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Step 1: Choose a quantum dot size – let’s use a 5 nm wide dot. Step 2: Use the formula for the first energy level of a particle in a 1‑D box: E₁ = (h²)/(8 m L²) where h = 6.626×10⁻³⁴ J·s, m (electron mass) = 9.11×10⁻³¹ kg, L = 5 nm = 5×10⁻⁹ m. Step 3: Plug in the numbers: E₁ = (6.626×10⁻³⁴)² / (8 × 9.11×10⁻³¹ × (5×10⁻⁹)²) E₁ ≈ (4.39×10⁻⁶⁷) / (8 × 9.11×10⁻³¹ × 2.5×10⁻¹⁷) E₁ ≈ (4.39×10⁻⁶⁷) / (1.82×10⁻⁴⁶) E₁ ≈ 2.41×10⁻²¹ J. Step 4: Convert to electron‑volts (1 eV = 1.602×10⁻¹⁹ J): E₁ ≈ 2.41×10⁻²¹ J / 1.602×10⁻¹⁹ J/eV ≈ 0.015 eV. Step 5: Determine the wavelength of light emitted when the electron falls back to the ground state: λ = hc/E ≈ (6.626×10⁻³⁴ × 3.00×10⁸) / (2.41×10⁻²¹) ≈ 8.24×10⁻⁵ m = 82.4 µm (infrared).