# Bound state transitions in QFT

## Main Question or Discussion Point

In non-relativistic QM, we spend a lot of time examining bound states--energy levels, spatial distributions, and all that. We can determine that an electron put into a Coulomb potential will have certain Hamiltonian eigenstates, which correspond to the orbitals of a hydrogen atom. However, in NRQM, all of these states are stable, whereas in real life an excited electron will eventually emit a photon and fall into a lower energy level. It makes sense that NRQM can't handle this phenomenon, because it has no way to discuss particle creation/destruction, so there's no way to describe energy leaving the system and dropping the electron to a different state.

Enter Quantum Field Theory. In QFT, it's very easy to describe particles being created/destroyed, so it seems as though it shouldn't have any problem discussing "excited electron -> electron + photon" transitions. However, most of the machinery I've seen thus far in QFT is focused around free-particle scattering--we spend a lot of time setting up Fock spaces for asymptotic in/out states, and computing the transition probabilities between them using perturbation theory. This makes sense, since most of our experiments come out of particle accelerators, but it doesn't seem like it helps much if you want to deal with bound states.

How, then, does one go about setting up a problem of this form? I know you can add an external potential to the Lagrangian in the form of a term like $V(x)\phi(x)$. I would imagine that one can then solve the equations of motion for the particle sort of like before, except that instead of having a solution that is a superposition of plane waves, you'll get a solution that's a superposition of bound states.

But where would you go from here? Normal perturbation theory relies on breaking down the Hamiltonian into combinations of free-field propagators, but that doesn't work here because the particle doesn't obey the free-field equation anymore. Is it possible to work out the equivalent of a propagator for an electron in an external potential, and then use that to compute Feynman diagrams for a photon interacting with it? I haven't been able to find any references that describe how to deal with a problem like this, but it seems like the sort of basic scenario that ought to have been examined by somebody at this point. I'd appreciate any information anybody has on how to deal with this sort of problem.