Please answer all parts of the questions and do not copy other online answers. They are incorrect.

13.3. Show that at low energies (ka + 0) the requirement (13.48) for the validity of the Born approximation becomes 2a .. ,.. a]; drV(r)kr uVoaz 52 ~ ((1 where V0 is the order of magnitude of the potential energy, a is the range of the potential, and we have neglected constants of order unity. By comparing this result with (13.49), argue that if the Born approximation is valid at low energies, it works at high energies as well. \fAt high energies (k -> co), the exponential and the sine in (13.48) oscillate rapidly and cut off the integral for r' 2 1/k. The condition (13.48) becomes in this case 2 ul 1/ k dr' V(r)) kr~ H VO 1/ k dr' kr ~ 2 0