Hydrogen atom in quantum field theory

You can't have the hydrogen atom as a state in QED with protons and electrons, because it is a bound state and hence it is nonperturbative. QED is inherently a perturbative theory. There are good reasons to believe that QED doesn't even exist nonperturbatively, unless it is associated with a broken phase of a non-abelian gauge theory.

You can, however, deduce the properties of the hydrogen atom from QED by making simplifying assumptions about the interaction:

1. You want to take the classical limit for the electromagnetic sector.
2. You want to assume that the Compton wavelength of the electron is much less than the typical scale of the atomic orbitals. This means you can treat the electron and proton as first-quantized particles without worrying about pair production.

You'll end up with the Dirac equation for the electron in Coulomb potential of the proton, which can be solved and leads to the well known result.

Small corrections to this result can also be deduced from perturbative QED. For example, the first-order correction to the photon propagator is responsible for the Lamb shift.