The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem – The Position Wave Function

Abstract

The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real setR with its corresponding real probability, and the imaginary setM with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex setC instead of the real setR. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the “real” laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally inC. Subsequently, it follows that chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. 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.