Wave–Particle Duality and the de Broglie Wavelength

Is Light a Wave or a Particle? Is an Electron?

Light spreads out and interferes like a wave, as Young's double-slit experiment shows. Yet in the photoelectric effect, it knocks electrons out of metals as if it came in tiny packets called photons. Stranger still, electrons, which we think of as particles, produce interference patterns as if they were waves. This dual behaviour, called wave–particle duality, is at the heart of quantum physics. This tutorial covers it as taught in PHY 102 and modern physics courses: the evidence for light as a wave and as a particle, photon energy and momentum, the de Broglie hypothesis and wavelength, electron diffraction, the double-slit experiment with single particles, the uncertainty principle, and applications such as the electron microscope. It builds on the tutorials on the photoelectric effect and on interference and diffraction.

The Evidence for Light as a Wave

Interference: light from two coherent slits produces bright and dark fringes, as in Young's experiment.
Diffraction: light spreads out after passing through narrow gaps, and a diffraction grating produces sharp spectra.
Polarisation: light can be polarised, which shows that it is a transverse wave.

Maxwell's theory in the 1860s explained light as an electromagnetic wave, and this view seemed complete.

The Evidence for Light as Particles

The photoelectric effect: electrons are emitted only if the light's frequency is above a threshold, however bright the light, and emission is instant. Einstein explained this in 1905 by treating light as photons, each with energy E = hf.
The Compton effect: when X-rays scatter off electrons, their wavelength increases. This is explained by photons colliding with electrons like billiard balls, transferring energy and momentum.
Line spectra: atoms emit and absorb light only in packets that match the gaps between their energy levels.

Photon momentum

Although photons have no mass, they carry momentum:

p = h ÷ λ (equivalently p = E/c)

Here h = 6.63 × 10⁻³⁴ J s is Planck's constant. For green light of wavelength 500 nm, p = 6.63 × 10⁻³⁴ ÷ 5.00 × 10⁻⁷ ≈ 1.3 × 10⁻²⁷ kg m/s. That is tiny, but "solar sails" in space use the steady push of sunlight to propel spacecraft.

The de Broglie Hypothesis

In 1924, Louis de Broglie made a bold suggestion: if waves can behave like particles, perhaps particles can behave like waves. He proposed that any particle with momentum p has a wavelength, now called its de Broglie wavelength:

λ = h ÷ p = h ÷ (mv)

For a charged particle of mass m and charge q, accelerated from rest through a potential difference V, its kinetic energy is qV, so p = √(2mqV), and:

λ = h ÷ √(2mqV)

Worked example 1: an electron

Find the de Broglie wavelength of an electron accelerated through 100 V. (m(e) = 9.11 × 10⁻³¹ kg, e = 1.60 × 10⁻¹⁹ C)

λ = 6.63 × 10⁻³⁴ ÷ √(2 × 9.11 × 10⁻³¹ × 1.60 × 10⁻¹⁹ × 100) ≈ 1.23 × 10⁻¹⁰ m, about 0.12 nm.

That is about the same as the spacing between atoms in a crystal (roughly 0.1–0.3 nm), so a crystal can act as a diffraction grating for electrons.

Worked example 2: a cricket ball

Find the de Broglie wavelength of a 0.16 kg cricket ball moving at 30 m/s.

λ = 6.63 × 10⁻³⁴ ÷ (0.16 × 30) ≈ 1.4 × 10⁻³⁴ m.

That is unimaginably small, far smaller than an atomic nucleus, so no gap or obstacle could ever diffract it. This is why we never notice the wave nature of everyday objects: their momentum is so large that their wavelength is far too tiny to detect.

Electron Diffraction

In 1927, Clinton Davisson and Lester Germer in the United States, and G. P. Thomson in Britain, confirmed de Broglie's idea. When a beam of electrons was fired at a crystal, or through a thin film of graphite or metal, the electrons came out in definite directions, forming a pattern of bright rings on a fluorescent screen, just like the diffraction of X-rays by crystals.

The measured wavelength agreed with λ = h/p.
Increasing the accelerating voltage increases the electrons' momentum, which reduces their wavelength, so the rings get smaller.

Diffraction is a purely wave property, so this was decisive evidence that electrons behave as waves. Neutrons, atoms and even large molecules have since been diffracted too.

One Particle at a Time

The double-slit experiment shows duality most dramatically. Electrons (or photons) are sent towards two slits one at a time, so slowly that only one is in the apparatus at any moment.

Each electron arrives at the screen as a single, localised dot, just like a particle.
But after thousands have arrived, the dots build up into an interference pattern of bright and dark fringes, just like a wave.
If detectors are placed at the slits to find out which slit each electron goes through, the interference pattern disappears.

