Curvature-dependent formalism, Schr?dinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

摘要

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, S _κ ~3 (κ > 0) and H _κ ~3 (κ < 0), is studied. The curvature κ is considered as a parameter and then the radial Schr?dinger equation becomes a κ-dependent Gauss hypergeometric equation that can be considered as a κ-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere S _κ ~3 (κ > 0) and the hyperbolic space H _κ ~3 (κ < 0). A comparative study between the spherical and the hyperbolic quantum results is presented.