= Solution
In <atomic units>, $\hbar=m_e=e=4\pi\varepsilon_0=1$, and the <hydrogen atom> has
$$
\boxed{\hat H\Psi=-\frac{1}{2n^2}\Psi,\qquad
\hat l^2\Psi=l(l+1)\Psi,\qquad \hat l_z\Psi=m\Psi,}
$$
$$
\boxed{\hat s^2\Psi=\frac34\Psi,\qquad \hat s_z\Psi=m_s\Psi.}
$$
The allowed <quantum numbers> are
$$
n=1,2,\ldots,\qquad l=0,1,\ldots,n-1,\qquad
m=-l,-l+1,\ldots,l,\qquad m_s=\pm\frac12.
$$
Here the fixed <electron spin> quantum number is $s=1/2$. The <energy eigenvalues> are in hartrees; restoring units gives $\hbar^2l(l+1)$ and $\hbar m$ for the orbital operators and the corresponding factors for <spin>. The spin-independent hydrogen <Hamiltonian> does not split the two <spin> projections.
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