Extracting Hidden Symmetry from the Energy Spectrum

TL;DR

This paper presents a method to identify hidden symmetries in quantum systems by analyzing energy spectra, demonstrated on a spin Hamiltonian and the hydrogen atom, revealing underlying symmetry groups.

Contribution

It introduces a general approach to uncover hidden symmetries from known eigenvalues and eigenstates, exemplified on specific quantum models.

Findings

Identified SU(2) symmetry in the spin Hamiltonian.

Revealed symmetry operators with clear physical meaning.

Applied method successfully to the hydrogen atom.

Abstract

In this paper we revisit the problem of finding hidden symmetries in quantum mechanical systems. Our interest in this problem was renewed by nontrivial degeneracies of a simple spin Hamiltonian used to model spin relaxation in alkali-metal vapors. We consider this spin Hamiltonian in detail and use this example to outline a general approach to finding symmetries when eigenvalues and eigenstates of the Hamiltonian are known. We extract all nontrivial symmetries responsible for the degeneracy and show that the symmetry group of the Hamiltonian is SU(2). The symmetry operators have a simple meaning which becomes transparent in the limit of large spin. As an additional example we apply the method to the Hydrogen atom.