The Lenz Vector and the Accidental Degeneracy

L.J. Weigert, M. E. Rose

A. A. Stahlhofen

S. P. Alliluev

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.

Harold V. McIntosh

Charles E. Burkhardt, Jacob J. Leventhal

We show that operators capable of converting certain spherical hydrogen atom eigenfunctions into others by raising the angular momentum quantum number ℓ can be constructed from components of the quantum mechanical Lenz vector operator. The derivation is performed using formalism no more sophisticated than that used to derive the properties of the angular momentum ladder operators in undergraduate quantum mechanics courses. The properties of these Lenz vector operators illustrate the accidental degeneracy of the hydrogen atom because they change the quantum number ℓ without affecting n.