In fact, an incoming plane wave along the direction can be expanded in Bessel functions.
the Bessel function becomes
The scattering potential will modify the plane wave, particularly the outgoing part.
To maintain the outgoing flux equal to the incoming flux,
the most the scattering can do is change the relative phase of the incoming an outgoing waves.
is called the phase shift for the partial wave of angular momentum
We can compute the differential cross section for scattering
In fact, for low energy scattering and short range potentials, the first term is often enough to solve the problem.
Jim Branson 2013-04-22