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

TL;DR

This paper develops a curvature-dependent formalism for the quantum harmonic oscillator on 3D spherical and hyperbolic spaces, deriving exact energy spectra and wavefunctions, and comparing results across different geometries.

Contribution

It introduces a unified approach to solve the Schrödinger equation on curved spaces, providing exact solutions for energy levels and wavefunctions in spherical and hyperbolic geometries.

Findings

Exact energy spectra obtained for both geometries

Wavefunctions expressed via hypergeometric functions

Comparative analysis highlights geometric effects on quantum states

Abstract

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($\kappa>0$) and $H_k^3$ ($\kappa<0$), is studied. The curvature $\k$ is considered as a parameter and then the radial Schr\"odinger equation becomes a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-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 $S_\k^3$ ($\kappa>0$) and the hyperbolic space $H_k^3$ ($\kappa<0$). A comparative study between the spherical and the hyperbolic quantum results is presented.