from previous post, the scattering matrix for non-spin case is:

for a spin case, and notices that the real space and spin apace are not the same but a general state is wriiten by this:

thus, the modification for the scattering matrix is:

and the differential cross section is:

for a particular spin state, it can be expanded into a combination of the eigen spin state χ.

thus, from a particular spin state to another particular spin state will be:

the scattering amplitude now becomes a matrix, we define:

and the d.c.s. becomes:

for example, spin 1/2 cases. the scattering amplitude matrix is:

we can see that the diagonal terms are non-spin flip, while the off diagonal terms are spin flipped. it should be clear that:

sub it into the d.s.c., we have:

for example, if we have an initial state is spin up, and we only measure the down state, thus:

the half is from the fact that, each only contribute , where s is the spin of target of incident beam. in the example, target spin is 0, incident beam is 1/2 or vice via.

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