The excited states of Hydrogen, like all atoms, have a non-zero lifetime due to interactions with the vacuum electromagnetic field. Rigorously speaking, these excited orbital states are not the true eigenstates of the combined atom-light system, as those eigenstates should not decay. For comparison, the $1s$ state, combined with zero photons in any mode, is truly the ground state of the combined atom-light system.
Formally, $\vert 1s;0\rangle$ is an eigenstate of the total Hamiltonian $$H_T=H_{atom}+H_{EM},$$ whereas $\vert 2p;0\rangle$ is not. The true excited eigenstates (I suppose) should be some superposition state of the excited orbital and excited photon number states.
$$\vert \psi_{2p} \rangle ?= \sum_i a_i\vert 1s;n_i\rangle + \sum_j b_j \vert 2p;n_j\rangle$$
My question is simple, can we write down the actual eigenstates of an excited level of the Hydrogen atom in the hydrogen orbital/photon number basis?
If these eigenstates are too complicated, can the nature of the true excited state be at least described in more detail? And, to be clear, I want to ignore fine, hyperfine, and spin structure unless it's necessary. Hopefully, the full machinery of QFT is not needed and some appropriately large volume $V$ can be used, but I am unsure about that.