Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problems 11.11-11.12.
We’ve looked at the Yukawa potential as an example of the variational principle, so here we’ll look at scattering by a Yukawa potential, using the first Born approximation. The Yukawa potential in its general form is
where and are constants. Since the potential is spherically symmetric, we can use the Born approximation in the form
We did the integral using Maple, but if you want to do it by hand, you can do it with two integrations by parts:
We can find the total cross section by integrating the differential cross section over solid angle:
Again, we did the integral using Maple. To do it by hand, we use the trig identity
followed by the substitution