Free particle and isotropic harmonic oscillator on a spheroidal surface: the Higgs-like approach

TL;DR

This paper studies the quantum dynamics of a free particle and isotropic harmonic oscillator on a spheroidal surface, revealing how non-sphericity causes energy level splitting and degeneracy lifting.

Contribution

It introduces a Hamiltonian approach for particles on spheroidal surfaces and applies perturbation theory to analyze energy spectra and eigenfunctions.

Findings

Non-sphericity induces an effective potential.

Energy levels of the oscillator are split due to spheroidal shape.

Degeneracy of energy levels is lifted by surface deviation.

Abstract

In this paper, we investigate the dynamics of both free particle and isotropic harmonic oscillator constrained to move on a spheroidal surface using two consecutive projections: a projection onto a sphere surface followed by the gnomonic projection onto a tangent plane to the spheroid. We obtain the Hamiltonian of the aforementioned systems in terms of the Cartesian coordinates of the tangent plane and then quantize it in the standard way. It is shown that the effect of non-sphericity of the surface can be treated as the appearance of an effective potential. By using the perturbation theory up to the first order in second eccentricity of the spheroid, we approximately calculate the eigenfunctions and eigenvalues of the free particle, as well as the isotropic harmonic oscillator on the spheroidal surface. We find that the deviation from the sphericity plays an important role in splitting the energy levels of the isotropic oscillator on a sphere, and lifting the degeneracy.