Effective Potential Model for Calculating Nuclear Corrections to the Energy Levels of Hydrogen

The present work is an attempt to re-evaluate the nuclear corrections to the energy levels of hydrogen by using an effective potential approach. The basic idea is to infer from electron-proton scattering a potential which may then be applied to the bound-state problem. In lowest order, the potential is chosen from the first-order Feynman diagram for the scattering. With this choice the Breit equation is obtained. It is then solved in an approximate way, in the non-relativistic limit of the proton, to obtain wave functions which are accurate enough for use in evaluating the effects of perturbations of the potential. The reduced mass corrections to the fine structure and the hyperfine structure levels are readily found. The effect on the hyperfine splitting of the distribution of the proton charge and magnetic moment is obtained by correcting the lowest-order potential to include the proton form factors. A further modification is needed in evaluating additional recoil corrections, of relative order $\frac{\alpha m}{M}$, to the fs and the hfs. This additional term accounts for the failure of the iteration of the lowest-order potential to reproduce the scattering obtained from the second-order Feynman diagrams. The $\frac{{\alpha }^{2}m}{M}$ contribution to the state-dependent mass corrections to the hfs is also analyzed within the context of this approach. All the corrections found are in complete agreement with previous results obtained by the Bethe-Salpeter (BS) equation, but the present method has the virtue of conceptual simplicity.