 Uniform-field spin precession

For a spin-half particle with $H=\alpha \mathbf{B}\cdot \mathbf{S}$ and $\mathbf{S}=\hslash \mathbf{\sigma }/2$, 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.