Question

Show that a photon can't transfer all its energy to a free electron. (Energy and impulse must be conserved). Note that this does not contradict the photo-electric effect because, in the latter case, the electron is not free but bound to the metal.

Answer 1

Let us consider a free electron which is at rest in a frame which we will use. In this frame, the initial total energy of electron and photon of mometum p, is
  
And if the photon is totally absorbed, it will transfer all its momentum to the electron and there will be no photon finally. So, the final energy of the system is
  
So, from energy conservation, we require,
  

  

But, this can't be true, because if the photon has a finite momentum p, then, this relation implies
  
i.e., the mass of the electron is zero. But, that is not the case. So, a photon can't transfer all its energy to a free electron.