Spin operator ($\hat{\mathbf{S}}$)

A spin operator describes intrinsic angular momentum. Its components obey angular-momentum commutators; for spin one half, ${\hat{S}}_j=\hslash {\sigma }_j/2$ in terms of the Pauli matrices.

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.

For a state initially $z$-up and precessing about $y$, measurement along $\mathbf{n}=(0,\sin \theta ,\cos \theta )$ has the displayed probabilities. eigenvectors can be chosen as $(\cos (\theta /2),i\sin (\theta /2){)}^T$ and $(i\sin (\theta /2),\cos (\theta /2){)}^T$. Their phases must be retained when taking inner products. Full $2\pi$ rotation changes their signs but leaves isolated-state probabilities unchanged.

For spin one-half, the Pauli representation makes every diagonal quadratic component a multiple of the identity. Thus $A{S}_z^2+B(S_x^2-S_y^2)=A{\hslash }^{2}I/4$ cannot split the doublet or cause spin reversal. This conclusion is special to this quadratic Hamiltonian and does not exclude linear Zeeman dynamics.