The2s12→2p12+onephotontransition in hydrogen and hydrogenlike ions

The $2s_{\frac{1}{2}}\to 2p_{\frac{1}{2}}+\mathrm{one}\mathrm{photon}$ transition rate is calculated and discussed for hydrogen and hydrogenlike ions. It is noted that the induced transition rather than the spontaneous transition is of primary importance since it is the basis of many of the precision Lamb-shift measurements. The lack of a calculation of the transition rate other than a heuristic nonrelativistic derivation which requires a nontrivial assumption motivates the calculation presented here based on the external field approximation to quantum electrodynamics. It is found that the heuristic answer is correct in lowest order. In this derivation we see that the $2s_{\frac{1}{2}}\to 2p_{\frac{1}{2}}+\mathrm{one}\mathrm{photon}$ transition gives an apparent contradiction to the often-stated remark that for the electric dipole matrix element there exist three equivalent representations, the "length," "velocity," and "acceleration" forms. The difficulties of an experimental determination of this transition rate using induced transitions in hydrogenlike ions are briefly noted as well as the somewhat different case of heavy muonic atoms where the spontaneous $2s_{\frac{1}{2}}\to 2p_{\frac{1}{2}}+\mathrm{one}\mathrm{photon}$ transition has been observed.