Worked example: An electron is trapped in a 1D box of width L = 1.0 nm (about 10 atoms). Find the ground-state energy and the photon emitted in the n = 2 → 1 transition. Step 1: E₁ = h²/(8mL²) = (6.626×10⁻³⁴)² / (8 × 9.11×10⁻³¹ × 10⁻¹⁸) J. Step 2: Numerator = 4.39×10⁻⁶⁷; denominator = 7.29×10⁻⁴⁸; E₁ = 6.02×10⁻²⁰ J = 0.376 eV. Step 3: E₂ = 4E₁ = 1.50 eV, so the 2→1 photon has energy ΔE = 3E₁ = 1.13 eV. Step 4: Wavelength λ = hc/ΔE = 1240/1.13 ≈ 1100 nm (infrared). Conclusion: a nanometre box gives eV-scale energies — shrink L to 0.3 nm and E₁ jumps nine-fold; this 1/L² scaling is why quantum dots change colour with size.