First excited hydrogen eigenspace

The $n=2$ eigenspace has basis

A perturbation $g{L}^2$ preserves the hydrogen eigenbasis and shifts $|n,\ell ,m⟩$ byIn the first excited eigenspace, the $2s$ state is unshifted and the three $2p$ states receive the common shift $2g{\hslash }^2$.

The operators $X_3$ and $P_3$ commute with $L_3$, so they preserve $m$. Both are odd under parity, so they connect only states of opposite orbital parity. Within the $n=2$ eigenspace, only the pair $|2,0,0⟩$ and $|2,1,0⟩$ can therefore have nonzero matrix elements.

Forset $a=U(t)b$ withThen $b$ evolves under the constant Hamiltonian