= Spin measurement
{title2=$P(\sigma_{\mathbf n}=\pm1)=(1\pm\mathbf n\cdot\mathbf r)/2$}
For a <spin one-half> system, an ideal <projective measurement> along a <unit vector> $\mathbf n$ measures $\sigma_{\mathbf n}=\mathbf n\cdot\boldsymbol\sigma$. Its <eigenvalues> are $\pm1$, corresponding to physical <spin angular momentum> components $\pm\hbar/2$, and its <orthogonal projections> are $\Pi_\pm=(I\pm\sigma_{\mathbf n})/2$. For <density matrix> $\rho=(I+\mathbf r\cdot\boldsymbol\sigma)/2$, the <Born rule> and the <Pauli matrix multiplication law> give the displayed <probabilities>. Thus a <spin measurement> along one direction tests only that component of the <Bloch vector>; measurements along three independent directions determine the entire <density matrix>.
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