The Paradigm of Complex Probability and Quantum Mechanics: The Quantum Harmonic Oscillator with Gaussian Initial Condition – The Position Wavefunction

What is it about?

In the current work, we extend and incorporate in the five-axioms probability system of Andrey Nikolaevich Kolmogorov set up in 1933 the imaginary set of numbers and this by adding three supplementary axioms. Consequently, any stochastic experiment can thus be achieved in the extended complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. The purpose here is to evaluate the complex probabilities by considering additional novel imaginary dimensions to the experiment occurring in the “real” laboratory. Therefore, the random phenomenon outcome and result in C = R + M can be predicted absolutely and perfectly no matter what the random distribution of the input variable in R is since the associated probability in the entire set C is constantly and permanently equal to one. Thus, the following consequence indicates that chance and randomness in R is replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge the chaotic factor in the probabilistic experiment. Moreover, I will apply to the established theory of quantum mechanics my original Complex Probability Paradigm (CPP) in order to express the quantum mechanics problem considered here completely deterministically in the universe of probabilities 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 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 23 previous papers.

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