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