Remarks on the Relativistic Kepler Problem. II. Approximate Dirac-Coulomb Hamiltonian Possessing Two Vector Invariants

Abstract

The Dirac-Coulomb Hamiltonian is shown to contain a "fine structure interaction" which, when removed, defines a new Hamiltonian differing from the Dirac-Coulomb Hamiltonian in order $\frac{{\left(\alpha Z\right)}^2}{\left|\kappa \right|}$. The solutions of this new Hamiltonian, as well as its complete set of invariant operators, are explicitly given. This "symmetric Hamiltonian" possesses a larger symmetry group than the $R_4$ group structure of the nonrelativistic Coulomb Hamiltonian. The simplicity of the complete orthonormal set of solutions of the symmetric Hamiltonian lends itself to several useful applications which are briefly indicated. The relation between the solutions of this new Hamiltonian and the Sommerfeld-Maue-Meixner-Furry wave functions is discussed.