De Broglie Matter Waves: Wavelength h/mv

Every particle is also a wave

In his 1924 doctoral thesis, Louis de Broglie made a proposal that seemed almost reckless at the time: if light, long understood as a wave, could also behave as discrete photon particles (as Einstein had shown for the photoelectric effect), then perhaps matter, long understood as particles, could also behave as waves. He assigned every particle a wavelength set by its momentum:

lambda = h / p = h / (m * v)

where h is Planck's constant, 6.626 x 10^-34 joule-seconds -- an extremely small number, and that smallness is the whole reason matter waves are invisible in daily life. Divide a tiny constant by the momentum of an everyday object and the resulting wavelength is unimaginably short; divide it by the momentum of an electron and the wavelength lands squarely in a range you can actually measure with a crystal lattice or a diffraction grating.

Confirmed within three years, by accident and on purpose

De Broglie's proposal was confirmed remarkably fast. In 1927, Clinton Davisson and Lester Germer, studying electron scattering off a nickel crystal at Bell Labs, noticed diffraction peaks at angles that made no sense for particles behaving as simple billiard balls -- but matched exactly what de Broglie's wavelength formula predicted for electron waves interfering off the crystal's regular atomic spacing. The same year, George Paget Thomson (son of J. J. Thomson, who had won a Nobel Prize for identifying the electron as a particle) independently fired electrons through thin metal foils and observed the same diffraction rings you get from X-rays hitting a powdered crystal. Both experiments won Nobel Prizes; the elder and younger Thomson's results, taken together, are often summarized as a father proving the electron is a particle and a son proving it is also a wave.

Why your wavelength is unmeasurably tiny

Plug a 70 kilogram person walking at 1 metre per second into the formula and lambda comes out around 10^-35 metres -- vastly smaller than a proton, let alone anything an instrument could resolve. Plug in an electron accelerated to a modest energy and lambda lands in the sub-nanometre range, comparable to the spacing between atoms in a crystal, which is precisely why electron diffraction works as a technique and why nobody has ever built an interferometer sensitive enough to see a person's matter wave. The de Broglie wavelength scales inversely with momentum, so heavy, fast objects always get pushed toward wavelengths far too small to matter, while light, slow particles keep wavelengths large enough to interfere visibly.

From a formula to a full wave equation

De Broglie's relation was a hypothesis about wavelength, not a complete dynamical theory -- it does not by itself say how the wave evolves in time or responds to a potential. Erwin Schrodinger took de Broglie's idea and built the differential equation that a matter wave must satisfy, arriving at the Schrodinger equation in 1926, which reproduces the de Broglie wavelength for a free particle as a special case (a plane wave with wavelength h/p) while also correctly predicting the discrete energy levels of the hydrogen atom, something the simple wavelength formula alone could never do.

Double-slit interference: the wave nature made visible

The clearest demonstration of matter-wave behaviour is the double-slit experiment performed with particles instead of light: fire electrons, neutrons, or even large molecules one at a time through two closely spaced slits, and an interference pattern of bright and dark fringes builds up on the detector screen, exactly as it would for light waves passing through the same slits. The fringe spacing is set by the de Broglie wavelength and the slit geometry, and the pattern appears even when particles are sent through one at a time, confirming that each individual particle's wave interferes with itself rather than with other particles.

Frequently asked questions

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