de Broglie Wavelength Calculator (λ = h/mv)
By Santaji GadepatilinUpdated October 30, 2026
Every moving particle behaves as a wave whose wavelength is Planck's constant divided by its momentum, λ = h/p. This de Broglie wavelength calculator gives the matter wavelength, momentum and kinetic energy of electrons, protons, neutrons or any object — the physics behind electron microscopes and electron diffraction.
How to use the de Broglie wavelength calculator
Choose the particle and enter its velocity. The calculator returns the momentum, the de Broglie wavelength in nanometres and the kinetic energy in electronvolts. To find the velocity needed for a given wavelength, clear the velocity and enter the wavelength.
de Broglie wavelength formula
λ = h ÷ p = h ÷ (m v)
KE = ½ m v² · λ = h ÷ √(2 m KE)
Example
An electron (9.109 × 10⁻³¹ kg) moving at 1 × 10⁶ m/s:
- p = 9.109 × 10⁻³¹ × 10⁶ = 9.1094 × 10⁻²⁵ kg·m/s
- λ = 6.626 × 10⁻³⁴ ÷ 9.1094 × 10⁻²⁵ = 0.7274 nm
- KE = ½ × 9.109 × 10⁻³¹ × (10⁶)² = 2.843 eV
Wavelength of electrons accelerated through a voltage
| Accelerating voltage | Non-relativistic λ | Relativistic λ |
|---|---|---|
| 10 V | 387.830 pm | 387.828 pm |
| 100 V | 122.643 pm | 122.637 pm |
| 1,000 V | 38.783 pm | 38.764 pm |
| 10,000 V | 12.264 pm | 12.205 pm |
| 100,000 V | 3.878 pm | 3.701 pm |
| 300,000 V | 2.239 pm | 1.969 pm |
Transmission electron microscopes run at 100–300 kV, giving wavelengths of a few picometres — far below the 400–700 nm of visible light.
Notes
- For an electron accelerated from rest through V volts, λ (nm) ≈ 1.226 ÷ √V (non-relativistic).
- Thermal neutrons (about 2,200 m/s) have λ ≈ 0.18 nm, ideal for neutron diffraction.
- The photon energy calculator does the equivalent conversion for light, where λ = h c ÷ E.
Frequently asked questions
What is the de Broglie wavelength formula?
λ = h ÷ p = h ÷ (m v), where h = 6.626 × 10⁻³⁴ J·s. Heavier or faster particles have shorter wavelengths.
What is the wavelength of an electron moving at 10⁶ m/s?
About 0.727 nm — comparable to the spacing of atoms in a crystal, which is why electrons diffract from crystals.
Why don't everyday objects show wave behaviour?
Their momentum is enormous by comparison. A 0.15 kg ball at 40 m/s has λ ≈ 1.1 × 10⁻³⁴ m, far too small to observe.
When do I need the relativistic formula?
Above about 10% of the speed of light. For electrons accelerated through more than a few kilovolts, use p = √(2 m E (1 + E ÷ 2mc²)); the table below shows the difference.