De Broglie Wavelength Calculator

What is the de Broglie wavelength?

In 1924 Louis de Broglie proposed that, just as light behaves both as a wave and as a particle, particles of matter such as electrons have wave properties too. A particle with momentum p has an associated wavelength of

λ = h / p

where h is the Planck constant (6.62607015 × 10⁻³⁴ J·s). For a particle that is slow compared with light, p = m × v and the formula becomes λ = h / (m × v).

From kinetic energy and from voltage

A slow particle with kinetic energy K has p = √(2 × m × K), so λ = h / √(2 × m × K). An electron accelerated through a voltage V gains an energy of e × V, and its wavelength is roughly λ (nm) = 1.2264 / √V.

Example

An electron accelerated through 100 V has a wavelength of 1.2264 / √100 = 0.1226 nm, or 1.23 Å. That is comparable to the spacing of atoms in a crystal, which is why electrons diffract from crystals. A 0.65 kg basketball moving at 10 m/s has a wavelength of 6.626 × 10⁻³⁴ / 6.5 = 1.02 × 10⁻³⁴ m, far too small to detect.

What changes at high speed?

As the speed approaches the speed of light, the momentum exceeds m × v: p = γ × m × v. This calculator always uses relativistic momentum and shows the classical result alongside. For a 108 keV electron the classical formula gives 3.73 pm, while the correct value is 3.55 pm.

Limits

The calculation is for a free particle. Masses are CODATA 2022 values. The voltage tab assumes the particle starts from rest and loses no energy on the way.