De Broglie Wavelength Calculator

Enter a particle's mass and velocity to find its de Broglie wavelength (λ = h/p), momentum, and Lorentz factor using non-relativistic or relativistic momentum.

Quick Facts

De Broglie relation
λ = h / p
Wavelength equals Planck's constant divided by momentum.
Planck's constant
h = 6.62607015 × 10⁻³⁴ J·s
An exact value fixed by the 2019 SI redefinition.
Non-relativistic momentum
p = m·v
Accurate when v is well below the speed of light c.
Relativistic momentum
p = γmv, γ = 1/√(1 − v²/c²)
Needed near c, e.g. accelerated electrons or particle beams.

Your Results

Calculated
De Broglie Wavelength
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λ = h / p
Momentum
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p, in kg·m/s
Wavelength in Angstroms
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1 Å = 10⁻¹⁰ m
Lorentz Factor
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γ = 1/√(1 − v²/c²)

Ready

Enter a particle mass, velocity, and momentum model, then press Calculate.

How to Calculate De Broglie Wavelength

In 1924, Louis de Broglie proposed that every moving particle has a wave associated with it, with wavelength λ = h / p, where h is Planck's constant (6.62607015 × 10⁻³⁴ J·s) and p is the particle's momentum. This extended the wave-particle duality already known for light to all matter, and the Davisson-Germer experiment confirmed it in 1927 by diffracting electrons off a nickel crystal. This calculator computes λ from a mass and velocity you supply, along with the resulting momentum and Lorentz factor.

Deriving the formula

De Broglie combined two known relations: Planck's E = hf for photon energy, and the relativistic photon momentum p = E/c. Substituting and using c = fλ gives λ = h/p for light. De Broglie's insight was to apply the same relation to matter by defining a particle's momentum as p = mv (or, at relativistic speeds, p = γmv with the Lorentz factor γ = 1/√(1 − v²/c²)). Rearranging p = h/λ gives the working formula λ = h/(mv) used by this calculator's non-relativistic mode.

Non-relativistic vs. relativistic momentum

For everyday speeds (v well under about 10% of the speed of light), p = mv is accurate to within roughly 0.5% and is the standard textbook formula. It breaks down for fast particles: electrons accelerated through 100 kV in a transmission electron microscope reach about 0.55c, where γ ≈ 1.2 — using p = mv there would understate momentum and overstate the wavelength by a similar margin. Switch to the relativistic model whenever v/c exceeds about 0.1, or whenever you know the particle was accelerated through a large voltage or found in a particle accelerator.

Real-world applications

  • Electron microscopy uses the short de Broglie wavelength of accelerated electrons (picometer scale) to resolve details far smaller than visible light (hundreds of nanometers) allows.
  • Neutron and electron diffraction are used in crystallography to map atomic structures, since particle wavelengths on the order of interatomic spacing produce interference patterns.
  • Quantum mechanics uses λ = h/p to explain why macroscopic objects (baseballs, cars) show no observable wave behavior: their wavelengths are absurdly smaller than any physical measurement.

Frequently Asked Questions

What is the de Broglie wavelength formula?
The de Broglie wavelength is λ = h / p, where h is Planck's constant (6.62607015 × 10⁻³⁴ J·s) and p is the particle's momentum. For everyday speeds, momentum is p = mv (mass times velocity), so λ = h / (mv). Every moving particle — not just photons — has an associated wavelength.
Why do electrons in an electron microscope need the relativistic formula?
Electron microscopes typically accelerate electrons to 60-300 keV, which corresponds to roughly 0.4c-0.8c. At those speeds the non-relativistic formula p = mv understates the true momentum, so you should use the relativistic momentum p = γmv (with γ = 1/√(1 − v²/c²)) to get an accurate wavelength.
Does a thrown baseball have a de Broglie wavelength?
Yes — every moving object does — but a 0.145 kg baseball thrown at 40 m/s has a de Broglie wavelength around 10⁻³⁴ m, vastly smaller than an atomic nucleus. That is why wave behavior is only observable for very light particles like electrons, neutrons, and atoms, not everyday objects.
How accurate is the Planck's constant value used here?
This calculator uses h = 6.62607015 × 10⁻³⁴ J·s, the exact defined value fixed by the 2019 SI redefinition of the kilogram. It carries no measurement uncertainty, so the precision of your result depends only on the mass and velocity you enter.

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