= Solution
With the <gamma matrices> derived in <5/a>, the momentum-space <Dirac equation> reads
$$
(E-\boldsymbol\sigma\cdot\mathbf p)\xi=m\eta,
\qquad (E+\boldsymbol\sigma\cdot\mathbf p)\eta=m\xi.
$$
For a positive-energy <particle> at rest, $E=m>0$ and $\mathbf p=0$, so $\xi=\eta$. The <Dirac adjoint> in this basis is $\bar u=(\eta^\dagger,\xi^\dagger)$, and the specified <normalization> gives
$$
\bar u u=\eta^\dagger\xi+\xi^\dagger\eta=2\xi^\dagger\xi=2m.
$$
Therefore, for any normalized two-component <spinor> $\chi$,
$$
\boxed{\xi(0)=\eta(0)=\sqrt m\,\chi,\qquad
u_W(0)=\sqrt m\begin{pmatrix}\chi\\\chi\end{pmatrix},\qquad \chi^\dagger\chi=1.}
$$
Equivalently, choosing $\chi_+=(1,0)^{\mathsf T}$ and $\chi_-=(0,1)^{\mathsf T}$ gives the two independent rest-frame <spin angular momentum>[spin states]. A strictly <massless particle>[massless] <particle> has no rest frame; this step requires $m>0$.
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