Spin one-half along an axis
= Spin one-half along an axis
{title2=$\mathbf n\mathbin{\cdot}\mathbf S$}
For $\mathbf n=(\sin\theta,0,\cos\theta)$, the normalized eigenstates of $\mathbf n\cdot\mathbf S$ are
$$
|\uparrow_\theta\rangle
=\cos\frac\theta2|\uparrow\rangle
+\sin\frac\theta2|\downarrow\rangle,
\qquad
|\downarrow_\theta\rangle
=-\sin\frac\theta2|\uparrow\rangle
+\cos\frac\theta2|\downarrow\rangle,
$$
with eigenvalues $+\hbar/2$ and $-\hbar/2$.