Analysis of the Dirac-Coulomb problem in the free-particle representation.

Abstract

The problem of an electron bound by a Coulomb field of strength $\ensuremath{\alpha}=Z{\ensuremath{\alpha}}^{\ensuremath{'}}({\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\approx}{137}^{\ensuremath{-}1})$ is considered in the free-particle representation of the Dirac equation in momentum space. It is shown explicitly how $O({\ensuremath{\alpha}}^{5})$ contributions to the energies for all $s$-wave states that come from the relativistic free-particle kinetic energy and from the modified Coulomb interaction are canceled by a contribution from electron-positron pair formation. The relativistic correction to the momentum-space Schr\odinger wave function is also shown to remove an unwanted $O({\ensuremath{\alpha}}^{6}\mathrm{ln}\ensuremath{\alpha})$ energy dependence that arises in lowest-order perturbation theory. The work shows how to perform more accurate calculations in semirelativistic quasipotential approaches that are commonly used in quark models.