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American Journal of Physics

SO(2) and the study of the infinite-well potential

Dennis Aebersold, Andrew Langerman

Published Mar 1, 1977• 4 citations
Publisher

Abstract

We identify the symmetry group of the one-dimensional infinite-well potential problem as SO(2). By studying the irreducible representations of the SO(2) group we obtain the eigenvalues and eigenkets for the problem.
DOI
10.1119/1.11008

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We will prove that if the order parameter belongs to the representation DDD which is the direct sum of two complex conjugate irreducible representations, $D=\ensuremath{\Delta}\ensuremath{\bigoplus}{\ensuremath{\Delta}}^{*}$, and if $\ensuremath{\Delta}$ has at least one quartic invariant, then the symmetry SO(2) appears in the parameter space. If, in addition, $\ensuremath{\Delta}$ and ${\ensuremath{\Delta}}^{*}$ are quasiequivalent then higher symmetry than SO(2) will appear in the parameter space. Since SO(2) is a continuous group, every fixed point of the renormalization-group transformations will be either invariant under SO(2) or part of a fixed line generated by SO(2) transformations on the fixed point.
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.