*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 p*ð Þ¼ j j *ϕ*ð Þ *p* 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]:

$$f(p) = \left|\phi(p)\right|^2 = \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \text{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right]$$

Where <sup>ℏ</sup> <sup>¼</sup> *<sup>h</sup>* <sup>2</sup>*<sup>π</sup>* is the reduced Planck constant and sincð Þ¼ *x* sin ð Þ *x <sup>x</sup>* is the cardinal sine *sinc* function.

Therefore, the wavefunction momentum cumulative probability distribution function (*CDF*) which is equal to *Pr*ð Þ *P* in **R** is:

$$P\_r(P) = F\left(p\_j\right) = P\_{rob}\left(P \le p\_j\right) = \int\_{-\infty}^{p\_j} \left|\phi(p)\right|^2 dp$$

$$= \int\_{-\infty}^{p\_j} \frac{L}{n\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \text{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp$$

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

$$P\_m(P)/i = \mathbf{1} - P\_r(P) = \mathbf{1} - F\left(p\_j\right) = \mathbf{1} - P\_{mb}\left(P \le p\_j\right) = P\_{mb}\left(P > p\_j\right)$$

$$= \mathbf{1} - \int\_{-\infty}^{p\_j} |\phi(p)|^2 dp = \int\_{p\_j}^{+\infty} |\phi(p)|^2 dp$$

$$= \mathbf{1} - \int\_{-\infty}^{p\_j} \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \text{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp$$

$$= \int\_{p\_j}^{+\infty} \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \text{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp$$

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

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

$$= i \left[ 1 - \int\_{-\infty}^{p\_j} |\phi(p)|^2 dp \right] = i \int\_{p\_j}^{+\infty} |\phi(p)|^2 dp$$

$$= i \left[ 1 - \int\_{-\infty}^{p\_j} \frac{L}{\pi \hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^2 \text{sinc}^2 \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] dp \right]$$

$$= i \int\_{p\_j}^{+\infty} \frac{L}{\pi \hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^2 \text{sinc}^2 \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] dp$$

Furthermore, the complex random number or vector in **C** ¼ **R** þ**M**which is *Z P*ð Þ is:

$$\begin{split} Z(P) &= P\_r(P) + P\_m(P) = P\_r(P) + i[1 - P\_r(P)] = F\left(p\_j\right) + i\left[1 - F\left(p\_j\right)\right] \\ &= P\_{mb}\left(P \le p\_j\right) + i\left[1 - P\_{mb}\left(P \le p\_j\right)\right] = P\_{mb}\left(P \le p\_j\right) + iP\_{mb}\left(P > p\_j\right) \\ &= \int\_{-\infty}^{\frac{p\_j}{2}} |\phi(p)|^2 dp + i\left[1 - \int\_{-\infty}^{\frac{p\_j}{2}} |\phi(p)|^2 dp\right] = \int\_{-\infty}^{\frac{p\_j}{2}} |\phi(p)|^2 dp + i\int\_{\frac{p\_j}{2}}^{+\infty} |\phi(p)|^2 dp \\ &= \int\_{-\infty}^{\frac{p\_j}{2}} \frac{L}{n\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \operatorname{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \\ &+ i\left[1 - \int\_{-\infty}^{\frac{p\_j}{2}} \frac{L}{n\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \operatorname{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp\right] \\ &= \int\_{-\infty}^{\frac{p\_j}{2}} \frac{L}{n\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \operatorname{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \\ &+ i\int\_{\frac{p\_j}{2}}^{+\infty} \frac{L}{n\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^2 \operatorname{sinc}^2\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \end{split}$$

Additionally, the degree of our knowledge which is *DOK P*ð Þ is:

In addition, the magnitude of the chaotic factor which is *MChf P*ð Þ is:

