Matter-wave evidence and de Broglie wavelength
Key idea: H2 Physics lessons on photons, matter waves, wavefunctions, uncertainty and atomic spectra.
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The core idea
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Big question: Why can a particle produce a diffraction pattern?
de Broglie's relation λ = h/p assigns a wavelength to matter. Electron diffraction confirms wave behaviour, while localised detections show particle-like events. Increasing momentum shortens wavelength, so appreciable diffraction needs structures with spacing comparable to λ.
Assign a de Broglie wavelength
A particle of momentum p has de Broglie wavelength λ = h/p. For a non-relativistic particle p = mv, so increasing momentum shortens wavelength. Diffraction becomes observable when wavelength is comparable with the structure spacing.
For an electron accelerated from rest through potential difference V, eV = p²/(2m) gives λ = h/√(2meV). This non-relativistic expression eventually fails at very high accelerating voltages.
Check your understanding: If electron momentum triples, what happens to de Broglie wavelength?
It becomes one third as large.
Interpret diffraction as probability evidence
Electron diffraction from a crystal produces peaks at angles associated with lattice spacing, just as waves do. Even when electrons arrive one at a time, many detections build the same distribution.
Each detection is localised, showing particle-like arrival, while the accumulated pattern is wave-like. The wave model predicts probabilities of outcomes, not a classical electron smeared into pieces at the screen.
Check your understanding: Why use a crystal to diffract electrons?
Atomic plane spacings are comparable with typical electron de Broglie wavelengths, making diffraction appreciable.
Key ideas to keep
- Matter wavelength depends on momentum, not simply speed.
- A diffraction pattern builds from many individual detection events.
- Wave and particle descriptions are complementary evidence, not classical paths switching back and forth.
See the reasoning
Worked example
Derive electron wavelength from accelerating voltage
Question: Electrons accelerated through V are non-relativistic. Derive their de Broglie wavelength.
Step 1: Turn electrical work into kinetic energy
Why: An electron accelerated from rest gains energy eV.
Working: eV = p²/(2m) in the non-relativistic model.
Step 2: Solve for momentum
Why: de Broglie wavelength depends directly on p.
Working: p = √(2meV).
Step 3: Apply de Broglie's relation
Why: The electron's matter wavelength is h/p.
Working: λ = h/√(2meV), so λ ∝ V⁻¹/².
Answer: eV = p²/(2m), so p = √(2meV) and λ = h/√(2meV). Increasing V decreases wavelength as V⁻¹ᐟ².
Check: Greater accelerating voltage gives greater momentum and therefore a shorter wavelength.
Use a hint if needed
Practise with support
Try this
Momentum triples. State the de Broglie wavelength factor.
Hint: Use inverse proportionality.
Check your answer
λ = h/p, so wavelength becomes one third.
Now work without the hint
Practise independently
Your turn
Explain electron diffraction and single-particle double-slit evidence, and calculate wavelength from a stated momentum.
Check your answer
A diffraction pattern and the gradual build-up of double-slit fringes from localised detections require wave-like probability amplitudes. Use λ = h/p with SI momentum; the evidence does not mean an electron is a classical material wave.
Avoid these traps
Common mistakes
Common mistake
An electron is either a classical particle or a classical wave.
What is wrong with this reasoning?
Show better thinking
Quantum evidence requires a quantum state with wave-like probability behaviour and localised detections.
Common mistake
de Broglie wavelength grows with momentum.
What is wrong with this reasoning?
Show better thinking
λ = h/p, so it decreases as momentum increases.
Write for the examiner
Exam guidance
Compare wavelength with aperture or lattice spacing to judge whether diffraction is observable.
Exam-style practice [6 marks]
A non-relativistic electron has kinetic energy 2.40 × 10⁻¹⁷ J. Find p and λ.
Plan before you answer
- Find non-relativistic momentum from K.
- Use λ = h/p.
- Keep the evidence interpretation separate.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
p = √(2mK) = 6.61 × 10⁻²⁴ kg m s⁻¹ and λ = h/p = 1.00 × 10⁻¹⁰ m.
Come back in three days
Check what stayed with you
Recall question
State the observation when electrons pass through a crystal with spacing comparable to λ.
Check the answer
A diffraction pattern is observed, supporting their wave nature.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes. Use ΔxΔp ≳ h in the form given in the syllabus. Infinite-square-well results apply to a one-dimensional well with ψ zero at both infinite walls and n = 1, 2, …. Photon and matter-wave evidence supports complementary quantum descriptions. X-ray production, solving the Schrödinger equation, finite barriers, tunnelling and scanning tunnelling microscopy are not required here.
- GCE A-Level H2 PhysicsTopic 19(d) / Topic 19(e) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027