De Broglie λ
λ = h / (mv) for non-relativistic momentum.
About this calculator
Louis de Broglie's 1924 hypothesis was that every moving particle has an associated wavelength, not just light — matter behaves as a wave with λ = h/p, where p is ordinary momentum (mass times velocity) and h is Planck's constant. This calculator plugs your entered mass and velocity straight into that relation, dividing Planck's constant by their product to get the wavelength in meters, then converts the same result to nanometers since atomic-scale wavelengths are easier to read in that unit. Because momentum sits in the denominator, heavier or faster particles get shorter wavelengths — a thrown baseball has a de Broglie wavelength so far below any measurable scale that its wave nature is permanently undetectable, while an electron moving at everyday lab speeds has a wavelength comparable to atomic spacing, which is exactly why electron diffraction experiments work and baseball diffraction experiments don't.
The formula used here is the non-relativistic approximation, valid as long as velocity stays well below the speed of light; the engine floors velocity at 1 m/s and mass at a tiny nonzero value purely to avoid dividing by zero, not because either bound has physical meaning. For particles moving at a meaningful fraction of light speed, momentum should be computed relativistically as p = γmv instead, which this simple formula does not do — using it at relativistic speeds will understate momentum and overstate wavelength.
Inputs
Results
λ (m)
0.000000001455
How to Use This Calculator
- Enter the particle mass in kg (electron = 9.109e-31 kg, proton = 1.673e-27 kg).
- Enter the particle velocity in m/s (must be significantly less than the speed of light for classical de Broglie).
- Review the calculated de Broglie wavelength in meters and nanometers.
- Compare wavelength to atomic scale (~0.1 nm) to assess whether wave behavior is significant.
- For relativistic particles, use the relativistic momentum p = gamma*m*v instead.
How the result changes with Mass (kg)
| Mass (kg) | λ (m) |
|---|---|
| 0 | 0.000000002906 |
| 0 | 0.00000000194 |
| 0 | 0.000000000967 |
| 0 | 0.000000000581 |
What each input means
- Mass (kg)
- Electron ~9.11e-31.
- v (m/s)
- Particle velocity in meters per second. Must be non-relativistic (much less than c) for this formula.
How this is calculated
Formula
λ = h / (m × v) — de Broglie wavelength from Planck's constant, mass, and velocityEngine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does a heavier particle get a shorter de Broglie wavelength at the same speed?
Wavelength is inversely proportional to momentum, and momentum is mass times velocity, so increasing mass while holding velocity fixed directly shrinks the wavelength. This is why a proton moving at the same speed as an electron has a de Broglie wavelength roughly 1,800 times shorter — its much larger mass dominates the momentum term.
Why can't I see wave behavior in everyday objects like a thrown ball?
A baseball's mass is so many orders of magnitude larger than an electron's that its de Broglie wavelength works out to a number vastly smaller than any distance that could ever be measured or that matters physically, such as the width of an atomic nucleus. Wave effects only become noticeable when the wavelength is comparable to the scale of whatever the particle is interacting with.
What counts as a 'non-relativistic' velocity for this formula to stay accurate?
As a rule of thumb, velocities below roughly 10% of the speed of light keep the classical momentum formula p = mv close enough to the relativistic value p = γmv that the difference is negligible for most purposes. Above that, the relativistic correction factor γ grows enough that this calculator's wavelength will start reading noticeably too long.
Why does electron diffraction work in a way that proton or baseball diffraction wouldn't?
Electrons at typical experimental speeds have de Broglie wavelengths on the order of atomic spacing in a crystal lattice, which is exactly the condition needed for a wave to diffract noticeably off that structure. Protons at the same speed have a much shorter wavelength due to their larger mass, and everyday objects have wavelengths so short that no physical structure could ever diffract them.
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