$$\begin{split} \text{MCdf}(P) &= [\text{MCf}(P)] = -2\text{P}P\_{\text{P}}(P) = -2\text{P}P\_{\text{P}}(P) \times [1 - P\_{\text{P}}(P)] \\ &= 2P\_{\text{P}}(P)[1 - P\_{\text{P}}(P)] = 2\text{P}\left(p\_{j}\right)[1 - P\left(p\_{j}\right)] \\ &= 2P\_{\text{vol}}\left(P \pm p\_{j}\right)\left[1 - P\_{\text{vol}}\left(P \pm p\_{j}\right)\right] = 2P\_{\text{vol}}\left(P \pm p\_{j}\right)P\_{\text{vol}}\left(P > p\_{j}\right) \\ &= 2\int\_{-\infty}^{p\_{j}} |\phi(p)|^{2} dp \times \left[1 - \int\_{-\infty}^{p\_{j}} |\phi(p)|^{2} dp\right] = 2\int\_{-\infty}^{p\_{j}} |\phi(p)|^{2} dp \times \int\_{p\_{j}}^{+\infty} |\phi(p)|^{2} dp \\ &= 2\int\_{-\infty}^{p\_{j}} \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^{2} \text{sinc}^{2}\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \\ &\times \left[1 - \int\_{-\infty}^{p\_{j}} \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^{2} \text{sinc}^{2}\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \right] \\ &= 2\int\_{-\infty}^{p\_{j}} \frac{L}{\pi\hbar} \left(\frac{n\pi}{n\pi + pL/\hbar}\right)^{2} \text{sinc}^{2}\left[\frac{1}{2}(n\pi - pL/\hbar)\right] dp \\ &\times \int\_{-\infty}^{+\infty$$

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

*Pc*<sup>2</sup> ð Þ¼ *<sup>P</sup>* f g ½ �þ *Pr*ð Þ *<sup>P</sup>* ½ � *Pm*ð Þ *<sup>P</sup> <sup>=</sup><sup>i</sup>* <sup>2</sup> <sup>¼</sup> f g ½ �þ *Pr*ð Þ *<sup>P</sup>* ½ � <sup>1</sup> � *Pr*ð Þ *<sup>P</sup>* <sup>2</sup> ¼ *F pj* h i � � <sup>þ</sup> <sup>1</sup> � *F pj* n o h i � � <sup>2</sup> ¼ *Prob P* ≤*pj* � � <sup>þ</sup> <sup>1</sup> � *Prob <sup>P</sup>*≤*pj* n o h i � � <sup>2</sup> ¼ *Prob P*≤ *pj* � � <sup>þ</sup> *Prob <sup>P</sup>* <sup>&</sup>gt;*pj* n o � � <sup>2</sup> ¼ ð *pj* �∞ j j *<sup>ϕ</sup>*ð*p*<sup>Þ</sup> <sup>2</sup> *dp* þ 1 � ð *pj* �∞ j j *<sup>ϕ</sup>*ð*p*<sup>Þ</sup> <sup>2</sup> *dp* 2 4 3 5 8 < : 9 = ; 2 ¼ ð *pj* �∞ j j *<sup>ϕ</sup>*ð*p*<sup>Þ</sup> <sup>2</sup> *dp* þ þð∞ *pj* j j *<sup>ϕ</sup>*ð*p*<sup>Þ</sup> <sup>2</sup> *dp* 8 >< >: 9 >= >; 2 ¼ þð∞ �∞ j j *<sup>ϕ</sup>*ð*p*<sup>Þ</sup> <sup>2</sup> *dp* 8 < : 9 = ; 2 ¼ ð *pj* �∞ *L π*ℏ *nπ nπ* þ *pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> 2 ð Þ *nπ* � *pL=*ℏ � �*dp* þ 1 � ð *pj* �∞ *L π*ℏ *nπ nπ* þ *pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> 2 ð Þ *nπ* � *pL=*ℏ � �*dp* 2 4 3 5 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; 2 ¼ ð *pj* �∞ *L π*ℏ *nπ nπ* þ *pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> 2 ð Þ *nπ* � *pL=*ℏ � �*dp* þ þð∞ *pj L π*ℏ *nπ nπ* þ *pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> 2 ð Þ *nπ* � *pL=*ℏ � �*dp* 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; 2 ¼ þð∞ �∞ *L π*ℏ *nπ nπ* þ *pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> 2 ð Þ *nπ* � *pL=*ℏ � �*dp* 8 < : 9 = ; 2 <sup>¼</sup> <sup>1</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>¼</sup> *Pc P*ð Þ

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

$$\begin{split} \mathcal{C}c^{2}(P) &= D\mathcal{O}K(P) - \mathcal{C}bf(P) = \left[P\_{r}(P)\right]^{2} + \left[P\_{m}(P)/t\right]^{2} - 2i\mathcal{P}\_{r}(P)P\_{m}(P) \\ &= \left[P\_{r}(P)\right]^{2} + \left[1 - P\_{r}(P)\right]^{2} + 2\mathcal{P}\_{r}(P)[1 - P\_{r}(P)] = \left\{P\_{r}(P) + \left[1 - P\_{r}(P)\right]\right\}^{2} \\ &= \left\{\int\_{-\infty}^{\frac{P\_{r}}{2}} |\phi(p)|^{2} dp + \left[1 - \int\_{-\infty}^{\frac{P\_{r}}{2}} |\phi(p)|^{2} dp\right] \right\}^{2} = \left\{\int\_{-\infty}^{\frac{P\_{r}}{2}} |\phi(p)|^{2} dp + \int\_{\frac{1}{P\_{r}}}^{\frac{+\infty}{+\infty}} |\phi(p)|^{2} dp \right\}^{2} \\ &= \left\{\int\_{-\infty}^{+\infty} |\phi(p)|^{2} dp\right\}^{2} \\ &= \mathbf{1}^{2} = \mathbf{1} = \text{Pr}(P) \end{split}$$

