= Spin-one transverse measurement after axial precession
{title2=$p_+=\cos^4(\theta/2),\quad p_0=\tfrac12\sin^2\theta,\quad p_-=\sin^4(\theta/2)$}
Prepare the $J_1=+\hbar$ <spin coherent state> and evolve with <Hamiltonian operator> $H=\omega J_3$ for time $t$. In the descending-$J_3$ basis of the <spin-one angular momentum matrices>, the initial vector is $(1,\sqrt2,1)^{\mathsf T}/2$ and the evolved vector is $(e^{-i\theta},\sqrt2,e^{i\theta})^{\mathsf T}/2$, where $\theta=\omega t$. The $J_1$ <orthonormal basis> has vectors $(1,\sqrt2,1)^{\mathsf T}/2$, $(1,0,-1)^{\mathsf T}/\sqrt2$ and $(1,-\sqrt2,1)^{\mathsf T}/2$. Taking their <inner products> with the evolved vector gives <probability amplitudes> $(1+\cos\theta)/2$, $-i\sin\theta/\sqrt2$ and $(\cos\theta-1)/2$. Their squared moduli give the <Born rule> <probabilities> in the title. At integer half-turns $\theta=\pi k$, the outcome is deterministically $(-1)^k\hbar$.
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