 Solution

Choose the growth direction as $z$ and the well as $0<z<L$. In the envelope function approximation, write a subband state as $A^{-1/2}{e}^{i{\mathbf{k}}_{\parallel }\cdot {\mathbf{r}}_{\parallel }}{F}_n(z)u_{j0}(\mathbf{r})$, where $u_{j0}$ is the band-edge cell-periodic Bloch factor. The infinite square well hasThe envelope has the same shape for electron and hole states: changing their effective masses changes the eigenenergies, but not these infinite-well eigenfunctions. With in-plane polarization, the leading interband momentum matrix element acts on the Bloch factors, while the slow envelopes giveorthonormality therefore yields the infinite-well interband selection rule $n^{\prime }=n$. A nonzero band-level matrix element is still required; the overlap rule does not replace polarization selection rules. Finite barriers can make electron and hole envelopes different, and interface asymmetry or band mixing can relax this ideal same-index rule.