Wave–Particle Duality
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Light and matter both act like waves and like particles. Light interferes (wave) but delivers energy in photons (particle). Electrons arrive as single dots (particle) but build up interference patterns (wave). Every moving object has a de Broglie wavelength, λ = h/p.
1. Matter waves
2. Why we don’t notice
Waves only show their wave behavior when they pass through openings or around objects about their own size.
An electron’s wavelength is about the spacing of atoms in a crystal, so crystals diffract electrons. A baseball’s wavelength is far smaller than anything it could pass through.
Electron microscopes use this: electrons’ tiny wavelengths let them see detail much finer than visible light can.
A photon and an electron with the same wavelength have the same momentum, p = h/λ.
3. Example, then try it
Find the de Broglie wavelength of an electron (9.11 × 10⁻³¹ kg) at 2.0 × 10⁶ m/s, and of a 0.145 kg baseball at 40 m/s.
Electron: λ = (6.63 × 10⁻³⁴) ÷ [(9.11 × 10⁻³¹)(2.0 × 10⁶)] ≈ 3.6 × 10⁻¹⁰ m, about atom-sized.
Baseball: λ = (6.63 × 10⁻³⁴) ÷ [(0.145)(40)] ≈ 1.1 × 10⁻³⁴ m, far too small to detect.
An electron starts from rest and is accelerated through 150 V. Find its kinetic energy, speed and de Broglie wavelength. Would it diffract off a crystal with atoms about 10⁻¹⁰ m apart?
Kinetic energy: K = qΔV = 150 eV = (150)(1.6 × 10⁻¹⁹) = 2.4 × 10⁻¹⁷ J.
Speed: v = √(2K/m) = √(2 × 2.4 × 10⁻¹⁷ ÷ 9.11 × 10⁻³¹) ≈ 7.3 × 10⁶ m/s.
Wavelength: λ = h/(mv) = (6.63 × 10⁻³⁴) ÷ [(9.11 × 10⁻³¹)(7.3 × 10⁶)] ≈ 1.0 × 10⁻¹⁰ m.
Diffraction: yes. Its wavelength matches the atom spacing.
Answer: 150 eV, 7.3 × 10⁶ m/s, 1.0 × 10⁻¹⁰ m; it diffracts.
4. Common mistakes
Why don’t we see a thrown baseball diffract through a doorway?
λ = h/(mv) is about 10⁻³⁴ m for a baseball, much too small compared with a doorway for any diffraction to show.
Ready to practice?
5 questions on Wave–Particle Duality, each with a hint and an explanation for every wrong answer.