724. Problem 39.41P (HRW) An electron of mass m and speed v undergoes a headon collision with a gammaray photon of energy , scattering the gammaray photon back in the direction of incidence. We have to verify that the energy of the scattered gammaray photon, as measured in the laboratory system, is

Solution: Click For PDF Version In answering this problem we will use relativistic mechanics and the property that a photon of frequency f has momentum in the direction of motion of the photon. Let the speed of the electron that is moving toward the photon, in the laboratory frame, be v, and the frequency of the incident photon be . Let us assume that photon and electron are back scattered. Let the speed of the electron after the headon collision be and that the frequency of the back scattered photon be . We will write the relativistic equations for energy and momentum conservation. Energy conservation equation is
Momentum conservation equation in headon collision of a photon and electron when they are back scattered is
We have two linear equations in two unknowns and . We solve these equations algebraically. We find that
The expression for the energy of the gammaray photon that is back scattered can be found from the equation for energy conservation. We have We now substitute the expression for and carry out algebraic simplifications. We find that
