On second-order magnetic hyperfine interactions in one-electron atoms: connections between the Schrodinger, Dirac and quantum electrodynamical perturbation calculations

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Abstract
Second-order hyperfine interactions are considered for a one-electron atom, treating the nucleus as a point magnetic dipole of infinite mass and charge Z. The dominant contribution to the hyperfine splittings in the 1s and 2s states is found to arise from the ultra- relativistic continuum. In the high momentum limit the Schrodinger calculation gives a linear divergence while the Dirac and QED calculations give a logarithmic divergence. This explains why the non-relativistic variational result of gregson et al. (1970) is too large by a factor of 102. The Dirac result is 8/9 of the QED result, (9/4)( alpha / pi )( mu m/M)ln( rho max), because of a missing retardation term. The Schrodinger and Dirac residuals are in good agreement.