= 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> has
$$
F_n(z)=\sqrt{\frac2L}\sin\frac{n\pi z}{L},\qquad n=1,2,\ldots.
$$
The 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 give
$$
\langle c,n',\mathbf k'|\hat p_x|v,j,n,\mathbf k\rangle\simeq\delta_{\mathbf k'\mathbf k}\langle u_c|\hat p_x|u_{v,j}\rangle\int_0^L F_{n'}^*(z)F_n(z)\,dz.
$$
<orthonormality> therefore yields \b[the <infinite-well interband selection rule> $n'=n$]. 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.
Back to article page
