Motion on constant curvature spaces and quantization using Noether symmetries

Area of Science:

• Mathematical Physics
• Quantum Mechanics
• Differential Geometry

Background:

• Quantizing nonlinear systems on manifolds presents significant challenges.
• Understanding systems on spaces with constant curvature is crucial in various physics domains.

Purpose of the Study:

• To develop a general method for quantizing metric nonlinear systems on manifolds of constant curvature.
• To provide an exact analytical solution for the system's energy spectrum and wave functions.

Main Methods:

• Utilizing a curvature-dependent procedure to determine Noether symmetries from the metric.
• Employing Lie differentiation of the metric to identify symmetries.
• Selecting a specific metric that allows solving the Schrödinger equation using hypergeometric functions.

Main Results:

• A general approach for quantizing nonlinear systems on constant curvature manifolds is established.
• The Schrödinger equation is solved exactly using hypergeometric functions.
• The energy spectrum and wave functions of the system are determined analytically.

Conclusions:

• The presented method offers an exact and generalizable framework for quantum system analysis on curved spaces.
• The use of hypergeometric functions provides a powerful tool for solving complex quantum mechanical problems.