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Quantum Dots – Energy Level Structure

4.0 nm
CdSe
n=1

Particle in a Spherical Box (Brus Model)

A quantum dot confines electrons in 3D. For a particle in a spherical infinite-well of radius R, energy levels follow from spherical Bessel zeros:

E_n,l = ℏ²α²_n,l / (2m*R²) where α_n,l are zeros of j_l
Brus equation: E_gap(R) = E_bulk + ℏ²π²/(2μR²) − 1.8e²/(4πεR)

The radial wavefunctions are j_l(α r/R). S-states (l=0) have no angular nodes; P-states (l=1) have one angular node. Smaller dots = higher confinement energy = blue-shifted emission (quantum confinement effect).

The colourful emission of quantum dots, now ubiquitous in QLED displays and biomedical imaging, is a direct consequence of this size-tunable quantization.