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Journal of Chemical Education

Accidental degeneracies of the particle in a box

CHARLES A. HOLLINGSWORTH

Published Dec 1, 1990• 1 citation
Publisher

Abstract

Describes the types of accidental degeneracies that occur for a single particle in a box with impenetrable walls.
DOI
10.1021/ed067p999

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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 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.