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Journal of Physics A Mathematical and General

On the drift in stochastic mechanics

Ian M Davies

Published Aug 21, 1989• 1 citation
Publisher

Abstract

For a large class of Hamiltonians the author expresses the stochastic mechanical drift in terms of a solution of the classical equation of motion and comments on the use of Ito equations to study how quantum mechanics approximates classical mechanics.
DOI
10.1088/0305-4470/22/16/009

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Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.
A new kinetic master equation for the generalized Langevin dynamics of a mechanical system undergoing radiative interaction with a broadband electromagnetic field in a squeezed state has been derived from a quantum stochastic differential equation constructed using the algebra of generators of two-quantum stochastic processes. The latter express the Hamiltonian of the radiative interaction in terms of the Ito increments of these processes. The motion of a harmonic oscillator and a free particle has been analyzed using the equation for the Wigner quasiprobability, which is next in the harmonic approximation of the original equation for an open system. Features of the system dynamics in a broadband squeezed field and a broadband field in a classical state have been revealed.
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