The problem of symmetry in quantum mechanics

Tapan Kumar Das

J. M. Jauch, Edward L. Hill

The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the $n$-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in $n$ dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.

J.P. Elliott, P. G. Dawber

Geoffrey Brooker

Abstract “Symmetry in quantum mechanics” shows how symmetry properties are helpful in taming problems in quantum mechanics. When two operators commute, wavefunctions may be prefabricated as eigenstates of the simpler operator. There is detailed application to spin–orbit interaction and to helium.

William Martin McClain

Xiaojing Jin, Boyang LIU

In the problem of inverse-square force, there exists a hidden SO(4) symmetry, which is manifested as a conserved quantity the Laplace-Runge-Lenz vector in the classical Kepler problem, and as the degeneracy of hydrogen atom energy levels beyond the three-dimensional rotational symmetry in quantum mechanics. This paper presents a more intuitive representation of this hidden symmetry through geometric imagery. Additionally, the paper derives in detail the relevant formulas for the raising and lowering operators corresponding to this symmetry in quantum mechanics.