Degeneracy of the Dirac Equation with Electric and Magnetic Coulomb Potentials

Abstract

Investigation is made of the symmetry and degeneracy of the Dirac equation for a Coulomb potential with a fixed center bearing both electric and magnetic charge. Seen from the viewpoint of classical mechanics, relativistic precession removes the accidental degeneracy of the nonrelativistic potential, and may be so severe as to lead to spiral rather than precessing elliptic orbits. The degeneracy may be restored by the introduction of a vector potential which combats the precession and leads to closed relativistic orbits. An angular momentum and a Runge vector are found for the ``symmetric'' potential for arbitrary values of electric and magnetic nuclear charges. A related symmetric Hamiltonian and constants of the motion may be constructed for the Dirac equation, which reduce to those of Biedenharn and Swamy in the absence of magnetic charge. Magnetic charge must be quantized—a requirement seen from the angular part of the wavefunction exactly as in the nonrelativistic problem. The Dirac Hamiltonian is singular for the lowest admissible angular momentum state, corresponding to the spiral orbits, when the magnetic charge is nonzero. The remaining states show an accidental doubling of degeneracy, whose presence may be deduced from an operator which reduces to that of Johnson and Lippman, or the algebra of Malkin and Manko, without the magnetic charge.