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The theoretical results of quantum mechanics (QM) have been verified by experiments, but the probabilistic Copenhagen interpretation is still controversial, and many counterintuitive phenomena are still difficult to understand. To trace the origin of probability in QM, we construct the amplitudes of a space-time position in a multiparticle quantum objective system and find that the probability in QM originates from the particle number distribution rate in a unit volume near position r at time t in the multiparticle quantum objective system. Based on the origin of probability, the quantization conditions of action and the principle of least action for an elementary particle (referred to as particle) , we find that when a particle moves in space-time, it periodically and alternately exhibit particle and wave state. We have derived the formula for the fringe positions in the double-slit experiment with particles, a new certainty relation and the uncertainty relation, obtain the localized and nonlocalized spatiotemporal range of the particle, the apparent trajectory of the particle motion. Based on this, through rigorous mathematical derivation and analysis, we propose new physical interpretations of the quantum superposition state, wave-particle duality, the double-slit experiment, the Heisenberg uncertainty principle, and the quantum tunnelling effect, and these interpretations are physically logical and not counterintuitive.

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.

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.

The theoretical results of quantum mechanics (QM) have been verified by experiments, but the probabilistic Copenhagen interpretation is still controversial, and many counterintuitive phenomena are still difficult to understand. To trace the origin of probability in QM, we construct the superimposed amplitudes of a space-time position in a multiparticle quantum objective system and find that the probability in QM originates from the particle number distribution rate in a unit volume near position r at time t in the multiparticle quantum objective system. Based on the origin of probability, We find that the state function of an elementary particle (hereinafter called particle for short) has precise physical meaning; that is, the particle periodically and alternately exhibits the particle state and wave state in time and space, obtain the localized and nonlocalized spatiotemporal range of the particle, the apparent trajectory of the particle motion. Based on this, through rigorous mathematical derivation and analysis, we propose new physical interpretations of the quantum superposition state, wave-particle duality, the double-slit experiment, the Heisenberg uncertainty principle, and the quantum tunnelling effect, and these interpretations are physically logical and not counterintuitive.

The finite time ruin probability in the classical surplus process setup with additional capital injections and withdrawals is investigated via the Quantum Mechanics Approach. The results are compared with the Picard–Lefevre Appell Polynomial approach and the traditional Markov Chain approach. In addition, several optimization problems in the insurance market are numerically solved by applying the Quantum Mechanics Approach.