1.2 Uncertainty Principle, de Broglie Relation and Wave–Particle Duality

1.2.1 Wave–particle duality ★★

• Light behaves as particles (photoelectric effect, Einstein 1905): electrons leave a metal only if ν exceeds a threshold ν₀, at once, however dim the light; their maximum kinetic energy rises linearly with ν and not with intensity. Each photon carries hν and gives it to one electron.SN

• Na (Φ = 2.28 eV): ν₀ = Φ/h = 5.51 × 10¹⁴ Hz (λ₀ = 544 nm); with 400 nm light (hν = 3.10 eV) KE_max = 0.82 eV. Compton effect: X-rays scattered by electrons come out with a longer wavelength, Δλ = (h/m_ec)(1 − cos θ) = 2.43 pm × (1 − cos θ): photons carry momentum h/λ.SN
• Matter behaves as waves: Davisson and Germer (1927) diffracted electrons from a nickel crystal; G.P. Thomson passed them through thin foils. The diffraction angles fit λ = h/p. Electron microscopes use this. Light and matter both show wave and particle behaviour: wave–particle duality.SN

1.2.2 de Broglie relation ★★★

• de Broglie (1924): any particle of momentum p has a wavelength h/p. For an electron through 100 V: λ = 0.123 nm, about an interatomic spacing, so crystals diffract electrons. An electron of λ = 0.10 nm moves at 7.27 × 10⁶ m s⁻¹ (KE 150 eV).SN
• A 0.15 kg ball at 40 m s⁻¹: λ = 1.1 × 10⁻³⁴ m, far smaller than any slit or nucleus: wave behaviour of everyday objects is never seen.SN
• Bohr orbits as standing waves: a stable orbit must hold a whole number of wavelengths, nλ = 2πr. With λ = h/mv this gives mvr = nh/2π: Bohr’s quantised angular momentum. For H, n = 1: λ = 2πa₀ = 0.332 nm, v = 2.19 × 10⁶ m s⁻¹.SN

1.2.3 Heisenberg uncertainty principle ★★★

• Statement: it is impossible to know both the position and the momentum of a particle along the same direction exactly at the same time; the product of their uncertainties is at least ħ/2. No limit applies to x and p_y. It is a property of matter, not of poor instruments CSE 2024 Q2(a) 10m CSE 2023 Q1(a) 10m.SN
• Why: a particle of definite p is a pure wave e^ikx spread over all space (Δx = ∞). To localise it we add waves of many wavelengths into a packet, and every added wavelength is another momentum. Narrow packet ⇔ wide spread of p. Formally, x̂ and p̂_x do not commute (see 1.1.6).SN
• Heisenberg’s γ-ray microscope: to see an electron sharply we need short-wavelength light, but such a photon has a large momentum h/λ and kicks the electron (Compton recoil) by an unknown amount.

• No electron in the nucleus: Δx = 10⁻¹⁴ m gives Δp ≥ 5.3 × 10⁻²¹ kg m s⁻¹, Δv ≥ 5.8 × 10⁹ m s⁻¹ > c (non-relativistic): an electron cannot be confined there.SN
• Zero-point energy: a particle in a box of length L has Δx ≤ L, so Δp ≥ ħ/2L > 0 and its kinetic energy cannot be zero. The exact lowest level is ; E = 0 would need ψ = 0 everywhere, which is not acceptable Geo 2025 Q1(l) 5m. Worked in 1.3 Particle in a box.SN
• ★ Beyond the syllabus: for any two operators ΔAΔB ≥ ½|⟨[Â, B̂]⟩|; ΔA = (⟨A²⟩ − ⟨A⟩²)^1/2. A Gaussian packet is the only shape that reaches the equality ΔxΔp = ħ/2.