There is a lot of general discussion about the connection between the QM description of photons and classical EM fields. I understand one needs to solve the quantized version of Maxwell's equations, and all of this is still a little beyond me.
For now, I would like to ask a more concrete question: let's say we have a hydrogen atom, and we are able to excite it to a specific energy level such that it will always emit photons of specific $\hbar\omega_1$.
Then I would say that the closest classical description would be that of a dipole radiation, of which the real part of the electric field can be written in far-file simply as $$\bar E = \bar A(r)\cos(\omega_2 t -kr)$$
Is it true to say that the frequency of classical oscillations of the dipole is the same as that of the monochromatic photon emission? I.e., does $\omega_1=\omega_2$?
I'm trying to get some inuition on the matter. On the one hand, it is tempting to view the oscillations as the classical point of view of some stream of emissions of a given frequency. But on the other hand, the photon emissions are not necessarily time correlated, so perhaps it is the collective properties of all the emission processes that determines the macroscopic oscillation frequency.
Which of the above (if any) is the correct interpretation?