Degeneracy in the particle-in-a-square problem

S. Pratik Khastgir

We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica$^{\circledR}$.

Kang-Kuen Ni

Pekka Pyykkö

G B Shaw

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.

J. M. Jauch, Edward L. Hill

Eduardo Suárez, Rubén Darío Santiago Acosta, R. Lemus

Abstract Symmetry considerations are taken into account when a particle in a square well potential is studied. This system may display natural degeneracy, accidental degeneracy or systematic accidental degeneracy depending on the depth of the potential. In order to obtain the solutions associated with an arbitrary potential an algebraic discrete variable representation approach based on Pöschl-Teller functions is proposed. It is proved that the geometrical group C 4v is the symmetry group of the system for the case of a finite potential barrier. A similar analysis is carried out for the rectangular square well potential with commensurate sides. In both cases the symmetry projection is crucial to simplify the calculations.