Wave-Particle Duality and the de Broglie Wavelength
Wave–particle duality is the property of light and matter to show both wave and particle behavior, and the de Broglie wavelength, λ = h/(mv), gives the wavelength associated with a particle of mass m moving at speed v. In brief: an electron moving at 1.0 × 106 m/s has a wavelength of 0.73 nm, comparable to the spacing of atoms in a crystal, whereas a baseball thrown at 40 m/s has a wavelength of about 10−34 m, which is far too small to detect.
Light was shown to behave as a particle in the photoelectric effect, and in 1924 Louis de Broglie suggested that matter, too, has wave properties. The idea explained why only certain orbits are allowed in atoms, and led to quantum mechanics. This article states the de Broglie relation, works through calculations for electrons, neutrons and large objects, gives a data table, describes the experiments that confirmed matter waves and shows how electron microscopes use them.
What is wave–particle duality?
Light shows wave properties (interference, diffraction) and particle properties (the photoelectric effect, the Compton effect). Electrons, which were seen as particles with a definite mass and charge, show diffraction and interference too. Neither picture alone describes the facts, and the behavior that is seen depends on the experiment. Photons have energy E = hν and momentum p = h/λ. De Broglie applied the same relation to particles.
What is the de Broglie relation?
For any particle with momentum p = mv:
λ = h / p = h / (mv)
Here h = 6.626 × 10−34 J·s, m is the mass in kg and v the speed in m/s. The wavelength decreases as the mass or the speed increases. For a particle accelerated through a potential difference V, the kinetic energy is eV, so p = √(2meeV) and, for an electron, λ ≈ 1.226 / √V nanometers (with V in volts), when V is small enough that relativistic effects can be ignored.
Wavelengths of different objects
| Object | Mass (kg) | Speed (m/s) | Wavelength |
|---|---|---|---|
| Electron | 9.11 × 10−31 | 1.0 × 106 | 0.73 nm |
| Electron (accelerated through 100 V) | 9.11 × 10−31 | 5.9 × 106 | 0.123 nm |
| Thermal neutron | 1.675 × 10−27 | 2200 | 0.180 nm |
| Helium atom at 300 K | 6.65 × 10−27 | 1370 | 0.073 nm |
| Baseball | 0.145 | 40 | 1.1 × 10−34 m |
| Walking person | 70 | 1.5 | 6.3 × 10−36 m |
Worked examples
Example 1: an electron
λ = h / (mv) = 6.626 × 10−34 / (9.109 × 10−31 × 1.0 × 106) = 7.27 × 10−10 m = 0.727 nm = 7.27 Å.
Example 2: a baseball
λ = 6.626 × 10−34 / (0.145 × 40) = 6.626 × 10−34 / 5.8 = 1.14 × 10−34 m. This is about 1019 times smaller than a proton (10−15 m), so wave effects of such an object cannot be observed.
Example 3: an electron accelerated through 100 V
- KE = eV = 1.602 × 10−19 C × 100 V = 1.602 × 10−17 J.
- v = √(2KE/m) = √(2 × 1.602 × 10−17 / 9.109 × 10−31) = 5.93 × 106 m/s.
- λ = h/(mv) = 6.626 × 10−34 / (9.109 × 10−31 × 5.93 × 106) = 1.23 × 10−10 m = 0.123 nm.
Example 4: speed needed for a given wavelength
What speed must a neutron (1.675 × 10−27 kg) have to have a wavelength of 0.200 nm? v = h / (mλ) = 6.626 × 10−34 / (1.675 × 10−27 × 2.00 × 10−10) = 1.98 × 103 m/s. Neutrons from a reactor, slowed down to thermal speeds, have wavelengths of this order, and are used to study crystals.
Example 5: the relativistic limit
For electrons accelerated through 100,000 V (as in a transmission electron microscope), the simple formula gives λ = 1.226 / √100,000 = 0.00388 nm, but the electrons move at about 0.55c, so a relativistic correction gives the more accurate value of 0.00370 nm (3.70 pm).
How does it explain the Bohr orbits?
Bohr had postulated that the angular momentum of the electron is quantized: mvr = nh/2π. De Broglie showed that this follows if the electron is a standing wave around the orbit: the circumference must contain a whole number of wavelengths, 2πr = nλ. With λ = h/mv this gives mvr = nh/2π. Only orbits that fit a whole number of waves are stable, and others cancel themselves by destructive interference, like the vibrations of a guitar string that must have nodes at both ends.
What experiments confirmed matter waves?
- Davisson and Germer (1927): electrons of 54 eV scattered from a nickel crystal gave a diffraction pattern with a maximum at the angle expected for a wavelength of 0.167 nm, in agreement with λ = 12.26 / √54 Å = 1.67 Å.
- G. P. Thomson (1927): electron diffraction by thin metal foils.
- Atoms and molecules: diffraction of helium atoms (Estermann and Stern, 1930), neutrons and, later, large molecules such as C60 (Zeilinger group, 1999).
- Double-slit experiment: electrons sent one at a time through two slits build up an interference pattern, which shows that each electron behaves like a wave that passes through both slits, but is detected at a single point as a particle.
Where are matter waves used?
- Electron microscopes: the resolution is limited by the wavelength, and electrons of picometer wavelength allow atoms to be imaged, in contrast to visible light with a wavelength of 400–700 nm.
- Electron and neutron diffraction: structure of crystals, surfaces and magnetic materials.
- Quantum chemistry: electrons in atoms and molecules are described by wave functions.
Who proposed the idea?
Louis de Broglie, a French physicist and a member of an aristocratic family, proposed matter waves in his doctoral thesis of 1924. Einstein supported the idea, and Erwin Schrödinger took it as the starting point for his wave equation in 1926. De Broglie received the Nobel Prize in Physics in 1929, and Davisson and Thomson shared the prize in 1937. Ernst Ruska built the first electron microscope in 1931, and he received the Nobel Prize in 1986.
Which mistakes are common?
- Using grams: the mass must be in kilograms.
- Thinking that large objects have no wavelength: they do, but it is far too small to observe.
- Mixing up the wave and the particle picture: an electron is not a small ball that wiggles.
- Forgetting the relativistic correction: it matters for fast electrons.
- Confusing the de Broglie wavelength with the size of the particle: they are different quantities.