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Journal of Physics A Mathematical Nuclear and General

Degeneracy in the particle-in-a-box problem

G B Shaw

Published Sep 1, 1974• 32 citations
Publisher

Abstract

The notions of systematic and accidental degeneracy are discussed for a quantum-mechanical particle in a two-dimensional box. Separability of coordinates is shown to admit extra systematic degeneracy over that expected from the point-group symmetry. The 'accidental' degeneracy for square and triangular boxes is discussed in detail and related to geometric symmetries-described by rings of operators, however, and not by a symmetry group.

DOI
10.1088/0305-4470/7/13/008

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The notions of systematic and accidental degeneracy are discussed for a quantum-mechanical particle in a two-dimensional box. Separability of coordinates is shown to admit extra systematic degeneracy over that expected from the point-group symmetry. The 'accidental' degeneracy for square and triangular boxes is discussed in detail and related to geometric symmetries-described by rings of operators, however, and not by a symmetry group.
In this paper we show that the point-group (geometrical) symmetry is insufficient to account for the degeneracy of the energy levels of the particle in a cubic box. The discrepancy is due to hidden (dynamical symmetry). We obtain the operators that commute with the Hamiltonian one and connect eigenfunctions of different symmetries. We also show that the addition of a suitable potential inside the box breaks the dynamical symmetry but preserves the geometrical one.The resulting degeneracy is that predicted by point-group symmetry.
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.
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.
A construction of Gaussian Klauder coherent states for systems with degenerate energy spectra is presented. Of special interest are the cases of truly accidental degeneracy that do not arise from a hidden or explicit symmetry. This is the case for a particle in a two-dimensional square box. In this case, the degeneracy results from the number of different ways an integer can be expressed as the sum of two squares. It is shown how to manage this situation and produce generalized coherent states that can mimic classical particle behavior for many collisions with the box walls.