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

TL;DR

This paper rigorously compares Fermi's golden rule with exact calculations for hydrogen's 2p-1s transition, showing the rule's high accuracy and providing a method to select the optimal cutoff in the dipole approximation.

Contribution

It offers a cutoff-independent regularization for the dipole approximation and demonstrates the rule's precision with deviations as low as 10^{-8} to 10^{-7}.

Findings

Fermi's golden rule closely matches exact dynamics.

Regularization improves dipole approximation accuracy.

Optimal cutoff frequency prescription enhances theoretical predictions.

Abstract

Focusing on the $2\mathrm{p}-1\mathrm{s}$ 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 ($2\mathrm{p}$) 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.