The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach

TL;DR

This paper investigates the quantum free particle on spherical and hyperbolic spaces by treating curvature as a parameter, analyzing geometric formalisms, quantization, and solving the curvature-dependent Schrödinger equation to find wavefunctions and spectra.

Contribution

It introduces a curvature-dependent formalism for quantum particles on curved spaces, deriving explicit solutions and spectra for spherical and hyperbolic geometries.

Findings

Discrete spectrum for positive curvature (spherical case)

Explicit wavefunctions related to orthogonal polynomials

Analysis of geometric and quantization formalism for curved spaces

Abstract

The quantum free particle on the sphere $S_\kappa^2$ ($\kappa>0$) and on the hyperbolic plane $H_\kappa^2$ ($\kappa<0$) is studied using a formalism that considers the curvature $\k$ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces ($S_\kappa^2$, $\IR^2$, $H_\kappa^2$) and with the the transition from the classical $\kappa$-dependent system to the quantum one using the quantization of the Noether momenta. The Schr\"odinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the $\kappa$-dependent Schr\"odinger equation. First the characterization of the $\kappa$-dependent `curved' plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if $\kappa>0$ then a discrete spectrum is obtained. The wavefunctions, that are related with a $\kappa$-dependent family of orthogonal polynomials, are explicitly obtained.