This question is about how the electromagnetic field interacts with matter.
Suppose we have an electromagnetic wave, let us say it is planar but not monochromatic. So the wave is described by an $L^1$ function, not even necessary period (but of course it can be). One can think of a planar wave of white light.
Suppose now there is a particle at the origin. At some time, let us define this time to be $t=0$, this particle absorbs the energy $E$ from the field, this then corresponds to the energy of a photon of frequency $\lambda$ by the Planck-Einstein law $E=hc/\lambda$.
Does the electron simply sit and wait until there has passed enough energy to be grabbed, and then probabilistically chooses to take the energy?
What does the momentaneous electromagnetic field look like? Where does this energy come from? Is it taken "uniformly" from the field (which in this case can extend indefinitely in space)? How does the particle determine where to take the energy from and when? Is there a way to determine the momentaneous EM-field before and after the absorption?
Or if we want to do away with a more complex wave pattern: then a concrete example would be to consider a simple monochromatic plane wave of frequency $\nu$ and an electron absorbing a photon of frequency $\nu$. I would imagine the monochromatic wave red-shifting, but how would this be described mathematically?
