Relativistic Correction to the Lamb Shift

Abstract

The relativistic corrections to the Lamb shift, i.e., terms of order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{5}m{c}^{2}$, are calculated. For this purpose, the Lamb shift is separated into one term in which the Coulomb potential acts only once, and another term in which it acts two or more times (Sec. II). The one-potential term is shown to be equal to the expression calculated in previous papers except for corrections of order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{6}$ (Sec. III), and a method is given by which these corrections could be evaluated if desired (Appendix). The many-potential term can be separated into a nonrelativistic part which is again equal to the term calculated in previous papers, and a relativistic term which can be calculated by considering the intermediate states as free (Sec. IV). The calculation of the latter term which, of course, involves the Coulomb potential exactly twice, is described in Sec. V. A correction to the vacuum polarization term which is of the same order, is evaluated in Sec. VI.The result for the relativistic correction is 7.13 Mc/sec, and is in agreement with the result of Karplus, Klein, and Schwinger which was obtained by an independent method. The result for the complete Lamb shift has been given in a recent paper by Salpeter. The small remaining discrepancy of 0.6 Mc/sec between theory and experiment might be due to the next order relativistic correction which should be of order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{6}\mathrm{ln}(Z\ensuremath{\alpha})$.