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

摘要

A nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates (r,phi) by making use of a curvature-dependent formalism. The curvature kappa is considered as a parameter and then the radial Schrodinger equation becomes a kappa-dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere S-2 (kappa>0) and the hyperbolic plane H-2 (kappa < 0). A comparative study between the spherical and the hyperbolic quantum results is presented. (C) 2007 American Institute of Physics.