= Uniform-field spin precession
{title2=$U(t)=\exp[-i\alpha Bt\,\mathbf n\cdot\boldsymbol\sigma/2]$}
For a spin-half particle with $H=\alpha\mathbf B\cdot\mathbf S$ and $\mathbf S=\hbar\boldsymbol\sigma/2$, exponentiating the <Pauli matrices> gives a rotation about the field axis. A field along $y$ evolves initial $z$-up <spin> to $(\cos(\alpha Bt/2),\sin(\alpha Bt/2))^T$. The sign of $\alpha$ sets the precession direction. <spin> parallel to the field is conserved even while transverse components vary.
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