One electron at a time

Send them through singly, so that each one is alone in the apparatus. Every arrival is a single dot. The dots assemble into the interference pattern anyway — and the pattern was fixed before the first one landed.

1909 – 198917 min

By the end you should be able to

Say what a single-particle buildup shows that a beam experiment cannot
Explain why the electron is definitely alone, using the arrival spacing
Distinguish what the experiment establishes from three things it does not

There is a loophole in the double slit as described so far, and it deserves a serious hearing. Everything up to now has used a beam — enormous numbers of electrons passing through together. And electrons repel one another. So perhaps the fringes are not one electron doing something odd, but a great many electrons pushing each other around and settling into a striped arrangement collectively. That is not a foolish objection. Charged particle beams genuinely do have collective behaviour — space-charge effects, plasma oscillations, self-organisation — and it is exactly the sort of thing that can produce regular structure out of nothing very mysterious. So the objection has to be closed, not waved away.

Start again from zero and watch. Then hide the prediction, and notice that the curve was never being built.

Loading the slits…

Arrivals are sampled one at a time from the two-slit distribution. Each is a single point. The histogram accumulates; the dashed curve does not — it is the distribution being sampled from, and it is fully determined before the first arrival.

Two true statements, and the gap between them

The dashed curve in the simulation is there to make one thing visible: it is not being built up. You can calculate it in advance — from the wavelength and the slit geometry — and it will be right. So hold these two together: Where any individual electron lands is unpredictable. Not unpredictable-in-practice. There is no known procedure, and on the standard account no possible procedure, that tells you where the next one goes. The distribution they are drawn from is predictable exactly. To as many decimal places as you care to measure. Both are true. Classical physics has nothing like this pairing — a dice throw is unpredictable because you lack information about the throw, and better information would fix it. Here, more information about the electron does not appear to be available, and the theory says there is none to have. Every interpretive dispute in quantum mechanics — Copenhagen, many-worlds, pilot waves, all of it — is an argument about what to make of the gap between those two sentences.

The experiment

Building the pattern from single electrons

Akira Tonomura, Junji Endo, Tsuyoshi Matsuda and Takeshi Kawasaki (following Merli, Missiroli and Pozzi, 1974) · 1989 · Hitachi Advanced Research Laboratory, Tokyo

The question: Do interference fringes survive when only one electron is in the apparatus at a time — or are they a collective effect of many electrons in a beam?
The apparatus: A field-emission electron microscope at 50 kV with an electron biprism — a fine charged filament between two earthed plates, which splits and recombines the beam and is far more efficient than physical slits. A two-dimensional position-sensitive detector recorded individual electron arrivals, each amplified and stored with its coordinates, so the accumulation could be watched and photographed at any stage. Arrival rate about 10³ per second.

Theory predicted

If the fringes are a many-electron effect, single arrivals should be distributed smoothly with no fringes. If interference is a single-particle phenomenon, individual arrivals should be points distributed according to the two-slit intensity — random individually, fringed in aggregate.

They measured

Individual arrivals appeared as single localised dots, apparently at random. With about 100 recorded, no pattern was discernible. By 3,000 the fringes were visible; by 20,000 they were sharp; by 70,000 the accumulated image was indistinguishable from a beam exposure.

How sure could they be? The decisive parameter is not precision but isolation: at 50 kV and 10³ s⁻¹, successive electrons are about 124 km apart in an instrument 1.5 m long. Each electron’s 12 ns flight is separated from the next by a millisecond — a margin of roughly 8 × 10⁴, which removes any possibility of electron–electron interaction.

Why it mattered

The interference is a property of each electron individually, not of the beam. G.I. Taylor had shown the equivalent for light in 1909, attenuating a source with smoked glass and exposing a plate for three months. Merli, Missiroli and Pozzi achieved single-electron buildup in Bologna in 1974, and were largely overlooked outside Italy. In 2002, readers of Physics World voted this the most beautiful experiment in physics, ahead of Galileo, Millikan, Newton and Rutherford.

It is worth pausing on that vote. In 2002 Physics World asked its readers to name the most beautiful experiment in physics. The single-electron double slit came first — ahead of Galileo's falling bodies, Millikan's oil drop, Newton's decomposition of sunlight with a prism, and Rutherford's gold foil. Which is a remarkable result for an experiment whose entire content is: dots appear one at a time, and the dots make stripes. There is no dramatic apparatus, no vast energy, no expedition. What it has is that the result is completely unambiguous and completely impossible to accommodate, and you can state both the setup and the finding in one sentence each.

You might think

The electron splits in two, goes through both slits, and reassembles.

Actually

Nobody has ever detected half an electron. Every arrival is one whole electron carrying the full elementary charge, at one place, registering one count. Charge is quantised and conserved; there is no such thing as half of it. Whatever crosses the apparatus does not divide — and if you place detectors at the slits, you find the electron at exactly one of them, every time, never both and never a fraction at each. The experiment does not license the splitting picture; it rules it out.

You might think

The electron is a small wave packet that spreads out, passes through both slits, and then collapses into a dot.

Actually

This is the most seductive of the wrong pictures, because it seems to explain everything. Two problems. First, it is a story about a physical object with a shape, and the C₆₀ result from two lessons ago rules that out directly: the de Broglie wavelength there was about 250 times smaller than the molecule, so there is nothing to spread with that wavelength. Second, the mathematics does not describe a lump in space — for two particles it lives in six dimensions, not three, and for a thousand particles in three thousand. Whatever the wavefunction is, it is not a shape that things have. (And the third common claim, that electrons interfere with one another, is precisely what this experiment was built to exclude.)

1909Taylor obtains diffraction fringes from an extremely faint source, with a three-month exposure.
1926Born gives the probabilistic reading of the wavefunction, in a footnote added in proof.
1961Jönsson performs the electron two-slit experiment with a beam.
1974Merli, Missiroli and Pozzi record single-electron buildup in Bologna. Largely overlooked.
1989Tonomura’s group produces the definitive images, 10 to 70,000 electrons.
2002Physics World readers vote it the most beautiful experiment in physics.

Where this leaves us

We have a rule that works and no account of what it describes. Amplitudes add along paths; the square gives probability; it never fails; nobody can say what the amplitude is. And there is one obvious thing to try. If the amplitude behaves like a wave — adding, cancelling, interfering — then write down a wave equation for it. Two people did that within about eighteen months of each other, in mathematical languages so different that neither initially recognised the other's work as physics. Heisenberg, aged 23 and recovering from hay fever on the island of Helgoland, produced arrays of numbers obeying strange multiplication rules, deliberately containing nothing visualisable at all — no orbits, no trajectories, no pictures. He was not sure what he had. Schrödinger, aged 38 and rather less ascetic about it, produced a wave equation over Christmas 1925 at a Swiss resort, in company that his biographers have never quite managed to identify. Each found the other's approach repellent, and said so in print. They were the same theory.