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

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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.