E_nl = ℏ²x_nl²/(2m*R²) — particle in a 3D spherical box; blueshift with size
Quantum dots confine electrons in all three dimensions, creating atom-like discrete spectra. Energy scales as E ∝ 1/R², so smaller dots absorb/emit higher-energy (bluer) photons. The lowest transition energy is E_gap + ℏ²x₁₀²/(2m*R²) where x₁₀ ≈ π is the first zero of j₀. CdSe dots (m* ≈ 0.13mₑ, Eg ≈ 1.74 eV) tunable from red to blue as R shrinks from 5→2 nm.