Short-time stochastic electron

Abstract

As in previous stochastic interpretations of quantum mechanics, the electron is treated as a modified Brownian particle. Here, however, the analysis is based on extensions of the short-time Ornstein-Uhlenbeck description of classical Brownian motion, rather than the approximate, long-time Einstein-Smoluchowski treatment utilized in the earlier work. It is shown that Schrödinger's equation with its proper probabilistic interpretation emerges as an asymptotic description of such a system. After reviewing relevant aspects of Brownian motion, the appropriate equation for the displacement distribution in the Ornstein-Uhlenbeck framework is derived. The Brownian system is then coupled to external potentials via a modified Lagrangian formulation, resulting in coupled equations which, except for transient terms, are shown to be equivalent to Schrödinger's equation. These transients imply that the short-time stochastic model, while reproducing the early successes of quantum mechanics, is, in principal, experimentally distinguishable from it.