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

$\left = \delta(p - p') + \frac{i}{2\pi m} \delta(E_{p'} - E ) f(p' \leftarrow p)$

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:

$\left|\psi\right> = \left| x \right> \otimes \left|\chi\right> = \left| x, \chi \right>$

thus, the modification for the scattering matrix is:

$\left = \delta(p - p') \delta_{\chi' \chi} + \frac{i}{2\pi m} \delta(E_{p'} - E ) f(p' , \chi' \leftarrow p, \chi)$

and the differential cross section is:

$\frac{d\sigma}{d\Omega}=|f(p',\chi' \leftarrow p,\chi)|^2$

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

$\left| \xi \right> = \sum \xi_{\chi} \left| \chi \right>$

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

$\frac{d\sigma}{d\Omega}=|\sum \xi'_{\chi'} f(p',\chi' \leftarrow p,\chi) \xi_{\chi}|^2$

the scattering amplitude now becomes a matrix, we define:

$F(p' \leftarrow p)=f_{\chi',\chi}(p' \leftarrow p)=f(p',\chi' \leftarrow p,\chi)$

and the d.c.s. becomes:

$\frac{d\sigma}{d\Omega}=|\xi'^\dagger F(p' \leftarrow p)\xi|^2$

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

$F(p' \leftarrow p) = \begin{pmatrix} f_{++} & f_{+-} \\ f_{-+} & f_{--} \end{pmatrix}$

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

$\xi_{out} = F(p' \leftarrow p) \xi_{in}$

$|\xi_{out}|^2 = \sum_{\chi} |\xi_{\chi' , out}|^2 = \sum | \sum f_{\chi', \chi} \xi_{\chi', out}|^2$

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

$\frac{d\sigma}{d\Omega}(p', \xi' \leftarrow p, \xi_{in}) =| \xi'^\dagger F(p'\leftarrow p) \xi_{in}|^2= | \xi'^\dagger \xi_{out}|^2$

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

$\frac{d\sigma}{d\Omega}(p', - \leftarrow p, +) =\frac{1}{2}|f_{+-}|^2$

the half is from the fact  that, each $\xi_{out} - \xi_{in}$ only contribute $\frac{1}{(2s_1+1) (2s_2+1)}$, 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.