= Solution
In weak-probe <optical absorption>, <Fermi's golden rule> creates an <electron-hole pair>, with negligible population depletion. Negligible photon momentum makes a <direct band gap> transition nearly vertical in crystal momentum:
$$
\alpha(\omega)\propto\frac1\omega\sum_{cv\mathbf k}|\mathbf e\cdot\mathbf p_{cv}|^2(f_v-f_c)\delta(E_c-E_v-\hbar\omega).
$$
For allowed transitions between three-dimensional parabolic bands, the <joint density of states> yields \b[$\alpha\propto\sqrt{\hbar\omega-E_g}/\omega$] above the gap. Selection rules and polarization enter the dipole matrix element. An <indirect band gap> needs an additional phonon to conserve crystal momentum, with phonon absorption/emission weights and shifted thresholds.
Electron-hole attraction produces <Wannier excitons> below the continuum, approximately $E_n=E_g-R^*/n^2$, with bright $s$-state oscillator strength decreasing as $n^{-3}$. It also enhances the continuum edge. Screening, disorder and phonons broaden these features; heavy/light valence bands can create distinct series. This linear-response picture includes Coulomb binding but excludes a resolved coherent photon–exciton anticrossing: that would require a strong light–matter coupling treatment. “Weak coupling” here is distinguished from neglecting the electron-hole interaction altogether.
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2014/paper-1-aqc2014-absorption-and-hybridization.png]
{title=Bound exciton lines below a continuum edge differ from coherent exciton–photon anticrossing}
{source=paper-1-aqc2014-absorption-and-hybridization.py}
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