Stochastic theory of quantum mechanics and the Schr\"odinger equation

TL;DR

This paper explores a stochastic approach to quantum mechanics based on Nelson's method, showing it differs from the Copenhagen interpretation and highlighting its limitations in magnetic fields.

Contribution

It demonstrates that the stochastic theory is not equivalent to standard quantum mechanics and clarifies the conditions under which the Schrödinger equation can be recovered.

Findings

Stochastic theory differs from Copenhagen interpretation.

Elementary solutions can be recovered in certain cases.

Schrödinger equation fails in magnetic fields.

Abstract

We have advocated in a previous paper (Godart M. arXiv: 1206.2917v2[quant-ph] ) a version of the stochastic theory of quantum mechanics. It is indirectly based on a method proposed by Nelson to associate a Markov process with any solution of the Schroedinger equation. The debate began very soon on the question to know if the new theory based on that stochastic procees was equivalent to the orthodox Copenhagen version. We conclude in this paper that the answer is in the negative in agreement with the opinion of several physicists. We show however that the elementary solutions of the Schroedinger equation can be recovered formally from the stochastic theory in a great number of particular cases. We confirm also that this equation is no longer valid when a magnetic field is at work and that it must then give way to some other equation.