An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

TL;DR

This paper derives classical solutions for harmonic oscillators on curved surfaces like the sphere and hyperbolic plane in polar coordinates, relates them to Cartesian solutions, and identifies the orthogonal polynomials in the quantum bound states.

Contribution

It provides a straightforward derivation of classical solutions in polar coordinates and clarifies the quantum wavefunctions' polynomial structure on curved geometries.

Findings

Classical solutions are explicitly derived in polar coordinates.

Relations between polar and Cartesian solutions are established.

The orthogonal polynomials in quantum wavefunctions are identified.

Abstract

A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.