Spontaneous light emission by atomic hydrogen: Fermi's golden rule without cheating

Abstract

Focusing on the 2p–1s transition in atomic hydrogen, we investigate through first order perturbation theory the time evolution of the survival probability of an electron initially taken to be in the excited (2p) state. We examine both the results yielded by the standard dipole approximation for the coupling between the atom and the electromagnetic field – for which we propose a cutoff-independent regularisation – and those yielded by the exact coupling function. In both cases, Fermi's golden rule is shown to be an excellent approximation for the system at hand: we found its maximal deviation from the exact behaviour of the system to be of order 10−8/10−7. Our treatment also yields a rigorous prescription for the choice of the optimal cutoff frequency in the dipole approximation. With our cutoff, the predictions of the dipole approximation are almost indistinguishable at all times from the exact dynamics of the system.