From previous post, the scattering amplitude matrix is :
let the in- density matrix be for our spin ½ ensemble is :
then the out-matrix is:
with some messy algebra, and focus out pull out the identity matrix and the Pauli’s matrix, we will get:
the form of out-polarization vector takes very complicate form. however, this is not bother in the differential cross section, coz it only care the trace.