By what mechanism is a photon emitted or absorbed in atomic electron state transitions? I understand atomic emission and absorption spectra well - photons of a specific energy can be emitted or absorbed by atoms, if that energy corresponds perfectly to the energy difference between two states of the electron of the atom - but I don't quite understand how the photons are absorbed and emitted during this transition. What process or mechanism underlies this phenomenon?
For Emission: Does it have something to do with the electron being accelerated during the transition, and the accelerating electron radiates a photon? If so, is this process random? What would cause the electron to suddenly drop energy level(s)? Where does the force/impetus for this acceleration come from?
For Absorption: Do the electric and magnetic fields of the photon apply a force to the electron when it interacts with the atom? If so, why do photons of only one energy apply this force, and all others have no effect on the atom?
 A: Neither the photon nor the electron are classical particles and there is no Newtonian picture of the process. Instead you have to imagine relativistic fields that describe the probabilities to detect photons and electrons in different spacetime points. Before the absorption there is a non-zero probability to detect the photon and the probabilities of the electron field are roughly those predicted by the Schroedinger equation for the low energy state of the atom. After the transition the probability to detect the photon is mostly gone and the electron distribution is now in a higher state. 
One has to be very careful even with this picture, since one can't do continuous measurements on this system without disturbing it. What these distributions really mean is that we prepare one photon, then perform one measurement on either the photon or the electron. We repeat this experiment many times and then we plot the probability distribution as a function of time. This would have to be a multidimensional plot because of the ways the parts of the quantum system interact. Unfortunately people are not very good at recognizing the finer details of such multi-dimensional phenomena. Whenever we talk about these probability distributions and we show images in books, the problem has already been greatly simplified for our own convenience. From the perspective of human perception it is probably next to impossible to visualize the entire process without some simplification or loss of information. 
A: The mechanism of interaction is very much the same as a radio wave interacting with an antenna. The "photon" is manifested as oscillations of the electric field, which drive the electron like a mass on a spring. The oscillation frequency is given by the difference between the initial state and the excited state, and you can track the oscillating charge motion by calculating the superposition of the two states.
The oscillating electron radiates electromagnetic energy just like any other antenna. Initially, because of the interaction of the in-phase nature of the radiated field with the incident field, there is actually a net absorption of energy; but eventually the incident field goes away and the atomic system then simply continues to oscillate, re-radiating any residual energy until it returns to the ground state.
You can also analyze this system in terms of "photons" using something called "Fermi's Golden Rule", but it all comes to exactly the same result in terms of what you can actually measure...amount of scattered radiation as a function of the incident field.
DISCLAIMER: I am a recognized crackpot whose opinions are routinely and massively downvoted by the experts in this forum who know much more than me.
A: Of course CuriousOne is completely right 

"Neither the photon nor the electron are classical particles and there is no Newtonian picture of the process. Instead you have to imagine relativistic fields that describe the probabilities to detect photons and electrons in different spacetime points. Before the absorption there is a non-zero probability to detect the photon and the probabilities of the electron field are roughly those predicted by the Schroedinger equation for the low energy state of the atom. After the transition the probability to detect the photon is mostly gone and the electron distribution is now in a higher state."

Though, I would like to interpret you question a little bit more as of 

"For Absorption: Do the electric and magnetic fields of the photon apply a force to the electron when it interacts with the atom? If so, why do photons of only one energy apply this force, and all others have no effect on the atom?"

And let me interpret it in a way as you would ask, if the photon 'can' practically 'move' the electron, or the atom at all.
There is only a few things I would like to add. 


*

*Although CuriousOne is completely right, and we have to use QM to understand absorption, but it is a fact though that EM waves do have an effect on the surface they 'hit'. 

*And the answer to the question is yes they could 'move' (but not the electron), but just the whole atom or object they 'hit'. 

*The EM wave will 'hit' the surface, we call this 'exert pressure'.
Reference: https://en.wikipedia.org/wiki/Radiation_pressure


*You can use QM and say that the total momentum of the system needs to be conserved, so the momentum of the photon will transfer into the higher energy level of the electron.

*Please note that as CuriousOne correctly stated you cannot see this as the absorbing electron being accelerated. In  fact, on a 'higher' orbit, the electron will 'move' slower. But it's total energy level is raised, and that now will include the photon's converted momentum. (But if the electron's momentum became 'smaller' then how can it's energy level be higher? In a classical view, it's potential energy level is higher on a higher orbit.)

*The only thing that you can think of as being accelerated is the whole atom or the whole object that the EM wave exerted pressure on. 
A: The question is "By what mechanism is a photon emitted or absorbed in atomic electron state transitions?"
The correct answer is that we have no idea about the MECHANISM.
We have statistical equations that describe probablity, these are GROSS measurements. This is like a population cencus identifying that there is a 2% probability that a house in a certain location will have an individual with red hair. It gives you no insight into how an individual got red hair (genetics, hair dye, wig?).
