Photons and Absorption Hello and thanks for reading my question:
Imagine we send one photon at an atom, and it happens to be the right frequency such that it gets absorbed fully by an electron in this atom. Obviously that photon is gone. However, in light that a photon is a perturbation of an E&M field, this implies that the E&M wave which represented the photon is gone. My question is how does the E&M wave dissapear? I have reasoned that it cannot dissappear instantaneously, as that would break SR. Moreover, the wave existed throughout the path between the electron which emitted it and the electron that is absorbed by. This leads to my confusion: if it does not dissappear instantaneously, then the photon still exists and is not absorbed. 
tl;dr: E&M Waves(Photons) are not localized, so how can they be absorbed.
 A: It's often confusing to think of a photon as a particle ... a little bit of something.  This is one of those cases.  For many purposes one imagines that a photon is an excitation of an electromagnetic "mode".    The mode is a spatial distribution describing where the excitation is occurring:  a wave function.  The excitation of the mode (the photon) exists everywhere the mode exists.  A simple example of a mode is a standing wave in a cavity with perfectly conducting walls.  But any EM wave field can be thought of as existing in one or more modes.  
One might think that the mode is empty if it has no excitations, that is, no photons.  But that's not really true as the zero point excitation of the mode guarantees that there is always something in the mode, even if that something can't be directly accessed. (The zero-point field does make its existence felt in other ways, such as triggering spontaneous emission.)
So the EM wave ... the mode ... does not disappear.  Even if the mode contains no photons, the mode exists along with its zero-point field. 
But the interaction with the atom occurs at one particular place: the location of the atom.  The mode loses one quantum of excitation, and the atom gains one, and the interaction looks very much like a particle has hit the atom and been absorbed.  Unfortunately, that enticing picture can lead to difficulties.
What about the finite speed of the EM wave?  The wavefunction of the photon (or the EM field) is just Maxwell's equations.  Just as with a classical wave, a change in the amplitude/excitation state propagates at the speed of light.  The analysis goes somewhat differently in QM, but the result is the same.
A: In the quantum mechanical level of the Schrodinger equation the atom has energy levels which the electron is allowed to occupy. Transitions from higher energy levels to lower ones result in the emission of a photon.
One can develop an intuition from  the more primitive version of quantization,   the Bohr atom, where the electron is constrained by the postulates to be in a quantized energy level. The Bohr atom was postulated because in classical electromagnetic theory described by Maxwell's equations an electron orbiting like a planet around a proton would emit continuously electromagnetic radiation until it fell on the proton. This was not observed. Instead spectra of light were observed , showing that light came chopped up,  in quanta of energy. Together with the photoelectric effect the presence of photons, particles of light, with zero mass and energy =h*nu were postulated and found consistently.
The electromagnetic wave of frequency nu is made up of a huge ensemble of such photons. There have been experiments shooting single photons at two slits, where the interference pattern is built up a dot at a time, showing the individual photon particle reaction with the screen and the collective wave nature.
You ask:

This leads to my confusion: if it does not dissappear instantaneously, then the photon still exists and is not absorbed.
E&M Waves(Photons) are not localized, so how can they be absorbed.

Thus  your confusion comes because you assign to the photon the complete electromagnetic wave. The single photon absorbed by the atom and raising the electron to a higher energy level is a tiny part of the electromagnetic wave and can behave as a particle transfering its energy and momentum at a time delta(t) compatible with the width of the line that absorbs it. When it behaves as a wave at the micro level of the atoms, it is a probability wave, as the two slit experiment shows. It has a probability of displaying the frequency of the classical wave it belongs to, in its interaction with matter at specific (x,y,z). The ensemble of photons is an energy wave in space, built up by the zillion individual photons.
So in the same sense that a billiard ball hits another and transfers its momentum/energy at a time interval delta(t) the photon behaves the same way and transfers its momentum/energy to the atom when absorbed. It is not the ensemble of photons making up the electromagnetic wave continuously impinging on matter that disappears. Just a tiny quantum of it , localized with a delta(t) and delta(x) delta(y) delta(z).
