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

Abstract

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.