On the reduction of the Dirac equation to non-relativistic form: the Fermi contact interaction in the hydrogen atom

Abstract

The second-order hyperfine energy in the 1s and 2s states of the hydrogen atom is calculated analytically using the Dirac equation for an electron in a Coulomb potential perturbed by a point nuclear magnetic moment. The (divergent) result is compared with the calculation using the non-relativistic form of the hamiltonian. It is emphasized that in this comparison all the contributions to the non-relativistic second-order energy must be included. It is shown that, at least for the Fermi contact interaction, the results are the same to order mc ²α⁶, so that for this case the reduction of the Dirac equation to non-relativistic form is not suspect, contrary to previous assertions.