Electrodynamic Displacement of Atomic Energy Levels. II. Lamb Shift

The vacuum fluctuations of the photon and pair fields modify the interaction of an electron with an electromagnetic field. The effects on the energy levels are conveniently described in terms of the mass operator and the vacuum polarization potential. An operator calculus for handling the mass operator is used to separate the contribution of low energy quanta, for which the external electromagnetic field may not be treated as weak. The remainder is given as a power series in the field. The additional displacement of order $Z\alpha$ with respect to the published values for the Lamb shift is calculated by specializing the external field to the Coulomb field of the nucleus and by taking the matrix element of the operators in an $S$-state of a hydrogen-like atom. It is found that the $\mathrm{nS}$-level is raised by $\frac{Z^5{\alpha }^4}{n^3}\mathrm{Ry}\left(1+\frac{11}{128}-\frac{1}{2}\ln 2+\frac{5}{192}\right).$ The theoretical value for the Lamb shift of an atom with infinitely heavy nucleus becomes 1058.42 Mc/sec. Effects of finite nuclear mass and size change this value to 1057.8 Mc/sec and 1058.9 Mc/sec for hydrogen and deuterium, respectively.