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:

for

and using:

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.

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