Why Quantum Dots Glow Different Colors

An electron trapped in a tiny crystal behaves like a wave on a string with fixed ends. Only certain standing waves fit, each with its own energy, and shrinking the crystal spreads those energies apart. That's why smaller quantum dots glow bluer.

How it works

Sturm–Liouville eigenvalue problems: eigenfunctions and orthogonality

In one sentence

A Sturm–Liouville problem $(p\,y')' - q\,y + \lambda w\,y = 0$ on $[a,b]$ with separated boundary conditions has real eigenvalues $\lambda_1 < \lambda_2 < \cdots \to \infty$ whose eigenfunctions are orthogonal with weight $w$ . For $-\psi'' = \lambda\psi$ with $\psi(0) = \psi(L) = 0$ , $\lambda_n = (n\pi/L)^2$ and $\psi_n = \sin(n\pi x/L)$ .

General formula

-\frac{{\htmlData{sym=2}{\hbar}}^2}{2{\htmlData{sym=3}{m}}}{\htmlData{sym=0}{\psi}}'' = {\htmlData{sym=1}{E}}{\htmlData{sym=0}{\psi}}, \quad {\htmlData{sym=0}{\psi}}(0) = {\htmlData{sym=0}{\psi}}({\htmlData{sym=4}{L}}) = 0 \implies {\htmlData{sym=0}{\psi}}_{\htmlData{sym=5}{n}} = \sin\frac{{\htmlData{sym=5}{n}}\pi x}{\htmlData{sym=4}{{L}}}, \quad {\htmlData{sym=1}{E}}_{\htmlData{sym=5}{n}} = \frac{{\htmlData{sym=2}{\hbar}}^2\pi^2{\htmlData{sym=5}{n}}^2}{2{\htmlData{sym=3}{m}}{\htmlData{sym=4}{L}}^2}

Fixed walls only allow waves that fit a whole number of half-wavelengths. Each fit is an eigenfunction, and how tightly it's squeezed sets its energy.

What the symbols mean

$\psi$
Wave function
The electron's standing wave inside the dot.
$E$
Energy
The energy of an allowed state.
$\hbar$
Reduced Planck constant
The quantum of action that sets the energy scale.
$m$
Electron mass
The electron's effective mass inside the crystal.
$L$
Dot size
The width of the box the electron is trapped in, a few nanometers.
$n$
Quantum number
How many half-waves fit across the dot.

Inside the dot, the electron's wave function $\psi$ obeys the Schrödinger equation $-\frac{\hbar^2}{2m}\psi'' = E\psi$ and must vanish at the walls: $\psi(0) = \psi(L) = 0$ .
That's a Sturm–Liouville eigenvalue problem (with $p = 1$ , $q = 0$ , $w = 1$ ): only the standing waves $\psi_n = \sin\frac{n\pi x}{L}$ fit, with energies $E_n \propto n^2/L^2$ .
The eigenfunctions are orthogonal, $\int_0^L \psi_m\psi_n\,dx = 0$ for $m \ne n$ , so any state is a unique mix of them, just like a Fourier series.
Light comes out when an excited electron drops back across the material's band gap. The confinement energy adds to the gap and grows like 1/L², so smaller dots emit higher-energy, bluer light. (This lab uses a simplified one-dimensional model of a cadmium selenide dot.)

Learn the math

The Method of Separation of VariablesPDE · Lesson 6, Separation of Variables Sturm–Liouville ProblemsPDE · Lesson 14, Beyond the Basics