Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

TL;DR

This paper applies group theory to analyze a 3D quartic anharmonic oscillator, predicting energy level degeneracies and aiding in solution methods based on symmetry considerations.

Contribution

It demonstrates how group theory can be used to predict degeneracies and construct symmetry-adapted basis sets for a complex quantum system.

Findings

Group theory predicts energy level degeneracies.

Symmetry-adapted basis sets facilitate solution methods.

Application of perturbation and variational methods with symmetry considerations.

Abstract

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with $O_{h}$ symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh-Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions . We show how to obtain suitable symmetry-adapted basis sets.