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Journal of Mathematical Physics

Uncertainty relations in stochastic mechanics

Simon Golin

Published Nov 1, 1985• 23 citations
Publisher

Abstract

Position–momentum uncertainty in Nelson’s stochastic mechanics [Phys. Rev. 150, 1079 (1966)] has previously been investigated by de la Peña-Auerbach and Cetto [Phys. Lett. A 39, 65 (1972)]. In this paper their result is generalized, and full equivalence between the uncertainty relations in stochastic mechanics and conventional quantum mechanics is established. Force–momentum uncertainty is also considered.

DOI
10.1063/1.526700

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Position–momentum uncertainty in Nelson’s stochastic mechanics [Phys. Rev. 150, 1079 (1966)] has previously been investigated by de la Peña-Auerbach and Cetto [Phys. Lett. A 39, 65 (1972)]. In this paper their result is generalized, and full equivalence between the uncertainty relations in stochastic mechanics and conventional quantum mechanics is established. Force–momentum uncertainty is also considered.
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It is shown that the Bohm equations for the phase SSS and squared modulus ρρρ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum psp_sps​ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat gradSSS and psp_sps​ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.