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

What is it about?

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 set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. 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 in C. 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.

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Why is it important?

Computing probabilities is all our work in the classical theory of probability. Adding new dimensions to our stochastic experiment is the innovative idea in the current paradigm which will make the study absolutely deterministic. As a matter of fact, the theory of probability is a nondeterministic theory by essence that means that all the random events outcome is due to luck and chance. Hence, we make the study deterministic by adding new imaginary dimensions to the phenomenon occurring in the “real” laboratory which is R, and therefore, a stochastic experiment will have a certain outcome in the complex probabilities set C. It is of great significance that random systems become completely predictable since we will be perfectly knowledgeable to predict the outcome of all stochastic and chaotic phenomena that occur in nature like for example in all stochastic processes, in statistical mechanics, or in the well-established field of quantum mechanics. Consequently, the work that should be done is to add the contributions of M which is the set of imaginary probabilities to the set of real probabilities R that will make the random phenomenon in C = R + M completely deterministic. Since this paradigm is found to be fruitful, then a new theory in prognostic and stochastic sciences is established and this is to understand deterministically those events that used to be stochastic events in R. This is what I coined by the term “The Complex Probability Paradigm” that was elaborated and initiated in my 25 previous papers.

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