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Quantum Degeneracy in Two Dimensional Systems

Debnarayan Jana

Published Jan 1, 2009• 1 citation
OpenAlex

Abstract

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.

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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.
A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum-mechanical formulations of the problems, and illustrates the care necessary in forming quantum-mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple-valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two-dimensional hydrogen atom.
A quantum particle in a impenetrable cubic well potential presents accidental degeneracy when the group is considered to be the symmetry group of the system. This degeneracy becomes natural when a new symmetry group, embedding the group, is proposed. This new group turns out to be the semidirect product , where is a two-dimensional compact continuous group whose generators correspond to linear combinations of the one-dimensional Hamiltonians. The systematic degeneracy is studied in detail, the new group is identified and its irreducible representations (irreps) are constructed by means of induction, an approach that allows the irreducibility and completeness to be assured. Pythagorean degeneracy as well as the one due to commensurable sides are not considered.
One-dimensional quantum systems that undergo spontaneous symmetry-breaking, having a symmetric (non-degenerate) and a broken-symmetry (doubly-degenerate) phase, have been intensely studied in different branches of physics. In most cases, the spontaneously-broken symmetry is parity. However, it is possible to obtain similar phases in systems without parity symmetry, through an antiunitary symmetry that implies a two-fold symmetry either on momentum or coordinate in the system's classical limit. To illustrate this phenomenon, we use a Kerr parametric oscillator (KPO) with one- and two-photon drives that, despite the breaking of parity symmetry, may have doubly-degenerate levels. Different realizations of squeezed KPOs convey a great deal of attention, as effective Hamiltonians for driven superconducting circuits and the occurrence of degeneracy in such systems could be of practical interest in their application to obtain protected qubits in parity-breaking setups. In addition to this, the reported spectral features strongly indicate the existence of additional symmetries in the system.
When the symmetry group of a quantum particle in an impenetrable cubic well potential is considered to be the group, systematic accidental degeneracy appears. This degeneracy becomes natural when a new symmetry group, embedding the group, is proposed. This new group turns out to be the semidirect product , where is a two-dimensional compact continuous group whose generators correspond to linear combinations of the one-dimensional Hamiltonians. The systematic degeneracy is studied in detail, the new group is identified and its irreducible representations are constructed by means of induction, an approach that allows the irreducibility and completeness to be assured. Similar to the hydrogen atom, we establish a one-to-one reciprocation between the energy and the new group irreducible representations. The impenetrable rectangular and square boxes are also analyzed as a reduction of symmetry from the cubic system. Pythagorean degeneracy as well as that due to commensurable sides is not considered.