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

$$\begin{split} \mathrm{Pc}^{2}(P) &= D \mathrm{OK}(P) + M \mathrm{Cl}\_{7}^{\circ}(P) = [P\_{r}(P)]^{2} + [P\_{m}(P)/i]^{2} + [-2i P\_{r}(P) P\_{m}(P)] \\ &= [P\_{r}(P)]^{2} + [1 - P\_{r}(P)]^{2} + 2P\_{r}(P)[1 - P\_{r}(P)] = \{P\_{r}(P) + [1 - P\_{r}(P)]\}^{2} \\ &= \left\{\int\limits\_{-\infty}^{P\_{j}} |\phi(p)|^{2} dp + \left[1 - \int\limits\_{-\infty}^{P\_{j}} |\phi(p)|^{2} dp\right]\right\}^{2} \\ &= \left\{\int\limits\_{-\infty}^{P\_{j}} |\phi(p)|^{2} dp + \left[\int\limits\_{P\_{j}} |\phi(p)|^{2} dp\right]^{2} = \left\{\int\limits\_{-\infty}^{+\infty} |\phi(p)|^{2} dp\right\}^{2} = \mathbf{1}^{2} = \mathbf{1} = \mathrm{Pc}(P) \end{split}$$

Hence, the prediction of all the wavefunction momentum probabilities of the random infinite potential well problem in the universe **C** ¼ **R** þ**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.

### *1.1.2.1 Simulations interpretation*

In **Figures 1, 6, 11, 16, 21, 26, 31, 36, and 37** 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, and 32** we can see also the graphs and the simulations of all the *CPP* parameters (*Chf*, *MChf*, *DOK*, *Pr*, *Pm*/*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.

**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* Pr *and* Pm*/*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* Pr *and* Pm *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* Pr *and* Pm*/*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* Pr *and* Pm *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* Pr *and* Pm*/*i *and* Pc *in terms of* P *and of each other for the wavefunction momentum probability distribution for* n *= 3.*

*The graphs of the probabilities* Pr *and* Pm *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* Pr *and* Pm*/*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* Pr *and* Pm *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* Pr *and* Pm*/*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* Pr *and* Pm *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* Pr *and* Pm*/*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* Pr *and* Pm *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* Pr *and* Pm*/*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* Pr *and* Pm *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.*

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 *Pc*<sup>2</sup> (*P*) = *DOK*(*P*) – *Chf*(*P*)=1= *Pc*(*P*) on the plane *P* = *Lb* = lower bound of *P*. This thick line starts at the point (*DOK* = 1, *Chf* = 0) when *P* = *Lb*, reaches the point (*DOK* = 0.5, *Chf* = 0.5) when *P* = 0, and returns at the end to (*DOK* = 1, *Chf* = 0) when *P=Ub* = 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=Ub*).

