Solution
= Solution
Put $c=\cos\omega t$, $s=\sin\omega t$. The real part of the off-diagonal coherence vanishes:
$$
\langle S_x\rangle=\frac{\hbar}{2}(c\,is-is\,c)=\boxed{0}.
$$
This also follows from $[H,S_x]=0$ and its zero initial expectation. Individual measurements still give $\pm\hbar/2$, each with <probability> $1/2$.