On the degeneracies of quantum systems

Mingchang Liu

Pengfei Zhang, Yu Chen

Kramers' theorem ensures double degeneracy in the energy spectrum of a time-reversal symmetric fermionic system with half-integer total spin. Here, we are trying to go beyond the closed system and discuss Kramers' degeneracy in open systems out of equilibrium. A natural way to extend the Kramers' degeneracy in open quantum systems is by the degeneracy of different spins' spectra together with the vanishing interspin spectrum. We find the violation of Kramers' degeneracy in time-reversal symmetric open quantum systems is locked with whether the system reaches thermal equilibrium. After a general coupling to an environment in a time-reversal symmetry-preserving way, the Kramers doublet experiences an energy splitting at a short time and then a recovery process. We verified the violation and revival of Kramers' degeneracy in a concrete model of interacting fermions and we find Kramers' degeneracy is restored after the local thermalization time. By contrast, for time-reversal symmetry $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}^{2}=1$, we find that although there is a violation and revival of spectral degeneracy for different spins, the inter-spin spectral function is always nonzero. We also prove that the degeneracy in spectral function protected by unitary symmetry can always be maintained.

J. M. Jauch, Edward L. Hill

Debnarayan Jana

Degeneracy is an important concept in physics and chemistry. Degeneracy and symmetry are closely connected. We distinguish between two types of degeneracy - symmetric or systematic and accidental one. In this paper, we illustrate the concept of degeneracy through some two dimensional quantum mechanical problems. We have also indicated the breaking of the degeneracy by suitable application of perturbing potentials.

Rajat K. Bhaduri

Stavros Fallieros, E. Hadjimichael

In the framework of quantum theory, we present one theorem and three corollaries regarding the direct connection between constants of motion of a physical system and degeneracies of its energy eigenvalues. It is shown that this connection emerges when there exist quantum operators which commute with the Hamiltonian, but not with each other. Further it is shown that if the commutator of these operators is a nonvanishing constant number then (a) all the eigenvalues of the system are degenerate, and (b) the degree of degeneracy is infinite. A number of examples are discussed including the parity degeneracy of the hydrogen atom and the infinite degeneracy of the Landau levels of a charged particle in a constant magnetic field.