**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

*H<sup>C</sup>*

**Figure 38.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 1*.*

#### **Figure 39.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>1</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 40.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 2*.*

#### **Figure 41.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>2</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 42.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 3*.*

#### **Figure 43.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>3</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 44.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 4*.*

#### **Figure 45.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>4</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 46.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 5*.*

#### **Figure 47.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>5</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 48.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 20*.*

#### **Figure 49.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>20</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

**Figure 50.** *The graphs of H<sup>R</sup> <sup>x</sup>* ,*H<sup>R</sup> <sup>x</sup>* ,*H<sup>C</sup> <sup>x</sup>* ,*NegH<sup>R</sup> <sup>x</sup> as functions of* X *for n* ¼ 100*.*

#### **Figure 51.**

*The graph of H<sup>M</sup> <sup>x</sup>* <sup>¼</sup> Re *<sup>H</sup><sup>M</sup> x* <sup>þ</sup> *<sup>i</sup>*Im *<sup>H</sup><sup>M</sup> x in red as functions of* <sup>X</sup> *for n* <sup>¼</sup> <sup>100</sup> *and for k* ¼ �1,0,1 *in the planes in yellow, in cyan, and in light gray, respectively.*

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 *Pc*<sup>2</sup> = *DOK* – *Chf* = *DOK* + *MChf* =1= *Pc*, that means that the simulation which behaved randomly and stochastically in the real set and universe **R** is now certain and deterministic in the complex probability set and universe **C** ¼ **R** þ**M**, and this after adding to the random experiment executed in the real universe **R** the contributions of the imaginary set and universe **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 **R**, **M**, and **C** by *Pr*, *Pm*, *Z* and *Pc* respectively. Consequently, at each value of *P*, the novel quantum mechanics and *CPP* parameters *Pr*, *Pm*, *Pm=i*, *DOK*, *Chf*, *MChf*, *Pc*, and *Z* are surely and perfectly predicted in the complex probabilities set and universe **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.
