= Solution
A <Wannier-Mott exciton> is a bound conduction-band <electron> and valence-band <hole> whose envelope extends over many <lattice> cells. That large radius permits an <effective mass> description and dielectric screening of the <electron>–<hole> attraction. With <electron> and <hole> masses $m_e^*,m_h^*$, define reduced mass $\mu=m_e^*m_h^*/(m_e^*+m_h^*)$ and center-of-mass mass $M=m_e^*+m_h^*$. The bulk relative-motion <Hamiltonian> is hydrogenic with an appropriate <relative permittivity> $\epsilon_r$:
$$
H_{\rm rel}=-\frac{\hbar^2\nabla^2}{2\mu}-\frac{e^2}{4\pi\epsilon_0\epsilon_r r},\qquad
a_X=\frac{4\pi\epsilon_0\epsilon_r\hbar^2}{\mu e^2},\qquad
R_X=\frac{\mu e^4}{2(4\pi\epsilon_0\epsilon_r)^2\hbar^2}.
$$
A <semiconductor> <quantum well> confines the <electron> and <hole> perpendicular to the plane while leaving approximately free in-plane center-of-mass motion. If the well width is small compared with the bulk <exciton Bohr radius>, their relative motion becomes quasi-two-dimensional. The ideal zero-width, same-dielectric limit is
$$
H_\rho=-\frac{\hbar^2\nabla_\rho^2}{2\mu}-\frac{e^2}{4\pi\epsilon_0\epsilon_r\rho},\qquad
\boxed{E_n=-\frac{R_X}{(n-1/2)^2},\quad n=1,2,\ldots.}
$$
Its ground-state binding is $4R_X$, compared with $R_X$ in three dimensions, and the ground envelope decays on the scale $a_X/2$. Stronger binding follows because confinement prevents the pair from separating in the third direction.
For a <finite-width quantum-well exciton>, average the Coulomb attraction over the normalized <electron> and <hole> confinement envelopes:
$$
V_{\rm eff}(\rho)=-\frac{e^2}{4\pi\epsilon_0\epsilon_r}
\int dz_e\,dz_h\,
\frac{|\chi_e(z_e)|^2|\chi_h(z_h)|^2}{\sqrt{\rho^2+(z_e-z_h)^2}}.
$$
This smooths the strict planar singularity and changes the binding. The ideal factor four is a useful reference, not a universal value for every real well. Finite barriers, dielectric mismatch, several confinement subbands and nonparabolic valence bands can require a coupled treatment.
Optical absorption creates a nearly zero in-plane center-of-mass wavevector <exciton>. For the lowest confinement subbands, its line lies approximately at
$$
\hbar\omega=E_g+E_{e,z}+E_{h,z}-E_b+\frac{\hbar^2K_\parallel^2}{2M}.
$$
The discrete bound-state peaks occur below the free <electron>–<hole> continuum. Dipole-allowed envelope states have nonzero relative <wavefunction> at zero separation; the absorption strength also contains the confinement overlap $|\int\chi_e^*(z)\chi_h(z)\,dz|^2$. Confinement can increase the overlap and the binding, making excitonic optical structure more prominent. A perpendicular electric field shifts and separates the confinement envelopes, changing the transition <energy> and usually reducing their optical overlap: this is the <quantum-confined Stark effect>.
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2015/paper-1-exciton-series.png]
{title=Ideal bulk and planar exciton binding series measured relative to their respective continuum edges}
\b[<Quantum wells> tune both the continuum threshold and the bound-state <energy>; an optical line shift is not automatically a change in binding alone.] The hydrogenic envelope model requires a Wannier-sized state, rather than an <exciton> localized to one <lattice> cell.
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