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Simple Generalization of Schrödinger's Equation

### Abstract

A possible generalization of quantum mechanics is examined by showing that the motion in phase space of a classical Brownian particle may be described by a complex probability amplitude depending on the phase‐space coordinates and the time, and obeying a Schrödinger‐like equation. However formal this result may seem, the usual dynamical operators may be defined whose physical meaning stems directly from the theory. An outstanding feature of the formalism is that ordinary quantum mechanics in configuration space may be recovered in a limiting process whereby the velocity variable, defined now through a statistical distribution, is eliminated. Therefore, it plays the role of a hidden variable. This result supports recent reinterpretations of von Neumann's theorem on the nonexistence of such variables in quantum mechanics and serves as a counterexample of the usual interpretation of his theorem.

© 1968 The American Institute of Physics

Received 07 August 1967
Published online 28 October 2003