Last time, we saw how symmetry fixed the form of scattering amplitude. Now, the rotation symmetry also impose the scattering amplitude as a series. to see this, we first need to know, the energy- angular momentum state is an eigen state of scattering matrix.
because of the conservation of energy and conservation of angular momentum by a central potential. and a momentum state can be expressed as:
thus, the scattering amplitude is :
if we set the incoming momentum lay on z-axis. thus:
thus, we have:
if we define the Partial-wave amplitude:
since the S is unitary, the eigen value should be simply . thus, we have:
this result means the scattering amplitude can be de-composited into Legendre polynomial. the total scattering cross section is:
by using the orthogonal properties,
then, we have a partial cross section.
from the last equality, we have a boundary for partial cross section:
this inequality is called unitarity condition.