= Solution
The first excited unperturbed level is twofold degenerate, spanned by $|1,2\rangle$ and $|2,1\rangle$, with energy $5h^2/(8ma^2)$. The perturbation is $q\varepsilon_Q(x-a/2)(y-a/2)$. Both diagonal matrix elements vanish, but the off-diagonal element does not:
$$
W_{12,21}=q\varepsilon_Q
\left[-\frac{16a}{9\pi^2}\right]^2
=\frac{256q\varepsilon_Qa^2}{81\pi^4}.
$$
Diagonalizing the $2\times2$ matrix with zero diagonal gives first-order shifts
$$
\boxed{\Delta E^{(1)}_\pm=
\pm\frac{256q\varepsilon_Qa^2}{81\pi^4}},
$$
with symmetric/antisymmetric combinations $(|1,2\rangle\pm|2,1\rangle)/\sqrt2$. Their order depends on the sign of $q\varepsilon_Q$; the lower branch has shift $-|W_{12,21}|$. Applying nondegenerate <perturbation theory> to either product state would miss the splitting.
Back to article page
