Position-momentum selection rules in the first excited hydrogen eigenspace
= Position-momentum selection rules in the first excited hydrogen eigenspace
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 $|2,0,0\rangle$ and $|2,1,0\rangle$ can therefore have nonzero matrix elements.