de Broglie Matter Waves
Louis de Broglie proposed that every moving object has a wavelength, not just light. A fast, heavy object has an incredibly tiny wavelength, but a light, slow particle like an electron has a wavelength large enough to make it diffract — bend and interfere — just like a wave passing through a crystal.
The formula
λ = h / p
- λ — wavelength (m): the matter wave's length associated with the moving object
- h — Planck's constant (J·s): a tiny fixed number, about 6.63 × 10⁻³⁴ joule-seconds
- p — momentum (kg·m/s): mass times velocity — how much motion the object carries
Worked example
An electron has a momentum of 6.63 times ten to the minus twenty-four kilogram meters per second. What is its de Broglie wavelength?
- h = 6.63 × 10⁻³⁴ J·s
- p = 6.63 × 10⁻²⁴ kg·m/s
- λ = h / p
- λ = (6.63 × 10⁻³⁴ J·s) / (6.63 × 10⁻²⁴ kg·m/s)
λ = 1 × 10⁻¹⁰ m
Test yourself
If the electron's momentum doubles to 1.326 × 10⁻²³ kilogram meters per second, what happens to its wavelength?
- Correct answer: It halves to 5 × 10⁻¹¹ m
- It doubles to 2 × 10⁻¹⁰ m
- It stays at 1 × 10⁻¹⁰ m
Right! Wavelength and momentum are inversely related: λ = h / p = (6.63 × 10⁻³⁴) / (1.326 × 10⁻²³) = 5 × 10⁻¹¹ m.
A thrown baseball also has a de Broglie wavelength. Why can't we ever see it diffract like the electron does?
- Baseballs don't actually have a wavelength
- Correct answer: Its wavelength is far too small to detect
- Baseballs move too slowly to have momentum
Exactly — a baseball's huge momentum makes h / p unimaginably tiny, far smaller than anything it could diffract around.
Where you see this
An electron microscope sees viruses and protein structures no optical microscope can resolve, for exactly one reason: accelerated electrons carry a wavelength thousands of times shorter than visible light, and wavelength sets the resolution. The machine is λ = h / p on a lab bench.
Common mistakes
The scaling trap: momentum sits on the bottom of the fraction, so doubling an electron's momentum HALVES its wavelength (the lesson's electron drops to 5 × 10⁻¹¹ m) — faster means smaller, not bigger. The second error is reserving waves for tiny things: a thrown baseball has a wavelength too, by exactly the same law; its momentum is just so enormous that the wavelength is unobservably small.
How it connects
de Broglie's flip completes the duality started in the photoelectric effect and Compton scattering: waves are particles, so particles are waves — and Bohr's mysterious rungs suddenly make sense as standing electron waves around the atom. The uncertainty principle, next, is the price tag attached to this wave nature.