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.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.

Physical ReviewApr 1, 1940369 citations

On the connections between the classical and quantum-mechanical Kepler problems

Joachim Dahl, Thomas Godsk Jørgensen

The Runge-Lenz vector, which accounts for the accidental degeneracy of the non-relativistic Kepler problem, has been the subject matter of many studies, both in quantum mechanics and in classical mechanics. Much less attention has been paid to the Johnson-Lippmann operator which accounts for the accidental degeneracy of the relativistic Kepler problem in Dirac`s quantum-mechanical description. In the present communication we discuss the properties of the Johnson-Lippmann operator. We show its relation to the non-relativistic Runge-Lenz vector and draw a connection to Sommerfield`s early discussion of the relativistic Kepler problem. This enables us, inter alia, to give an explanation of the apparent coincidence of the energy expressions of the two theories.The Runge-Lenz vector, which accounts for the accidental degeneracy of the non-relativistic Kepler problem, has been the subject matter of many studies, both in quantum mechanics and in classical mechanics. Much less attention has been paid to the Johnson-Lippmann operator which accounts for the accidental degeneracy of the relativistic Kepler problem in Dirac`s quantum-mechanical description. In the present communication we discuss the properties of the Johnson-Lippmann operator. We show its relation to the non-relativistic Runge-Lenz vector and draw a connection to Sommerfield`s early discussion of the relativistic Kepler problem. This enables us, inter alia, to give an explanation of the apparent coincidence of the energy expressions of the two theories.

Bulletin of the American Physical SocietyMay 1, 19930 citations

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.