Stochastic mechanics and the Ehrenfest relations. Memorandum no. 628

The Ehrenfest relations in quantum mechanics maintain that the acceleration of the mean position of a particle in configuration space equals the expectation of the force acting on the particle. The proof of this equality depends on the form of the position and momentum operators. It is assumed that the position of this particle can be represented as a stochastic process and using a symmetric definition of the derivative within the expectation, it is demonstrated that the acceleration of the mean equals the expectation of the mean acceleration operator commonly found in stochastic mechanics. The subsequent requirement that this mean acceleration equals the force for every possible position of the particle reproduces the stochastic analog of the Newton equation introduced by Nelson in the theory of stochastic quantization. 12 refs.; 13 schemes