= Solution
The absorptive <dielectric function> counts optically accessible <electron-hole pairs>. In the electric dipole approximation the perturbation is proportional to $\mathbf a\cdot\hat{\mathbf p}$, with $\mathbf a$ the <light> <polarization> and $\hat{\mathbf p}$ the electronic <momentum operator>. The squared <matrix element> between occupied valence and empty conduction states is the transition strength; it includes <polarization> and band-symmetry selection rules. The sums enumerate initial and final subbands. <photon> <momentum> is negligible on the crystal-momentum scale, so the transition is approximately vertical in the in-plane wavevector $\mathbf k$.
For each subband pair, $C$ is the constant-transition-energy contour in the two-dimensional <Brillouin zone>. Its line element is $dl$, and the reciprocal gradient weights the states satisfying <energy> conservation. Specifically, the <level-set representation of the joint density of states> is
$$
\int d^2k\,\delta\!\left(E_{n'}(\mathbf k)-E_n(\mathbf k)-\hbar\omega\right)=\int_C\frac{dl}{|\nabla_{\mathbf k}(E_{n'}-E_n)|}.
$$
A small <energy> gradient means many nearly resonant states; a vanishing gradient can produce a <Van Hove singularity> or an edge. The displayed expression suppresses overall optical prefactors and assumes the <momentum> <matrix element> is approximately constant along each contour. More generally it stays inside that integral. Occupation factors $f_v-f_c$ are also implicit; at low <temperature> in an undoped well the relevant valence states are full and the conduction states empty. This is the independent-particle continuum contribution from <Fermi's golden rule>. Bound <excitons> add optical resonances below the corresponding free-pair thresholds.
The <GaAs four-band model> is rooted in bonding and <spin-orbit coupling>. Near the zone centre, the conduction edge is predominantly s-like and antibonding, while the upper valence manifold is predominantly p-like. Combining <orbital angular momentum> $l=1$ with <spin> $s=1/2$ splits the six p-derived states into a $j=3/2$ quartet and a lower $j=1/2$ doublet. The latter is the <split-off valence band>. Within the upper quartet, the $m_j=\pm3/2$ and $m_j=\pm1/2$ branches, defined relative to a propagation or quantization direction, have different curvatures and are called <heavy holes> and <light holes>. They coincide at the unstrained bulk zone centre but need not remain degenerate in a well. The conduction edge is a <spin> doublet. Thus “four bands” counts conduction, heavy-hole, light-hole and split-off branches, including their <spin> multiplicities; it does not count four individual states. Positive hole masses describe the downward valence curvature, whereas the conduction curvature is upward.
For the final <exciton> comparison, define the <reduced mass> $\mu_j$ and screened <exciton> Rydberg $R_j^*$ by
$$
\frac1{\mu_j}=\frac1{m_e^*}+\frac1{m_{h,j}^*},\qquad R_j^*=\frac{\mu_j}{m_e}\frac{13.6\,\mathrm{eV}}{12.8^2}.
$$
The resulting $(\mu_j/m_e,R_j^*)$ are approximately $(0.05948,4.937\,\mathrm{meV})$ for <heavy holes>, $(0.03646,3.027\,\mathrm{meV})$ for <light holes> and $(0.04631,3.844\,\mathrm{meV})$ for split-off holes. In the ideal <two-dimensional exciton> model, the first two optically active internal states lie $4R_j^*$ and $4R_j^*/9$ below their continuum edge. Their separation is $32R_j^*/9$, independent of a common threshold offset. The corresponding three-dimensional separation would be only $3R_j^*/4$. The listed peak spacings specified for this theoretical spectrum give
$$
\begin{array}{c|c|c|c}
\text{subband family}&\text{listed successive pair spacings / meV}&\text{2D prediction / meV}&\text{3D prediction / meV}\\\hline
\mathrm{hh},\ n=1,2,3&18,\ 18,\ 17&17.56&3.70\\
\mathrm{lh},\ n=1,2,3&11,\ 11,\ 10&10.76&2.27
\end{array}
$$
Here each pair comprises the stronger lower-energy peak and its weaker nearby companion, in the order associated with the continuum edges calculated in part (c). \b[The approximately 18-meV and 11-meV pair splittings support the two-dimensional series and are incompatible with the three-dimensional series at the supplied masses and permittivity.] Equivalently, for a two-dimensional pair with energies $E_1,E_2$, the inferred edge is $E_2+(E_2-E_1)/8$; the observed pair splittings predict ground-state bindings around 20 and 12 meV, close to $4R_j^*$.
Absolute agreement should not be overstated. For example, the stated square-well parameters give the first heavy-hole edge $1.56390$ eV and the first light-hole edge $1.59161$ eV. Their ideal two-dimensional peak predictions are approximately $(1.54415,1.56170)$ eV and $(1.57950,1.59027)$ eV, respectively. The listed peaks are lower by about 1–2 meV across the families; rounded parameters and this simple effective-mass model do not reproduce every absolute <energy> exactly. No gap, width or mass has been silently adjusted to fit them. The peak-spacing comparison remains useful because it avoids a common edge offset. The split-off transitions predicted by the four-band data are discussed separately in part (c), rather than assigned fictitious extra peaks.
At 300 K, $k_BT\simeq25.9$ meV, whereas the ideal ground-state binding energies are only about $19.7$, $12.1$ and $15.4$ meV; excited states are still more weakly bound. \b[None of these series is expected to remain a clearly resolved, robust bound-exciton peak at room <temperature> in this simple estimate.] Thermal ionization and <phonon> broadening work against their visibility. This is a scale argument, not a theorem that the <exciton> population or all Coulomb-enhanced absorption becomes exactly zero: actual observability also requires temperature-dependent gaps, linewidths, screening and carrier density, which are not supplied.
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