Separability of the Planar $1/\rho^{2}$ Potential In Multiple Coordinate Systems

TL;DR

This paper investigates the separability of the Schrödinger equation for a specific $1/\rho^{2}$ potential in multiple coordinate systems, revealing classical and quantum features like bound states and degeneracies.

Contribution

It demonstrates that the $1/\rho^{2}$ potential is separable in both cylindrical and parabolic coordinates, and analyzes the resulting classical and quantum properties.

Findings

Bound classical orbits always close in this potential.

Schrödinger equation separates into three equations in parabolic coordinates.

Two of these equations resemble Coulomb radial equations with opposite potentials.

Abstract

With a number of special Hamiltonians, solutions of the Schr\"{o}dinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including the isotropic harmonic oscillator and the Coulomb potential. Multiply separable Hamiltonians exhibit a number of interesting features, including "accidental" degeneracies in their bound state spectra and often classical bound state orbits that always close. We examine another potential, for which the Schr\"{o}dinger equation is separable in both cylindrical and parabolic coordinates: a $z$-independent $V\propto 1/\rho^{2}=1/(x^{2}+y^{2})$ in three dimensions. All the persistent, bound classical orbits in this potential close, because all other orbits with negative energies fall to the center at $\rho=0$. When separated in parabolic coordinates, the Schr\"{o}dinger equation splits into three individual equations, two of which are equivalent to the radial equation in a Coulomb potential---one equation with an attractive potential, the other with an equally strong repulsive potential.