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Journal of Mathematical Physics

Kinetic equations for the quantized motion of a particle in a randomly perturbed potential field

Ioannis M. Besieris, F. D. Tappert

Published Dec 1, 1973• 30 citations
Publisher

Abstract

Within the framework of the first-order smoothing approximation and the long-time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ-correlated (in time), homogeneous, wide-sense stationary, Gaussian process.

DOI
10.1063/1.1666255

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Within the framework of the first-order smoothing approximation and the long-time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ-correlated (in time), homogeneous, wide-sense stationary, Gaussian process.
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From the equation of motion for a radiating charged particle embedded in the zero-point radiation field we construct a stochastic Liouville equation which serves to derive, by a smoothing process, a Fokker–Planck-type equation with infinite memory. We show that an exact alternative form of this phase-space equation is the Schrödinger equation in configuration space, with radiative corrections. In the asymptotic, radiationless limit (when the radiative corrections became negligible), the phase-space density reduces to Wigner’s distribution, thus confirming Weyl’s rule of correspondence. We briefly discuss several other implications of stochastic electrodynamics which are relevant for quantum theory in general.
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We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.
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