The question asks me to prove that the photoelectric effect cannot occur with a free electron. ie. one not bound to an atom. A hint is also provided: Consider the reference frame in which the total momentum of the electron and incident photon are zero.
The Attempt at a Solution
I've been thinking about how to prove this for like a week, but i cannot figure out how to do so. I know that in the case of a free electron, we will have a work function potential of 0V. As for the hint, I cannot really figure out what it is trying to get me to do. In the zero momentum frame, we are moving at a velocity that gives the electron a momentum equal to the momentum of the photon. This direction is the same direction that the photon moves in. The electron momentum will be the same, since p=E/c, which is a constant after an energy is predetermined in all frames (right?). After the collision (or absorption in this case), in this frame, the electron should stay stationary, since this is the zero momentum frame. If I were to move back to the lab frame, I would see the electron moving in the same direction as the photon was moving.
This actually brings me to more confusion. Why don't we just get Compton scattering at this point? Why should the photon be absorbed by the electron anyways?
If anyone could please push me into the right direction I would appreciate it greatly!