de Broglie wavelength
/ de Broglie = duh BROY /
Light was long known to be a wave, and quantum physics then showed it also comes in particle-like packets. In 1924 Louis de Broglie asked the bold mirror-image question: if light waves can act like particles, can particles like electrons act like waves? His answer was yes, and he gave a precise recipe for the wavelength any moving particle should have. Every chunk of matter, he proposed, has a wave associated with it.
The de Broglie wavelength is the wavelength of a particle's matter wave, and the rule is simple: wavelength equals Planck's constant divided by momentum (lambda = h / p). Faster, heavier, more energetic objects have shorter wavelengths; slow, light ones have longer wavelengths. Because Planck's constant is so tiny, the wavelength of anything large is hopelessly small — a thrown baseball has a wavelength far below the size of an atomic nucleus, which is why we never see a baseball diffract. But an electron, being extremely light, can have a wavelength comparable to the spacing of atoms in a crystal, and then its wave nature becomes obvious.
This idea is the practical engine of much of particle physics and beyond. It explains why electrons fired at a crystal produce a diffraction pattern, confirming that matter really does wave, and it is the working principle of the electron microscope, which resolves far finer detail than light because fast electrons have far shorter wavelengths. In particle physics it gives the precise meaning of the rule that probing tiny distances needs high energy: to reach a small distance you need a small de Broglie wavelength, which means large momentum, which means high energy. The de Broglie relation is the dial that converts a beam's energy into its resolving power.
Fire a beam of electrons at a thin crystal and they spread into a ring pattern just like X-rays do — proof that the electrons travelled as waves, with a wavelength matching de Broglie's lambda = h / p exactly.
Matter waves are real: an electron's wavelength is just Planck's constant over its momentum.
Everything has a de Broglie wavelength, but for macroscopic objects it is so absurdly tiny that no wave effect is ever observable; the wave nature only shows for very light, fast particles.