Last updated: 2026-06-15

de Broglie Wavelength

In 1924, French physicist Louis de Broglie proposed that matter, like light, possesses a wave–particle dual nature. Every moving particle with momentum $p$ has an associated wavelength given by:

where $h = 6.626 \times 10^{-34}\ \text{J·s}$ is Planck's constant, $m$ is the particle's mass in kilograms, and $v$ is its speed in metres per second. This result, confirmed experimentally in 1927 by electron diffraction, is foundational to quantum mechanics: it means that electrons, protons, and even atoms exhibit interference and diffraction just as light does.

This calculator computes the de Broglie wavelength and momentum of a particle from its mass and speed.

Formula

Because Planck's constant is so small, $\lambda$ is negligibly small for everyday objects but becomes comparable to atomic spacings for electrons and other sub-atomic particles.

Worked Example

An electron (mass $m_e = 9.109 \times 10^{-31}\ \text{kg}$) moves at $v = 1 \times 10^6\ \text{m/s}$ — about 0.3% of the speed of light, well within the non-relativistic regime.

This wavelength is on the same scale as atomic separations in a crystal (~0.1–1 nm), which is why diffraction of electrons can reveal atomic structure — the principle behind electron microscopy and low-energy electron diffraction (LEED).

Scale of the Effect

The tennis ball's wavelength is smaller than a proton by twenty orders of magnitude — utterly undetectable, which is why quantum wave behaviour plays no role in everyday life.

Relativistic Note

The formula $\lambda = h/(mv)$ uses the classical momentum $p = mv$. For particles moving at speeds approaching the speed of light ($v \gtrsim 0.1c$), the relativistic momentum $p = \gamma mv$ should be used instead, where $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the Lorentz factor. For the electron at $10^6\ \text{m/s}$ in the example above, $v/c \approx 0.003$, so the relativistic correction is less than 0.001% — negligible. At $v = 0.9c$, however, $\gamma \approx 2.29$ and the correction is significant.

Applications

The de Broglie wavelength is not merely theoretical. Transmission electron microscopes (TEMs) accelerate electrons to tens or hundreds of kilovolts, giving them sub-ångström de Broglie wavelengths and the ability to image individual atoms. Neutron diffraction uses thermal neutrons whose wavelengths match interatomic spacings to probe crystal and magnetic structures. In quantum computing, controlling the wave nature of particles is central to maintaining coherence in qubits.