Wave-particle duality

Physics· Unit 26: Quantum physics· 25 min read

1. Key Concept of Wave-Particle Duality★★☆☆☆⏱ 8 min

🚫 No Calculator

📘 Definition

Wave-particle duality

The principle that every quantum entity (e.g. photon, electron) exhibits both wave-like and particle-like properties. The property observed depends on the type of experiment performed.

Example:

Light acts as a wave in diffraction experiments, and as a particle in the photoelectric effect

For centuries, scientists debated whether light was fundamentally a wave or a particle. Early experiments supported opposing theories, but quantum physics resolved the debate by showing all entities have both properties.

Light demonstrates particle properties: photoelectric effect, Compton scattering
Light demonstrates wave properties: double-slit interference, diffraction
Electrons demonstrate particle properties: cloud chamber tracks, electric/magnetic deflection
Electrons demonstrate wave properties: electron diffraction through crystalline lattices

📐 Worked Example

Classify each observation as demonstrating wave or particle properties: (a) Photoelectric effect, (b) Electron diffraction, (c) Interference of light

1
The photoelectric effect shows light transfers energy in discrete packets (photons), so this demonstrates:
2
Particle property of light
3
Diffraction is an effect that only occurs for waves, so this demonstrates:
4
Wave property of electrons
5
Interference is a characteristic wave effect, so this demonstrates:
6
Wave property of light

2. De Broglie Hypothesis★★★☆☆⏱ 9 min

✓ Calculator OK

📘 Definition

de Broglie wavelength

\lambda

The wavelength of the matter wave associated with a moving particle with momentum $p$ , given by the de Broglie relation.

\lambda = \frac{h}{p} = \frac{h}{mv}

In 1924, Louis de Broglie proposed that just as light has dual properties, all moving matter has an associated wave. This hypothesis was confirmed by Davisson and Germer's electron diffraction experiment 3 years later.

Macroscopic objects have very large momentum, so their de Broglie wavelength is far too small to observe any wave effects. Only subatomic particles like electrons have small enough momentum to produce measurable wavelengths.

📐 Worked Example

Calculate the de Broglie wavelength of an electron moving at $2.0 \times 1 0^{6}$ m s $^{- 1}$ . Mass of electron = $9.11 \times 1 0^{- 31}$ kg, $h = 6.63 \times 1 0^{- 34}$ J s.

1
First calculate the momentum of the electron:
2
$p = mv = (9.11 \times 10^{-31}) \times (2.0 \times 10^6) = 1.822 \times 10^{-24} \text{ kg m s}^{-1}$
3
Substitute into the de Broglie relation:
4
$\lambda = \frac{h}{p} = \frac{6.63 \times 10^{-34}}{1.822 \times 10^{-24}} \approx 3.6 \times 10^{-10} \text{ m}$
5
Final answer: $λ = 3.6 \times 1 0^{- 10}$ m, which is comparable to the spacing between atoms in a crystal.

3. Experimental Evidence: Electron Diffraction★★★☆☆⏱ 8 min

✓ Calculator OK

When a beam of electrons is accelerated through a potential difference towards a thin crystalline graphite foil, a pattern of concentric rings is observed on a screen behind the foil. This is a diffraction pattern, which can only be produced by waves.

📐 Worked Example

Electrons are accelerated through 100 V. Show that their de Broglie wavelength is approximately $1.2 \times 1 0^{- 10}$ m. $e = 1.6 \times 1 0^{- 19}$ C, $m_{e} = 9.1 \times 1 0^{- 31}$ kg, $h = 6.63 \times 1 0^{- 34}$ J s.

1
Kinetic energy gained by the electron equals electric potential energy lost:
2
$\frac{1}{2}m_e v^2 = eV$
3
Rearrange to solve for momentum $p = m_{e} v$ :
4
$p^2 = m_e^2 v^2 = 2 m_e eV \implies p = \sqrt{2 m_e eV}$
5
Substitute values to find momentum:
6
$p = \sqrt{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19} \times 100} \approx 5.4 \times 10^{-24} \text{ kg m s}^{-1}$
7
Calculate de Broglie wavelength:
8
$\lambda = \frac{h}{p} = \frac{6.63 \times 10^{-34}}{5.4 \times 10^{-24}} \approx 1.2 \times 10^{-10} \text{ m}$
9
This matches the required value, as required.

4. Common Pitfalls

Wrong move:

Claiming entities switch between being a wave and a particle

Why:

Wave-particle duality means entities inherently have both properties, not that they change identity

Correct move:

State that only one property is observed in a given experiment, but both exist inherently

Wrong move:

Thinking the de Broglie relation only applies to electrons

Why:

All moving particles (from neutrinos to footballs) have a de Broglie wavelength

Correct move:

Remember that large mass objects have negligible wavelengths so wave effects are unobservable

Wrong move:

Using the photon wavelength relation $λ = h c / E$ for matter

Why:

This relation only applies to massless photons, not particles with mass

Correct move:

Always use $λ = h / p$ for any moving particle with mass

Wrong move:

Forgetting that diffraction requires wavelength comparable to grating spacing

Why:

Exam questions often ask why electron diffraction is observed with crystals but not large slits

Correct move:

Recall that interatomic spacing in crystals matches the de Broglie wavelength of accelerated electrons

5. Quick Reference Cheatsheet

Entity	Key Property	Relation/Evidence
Light	Particle	$E = h f$ , Photoelectric effect
Light	Wave	Double-slit interference, Diffraction
Electrons	Particle	$p = m v$ , Cloud chamber tracks
Electrons	Wave	Electron diffraction, $λ = h / p$
Any moving matter	Dual property	$λ = \frac{h}{p}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 2
Calculate de Broglie wavelength
2023 · 1
Identify evidence for duality
2021 · 4
Explain electron diffraction

Going deeper

simulationElectron diffraction simulationInteractive demonstration of duality

What's Next

Wave-particle duality is the foundation of all quantum mechanics, underpinning modern technologies from electron microscopy to semiconductor design. This concept prepares you for further study of quantum energy levels, wavefunctions, and Heisenberg's uncertainty principle, all core topics in CIE A-Level quantum physics. The de Broglie relation is regularly tested in both multiple choice and structured questions, often connected to kinetic energy and accelerating potential difference. Mastery of this concept is essential for all higher-level quantum physics topics.