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Fire electrons at a barrier with two narrow slits, one electron at a time, seconds apart, so that only one is ever inside the apparatus. Each arrives at the screen as a single dot in a single place — a particle, unmistakably. But let the dots pile up over an hour and they form interference fringes: bright bands and dark bands, exactly what a wave passing through both slits at once would make. There is no other electron to interfere with. So what did each one do on the way?
Fire electrons at a barrier with two narrow slits, one electron at a time, seconds apart, so that only one is ever inside the apparatus. Each arrives at the screen as a single dot in a single place — a particle, unmistakably. But let the dots pile up over an hour and they form interference fringes: bright bands and dark bands, exactly what a wave passing through both slits at once would make. There is no other electron to interfere with. So what did each one do on the way?
Richard Feynman called this "the only mystery" of quantum mechanics, and said it contains the whole of it. The experiment was first done with electrons by Claus Jönsson in 1961 and famously repeated one particle at a time by Akira Tonomura's group in 1989, whose film shows the fringes assembling out of what starts as random speckle. The result is not a curiosity about electrons: it has since been repeated with neutrons, atoms, and molecules of over two thousand atoms.
An electron reaching a point on the screen has two indistinguishable ways of getting there — through the top slit or the bottom one. Quantum mechanics says you add the amplitudes for those routes and square the total, rather than adding two probabilities. The path difference between the routes is , so the amplitudes arrive in phase wherever and exactly out of phase halfway between, giving . For small angles the bright fringes are evenly spaced on the screen at . Each slit has a finite width , so this is modulated by that slit's own diffraction envelope, and any order where coincides with an envelope zero goes missing.
The arithmetic is the same as Young's slits with light, and that is exactly the point: matter obeys the interference formula that was invented for waves. What is new is what the formula is a statement about. With light you can read as the brightness at each point. With one electron in the apparatus there is no brightness — there is one dot — and is the probability of that dot appearing there. The fringes are the shape of that probability, drawn out by repetition, which is why a hundred electrons show it and one cannot. **All forms.** gives the fringe angles exactly; is the small-angle version, valid when , which for electrons it overwhelmingly is. Rearranged, turns the experiment into a way of measuring the electron's wavelength, and combined with it measures Planck's constant. **Limiting case.** Push up and the fringes crowd together as until they are finer than the detector can resolve, and the pattern washes out into the smooth blob you would expect from particles. Nothing quantum has switched off; the interference has simply become too fine to see. That is why the effect is invisible for everyday objects: a grain of sand has m, so its fringes are spaced some m apart. **Connect it.** This is de Broglie's being cashed in. The double slit does not add a new law — it takes the wavelength that predicts and puts it somewhere you can measure it with a ruler, which is how Davisson, Germer and Jönsson turned a hypothesis into a number.