Quantization of a Particle on a Two-Dimensional Manifold of Constant Curvature

TL;DR

This paper explores the quantization of a particle on a 2D manifold with constant curvature, deriving the quantum equations, and analyzing their solutions and spectra.

Contribution

It introduces a method to quantize classical systems on curved manifolds using Noether momenta and provides detailed solutions for the resulting Schrödinger equation.

Findings

Schrödinger equation is separable on the manifold

Spectrum and eigenfunctions are explicitly determined

Quantization via Noether momenta is effective on curved spaces

Abstract

The formulation of quantum mechanics on spaces of constant curvature is studied. It is shown how a transition from a classical system to the quantum case can be accomplished by the quantization of the Noether momenta. These can be determined by Lie differentiation of the metric which defines the manifold. For the metric examined here, it is found that the resulting Schrodinger equation is separable and the spectrum and eigenfunctions can be investigated in detail.