Motion on Constant Curvature Spaces and Quantization Using Noether Symmetries

TL;DR

This paper introduces a method for quantizing systems on constant curvature manifolds using Noether symmetries, enabling exact solutions for energy spectra and wave functions in quantum mechanics.

Contribution

It presents a curvature-dependent quantization approach leveraging Noether symmetries, allowing exact solutions on constant curvature spaces.

Findings

Exact solutions for the Schrödinger equation in hypergeometric functions

Determination of energy spectra and wave functions

Application of symmetry-based quantization method

Abstract

A general approach is presented for quantizing a metric nonlinear system on a manifold of constant curvature. It makes use of a curvature dependent procedure which relies on determining Noether symmetries from the metric. The curvature of the space functions as a constant parameter. For a specific metric which defines the manifold, Lie differentiation of the metric gives these symmetries. A metric is used such that the resulting Schrodinger equation can be solved in terms of hypergeometric functions. This permits the investigation of both the energy spectrum and wave functions exactly for this system.