The problem is re-examined from the point of view of the Hamiltonian applicable to two charged particles and accounting for effects of order $\frac{v^2}{c^2}$, where $v$ is the velocity. The work differs from that of Bechert and Meixner in that the energy of the Dirac's electron in a central field is taken as the reference point. The solution is carried out in terms of an eight-component rather than a four-component approximation to the 16-component wave function. The result is the same for practical purposes as that of Darwin for the prequantum-mechanical problem and of Bechert and Meixner for the four-component approximation. The energy formula is affected only by the original Bohr reduced mass correction to the term value and also by a term which is independent of the particular fine structure component and depends only on the principal quantum number. Bethe's electrodynamic shift is, therefore, not obscured to within terms of relative order ${\alpha }^2\left(\frac{m}{M}\right)$ by effect of nuclear motion.

• Received 26 July 1948

DOI:https://doi.org/10.1103/PhysRev.74.1278