The quantum harmonic oscillator on the sphere and the hyperbolic plane: κ-dependent formalism, polar coordinates, and hypergeometric functions

Abstract

A nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates (r,ϕ) by making use of a curvature-dependent formalism. The curvature κ is considered as a parameter and then the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere S2 (κ>0) and the hyperbolic plane H2 (κ<0). A comparative study between the spherical and the hyperbolic quantum results is presented.