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Part (a) givesHence both operators preserve the magnetic quantum number $m$. They are also odd under parity, while $|n,\ell ,m⟩$ has parity $(-1{)}^{\ell }$. Their matrix elements between states of equal parity vanish. The position-momentum selection rules in the first excited hydrogen eigenspace therefore leave only the coupling betweenThe $m=\pm 1$ states decouple, and all diagonal matrix elements vanish.

In the ordered basis $(|s⟩,|p⟩)$, remove the common energy $E_2$ and writeLetThe rotating-frame amplitudes obey a constant Schrödinger equationPutSince $H_{\mathrm{eff}}^2=q^{2}I$,For the initial amplitudes $(A_0,A_1)$, the exact coefficients are thereforeandRestoring the common unperturbed phase, the solution isThis is the rotating-frame solution for a harmonically driven two-level system.