Quantum Confinement in Nanostructures

Confine a particle in an infinite square well of side $L$ and its energy is quantised, $E_n=\hbar^2\pi^2 n^2/(2mL^2)$ : the levels go as $n^2$ , so $E_2-E_1=3E_1$ , and the ground-state confinement energy (the gap) scales as $1/L^2$ , growing as the box shrinks and vanishing in the bulk limit. The left panel shows the well, its levels and the wavefunctions $\psi_n(x)=\sin(n\pi x/L)$ . Which directions are confined sets the dimensionality, and the density of states takes a qualitatively different shape in each case: $g(E)\propto E^{1/2}$ for the 3D bulk, a staircase for the 2D quantum well, $(E-E_c)^{-1/2}$ van Hove spikes for the 1D wire, and discrete delta peaks for the 0D dot. The right panel is that DOS with the optical-absorption onset (the effective gap) marked. Everything is closed form (gate-tested).

Figure 1. Left: the infinite-square-well levels

E_n\propto n^2

and wavefunctions. Right: the density of states,

E^{1/2}

(3D bulk), step (2D well),

(E-E_c)^{-1/2}

(1D wire) or delta peaks (0D dot), with the absorption onset marked. Method: closed-form particle-in-a-box energies and dimensionality-resolved DOS (gate-tested sim.js), Canvas2D; the live readouts are

E_1

E_2-E_1

, the gap and the absorption onset.

dimensionality

box size L2.00

eff. mass m1.00

WHAT TO TRY

Shrink the box size L and watch the scaling panel: every level climbs the 1/L^2 curve, the n=1 to n=2 gap widens and the emission swatch shifts bluer. That blue shift is exactly how a smaller quantum dot glows a bluer colour.
Watch the wavefunctions oscillate in the well: each psi_n carries n half-waves and evolves at its own rate omega_n proportional to E_n, so higher levels visibly wiggle faster while their probability density stays fixed.
Change the dimensionality from 3D bulk to 2D well to 1D wire to 0D dot: the density of states morphs from a smooth square-root edge to a staircase to a van Hove sawtooth to a comb of degenerate delta peaks.
Raise the effective mass m: heavier carriers lower the confinement energy, since E_n scales as 1/m. Light electrons in a small dot are confined hardest.