De Broglie Wavelength Calculator
Enter a particle's mass and speed to get its de Broglie (matter) wavelength in nanometres and metres — and see why fast, light particles behave like waves.
Nanometres and metres at once
Enter the mass and velocity and this de Broglie wavelength calculator returns the wavelength λ = h / (m·v) in nanometres and in metres together.
Use SI units
Mass in kilograms and velocity in metres per second give the wavelength in metres — an electron's rest mass is about 9.109e-31 kg.
What is the de Broglie wavelength?
The wave that goes with a particle
The de Broglie wavelength calculator answers a strange and beautiful question: every moving particle also behaves like a wave, and this is the length of that wave. Louis de Broglie proposed in 1924 that matter — electrons, protons, even baseballs — carries a wavelength equal to the Planck constant divided by its momentum. For everyday objects the result is far too small to notice, but for tiny, fast particles like electrons it is large enough to produce real interference and diffraction, the discovery that launched quantum mechanics. Give the calculator a mass in kilograms and a velocity in metres per second and it returns the wavelength in nanometres and metres.
Enter a mass in kilograms and a speed in metres per second to get the de Broglie wavelength in nanometres and metres instantly.
The de Broglie wavelength is the Planck constant divided by the particle's momentum, where momentum is mass times velocity.
λ = h / (m × v)The Planck constant h is a tiny fixed number (6.626 × 10⁻³⁴ J·s), so the wavelength only becomes appreciable when the momentum m·v is also tiny — that is, for very light particles. A heavier or faster particle has more momentum and therefore a shorter wavelength.
Suppose an electron (mass 9.109 × 10⁻³¹ kg) is moving at 10,000,000 m/s.
Find the momentum
9.109e-31 × 1e7 = 9.109e-24 kg·m/s — mass times velocity.
Divide Planck's constant by it
6.62607015e-34 / 9.109e-24 = 7.274e-11 m — the wavelength in metres.
Convert to nanometres
7.274e-11 × 1e9 = 0.0727 nm — about the spacing of atoms in a crystal, which is why electrons diffract through them.
The formula is exact, but a couple of practical points are worth keeping in mind.
Non-relativistic momentum and consistent units
This calculator uses the classical momentum m·v, which is accurate for speeds well below the speed of light; near light speed you need the relativistic momentum instead. It also assumes a particle with non-zero mass, so it does not apply to photons. Keep your units consistent — kilograms for mass and metres per second for velocity — or the wavelength will be wrong.