De Broglie Wavelength Calculator
Find the de Broglie wavelength of an electron, proton, neutron, or any mass.
What it does
Every mass in motion carries a wave, and its wavelength is Planck's constant divided by momentum. De Broglie proposed the relation in 1924; electron diffraction proved it three years later.
The formula needs only two ingredients. Momentum can come straight from mass times velocity, or from kinetic energy through p = sqrt(2 m E), which is the natural form for particles accelerated through a voltage.
The defaults show how quantum the small scale is. An electron drifting at ten million meters per second carries a wavelength of 7.27e-11 m, about the spacing of atoms in a crystal, which is exactly why crystals diffract electron beams.
Energy changes the picture quickly. The same electron at 100 eV has a wavelength of 1.23e-10 m, and at everyday speeds a thrown baseball would carry a wave some 1.14e-34 m wide, undetectable by any conceivable ruler.
The tool handles electrons, protons, neutrons, or any custom mass. The math is the nonrelativistic form, which stays accurate below about a tenth of light speed; faster particles need the relativistic momentum correction.
How to use
- Pick whether you know velocity or kinetic energy.
- Choose electron, proton, neutron, or a custom mass.
- Enter the velocity in m/s or the energy in eV.
- Read the wavelength in meters and picometers.
- Check the momentum the tool used.
- Sanity check speed against light: relativistic cases need another formula.
Formula
- lambda
- De Broglie wavelength in meters.
- h
- Planck constant, 6.62607015e-34 J·s, exact in SI since 2019.
- p
- Momentum in kilogram meters per second.
- m
- Particle mass in kilograms: electron 9.109e-31, proton 1.673e-27, neutron 1.675e-27.
- E
- Kinetic energy in joules; the tool converts from eV at 1.602e-19 J each.
Picometers suit atomic scale results: one picometer is 1e-12 m, roughly a hundredth of a typical atomic radius. The kinetic energy form assumes nonrelativistic physics, where E = p² / 2m holds. For an electron above about 10 percent of light speed, about 100 keV, the relativistic momentum grows faster and the true wavelength shrinks beyond what this formula reports.
Worked example
Example: electron at 10,000,000 m/s
Electron, velocity mode, 1e7 m/s (defaults).
- Momentum: p = 9.109e-31 x 1e7 = 9.11e-24 kg m/s.
- Wavelength: lambda = 6.626e-34 / 9.11e-24 = 7.27e-11 m.
- In picometers: 72.7 pm, under the diameter of a hydrogen atom.
- Switch to energy mode at 100 eV: 1.23e-10 m, 122.6 pm.
- Both land in the range crystals diffract, which is how the wave nature was first measured.
Reference table
| Case | Wavelength |
|---|---|
| Electron at 1,000,000 m/s | 7.27e-10 m (727.4 pm) |
| Electron at 100 eV | 1.23e-10 m (122.6 pm) |
| Electron at 54.4 eV | 1.66e-10 m (166.3 pm) |
| Neutron at 2,200 m/s | 1.80e-10 m (179.8 pm) |
| Baseball, 145 g at 40 m/s | 1.14e-34 m (1.14e-22 pm) |
Limits
- Nonrelativistic formula only; above about 0.1c use relativistic momentum.
- Free particle; bound states need quantum mechanics proper.
- Ignores wave packet spreading over time.
- Custom mass range stops at 1e6 kg in the tool.
Common mistakes
- Feeding energy in joules while the field reads eV.
- Using grams for mass when the formula wants kilograms.
- Expecting a thrown object to show measurable diffraction.
- Ignoring relativity for electron energies above 100 keV.
- Mixing modes: velocity and energy describe the same particle differently, not in parallel.
Try it yourself
Quick checks. Answers sit tucked under each question.
Electron at 1e6 m/s?
7.27e-10 m, 727.4 pm.
Electron with 100 eV of kinetic energy?
1.23e-10 m, 122.6 pm.
Neutron at 2,200 m/s?
1.80e-10 m, 179.8 pm, the thermal neutron scale.
Questions
What is the de Broglie wavelength?
The wavelength associated with any moving mass, equal to Planck's constant over momentum. It explains electron diffraction and the quantization of orbits.
Why can't I see my own wavelength?
Planck's constant is minuscule. A 70 kg person walking at 1 m/s carries a wave about 1e-35 m wide, twenty orders of magnitude below any physical probe.
When do I need the relativistic version?
Above roughly 10 percent of light speed, momentum grows faster than mv predicts, so the wavelength falls below this formula's answer. Electron microscopes at hundreds of keV need the correction.
Why do electron microscopes use this?
Accelerated electrons carry wavelengths thousands of times shorter than visible light, so they resolve structures light cannot, down to individual atoms.
Is this the same h as in photoelectric effect?
Yes. The same Planck constant ties photon energy to frequency and particle momentum to wavelength, the two halves of wave-particle duality.
What is the check value for a neutron?
A thermal neutron near 2,200 m/s carries about 1.80e-10 m, matching interatomic spacings, which is why neutron diffraction maps crystal structures so well.
Sources
- OpenStax University Physics volume 3: matter waves and key equations
- NIST CODATA: fundamental physical constants