2}} \mathbf{x} \right\} \mathbf{x} \left\{ 1 - \frac{2}{L} \sin^{2} \left[ k\_{n} \left( \mathbf{x} - \mathbf{x}\_{\varepsilon} + \frac{L}{2} \right) \right] \right\} d\mathbf{x} \\ &= i \left\{ \int\_{-\frac{L}{2}}^{\mathbf{x} + \frac{L}{2}} \mathbf{x} \mathbf{x} - \int\_{-\frac{L}{2}}^{\mathbf{x} + \frac{L}{2}} \mathbf{x} \sin^{2} \left[ k\_{n} \left( \mathbf{x} - \mathbf{x}\_{\varepsilon} + \frac{L}{2} \right) \right] d\mathbf{x} \right\} = i \left\{ \left[ \frac{\mathbf{x}^{2}}{2} \right]\_{\mathbf{x} - \frac{L}{2}}^{\mathbf{x} + \frac{L}{2}} - \langle \mathbf{x} \rangle\_{R} \right\} \\ &= i \left\{ \left[ \left( \mathbf{x}\_{\varepsilon} + \frac{L}{2} \right)^{2} - \frac{\left( \mathbf{x}\_{\varepsilon} - \frac{L}{2} \right)^{2}}{2} \right] - \mathbf{x}\_{\varepsilon} \right\} \\ &= i \langle \mathbf{x}\_{\varepsilon} L - \mathbf{x}\_{\varepsilon} \rangle = i \mathbf{x}\_{\varepsilon} (L - 1) \end{split} $$

To simplify, consider here and in what follows that *xc* ¼ 0 ⇔ h i *x <sup>R</sup>* ¼ 0 and h i *x <sup>M</sup>* ¼ 0.

Moreover,

Var*x*,*<sup>M</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>M</sup>* � h i *x* 2 *M* ¼ þ ð∞ �∞ *<sup>x</sup>*<sup>2</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i *dx* 8 < : 9 = ; � þ ð∞ �∞ *x i* 1 � j j *ψ*ð Þ *x* <sup>2</sup> n o h i *dx* 8 < : 9 = ; 2 ¼ *i* ð *xc*þ*<sup>L</sup>* 2 *xc*�*<sup>L</sup>* 2 *<sup>x</sup>*<sup>2</sup> <sup>1</sup> � <sup>2</sup> *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � � � *dx* � <sup>0</sup> ¼ *i* ð *xc*þ*<sup>L</sup>* 2 *xc*�*<sup>L</sup>* 2 *x*2 *dx* � ð *xc*þ*<sup>L</sup>* 2 *xc*�*<sup>L</sup>* 2 *<sup>x</sup>*<sup>2</sup> <sup>2</sup> *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � � � *dx* 8 >>>< >>>: 9 >>>= >>>; ¼ *i* ð þ*L* 2 �*L* 2 *u*2 *du* � Var*<sup>x</sup>*,*<sup>R</sup>* 8 >>>< >>>: 9 >>>= >>>; <sup>¼</sup> *<sup>i</sup> <sup>u</sup>*<sup>3</sup> 3 � �þ*<sup>L</sup>* 2 �*L* 2 � Var*<sup>x</sup>*,*<sup>R</sup>* ( ) <sup>¼</sup> *<sup>i</sup> <sup>L</sup>*<sup>3</sup> <sup>12</sup> � *<sup>L</sup>*<sup>2</sup> 12 <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � � � <sup>¼</sup> *<sup>i</sup> <sup>L</sup>*<sup>2</sup> 12 *<sup>L</sup>* � <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � � � � �

In the probability set and the universe **C** ¼ **R** þ**M**, we have from *CPP*:

