Energy Levels and Lorentz Invariance

Abstract

AN atom is in an excited state S′ with energy E′. It falls to a state S with lower energy E and emits a photon with frequency v. According to a basic postulate of quantum mechanics, this frequency is related to the energies by The theoretical calculation of E and E′ is carried out in the rest-frame of the atom, and so it would appear that v in equation (1) is the frequency observed in that frame. Conservation of momentum, however, demands that the atom recoils when the photon is emitted, and so there is not one rest-frame but two—the rest-frame of S′ and the rest-frame of S. It is true that the recoil is small and consequently the two rest-frames are nearly the same. But in spectroscopy wavelengths are sometimes given with 8-figure accuracy, and it is interesting to inquire whether such (or higher) accuracy is consistent with neglect of the small difference between the two rest-frames.