Exponential of a Pauli vector
= Exponential of a Pauli vector
{title2=$e^{-ia\mathbf n\cdot\boldsymbol\sigma}=I\cos a-i\mathbf n\cdot\boldsymbol\sigma\sin a$}
For a real <unit vector> $\mathbf n$, the <Pauli matrix multiplication law> gives $(\mathbf n\cdot\boldsymbol\sigma)^2=I$. Separating the even and odd powers in the <matrix exponential> proves the displayed identity. For <spin one-half>, the physical rotation through <angle> $\theta$ has $a=\theta/2$.