h i *x <sup>C</sup>* ¼ þ ð∞ �∞ *xz x* ½ � ð Þ *dx* ¼ þ ð∞ �∞ *x* j j *ψ*ð Þ *x* <sup>2</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i *dx* ¼ þ ð∞ �∞ *x*j j *ψ*ð Þ *x* 2 *dx* þ þ ð∞ �∞ *xi* 1 � j j *ψ*ð Þ *x* <sup>2</sup> h i*dx* ¼ ð *xc*þ*<sup>L</sup>* 2 *xc*�*<sup>L</sup>* 2 *x* 2 *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � *dx* <sup>þ</sup> *<sup>i</sup>* ð *xc*þ*<sup>L</sup>* 2 *xc*�*<sup>L</sup>* 2 *<sup>x</sup>* <sup>1</sup> � <sup>2</sup> *<sup>L</sup>* sin <sup>2</sup> *kn <sup>x</sup>* � *xc* <sup>þ</sup> *L* 2 � � � � � � *dx* ¼ h i *x <sup>R</sup>* þ h i *x <sup>M</sup>* ¼ *xc* þ *ixc*ð Þ¼ *L* � 1 *xc*½ �¼ 1 þ *i L*ð Þ � 1 0 for *xc* ¼ 0 Var*<sup>x</sup>*,*<sup>C</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>C</sup>* � h i *x* 2 *<sup>C</sup>* ¼ þ ð∞ �∞ *x*2 ½ � *z x*ð Þ *dx* 2 4 3 5 � h i *x <sup>R</sup>* þ h i *x <sup>M</sup>* � �<sup>2</sup> ¼ þ ð∞ �∞ *<sup>x</sup>*<sup>2</sup> j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup> n o h i *dx* 2 4 3 5 � h i *x <sup>R</sup>* þ h i *x <sup>M</sup>* � �<sup>2</sup> ¼ þ ð∞ �∞ *x*2 j j *ψ*ð Þ *x* 2 *dx* þ þ ð∞ �∞ *x*2 *i* 1 � j j *ψ*ð Þ *x* <sup>2</sup> h i*dx* 2 4 3 5 � h i *x <sup>R</sup>* þ h i *x <sup>M</sup>* � �<sup>2</sup>

*Applied Probability Theory - New Perspectives, Recent Advances and Trends*

<sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>R</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � � *M* � � � h i *<sup>x</sup> <sup>R</sup>* <sup>þ</sup> h i *<sup>x</sup> <sup>M</sup>* � �<sup>2</sup> <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>R</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � � *M* � � � h i *<sup>x</sup>* <sup>2</sup> *<sup>R</sup>* <sup>þ</sup> h i *<sup>x</sup>* <sup>2</sup> *<sup>M</sup>* þ 2h i *x <sup>R</sup>*h i *x <sup>M</sup>* h i <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>R</sup>* � h i *<sup>x</sup>* <sup>2</sup> *R* h i <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � � *<sup>M</sup>* � h i *<sup>x</sup>* <sup>2</sup> *M* h i � <sup>2</sup>h i *<sup>x</sup> <sup>R</sup>*h i *<sup>x</sup> <sup>M</sup>* <sup>¼</sup> Var*<sup>x</sup>*,*<sup>R</sup>* <sup>þ</sup> Var*<sup>x</sup>*,*<sup>M</sup>* � <sup>2</sup>h i *<sup>x</sup> <sup>R</sup>*h i *<sup>x</sup> <sup>M</sup>* ¼ *L*2 12 <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � <sup>þ</sup> *<sup>i</sup> <sup>L</sup>*<sup>2</sup> 12 *<sup>L</sup>* � <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � � � � � � 2 0ð Þð Þ 0 ¼ *L*2 12 <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � <sup>þ</sup> *<sup>i</sup> <sup>L</sup>*<sup>2</sup> 12 *<sup>L</sup>* � <sup>1</sup> � <sup>6</sup> *n*<sup>2</sup>*π*<sup>2</sup> � � � � � �

The following tables (**Tables 1**–**4**) compute the position distribution characteristics for *xc* ¼ 0, *L* ¼ 200, and *n* ¼ 1,2,3,20.


#### **Table 1.**

*The position distribution characteristics for xc* ¼ 0*, L* ¼ 200*, and n* ¼ 1*.*


#### **Table 2.**

*The position distribution characteristics for xc* ¼ 0*, L* ¼ 200*, and n* ¼ 2*.*


**Table 3.**

*The position distribution characteristics for xc* ¼ 0*, L* ¼ 200*, and n* ¼ 3*.*


#### **Table 4.**

*The position distribution characteristics for xc* ¼ 0*, L* ¼ 200*, and n* ¼ 20*.*

For *n* ≫ 1 (large *n*) and with *xc* ¼ 0 we get:

$$\text{Var}\_{\mathbf{x}, \mathcal{R}} \rightarrow \frac{L^2}{12} = 3.3333\dots \mathbf{e} + 0\mathbf{3},$$

$$\text{Var}\_{\mathbf{x}, \mathcal{M}} \rightarrow i \left\{ \frac{L^2(L-1)}{12} \right\} = i \times 6.6333\dots \mathbf{e} + 0\mathbf{5}$$

$$\text{Var}\_{\mathbf{x}, \mathcal{C}} \rightarrow \frac{L^2}{12} + i \left\{ \frac{L^2(L-1)}{12} \right\} - 2(0)(\mathbf{0}) = 3.3333\dots \mathbf{e} + 0\mathbf{3} + i \times 6.6333\dots \mathbf{e} + 0\mathbf{5}$$
