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Wave-particle duality

The photoelectric effect caught light behaving as particles. Electron diffraction catches matter behaving as waves. Neither label survives on its own, and one small equation, lambda equals h over mv, connects the two worlds.

Year 12AQA 3.2.2.4

Builds on The photoelectric effect.

IN THIS TOPIC

  • State the two-way evidence: electron diffraction for the wave nature of particles, the photoelectric effect for the particle nature of light.
  • Use the de Broglie wavelength λ = h/mv.
  • Explain how and why the amount of diffraction changes when a particle's momentum changes.

WHAT YOU PROBABLY THINK

Light is a wave, and electrons are particles.

The evidence runs both ways

By the photoelectric effect, light, the textbook wave, delivers energy in particle-like photons. The reverse ambush came from electrons. Fire a beam of them through a thin sheet of graphite and they land on the screen not as a single spot but as concentric rings: a diffraction pattern, the signature behaviour of waves.

Electrons fired through a thin crystal land in diffraction rings: particles behaving as wavesthin graphiteelectronsrings: an interference pattern
FIG. 1Electrons through thin graphite produce rings on the screen: diffraction, from things with mass and charge.

So electron diffraction shows that particles possess wave properties, and the photoelectric effect shows that electromagnetic waves have a particulate nature. AQA wants exactly that two-way sentence, and does not expect the details of any particular diffraction method.

The de Broglie wavelength

de Broglie's proposal gave the wave a wavelength, set by the particle's momentum:

λ = hmvON YOUR DATA SHEET

where mv is the momentum. The Planck constant's smallness explains the world's apparent normality: everyday objects carry so much momentum that their wavelengths are unmeasurably tiny, while an electron's small momentum gives a wavelength around the spacing of atoms, exactly the scale needed to diffract off a crystal lattice.

The de Broglie wavelength shrinks as momentum grows, so faster particles diffract lesssmall momentum: long wavelengthlarge momentum: short wavelengthλ = h / mv: more momentum, less diffraction
FIG. 2Two de Broglie waves: small momentum, long wavelength; large momentum, short wavelength. λ = h/mv.

The equation also predicts what happens when you change the beam. Accelerate the electrons harder and their momentum rises, so λ shrinks, and since diffraction is strong only when the wavelength is comparable to the gap, the pattern's rings tighten toward the centre: more momentum, less diffraction. That prediction is testable at the turn of a voltage knob, and it holds.

Ideas on probation

Duality is also AQA's case study in how physics changes its mind. The wave picture of light stood for a century before the photoelectric effect broke it; the particle picture of electrons lasted barely thirty years. New claims of this size earn acceptance only through peer review and independent validation by the scientific community, and the lesson of duality is that even the most settled classification stays open to evidence.

THE EXAM BIT

  • The evidence pairing must point the right way: electron diffraction shows particles behaving as waves; the photoelectric effect shows light behaving as particles. Swapping them scores zero.
  • In λ = h/mv the denominator is the momentum. For an electron given kinetic energy, find v from ½mv2 first, then multiply by m.
  • “Explain how the pattern changes if the accelerating pd increases”: momentum rises, λ = h/mv falls, diffraction decreases, rings move inward. Four links, in order.
  • Diffraction is significant when λ is comparable to the gap or spacing. That comparison sentence is usually a mark on its own.
  • Why no wave effects for a thrown ball: enormous momentum, so λ is immeasurably small compared with any real gap.

CHECK YOURSELF

An electron (m = 9.11 × 10−31 kg) moves at 3.3 × 106 m s−1. Find its de Broglie wavelength, and state what happens to the diffraction pattern if the electrons are accelerated to a higher speed.

Show a hint

Momentum first. Then remember what diffraction needs from a wavelength.

Show the answer

Momentum: mv = 9.11 × 10−31 × 3.3 × 106 = 3.0 × 10−24 kg m s−1.

Wavelength: λ = hmv = 6.63 × 10−34 / 3.0 × 10−24 = 2.2 × 10−10 m, about an atomic spacing, which is why crystals diffract electron beams so well.

At higher speed the momentum grows, λ shrinks, and the electrons diffract less: the rings contract toward the centre.

Everything carries both behaviours.

Momentum decides which one you see.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.