The interaction with the transverse modes of the vacuum electromagnetic field has been shown to yield a radiative (or self-energy) shift of the ground state of a two-level atom in Section 6.4. In this appendix we will show that the same effect occurs also in the energy of the states of a multilevel atom of the hydrogenic kind, which can be modelled by an electron bound to a fixed nucleus of charge Ze. This self-energy shift will turn out to depend on the form of the wavefunction of the state of the electron, leading to the possibility of lifting some of the accidental degeneracies which occur in hydrogenic atoms. Indeed the first experimental observation of this effect is related to the lifting of the well-known 2s-2p degeneracy in atomic hydrogen, and it is due to Lamb and Retherford (1947). Its nonrelativistic QED explanation, on the other hand, is due to Bethe (1947), and this appendix is simply a short account of his theory.

Consider a bare one-electron atom, whose energy levels and corresponding wavefunctions are denoted by En and un(x) = 〈x | n〉. Here n indicates the triplet of quantum numbers N, L, M and the electron spin is disregarded.