The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem - The Momentum Wavefunction and the Wavefunction Entropies

What is it about?

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R + M independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be determined perfectly in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. 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?

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem in quantum mechanics considered here, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in C = R +M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic and mainly in quantum mechanics is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my nineteen previous papers.

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