A bold symmetry
By the early 1920s physicists had grudgingly accepted that light — long thought a pure wave — also comes in particle-like lumps, the photons of the last guide. A young French aristocrat named Louis de Broglie, writing his doctoral thesis, asked a question so simple it sounds almost cheeky: if waves can act like particles, why should the reverse not also be true? Why should not particles — electrons, atoms, anything with mass — also act like waves? Nature, he felt, loves symmetry. This single hunch is the de Broglie hypothesis, and it turned out to be one of the great correct guesses in the history of science.
How long is the wave?
A wave's most basic measurement is its wavelength — the distance from one crest to the next. De Broglie's genius was to give matter a precise wavelength with a startlingly tidy rule: the faster and heavier something moves — that is, the more momentum it carries — the shorter its wavelength. Fast, heavy things have stubby little waves; slow, light things have long sprawling ones. The exact relationship is the de Broglie wavelength, and it hangs on a single tiny number of nature.
That tiny number is Planck's constant, usually written h. It is the fundamental "grain size" of the quantum world, and it is breathtakingly small. Because it sits on top of the fraction that gives the wavelength, the wavelength of anything ordinary comes out fantastically small. The formula is honestly just one line:
wavelength = h / momentum h = Planck's constant (a tiny fixed number) momentum = mass x speed (how much "oomph" the motion carries) big momentum -> short wave small momentum -> long wave
Do the back-of-envelope numbers
Numbers make this concrete and, frankly, fun. Take a slow electron, the kind inside an atom: its de Broglie wavelength comes out roughly the size of an atom itself. That is enormous luck for us — it means the regular spacing of atoms in a crystal is just right to act as a grating and reveal the electron's wave. Now take yourself, walking at a stroll: your wavelength is something like a hundred billion billion billion times smaller than a single atomic nucleus. There is no slit, no grating, no object anywhere in the universe fine enough to make you diffract. You have a wavelength. You will simply never see it.
This is the clean answer to the nagging worry from the last guide. "Why is the everyday world not visibly wavy?" is not a mystery — it is arithmetic. Wave behaviour only shows up when the wavelength is comparable to the obstacles around it. Shrink the wavelength a billion-billion-fold and the waviness vanishes from view, even though it is still formally there. The boundary between quantum weirdness and everyday solidity is set, more than anything, by the smallness of h.
From hunch to hard fact
A beautiful guess is still only a guess until nature signs off. Within a few years it did — twice, by accident and by design. In the United States, Clinton Davisson and Lester Germer were bouncing electrons off a nickel crystal when a lab mishap recrystallized their sample into a clean grating. Suddenly the scattered electrons came back in sharp peaks at exactly the angles you would expect if they were waves of de Broglie's predicted length. This is the Davisson-Germer experiment, and it is one of those rare cases where a happy accident confirmed a major theory.
Once you can make electrons spread and overlap into patterns, you have electron diffraction — the wave behaviour of matter, on tap. It is not a museum curiosity: electron microscopes exploit the tiny wavelength of fast electrons to see far finer detail than light ever could. And the experiment has since been pushed up the size ladder, with whole atoms and even large molecules made to interfere as waves. De Broglie's cheeky symmetry is now everyday laboratory fact: matter, all of it, ripples.