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Physical Review A

Thermal-mixture Kepler system in stochastic mechanics

Tetsuya Misawa

Published Nov 1, 1991• 1 citation
Publisher

Abstract

An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in ${\mathit{openR}}^{\mathit{n}}$ (n=2,3,5) is derived from one associated with the harmonic oscillator in ${\mathit{openR}}^{\mathit{N}}$ (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.

DOI
10.1103/physreva.44.6313

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An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in openRn{\mathit{openR}}^{\mathit{n}}openRn (n=2,3,5) is derived from one associated with the harmonic oscillator in openRN{\mathit{openR}}^{\mathit{N}}openRN (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.
The microscopic model of Ullersma (1966) for a harmonic oscillator in contact with a thermal reservoir is quantised in the framework of Nelson's stochastic mechanics. Eliminating the degrees of freedom of the thermal reservoir the stochastic process of the quantum Brownian oscillator is obtained as the sum of the thermal contribution and the quantum (zero-temperature) contribution. The properties of the quantum fluctuations are studied in detail and the equation of motion is derived, obtaining the first example of a non-Markovian Nelson process.
We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson's stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.