Nonrelativistic Stochastic Quantum Mechanics

Bernard H. Lavenda, Enrico Santamato

R. Gagnon

Ramabhadra Vasudevan

Silviu Olariu

In this work we analyze the relation between the stochastic representation of nonrelativistic quantum mechanics and the Newtonian stochastic mechanics. In order to understand how the nodes of a wave function are accounted for in stochastic quantum mechanics, we solve the Fokker-Planck equation associated with the quantum-mechanical density ${\ensuremath{\rho}}_{0}$=${x}^{2}$ and current ${j}_{0}$=0, and find that the stochastically perturbed particle does not cross the nodal surface x=0 of the density ${\ensuremath{\rho}}_{0}$. Further we consider the classical motion of a particle subject to a stochastic perturbation, in the absence of conventional external forces and friction, and moreover we consider the classical stochastic motion of a harmonically bound particle in the presence of a friction force proportional and opposite to the velocity of the particle, to show that although the quantum-mechanical equations for the probability density and current can be put in a form resembling Newton's second law, the actual significance of those equations is not related to Newtonian stochastic mechanics. From the fact that, in the stochastic representation of quantum mechanics, the particles have continuous trajectories, we infer that the interactions would possess, in this theory, a global character. We finally show that the property of reversibility in stochastic quantum mechanics depends on the existence of suitably chosen compensating drifts in the mean forward velocity or acceleration of the particle under consideration.

EDWARD B. NELSON

Stochastic mechanics is an interpretation of nonrelativistic quantum mechanics in which the trajectories of the configuration, described as a Markov stochastic process, are regarded as physically real. The natural stochastic generalization of classical variational principles leads to a derivation of the Schrödinger equation. A brief review of the successes and failures of the theory is given, with references.