**1. Introduction**

There are several names for this idealized and highly artificial potential, prominent among them: *The infinite square-well potential*, *the infinite potential* [1–3]. However, it is the phrase *particle in an escape-proof box* that is more likely to be intuitively appealing. Very simply it tells us about a particle moving inside the box as a *free particle* except at the box walls, which are postulated to be impenetrable by definition. The particle in an escape proof box is a sleek, easy-on-the-mathematics model for initiating students into quantum mechanics, with the added advantage that it is one of the few sectors within quantum mechanics where the Schrödinger equation can be solved analytically without resorting to approximation techniques. In the context of this simple potential, students typically find their first very intuitive understanding of the meaning of bound states, boundary conditions, stationary states, and energymomentum quantization. It is even an introduction to quantum tunneling by emphasizing by contrast *why a particle in a box cannot tunnel out of the box!* The infinite square well is then an easy introduction to a more general understanding of the time independent Schrödinger equation for bound states in more sophisticated potentials, *where the quantum tunneling phenomenon is exhibited.* It is easy stepping-stones away from this *first* potential to the more complicated structures, such as the simple harmonic oscillator, which plays a seminal role in quantum field theory. In a clear and present sense, the quantum adventure can fairly be said to begin with this humble but very remarkable particle in an escape-proof box conception. However, *genius in simplicity* is another watchword for this potential. Remarkably, from such simplicity, one is also able to extract an enormous amount of excellent physics. Never mind that there are no actual confining forces in the world that are infinitely strong, physicists successfully deploy the square well potential to model complicated physics all the time, witness the infinite square well potential, which was used by physicist Sommerfeld to model his electron gas theory, where he construed the moving electrons as free particles confined to an escape-proof box. And again, the particle in a box is also deployed to model and investigate a myriad of other complex physical systems – the Hexatriene molecule, among others, as well as in fabricated semiconductor layers. As Cartwright notes, "Of course, this is not a true description of the potentials that are actually produced by the walls and the environment. But is not exactly false either. It is just the way to achieve the results in the model that the walls and environment are supposed to achieve in reality. The infinite potential is a good piece of stage setting." [3] True, the particle in the escape proof-box is by definition a highly contrived and idealized model. Consequently, this important and well-known problem in quantum mechanics will be related to my complex probability paradigm (*CPP*) in order to express it totally deterministically.

In the end, and to conclude, this research work's first chapter is organized as follows: After the introduction in section 1, the purpose and the advantages of the present work are presented in section 2. Afterward, in section 3, the extended Kolmogorov's axioms, and hence, the complex probability paradigm with their origin nal parameters and interpretation, will be explained and summarized. Moreover, in section 4, we will explain briefly the one-dimensional case of the infinite square well problem considered in this work. Additionally, in section 5, the new paradigm will be

related to the particle in a box problem after applying *CPP* to the position wave function; hence, some corresponding simulations will be done, and afterward, the characteristics of this stochastic distribution will be computed in the probabilities sets **R**, **M**, and **C**. Finally, we conclude the work by doing a comprehensive summary in section 6, and then present in section 7 the list of references cited in the current chapter. Furthermore, in the following second chapter, the new paradigm will be related to the particle in a box problem after applying *CPP* to the momentum wave function of the problem; hence, some corresponding simulations will be done, and afterward, the characteristics of this stochastic distribution will be computed in the probabilities sets **R**, **M**, and **C**. Also, in the following chapter, *CPP* will be used to extend and verify Heisenberg uncertainty principle in **R**, **M**, and **C**. In addition, we will calculate and determine the position and the momentum wave functions entropies in **R**, **M**, and **C**.
