= Solution
In the <orthonormal> $S_z$ <eigenbasis> $(\alpha,\beta)$, the <spin one-half> operator is $S_x=(\hbar/2)\begin{pmatrix}0&1\\1&0\end{pmatrix}$. An eigenvector $a\alpha+b\beta$ with <eigenvalue> $s_x$ obeys $b=(2s_x/\hbar)a$ and $a=(2s_x/\hbar)b$. A nonzero vector requires $(2s_x/\hbar)^2=1$. Choosing normalized <quantum phases> gives
$$
\boxed{s_x=+\frac\hbar2,\quad \chi_+=\frac{\alpha+\beta}{\sqrt2};\qquad s_x=-\frac\hbar2,\quad \chi_-=\frac{\alpha-\beta}{\sqrt2}.}
$$
These <spin eigenstates> are <orthogonal> and have unit <norm> because $\alpha$ and $\beta$ do.
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