What this calculator does
De Broglie proposed that every moving particle has a wavelength, given by Planck's constant divided by its momentum. It sounds like an abstraction until you notice that electron diffraction works exactly as the idea predicts, which is what electron microscopes rely on.
The reason the effect is invisible in daily life is scale. Planck's constant is so tiny that anything with everyday mass has a wavelength far below any measurable size, so its wave nature never shows.
The formula
Divide Planck's constant by momentum, which is mass times speed. The result is in metres and is converted to nanometres and picometres, which are the useful units at this scale.
| Term | Meaning |
|---|---|
| De Broglie wavelength (λ) | The wavelength associated with a particle's momentum. |
| Momentum (p) | Mass times velocity. Larger momentum means a shorter wavelength. |
| Wave-particle duality | The principle that matter and light both show wave and particle behaviour depending on what is measured. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The particle's mass in kilograms. An electron is 9.109 × 10⁻³¹. |
| Speed (m/s) | The particle's speed in metres per second. |
When to use it
Understanding electron microscopes
Electrons have far shorter wavelengths than visible light, which is why they resolve far finer detail.
Seeing why everyday objects show no wave behaviour
Running the calculation for an ordinary object gives a wavelength so small that no experiment could ever detect it.
Working a quantum mechanics problem
The de Broglie relation is the usual starting point for questions about matter waves and diffraction.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does an electron's wavelength change with speed?
An electron at a range of speeds.
| Speed | Wavelength (nm) | Momentum |
|---|---|---|
| 1.1 million m/s | 0.66126319 nm | 1.0020e-24 kg·m/s |
| 2.2 million m/s | 0.3306316 nm | 2.0041e-24 kg·m/s |
| 4.4 million m/s | 0.1653158 nm | 4.0081e-24 kg·m/s |
Questions
Why do everyday objects show no wave behaviour?
Because their momentum is enormous compared with Planck's constant. A cricket ball at 40 m/s has a wavelength around 10⁻³⁴ metres, which is smaller than any length that has physical meaning, so there is nothing to observe.
Has this actually been demonstrated?
Yes, repeatedly. Electron diffraction was confirmed within a few years of the proposal, and the same behaviour has since been shown for neutrons, atoms and even large molecules.
Why do electron microscopes resolve more than light ones?
Because resolution is limited by wavelength, and fast electrons have wavelengths thousands of times shorter than visible light. That is the entire advantage.
Does the wavelength depend on the particle type?
Only through its mass and speed, which together give momentum. Two different particles with the same momentum have exactly the same de Broglie wavelength.
For the energy of a light quantum rather than a matter wave, see the photon energy calculator. For the limit on knowing position and momentum together, see the Heisenberg uncertainty calculator.