In the cubes (**Figures 4, 9, 14, 19, 24, 29**, and **34**), we can notice the simulation of the real probability *Pr*(*P*) in **R** and its complementary real probability *Pm*(*P*)/*i* in **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 *Pc*<sup>2</sup> (*P*) = *Pr*(*P*) + *Pm*(*P*)/*i* =1= *Pc*(*P*) on the plane *P* = *Lb* = lower bound of *P*. 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*(*P*) in the plane *Pr*(*P*) = *Pm*(*P*)/*i* in light gray. This curve starts at the point (*Pr* = 0, *Pm*/*i* = 1, *P* = *Lb* = lower bound of *P*), reaches the point (*Pr* = 0.5, *Pm*/*i* = 0.5, *P* = 0), and gets at the end to (*Pr* = 1, *Pm*/*i* = 0, *P=Ub* = upper bound of *P*). The blue curve represents *Pm*(*P*)/*i* in the plane in cyan *Pr*(*P*) + *Pm*(*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 *Pr*(*P*) = *Pm*(*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 **C** ¼ **R** þ**M** as a function of the real probability *Pr*(*P*) = Re(*Z*) in **R** and of its complementary imaginary probability *Pm*(*P*) = *i* � Im(*Z*) in **M**, and this in terms of the random variable *P* for the infinite potential well problem wavefunction momentum probability distribution. The red curve represents *Pr*(*P*) in the plane *Pm*(*P*) = 0 and the blue curve represents *Pm*(*P*) in the plane *Pr*(*P*) = 0. The green curve represents the complex probability *Z*(*P*) = *Pr*(*P*) + *Pm*(*P*) = Re(*Z*) + *i* � Im(*Z*) in the plane *Pr*(*P*) = *iPm*(*P*) + 1 or *Z*(*P*) plane in cyan. The curve of *Z*(*P*) starts at the point (*Pr* = 0, *Pm* = *i*, *P=Lb* = lower bound of *P*) and ends at the point (*Pr* = 1, *Pm* = 0, *P=Ub* = upper bound of *P*). The thick line in cyan is *Pr*(*P=Lb*) = *iPm*(*P=Lb*) + 1 and it is the projection of the *Z*(*P*) curve on the complex probability plane whose equation is *P=Lb*. This projected thick line starts at the point (*Pr* = 0, *Pm* = *i*, *P=Lb*) and ends at the point (*Pr* = 1, *Pm* = 0, *P=Lb*). Notice the importance of the point corresponding to *P* = 0 and *Z* = 0.5 + 0.5*i* when *Pr* = 0.5 and *Pm* = 0.5*i*.

## *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: h i *p* ¼ þ Ð∞ �∞ *p*j j *ϕ*ð Þ *p* 2 *dp* ¼ þ Ð∞ �∞ *p <sup>L</sup> π*ℏ *nπ nπ*þ*pL=*ℏ � �<sup>2</sup> sinc<sup>2</sup> <sup>1</sup> <sup>2</sup> ð Þ *<sup>n</sup><sup>π</sup>* � *pL=*<sup>ℏ</sup> � �*dp***.**

For the steady state particle in a box, it can be shown that the average momentum is always h i *p* ¼ 0 regardless of the state of the particle. In the probability set and universe **R**, we have:

$$
\langle p \rangle\_R = \langle p \rangle = 0
$$

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

$$\begin{split} \mathsf{Var}\_{p,\mathsf{R}} &= \mathsf{Var}(p) = \langle p^2 \rangle\_{\mathsf{R}} - \langle p \rangle\_{\mathsf{R}}^2 = \int\_{-\infty}^{+\infty} p^2 |\phi(p)|^2 dp - \mathsf{0} \\ &= \int\_{-\infty}^{+\infty} p^2 \left\{ \frac{L}{\pi \hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^2 \mathrm{sinc}^2 \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] \right\} dp = \left( \frac{\hbar m\pi}{L} \right)^2 \end{split}$$

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

$$\begin{split} \langle p \rangle\_{M} &= \int\_{-\infty}^{+\infty} p \left\{ i \left[ 1 - |\phi(p)|^{2} \right] \right\} dp = i \int\_{-\infty}^{+\infty} p \left\{ 1 - \frac{L}{\pi\hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^{2} \text{sinc}^{2} \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] \right\} dp \\ &= i \left\{ \int\_{-\infty}^{+\infty} p dp - \int\_{-\infty}^{+\infty} p \left\{ \frac{L}{\pi\hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^{2} \text{sinc}^{2} \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] \right\} dp \right\} \\ &= i \left\{ \left[ \frac{p^{2}}{2} \right]\_{-\infty}^{+\infty} - \langle p \rangle\_{R} \right\} = i \left\{ \left[ \frac{p^{2}}{2} \right]\_{-\text{U}\_{b}}^{\text{U}\_{b}} - \langle p \rangle\_{R} \right\} = i \{0 - 0\} = 0 \end{split}$$

$$\begin{split} \mathsf{Var}\_{p,M} &= \langle p^2 \rangle\_M - \langle p \rangle\_M^2 \\ &= \int\_{-\infty}^{+\infty} p^2 \left\{ i \left[ 1 - |\phi(p)|^2 \right] \right\} dp - 0 \\ &= i \int\_{-\infty}^{+\infty} p^2 \left\{ 1 - \frac{L}{\pi \hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^2 \text{sinc}^2 \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] \right\} dp \\ &= i \left\{ \int\_{-\infty}^{+\infty} p^2 dp - \int\_{-\infty}^{+\infty} p^2 \left\{ \frac{L}{\pi \hbar} \left( \frac{n\pi}{n\pi + pL/\hbar} \right)^2 \text{sinc}^2 \left[ \frac{1}{2} (n\pi - pL/\hbar) \right] \right\} dp \right\} \\ &= i \left\{ \int\_{-\infty}^{+\infty} p^2 dp - \text{Var}\_{p,R} \right\} = i \left\{ \left[ \frac{p^3}{3} \right]\_{-\infty}^{+\infty} - \text{Var}\_{p,R} \right\} \to i \left\{ +\infty - \left( \frac{\hbar n\pi}{L} \right)^2 \right\} dp \\ &\to +\infty \end{split}$$

