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Journal of Physics A Mathematical and Theoretical

Symmetry group of an impenetrable cubic well potential

Ariatna Hernández-Castillo, R. Lemus

Published Nov 1, 2013• 8 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/46/46/465201

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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 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.
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.
This paper solves the wavefunctions and energy level formulas of two types of three-dimensional infinite deep cubic potential wells and three-dimensional infinite deep spher-ical potential wells, and studies their degeneracy to elucidate how symmetry governs energy level degeneracy in quantum systems. For the cubic potential well, the potential energy inside the cube is zero, and outside the cube is infinite. Solving the Schrödinger equation shows that the energy is determined by the total of the squares of three quantum numbers (nx, ny, nz). Degeneracy occurs when different combinations of quantum numbers produce the same sum of squares, which reflects cubic symmetry, i.e., the equivalence of the x, y, and z directions, such that permuted states are physically identical. For the spherical potential well, the poten-tial energy inside the sphere is zero, and outside the sphere is infinite. The wavefunction can be devided into angular and radial parts. The energy depends on the radial quantum number n and the angular momentum quantum number l, and degeneracy arises from the magnetic quantum number m, which ranges from −l to l. Since the potential energy of the spherical potential well is independent of direction, it has spherical symmetry, which leads to degen-eracy of magnitude 2l + 1-fold. To summarize, the degeneracy of the cubic potential well originates from its cubic symmetry, while that of the spherical potential well originates from its spherical symmetry. This highlights the central role of symmetry in quantum energy level degeneracy.