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:

using:

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.

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