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Journal of Physics Conference Series

Symmetry group of a particle in an impenetrable cubic well potential

Ariatna Hernández-Castillo, R. Lemus

Published May 12, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/512/1/012025

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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.
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.
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.
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.