Semiclassical wave functions and semiclassical dynamics for the Kepler/Coulomb problem

Neate, Andrew and Truman, Aubrey (2014). Semiclassical wave functions and semiclassical dynamics for the Kepler/Coulomb problem. Journal of Physics A: Mathematical and Theoretical, 47(22), article no. 225302.

DOI: https://doi.org/10.1088/1751-8113/47/22/225302

Abstract

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.

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