Wave-Particle Duality and Matter Waves
Wave-particle duality is the property of light and matter to show both wave-like and particle-like behavior; Louis de Broglie proposed in 1924 that every moving particle has an associated wavelength λ = h ÷ p = h ÷ mv, which was confirmed by the diffraction of electrons by crystals. In brief: light behaves as a wave in interference and diffraction, and as photons (E = hf, p = h ÷ λ) in the photoelectric effect and Compton scattering. Electrons, neutrons, atoms and even large molecules diffract and interfere. The de Broglie wavelength of everyday objects is far too small to detect, which is why we do not see wave behavior in the macroscopic world.
The double-slit experiment with single electrons is one of the strangest in physics: each electron arrives at a single point on the screen, like a particle, but after many electrons the pattern is an interference pattern of waves. This guide explains the duality of light, the de Broglie hypothesis and its experimental confirmation, the electron microscope, compares waves and particles in a table, works through wavelength calculations and corrects common mistakes.
How can light be both a wave and a particle?
Light shows interference, diffraction and polarization, which need a wave description (wavelength λ and frequency f). But in the photoelectric effect and the Compton effect, it exchanges energy and momentum in discrete amounts, as particles called photons: E = hf = hc ÷ λ and p = E ÷ c = h ÷ λ. Neither picture alone describes all the properties of light. In quantum theory, the wave function describes the probability of finding the photon at different positions: the wave governs where a photon is likely to arrive, and the detection of a photon occurs at a single point. The wave or particle character that shows up depends on the experiment that is performed, which is a statement of the complementarity principle of Niels Bohr: wave and particle aspects are complementary, and an experiment cannot show both at the same time.
What is the de Broglie hypothesis?
Since light, which was considered a wave, also has particle properties with p = h ÷ λ, Louis de Broglie suggested in his 1924 doctoral thesis that matter also has wave properties: a particle with momentum p has a wavelength
λ = h ÷ p = h ÷ mv.
For a particle accelerated from rest through a potential difference V (nonrelativistic), the kinetic energy is eV = p2 ÷ 2m, so λ = h ÷ √(2meV) = 1.226 nm ÷ √V (with V in volts) for an electron. The wavelength is smaller for larger masses and higher speeds. Matter waves also explain the Bohr model: an electron orbit is stable if its circumference holds a whole number of wavelengths, 2πr = nλ, which gives the quantization of angular momentum, L = nħ. The wave nature of particles is the starting point of wave mechanics (Schrödinger, 1926).
What is the experimental evidence?
- Davisson-Germer experiment (1927): Clinton Davisson and Lester Germer scattered electrons of 54 eV from a nickel crystal and observed a maximum at a particular angle, as with X-ray diffraction, with a wavelength agreeing with the de Broglie formula. George Paget Thomson independently observed electron diffraction by thin films. Davisson and Thomson shared the 1937 Nobel Prize in Physics.
- Neutron and atom diffraction: thermal neutrons diffract from crystals (used to study the structure of materials); helium atoms and large molecules such as C60 fullerene (1999) and molecules with hundreds of atoms (2010s) show interference.
- Double slit with single electrons: Claus Jönsson (1961), and the single-electron build-up of a pattern by Tonomura (1989).
- Electron microscopy: electrons accelerated through tens of kV have wavelengths of a few picometers, much shorter than light, so the electron microscope resolves objects down to atomic scales.
Wave and particle behavior compared
| Entity | Wave-like behavior | Particle-like behavior |
|---|---|---|
| Light | Interference, diffraction, polarization | Photoelectric effect, Compton scattering |
| Electrons | Diffraction (Davisson-Germer), double-slit interference | Track in a cloud chamber, detected as single points, charge e, mass m |
Worked example: de Broglie wavelengths
- Electron of 100 eV. p = √(2mE) = √(2 × 9.11 × 10−31 × 100 × 1.602 × 10−19) = 5.40 × 10−24 kg·m/s. λ = h ÷ p = 6.626 × 10−34 ÷ 5.40 × 10−24 = 1.23 × 10−10 m = 0.123 nm, similar to the spacing of atoms in a crystal (0.2 nm), so diffraction is possible.
- Davisson-Germer. Electrons accelerated through 54 V: λ = 1.226 nm ÷ √54 = 0.167 nm.
- Baseball. A 0.145 kg ball at 40 m/s: p = 5.8 kg·m/s, λ = 6.626 × 10−34 ÷ 5.8 = 1.1 × 10−34 m, about 1019 times smaller than the size of an atomic nucleus. Wave effects are unobservable.
- Thermal neutron. Energy 0.025 eV (room temperature), mass 1.675 × 10−27 kg: λ = h ÷ √(2mE) = 0.18 nm, suitable for studying crystals.
- Photon momentum. A green photon of 550 nm has p = h ÷ λ = 6.626 × 10−34 ÷ 5.5 × 10−7 = 1.2 × 10−27 kg·m/s. The electron in item 1 has 4,500 times more momentum.
- Electron microscope. For electrons at 100 kV, relativistic effects are small but not negligible: λ ≈ 3.7 pm (0.0037 nm), compared with 500 nm for visible light. In practice, lens aberrations limit the resolution to about 0.1 nm in many instruments.
How did the idea develop?
Newton favored a particle theory of light, and Huygens a wave theory. Young and Fresnel proved the wave nature (1801 to 1820), and Maxwell identified the waves as electromagnetic (1865). Planck’s quantum (1900) and Einstein’s photon (1905) revived particle aspects, and Compton’s X-ray scattering (1923) confirmed that photons carry momentum. De Broglie’s thesis (1924) extended duality to matter; Einstein supported it. Davisson and Germer’s experiment (1927) confirmed it, and Erwin Schrödinger and Werner Heisenberg formulated quantum mechanics in 1925 to 1926. De Broglie received the Nobel Prize in Physics in 1929.
Which mistakes are common?
Mistake 1: Thinking that an electron is sometimes a wave and sometimes a particle in the classical sense
It is a quantum object that shows wave-like or particle-like properties depending on the measurement. It is not a tiny ball that wiggles.
Mistake 2: Using the speed of light for matter waves
λ = h ÷ mv uses the speed of the particle.
Mistake 3: Expecting to see the wavelength of everyday objects
h is so small that λ is far below any measurable scale for macroscopic masses.
Mistake 4: Confusing the wave function with a physical wave in a medium
The matter wave describes probabilities. It is not a vibration of anything material.