Explain wave-particle evidence and use de Broglie wavelength
Compare wave and particle evidence for radiation and matter, then calculate a particle's de Broglie wavelength.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 22. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use wave frequency and wavelength, energy, momentum, SI prefixes, electronvolts and scientific notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which observation supports the wave nature of light?
What you will learn
- Relate photoelectric effect to particle evidence and interference/diffraction to wave evidence
- Interpret electron diffraction qualitatively
- Use λ = h/p for a moving particle
Radiation and matter show complementary evidence
The photoelectric effect supports a particulate model of electromagnetic radiation: energy arrives in photons, and an individual photon must meet the work-function threshold. Interference and diffraction of light support its wave nature because they arise from superposition and spreading of waves.
The evidence is not a claim that light is sometimes literally one classical object and sometimes another. The wave and photon descriptions account for different observations. In this syllabus, use the evidence to recognise the wave-particle duality of electromagnetic radiation.
Moving particles have an associated wavelength
Electron diffraction provides evidence that particles can show wave behaviour. When a beam of electrons passes through a regular crystal structure, the scattered electrons form a diffraction pattern; the pattern is interpreted using wave interference from the crystal spacing.
The de Broglie wavelength associated with a moving particle is λ = h/p, where p is the particle momentum. A particle with greater momentum has a shorter associated wavelength. Use h = 6.63 × 10⁻³⁴ J s and momentum in kg m s⁻¹ to obtain λ in metres.
Pause and explain
What is the de Broglie wavelength of a particle with momentum p?
Worked example
Calculate the de Broglie wavelength of a particle with momentum 3.32 × 10⁻²⁴ kg m s⁻¹.
Show the worked solution
- Use λ = h/p.
- Substitute Planck's constant and momentum.
- State the wavelength in metres.
Answer λ = 6.63 × 10⁻³⁴/(3.32 × 10⁻²⁴) ≈ 2.00 × 10⁻¹⁰ m.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A particle has momentum 6.63 × 10⁻²⁴ kg m s⁻¹. Calculate its de Broglie wavelength.
Give me a hint
Divide h = 6.63 × 10⁻³⁴ J s by the momentum.
Compare my reasoning
- λ = h/p.
- λ = 6.63 × 10⁻³⁴/(6.63 × 10⁻²⁴).
- λ = 1.0 × 10⁻¹⁰ m.
λ = 1.0 × 10⁻¹⁰ m.
Look for these in your work
- Relation selected
- Substitution shown
- Wavelength and unit stated
Two electrons have momenta p and 3p. Compare their de Broglie wavelengths.
Give me a hint
Use the inverse dependence λ = h/p.
Compare my reasoning
- For the first electron, λ₁ = h/p.
- For the second, λ₂ = h/(3p).
- Thus λ₂ = λ₁/3.
The electron with momentum 3p has one-third the wavelength.
Look for these in your work
- Inverse relationship applied
- Momentum ratio substituted
- Wavelength ratio stated
An electron beam produces rings after passing through a thin crystal. Explain what the pattern shows about the electrons.
Give me a hint
Interpret the regular crystal as a structure that can diffract waves.
Compare my reasoning
- The crystal has regularly spaced atomic planes.
- The scattered electrons form a diffraction pattern due to interference associated with their de Broglie wavelength.
- This is evidence that moving electrons have wave behaviour.
The rings are electron-diffraction evidence for the wave nature of moving particles.
Look for these in your work
- Crystal spacing recognized
- Diffraction interpreted
- Wave evidence stated
Common mistakes
- Calling photoelectric emission evidence for light waves
- Claiming electron diffraction is evidence for electrons as particles only
- Using p/h instead of h/p
- Mixing momentum units with a wavelength in metres
What can you explain now?
Which observation supports the wave nature of light?
Compare with the explanation
Interference or diffraction
Wave superposition produces interference and diffraction patterns.
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Make it stick
Which observations support wave behaviour of light, particle behaviour of light, and wave behaviour of electrons?
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