Assume a low energy beam is incident upon a small, hard sphere of radius
.
We will assume that
so that only the
partial wave is significantly affected by the sphere.
As with the particle in a box, the boundary condition on a hard surface is that the wavefunction is zero.
Outside the sphere, the potential is zero and the wavefunction solution will have reached its form for large
.
So we set

For very low energy,
and

The total cross section is then
which is 4 times the area of the hard sphere.

Jim Branson
2013-04-22