= Solution
Since $\mathbf S=(\hbar/2)\boldsymbol\sigma$,
$$
\mathbf n\mathbin{\cdot}\mathbf S
=\frac\hbar2
\begin{pmatrix}
\cos\theta&\sin\theta\\
\sin\theta&-\cos\theta
\end{pmatrix}.
$$
Its normalized <spin one-half along an axis>[eigenstates] are
$$
\boxed{|\uparrow_\theta\rangle
=\cos\frac\theta2|\uparrow\rangle
+\sin\frac\theta2|\downarrow\rangle},
\qquad E_\uparrow=\frac\hbar2,
$$
and
$$
\boxed{|\downarrow_\theta\rangle
=-\sin\frac\theta2|\uparrow\rangle
+\cos\frac\theta2|\downarrow\rangle},
\qquad E_\downarrow=-\frac\hbar2.
$$
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