The de Broglie wavelength

Light shows both wave and particle behaviour — interference patterns on one hand, photons with definite momentum on the other. De Broglie's proposal, made before any direct evidence existed for it, was that this duality is not special to light: every particle with momentum has an associated wavelength, given by the same relation photons already obeyed, λ = h/p. A moving electron is exactly as much a wave as a photon is, tied to its motion through momentum rather than through anything unique to electromagnetism.

Because Planck's constant h is extraordinarily small, this wavelength is tiny for anything with everyday momentum. A thrown baseball's de Broglie wavelength is dozens of orders of magnitude smaller than the size of a proton — utterly unmeasurable and physically irrelevant to any experiment ever performed on it, which is exactly why baseballs never visibly diffract or interfere. An electron, by contrast, has such a small mass that even modest momenta give it a wavelength comparable to the spacing between atoms in a crystal, which is large enough to produce genuinely observable diffraction patterns when a beam of electrons passes through a crystalline sample — the direct experimental confirmation of de Broglie's proposal, and today the working principle behind electron microscopy.

The inverse relationship between wavelength and momentum is worth sitting with, because it runs opposite to the everyday intuition that "faster should mean a bigger wave." Higher momentum means a shorter wavelength, not a longer one — accelerating a particle compresses its associated wave rather than stretching it, exactly the reverse of what raising a wave's amplitude or speed does to an ordinary mechanical wave.