Physics 9702 · Paper 4

Wave–Particle Duality

If waves can act like particles, can particles act like waves? Fire electrons at a crystal and the answer — a diffraction pattern — is yes.

Electrons that diffract

Electrons make diffraction rings — proof they travel as waves. Raise the accelerating voltage: the electrons gain momentum, their de Broglie wavelength λ = h/p shrinks, and the rings pull in closer. Diffraction (a wave behaviour) settling a matter question is the heart of wave–particle duality.

A beam of electrons through thin graphite makes a ring diffraction pattern on a screen. Diffraction is a wave behaviour, so the pattern shows that electrons can behave as waves (9702/42/O/N/24 Q9(a)) — the direct evidence for the wave nature of matter.

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

Symbols

$\lambda$ = de Broglie wavelength of the particle (m)
$p = mv$ = momentum of the particle (N s)
$h$ = the Planck constant (J s)

The is the wavelength associated with a moving particle — the banked one-liner (9702/41/M/J/23 Q7(a)) (9702/41/M/J/25 Q8(a)). Bigger momentum means shorter wavelength.
The rings tighten when you turn up the accelerating voltage, and the recycled explanation chains three links (9702/41/M/J/23 Q7(c)(ii)) (9702/42/O/N/24 Q9(c)): greater p.d. → greater momentum → smaller de Broglie wavelength → smaller diffraction angle, so the rings close in.
For an electron accelerated through $V$ , $qV = \tfrac{1}{2}mv^2$ gives the momentum $p = \sqrt{2qVm}$ (9702/42/O/N/24 Q9(b)). A plot of $p$ against $1/\lambda$ is a straight line through the origin with gradient $h$ .

Worked example

An electron's wavelength (2025 paper)

Find the de Broglie wavelength of an electron moving at $4.9\times10^{7}$ m s−¹ (9702/41/M/J/25 Q8(b)).

$\lambda = \dfrac{h}{mv} = \dfrac{6.63\times10^{-34}}{9.11\times10^{-31} \times 4.9\times10^{7}} = 1.5\times10^{-11}\ \text{m}$ .

Answer

\lambda = 1.5\times10^{-11}\ \text{m}

About the spacing of atoms in a crystal — which is exactly why crystals diffract electrons.

Duality both ways

Light interacts with matter as a particle (the photoelectric effect, line spectra) but travels and spreads as a wave (diffraction, interference). Electrons are the mirror image: normally particles, they diffract as waves. The Planck constant

h

stitches the two together — it sets the photon's energy

E = hf

and the particle's wavelength

\lambda = h/p

alike.

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