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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Stochastic mechanics and the Kepler problem

Piotr Garbaczewski

Published May 15, 1986• 3 citations
Publisher

Abstract

The stochastic mechanics of Nelson and Guerra is formulated for the hydrogen atom. We demonstrate that this simple quantum system can be described in terms of three independent Gaussian Markov processes which are driven (controlled) by the classical Kepler problem. It reveals a manifest connection between the classical and quantized versions of the Kepler problem.

DOI
10.1103/physrevd.33.2916

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American Journal of Physics•Apr 22, 2025•1 citation
The stochastic mechanics of Nelson and Guerra is formulated for the hydrogen atom. We demonstrate that this simple quantum system can be described in terms of three independent Gaussian Markov processes which are driven (controlled) by the classical Kepler problem. It reveals a manifest connection between the classical and quantized versions of the Kepler problem.
The expression “quantization of a mechanical system” is somewhat vague and this is not the place to give an exact and precise definition (which, probably’, can not be given at all). So we take it for granted that the reader is familiar with the fascinating subtleties of quantum mechanics and only recall that, from the physical point of view, quantizing the Kepler Problem means to find the spectrum of the energy of the hydrogen atom and the probability distribution of the electron. We will try, in Part II of this book, to make mathematically well defined the term “quantization”, while, in this chapter, we review the historical methods for quantizing the problem, and, more precisely, the Schrödinger, the Pauli and the Fock methods.
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.
Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton's equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.
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.