Stochastic interpretation of quantum mechanics

Abstract

The paper addresses the problem of expressing (describing and interpreting) quantum mechanics in terms of probability theory. The solution is demonstrated through computational results based on certain principles, (viewpoints and postulates). In particular, the expectation value of a quantity corresponding to a quantum mechanical operator in the momentum representation is expressed via a complex quasi-probability function. A relation for the quasi-probability function is obtained for the case where the state of the system is described by a density matrix. The numerical representation of the Hamiltonian operator is examined, both in forms that correspond to and deviate from the classical model. Averaged expressions with respect to the quasi-probability function are derived for the stationary case. The problem of describing a stochastic process occurring in complex phase space using a quasi-probability function was addressed by means of a differential equation for the density matrix. It was proven that this equation reduces to the Liouville equation for quantum phenomena in the limit of a vanishing Planck constant, and that the quasi-probability function transforms into an ordinary probability distribution (a specific positive function). Conditions were established under which the equation for the probability density transitions into an approximate complex equation structurally analogous to the Fokker–Planck equation for a diffusive Markov process. Stochastic differential equations corresponding to this complex equation were presented, which define the connections between random processes and the quasi-probability function and incorporate all necessary information for quantum-mechanical calculations. Finally, a system of motion equations in the quasi-classical approximation and stochastic differential equations relevant to the general case and characteristics of the wave function were derived. All computational results and theoretical assertions were thoroughly discussed, and the necessary conclusions were drawn.