For the scattering problem, the energy is greater than zero.
We must choose the Bessel function in the region containing the origin.
Scattering from a Spherical Well *
, we can have a linear combination of functions.
Matching the logarithmic derivative, we get
Recalling that for
and that our formula with the phase shift is
we can identify the phase shift easily.
We need to use the boundary condition to get this phase shift.
, we get
We can now get the phase shift.
With just the
term, the differential scattering cross section is.
The cross section will have zeros when
There will be many solutions to this and the cross section will look like diffraction.