Symmetry in quantum mechanics and symmetry groups

Paul F. McMillan, Anthony C. Hess

A. Cabo, J. L. Lucio

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.

Florian A. Scheck

Gianni Cassinelli, Ernesto De Vito, PEKKA J. LAHTI +1 more

Various mathematical formulations of the symmetry group in quantum mechanics are investigated and shown to be mutually equivalent. A new proof of the theorem of Wigner on the symmetry transformations is worked out.