The big idea: An electron microscope can image a virus far too small for any light microscope — because its speeding electrons behave like waves with a tiny wavelength. Every moving particle has this de Broglie wavelength, set by its momentum p: the bigger the momentum, the shorter the wavelength (λ = h/p).
- de Broglie wavelength (m)
- Planck constant, 6.63×10⁻³⁴ J s
- momentum of the particle (kg m s⁻¹)
An electron (mass 9.11 × 10⁻³¹ kg) moves at 2.0 × 10⁶ m s⁻¹. Find its de Broglie wavelength.
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Electrons make diffraction patterns: Fire a beam of electrons at a thin crystal and you get a diffraction pattern of rings — exactly what waves do when they pass through gaps about the size of their wavelength.
The atomic spacing in the crystal (~10⁻¹⁰ m) matches an electron's de Broglie wavelength, so the effect shows up. This is direct proof that particles have a wave nature.
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Big, everyday objects have huge momentum, so their de Broglie wavelength is far too small to ever notice. Wave behaviour only shows up for tiny particles like electrons.
A 0.16 kg cricket ball is bowled at 40 m s⁻¹. Find its de Broglie wavelength, and comment.
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You can't have it both ways: Because particles behave as waves, you cannot know both a particle's position and its momentum exactly at the same time. The more precisely you pin down one, the less precisely you can know the other.
- uncertainty in position (m)
- uncertainty in momentum (kg m s⁻¹)
- Planck constant, 6.63×10⁻³⁴ J s
An electron's position is known to within Δx = 1.0 × 10⁻¹⁰ m. Find the minimum uncertainty in its momentum.
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How this is tested — matter waves are HL only (E.2) and split cleanly by paper:
Paper 1A
- A quick λ = h/p.
- Or 'why do electrons diffract but cricket balls don't?'
Paper 2
- Find a de Broglie wavelength (often after the momentum first).
- Or a minimum uncertainty from Δx·Δp ≥ h/4π.
The classic trap: Find the momentum p = mv first, then λ = h/p. A bigger momentum gives a shorter wavelength — the relation is inverse.
Three easy marks: (1) Find the momentum p = mv first, then λ = h/p. (2) A bigger momentum → a shorter wavelength. (3) Diffraction needs a gap about the size of the wavelength.
A proton has a momentum of 3.0 × 10⁻²¹ kg m s⁻¹. Determine its de Broglie wavelength.
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