Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 35C / c / Solution

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 $\mathbf{n}$.