Derrick Hylton

Recent measurements of the n = 2 Lamb shift for hydrogen have uncertainties ranging from 4-40 ppm. It has been suggested that theoretical calculations at this level of accuracy should include nuclear size corrections to the self-energy and vacuum polarization. However, there have been questions concerning the validity of approximations in calculating this effect. The present work checks the validity of two approximate methods in calculating the nuclear size effects for hydrogenic atoms (specifically for Z = 1 or 80). These methods are relativistic perturbation theory up to and including second-order and a nonrelativistic approach based on the short distance approximations of Zemach. The validity of the approximate methods is checked by comparing the result of calculating the nuclear size correction to the lowest order vacuum polarization (Uehling potential) via the approximate methods to the result of an exact calculation. The exact calculation is based upon a numerical solution of the Dirac equation. In order to perform the numerical calculation in perturbation theory, the reduced Dirac Green function for the Coulomb potential is rewritten in a form suitable for numerical calculations. The nonrelativistic calculation is carried out analytically. Perturbation theory and the nonrelativistic approximation are consistent with the exact calculation at Z = 1, but not at Z = 80. After checking the validity of the approximate methods, the order of magnitude of the nuclear size correction to the lowest order self-energy is obtained. This is accomplished by treating the renormalized level shift due to the low virtual photon energy part of the self-energy in first-order nonrelativistic perturbation theory. The perturbing potential is due to the finite nuclear size. The result is that the order of magnitude of the nuclear size correction to the lowest order self-energy and vacuum polarization for n = 1, is $\alpha$($Z\alpha$)$\sp5$($R$/$\lambda\sb{e}$)$\sp2m\sb{e}c\sp2$, where R is the rms radius of the nucleus. This amounts to about 0.01 ppm in hydrogen. One would expect a correction on the same order for the n = 2 case.