Atomic transitions in QED

[KeepWondering]

In the straight-forward derivation of the Kramer-Heisenberg formula describing the photon-atom scattering cross section (to second order perturbation theory, see e.g. the path leading to https://quantummechanics.ucsd.edu/ph130a/130_notes/node470.html), the finite lifetimes of the intermediate states and their natural linewidth, respectively, do not (explicitly?!) show up. As their neglection would cause an infinite coupling for resonant scattering, the linewidths are included subsequently and manually (after the rigorous derivation) without justification.

One possible justification could be, that the linewidth show up when the derivation includes higher order perturbation theory. But I doubt this.

Another approach could be linked to the question: Which states are actually included in the generic "sum over all intermediate states"? Does it include just all "classical" atomic eigenstates? Or rather any discrete energy levels/states from the ground state to the ionisation energy threshold possibly discretised by ΔE = h/(2π τ_planck)?
This would perfectly solve the problem because the linewidth is already included in the sum since the very beginning of the rigorous derivation. In consequence, we/I/the notation however would have to change from the comfortable concept that there are some "special" atomic energy states due to the quantum numbers to the concept of purely dynamic intra-atomic behaviour where all energy steps n*h/(2π τ_planck) are basically allowed and in principle identical in their physical nature except of their different lifetimes.

Can someone provide a justification?

Many thanks in advance!

 

[BigJDubs]

One possible justification for the inclusion of the linewidths in the Kramer-Heisenberg formula is that they are a natural consequence of the quantum mechanical nature of the atom. In particular, due to Heisenberg's uncertainty principle, the energy levels of an atom are not perfectly discrete but instead are spread out in energy around their nominal values. This is because the electrons in an atom can never be completely localized and so their momentum is always uncertain. This means that each energy level can be thought of as having a finite width, or linewidth. When calculating the photon-atom scattering cross section, this linewidth must be taken into account since it affects the probability of the photon being scattered at any given energy. Thus, by including the linewidths in the calculation, one is taking into account the fact that the energy levels of the atom are not perfectly discrete but instead have a finite spread in energy.