What are the detailed steps involved in the emissions of a photon, for example, when an electron drops to a lower energy level? How well do we understand the production of the photon in this example? Any good reference material I can study?
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The best account for photon emission when an electron drops to a lower eigenstate is the Wigner-Weisskopf Model for spontaneous emission, see this paper from the Photonics group at ETH Zürich and the co-efficients for this model can be calculated by standard quantum electrodynamics. I explain this model further and give references in my answer here.
A summary of what is going on: when one solves the first quantized Dirac or Schrödinger equation to calculate orbitals in an atom or molecule, one is assuming that the atom / molecule is sundered from the rest of the World, i.e. it has no interaction with its surroundings. So the "eigenstates" thus calculated are only energy eigenstates of the Hamiltonian (and thus stationary) in this highly idealized, one atom / molecule universe.
However, the electrons in real atoms / molecules are always coupled to the electromagnetic field. So the "eigenstates" as calculated above are no longer true eigenstate of the whole, coupled system (atom/molecule + the second quantized electromagnetic field), which is why the transition happens. The eigenstates of this system are quantum superpositions of the atom / molecule in its excited state with free photons in the field modes, and the superposition weights for the excited atom state in these new eigenstates are very small compared to those for the free photon weights. Once we "switch on" the electromagnetic field, the system will thus smoothly but inexorably evolve to one where the photon has been radiated. This coupling also begets the Lamb Shift: the splitting of energies of so-called "eigenstates" which are degenerate (have the same energy) in the isolated atom Dirac model.
Selene Routley
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