Symmetry and Degeneracy

S. P. Alliluev

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.

Michael Berry

R. Shankar

Yuanbei Sun

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.

Clay Córdova, Diego García-Sepúlveda, Nicholas Holfester

A bstract We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ 2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.