The correspondence limit of the atomic elliptic state

Department of Mathematics, Swansea University - Singleton Park, Swansea, SA2 8PP, Wales, UK, EU

Corresponding author: a.d.neate@swansea.ac.uk

Received: 30 July 2008
Accepted: 19 September 2008

Abstract

We consider the wave function for the atomic elliptic state using Nelson's stochastic mechanics. The Bohr correspondence limit is taken to reveal a limiting Nelson diffusion. We demonstrate that this limiting diffusion is a small random perturbation of a deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This shows how to derive Kepler's laws of motion in a quantum-mechanical setting. We show that the generator of the limiting Nelson diffusion has a spectral gap and thereby give an explicit rigorous estimate for the rate of convergence to the elliptical orbit. We discuss possible applications of these results to the long time limit of the quantum particle density for a semiclassical system in a Coulomb potential and also to the formation of planets from a protosolar nebula.