Q1(a). Light emitted by a particle in a box; the uncertainty principle

Answer in brief

(i) λ = 8mL²c/3h = 1.09 × 10⁻⁷ m = 109 nm (ultraviolet); (ii) Δx·Δpₓ ≥ h/4π: position and momentum cannot both be exact.

Key points

• Levels Eₙ = n²h²/8mL²; with m = 1.0 × 10⁻³⁰ kg and L = 3.0 × 10⁻¹⁰ m, E₁ = 6.10 × 10⁻¹⁹ J.
• Energy given out: ΔE = E₂ − E₁ = 3E₁ = 1.83 × 10⁻¹⁸ J; the photon carries this energy, λ = hc/ΔE = 109 nm.
• It is impossible to know both the position and the momentum of a particle along the same direction exactly at the same time.
• A particle of momentum p behaves as a wave of λ = h/p; locating it needs a wave packet of many wavelengths, so sharpening x blurs pₓ.
• The limit is a property of matter, not of poor instruments; a confined particle has zero-point energy.