= Solution
The unperturbed $n=2$ level \b[is degenerate]. The useful point is that a central perturbation is already diagonal within its usual orbital <eigenbasis>:
$$
\langle nlm|br|nl'm'\rangle
=b\left(\int_0^\infty r^3R_{nl}^*R_{nl'}\,dr\right)\delta_{ll'}\delta_{mm'}.
$$
Angular <orthogonality> removes the possible mixing of $2s$ and $2p$, and also the mixing of different <magnetic sublevels>. Therefore the <matrix> required by <degenerate perturbation theory> needs no further diagonalization: its diagonal entries are precisely the expectation values above. It would be incorrect to justify the calculation by claiming that the unperturbed spectrum was nondegenerate.
For $b=0$ the <energy> needs only $n$. For a nonzero central confinement term the <energy> generally needs \b[$n,l$], but remains independent of $m$. The <magnetic quantum number> is still needed to specify an individual orbital <eigenstate>, rather than just its <energy>.
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