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All lessons Modern Physics26 min

de Broglie Waves and Wave-Particle Duality

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← Phase TransitionsThe Double-Slit Experiment →
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01

Hook

Electrons — tiny bits of matter — fired one at a time at a double slit produce an interference pattern. But interference is a wave phenomenon. How can a particle interfere with itself? Is an electron a particle or a wave — and what does that question even mean?

02

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Spoilers

de Broglie Waves and Wave-Particle Duality — summary and key formula

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The question

Electrons — tiny bits of matter — fired one at a time at a double slit produce an interference pattern. But interference is a wave phenomenon. How can a particle interfere with itself? Is an electron a particle or a wave — and what does that question even mean?

In 1924, Louis de Broglie proposed that all matter has an associated wavelength. This wave-particle duality is not a limitation of our measurements — it is a fundamental property of quantum objects. The electron really is both.

The key idea

Louis de Broglie proposed that all matter — not just light — has wave properties. A particle with momentum p has an associated wavelength λ = h/p. This de Broglie wavelength is observable only when it is comparable to the size of the structures the particle interacts with.

De Broglie's hypothesis was confirmed by the Davisson-Germer experiment in 1927: electrons fired at a crystal produced diffraction patterns — exactly as X-rays do. The electron beam behaved as a wave with wavelength predicted by λ = h/p. For electrons in atoms, de Broglie waves explain why only certain orbits are allowed (Bohr model): only orbits where the circumference equals a whole number of wavelengths are stable (standing electron waves). This quantises energy levels. Electron microscopes exploit de Broglie waves: electrons accelerated to 100 keV have λ ≈ 0.004 nm — thousands of times shorter than visible light. This allows imaging of individual atoms. The wave-particle duality is not an analogy — quantum objects are genuinely described by wave functions whose square gives probability density.

The formula

λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}λ=ph​=mvh​
  • ·λ = de Broglie wavelength (m)
  • ·h = Planck's constant (6.626×10⁻³⁴ J·s)
  • ·p = momentum (kg·m/s)
  • ·m = mass (kg)
  • ·v = velocity (m/s)