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

h i *p <sup>C</sup>* ¼ þ ð∞ �∞ *pz p* ½ � ð Þ *dp* ¼ þ ð∞ �∞ *p* j j *ϕ*ð Þ *p* <sup>2</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ϕ</sup>*ð Þ *<sup>p</sup>* <sup>2</sup> n o h i *dp* ¼ þ ð∞ �∞ *p*j j *ϕ*ð Þ *p* 2 *dp* þ þ ð∞ �∞ *pi* 1 � j j *ϕ*ð Þ *p* <sup>2</sup> h i*dp* ¼ h i *p <sup>R</sup>* þ h i *p <sup>M</sup>* ¼ 0 þ *i*ð Þ¼ 0 0 Var*<sup>p</sup>*,*<sup>C</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> � � *<sup>C</sup>* � h i *p* 2 *<sup>C</sup>* ¼ þ ð∞ �∞ *p*2 ½ � *z p*ð Þ *dp* 2 4 3 5 � h i *p <sup>R</sup>* þ h i *p <sup>M</sup>* � �<sup>2</sup> ¼ þ ð∞ �∞ *<sup>p</sup>*<sup>2</sup> j j *<sup>ϕ</sup>*ð Þ *<sup>p</sup>* <sup>2</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> � j j *<sup>ϕ</sup>*ð Þ *<sup>p</sup>* <sup>2</sup> n o h i *dp* 2 4 3 5 � h i *p <sup>R</sup>* þ h i *p <sup>M</sup>* � �<sup>2</sup> ¼ þ ð∞ �∞ *p*2 j j *ϕ*ð Þ *p* 2 *dp* þ þ ð∞ �∞ *p*2 *i* 1 � j j *ϕ*ð Þ *p* <sup>2</sup> h i*dp* 2 4 3 5 � h i *p <sup>R</sup>* þ h i *p <sup>M</sup>* � �<sup>2</sup> <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> � � *<sup>R</sup>* <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> � � *M* � � � h i *<sup>p</sup> <sup>R</sup>* <sup>þ</sup> h i *<sup>p</sup> <sup>M</sup>* � �<sup>2</sup> <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> � � *<sup>R</sup>* <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> � � *M* � � � h i *<sup>p</sup>* 2 *<sup>R</sup>* þ h i *p* 2 *<sup>M</sup>* þ 2h i *p <sup>R</sup>*h i *p <sup>M</sup>* h i <sup>¼</sup> *<sup>p</sup>*<sup>2</sup> � � *<sup>R</sup>* � h i *p* 2 *R* h i <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> � � *<sup>M</sup>* � h i *p* 2 *M* h i � <sup>2</sup>h i *<sup>p</sup> <sup>R</sup>*h i *<sup>p</sup> <sup>M</sup>* ¼ Var*<sup>p</sup>*,*<sup>R</sup>* þ Var*<sup>p</sup>*,*<sup>M</sup>* � 2h i *p <sup>R</sup>*h i *p <sup>M</sup>* ! <sup>ℏ</sup>*n<sup>π</sup> L* � �<sup>2</sup> þ ∞ � 2 0ð Þð Þ 0 ! þ∞

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


#### **Table 1.**

*The momentum distribution characteristics for L* ¼ 200*, h* ¼ 1*, and n* ¼ 1*.*


#### **Table 2.**

*The momentum distribution characteristics for L* ¼ 200*, h* ¼ 1*, and n* ¼ 2*.*


#### **Table 3.**

*The momentum distribution characteristics for L* ¼ 200*, h* ¼ 1*, and n* ¼ 8*.*

The following tables (**Tables 1**–**4**) compute the momentum distribution characteristics for *L* ¼ 200, *h* ¼ 1, and *n* ¼ 1,2,8,10000.

For *<sup>n</sup>* <sup>≫</sup> 1 (large *<sup>n</sup>*) we get: Var*<sup>p</sup>*,*<sup>R</sup>* <sup>¼</sup> <sup>ℏ</sup>*n<sup>π</sup> L* <sup>2</sup> ! þ∞.