The modern interpretation is that each particle is described by a wave function. The wave passes through both slits and interferes, and the square of its amplitude gives the probability of finding the particle at each point. The wave describes where the particle is likely to be detected, and each detection happens at a single point.

The Uncertainty Principle

A consequence of wave behaviour is Heisenberg's uncertainty principle (1927): the position and the momentum of a particle cannot both be known exactly at the same time.

Δx × Δp ≥ h ÷ (4π)

This is not a limitation of our instruments. It is a basic property of nature: a wave that is confined to a small region must contain a wide spread of wavelengths, and so a wide spread of momenta.

Worked example 3

An electron is confined to a region about the size of an atom, Δx = 1.0 × 10⁻¹⁰ m. Find the minimum uncertainty in its momentum and its speed.

Δp ≥ 6.63 × 10⁻³⁴ ÷ (4π × 1.0 × 10⁻¹⁰) ≈ 5.3 × 10⁻²⁵ kg m/s.

Δv = Δp ÷ m ≥ 5.3 × 10⁻²⁵ ÷ 9.11 × 10⁻³¹ ≈ 5.8 × 10⁵ m/s.

So an electron inside an atom cannot sit still: it must be moving at hundreds of kilometres per second. This is one reason why atoms do not collapse, and why electrons are described by orbitals (clouds of probability) instead of definite orbits.

Applications of Matter Waves

The electron microscope: the detail any microscope can resolve is limited by the wavelength it uses. Visible light (around 500 nm) cannot show anything much smaller than about 200 nm. Electrons accelerated through 50 kV have a de Broglie wavelength of about 0.005 nm (5.5 × 10⁻¹² m, using the non-relativistic formula), so electron microscopes can show viruses, cell structures and even rows of atoms.
Electron and neutron diffraction reveal the arrangement of atoms in crystals, metals and biological molecules.
Semiconductor devices and lasers are designed using quantum mechanics, which grew directly out of wave–particle duality.

Wave or Particle? Summary

Light and matter both show wave behaviour (interference and diffraction) when they travel.
They both show particle behaviour (localised dots, and energy and momentum in packets) when they are emitted, absorbed or detected.
Which behaviour appears depends on the experiment. Neither the pure wave picture nor the pure particle picture is complete. Bohr called these complementary descriptions of the same reality.
The two pictures are linked by Planck's constant: E = hf and p = h/λ.

Common Mistakes with Wave–Particle Duality

Thinking only light has a dual nature. All matter has a de Broglie wavelength.
Forgetting to use momentum in λ = h/p. For a particle accelerated through V, first find p = √(2mqV).
Getting the relationship backwards. Higher momentum means a shorter wavelength.
Treating the uncertainty principle as measurement error. It is a fundamental limit, not a problem with equipment.
Using the non-relativistic formula for electrons accelerated through very high voltages. Above about 100 kV, the relativistic momentum should be used.

Wave–Particle Duality: Practice Questions

Name one experiment that shows the wave nature of light and one that shows its particle nature.
Find the momentum of a photon of wavelength 600 nm.
Find the de Broglie wavelength of an electron moving at 2.0 × 10⁶ m/s.
What happens to the diffraction rings in an electron diffraction tube when the accelerating voltage is increased? Explain.
Why can electron microscopes resolve much finer detail than light microscopes?
A proton and an electron have the same speed. Which has the longer de Broglie wavelength?

Answers: 1. Wave: Young's double-slit experiment (or diffraction). Particle: the photoelectric effect (or the Compton effect). 2. p = 6.63 × 10⁻³⁴ ÷ 6.0 × 10⁻⁷ ≈ 1.1 × 10⁻²⁷ kg m/s. 3. λ = 6.63 × 10⁻³⁴ ÷ (9.11 × 10⁻³¹ × 2.0 × 10⁶) ≈ 3.6 × 10⁻¹⁰ m. 4. They get smaller. A higher voltage gives the electrons more momentum, so their wavelength is shorter, and they diffract through smaller angles. 5. Fast electrons have a much shorter wavelength than visible light, and the smallest detail that can be resolved is about the size of the wavelength. 6. The electron. With the same speed, it has much less momentum, so λ = h/(mv) is larger.

Conclusion

Light and matter show both wave and particle behaviour. Light interferes and diffracts as a wave, but in the photoelectric and Compton effects it acts as photons with energy E = hf and momentum p = h/λ. De Broglie proposed that every particle has a wavelength λ = h/p, and electron diffraction confirmed it: an electron accelerated through 100 V has a wavelength of about 0.12 nm, comparable to atomic spacings. Single particles arrive as dots but build up interference patterns, described by probability waves, and the uncertainty principle ΔxΔp ≥ h/4π sets a fundamental limit on what can be known. These ideas underpin electron microscopes and all of quantum physics.