Quantum mechanics as a statistical theory

TLDR

The paper is organized into three parts. The study aims to interpret quantum mechanics as a non‑deterministic statistical theory. The authors formulate phase‑space distributions of all dynamical variables in terms of quantum wave vectors, derive their stochastic time‑evolution equations from quantum dynamics, and extend the framework to quantum statistics of large assemblies. They show that this phase‑space approach is equivalent to the operator formalism, can replace the Schrödinger equation for problems such as wave‑packet evolution and collision dynamics, and discuss its limitations, uniqueness, and prospects for experimental verification.

Abstract

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.