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The angular momentum commutation relations implyThus $S_{\pm }|s,\sigma ⟩$ has the same total-spin eigenvalue and magnetic quantum number $\sigma \pm 1$. Its squared norm follows fromChoosing the conventional positive phases gives the spin ladder operator formula

For $s=1$, in the ordered basis $|1,1⟩,|1,0⟩,|1,-1⟩$,Since $S_x=(S_{+}+S_{-})/2$,

The incident state is the spin coherent state obtained by rotating $|1,1⟩$ through $\theta$ about the $y$ axis. Its amplitudes in the $S_z$ basis areup to an irrelevant phase convention. The Born rule therefore gives

The required state is the maximal-spin coherent state pointing at azimuth $\phi$ in the equatorial plane. In the same ordered $S_z$ basis it isIts norm is $(1+3+3+1)/8=1$. Direct multiplication using the matrices in part (b), or rotation of $|\hat{\mathbf{z}},3/2⟩$, givesIt therefore yields the value $3\hslash /2$ with probability one when spin is measured along $\mathbf{n}$.