Please answer ALL PARTS of the question. Equation 13.75 in Townsend Quantum Mechanics is given for reference.

Comparing this result with the expression (13.70) for the scattering amplitude, we see that a = 5151111 f(0) (13.75) where we have taken advantage of the fact that the Legendre polynomials satisfy P;(1) = 1. Equation (13.75) is known as the optical theorem and is a reection of the fact that, as we discussed earlier. the very existence of scattering requires scattering in the forward direction in order to interfere with the incident wave and reduce the probability current in the forward direction. 2 A particle of mass m with momentum p = his scatters elastically from the potential V(r) = 5% 6(zr - a.) due to a very massive particle. You may assume that a is positive, but should not assume that e is necessarily small. a) Under what conditions will the Born approximation for this system be valid? b) Assuming the validity of the Born approximation, calculate the differential cross section for scattering m through an angle 9. Make certain that your final expression shows the complete dependence on 6. c) Find the total cross section for scattering in the low energy limit. d) Another way to find the total cross section is to use the optical theorem (Eq. 13.75). You will get a different answer than in part c) What does the parametric form of your answer in part 0) tell you about what is needed to reconcile these two methods for finding the total cross