Fourth-Order Radiative Corrections to Atomic Energy Levels. II

A rigorous method is described for treating the problem of the hydrogen-like atom in quantum electro-dynamics by the separation of integrals into parts which can be evaluated with relativistic and nonrelativistic approximations. The method is based on that developed by Baranger, Bethe, and Feynman for the second-order problem, and is applied here to the problem of the fourth-order radiative energy level displacement. It is shown that the result inferred by Weneser, Bersohn, and Kroll from the study of fourth-order radiative corrections to elastic scattering by a given external potential is correct to order ${\alpha }^2{\left(\alpha z\right)}^{4}m{c}^2$ for a hydrogenic atom. It is also shown that the $\alpha Z$ corrections to this result can be obtained using this method, just as was done in the second-order problem by Baranger, Bethe, and Feynman.