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# Advances in Theoretical and Mathematical Physics

## Volume 9 (2005)

### Number 4

### Degenerate eigenvalues for Hamiltonians with no obvious symmetries

Pages: 593 – 602

DOI: http://dx.doi.org/10.4310/ATMP.2005.v9.n4.a4

#### Authors

#### Abstract

Certain Hamiltonians, based on two coupled quantum mechanical\break spins, exhibit degenerate eigenvalues despite having no obvious non-Abelian symmetries. Operators acting to permute the degenerate states do not have a simple form when expressed as polynomials of the generators of rotations for the respective spins. As observed in [3], one such Hamiltonian helps explain resonances in the spin relaxation rate of optically pumped ${\rm Rb}_2$, as a function of applied magnetic field. We give an explanation of why the degeneracies exist, based on properties of the commutator and anti-commutator of the Hamiltonian and its image under magnetic field reversal.