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§ Phase vii · Wavepacket Dynamics · Chapter 02

The double slit, revisited

Two narrow slits, a screen, and a beam so faint that the particles arrive one by one. Each electron lands at a single point on the detector, a hard particle event. But wait until ten thousand have landed and the points organize themselves into bright and dark bands that no classical particle picture can explain. This is the experiment Richard Feynman called the only mystery, and it deserves a second look now that we know how to read a wavepacket.

In the spring of 1801, a 28-year-old English polymath named Thomas Young stood in front of the Royal Society in London and proposed something that most of the audience found mildly heretical. Isaac Newton, dead three quarters of a century but still the patron saint of British physics, had taught that light was made of tiny corpuscles flying in straight lines. Young, who was a physician by training and could already read thirteen languages by the time he was fourteen, suggested otherwise. Light, he said, was a wave. Two waves passing through the same region of space could either reinforce each other or cancel out, the same way two ripples on a pond can meet and either bulge higher or flatten to nothing. To prove the point he aimed a pinhole of sunlight at a card with two slits cut in it and let what came through fall on a wall. There, in alternating stripes of bright and dark, was the signature of interference.

Young’s stripes were the first clear evidence of the wave nature of light. They were not, in 1801, fully convincing to everyone. Newton’s reputation cast a long shadow, and Young’s writing was famously dense (one reviewer called it “destitute of every species of merit”). The case took decades to settle. By the time Fresnel’s mathematical theory of diffraction landed in Paris in 1819 and Maxwell’s equations of electromagnetism arrived in 1865, the wave theory had won. Light was a vibration of the electromagnetic field, and the two-slit pattern was its fingerprint.

Then physics dropped a complication. In 1905, as we saw in Phase i, Einstein argued that light was also a stream of discrete energy packets, and by the 1920s, after de Broglie and the matter-wave hypothesis, it was clear the same trick ran the other way. If light could behave like particles, then particles could behave like waves. In 1927 Clinton Davisson and Lester Germer at Bell Labs, almost by accident, scattered electrons off a nickel crystal and found the same diffraction pattern X-rays produced from the same surface. Electrons, undeniably particles in J. J. Thomson’s cathode-ray tubes, were also waves. The double-slit experiment was no longer a story about light. It was a story about everything.

The natural question, once electrons had been added to the guest list, was whether each electron passes through both slits like a wave or through one slit like a particle. The textbooks gave a tidy answer (both, somehow, until you measure), but no one had yet seen the experiment done with the beam turned down to a literal trickle. The technology did not exist. To send electrons one at a time meant a beam so dim that the next electron would not leave its filament until the previous one had already crashed into the screen, and you needed a detector sensitive enough to record a single hit. For most of the 20th century, that combination of patience and instrumentation was not available.

The first move came in 1961. A young German physicist named Claus Jönsson, working in Tübingen for his doctoral thesis, did what no one had managed since Young: he built a clean two-slit interferometer for electrons. Jönsson cut slits about half a micrometer wide and a few micrometers apart in a thin copper foil, fired a coherent electron beam at them, and photographed the fringes that formed on a fluorescent screen downstream. The pattern was the predicted one. The wavelength worked out to the de Broglie value for his beam. Jönsson’s experiment was a tour de force of low-budget craftsmanship (the slits were cut with a galvanic deposition process he developed himself, the apparatus assembled in a basement at the Physikalisches Institut). It settled, for any remaining skeptics, that electrons obey the same diffraction formula as light.

But Jönsson’s beam, while coherent, was not single. Many electrons traversed the apparatus at any moment. The pattern that formed on the screen could, in principle, have been due to electrons interfering with each other, the way two laser beams in a hall interfere. To rule that out you had to do something more delicate. You had to send the electrons one at a time, with a long gap between them, and watch the pattern build up dot by dot. If the pattern still appeared, then each electron must have somehow interfered with itself, and the wave-particle question would be sharpened to the point where it could not be dodged.

The decisive single-electron experiment was done in 1989 in Hatoyama, Japan, by a team at the Hitachi Advanced Research Laboratory led by Akira Tonomura. Tonomura was an extraordinary instrument builder. He had spent the 1970s developing electron holography, a technique that needs an electron source so coherent it could only be made by field emission from a sharpened tungsten tip in ultra-high vacuum. His source spat out roughly a thousand electrons per second through an electron biprism (Möllenstedt’s old trick again, doing the work of the two slits), into a detector that could register single electrons as bright points. The mean separation between successive electrons in the apparatus was about 150 kilometers, which is to say that when one electron entered the interferometer the previous one had already long since hit the detector and a third one was still inside its filament waiting to be born. There was no possibility of two electrons being inside the machine at the same time.

Tonomura’s team filmed the screen. The result is the most quoted moving image in the foundational history of quantum mechanics. After ten electron events the screen looks like noise. After a hundred, still noise. After a thousand, a faint banding starts to appear in the statistics. After ten thousand, the bands have sharpened into the unmistakable pattern of interference, with alternating columns of high and low density spaced exactly as the de Broglie wavelength and the slit geometry demand. Each electron has landed at a single, definite point. None of them have split, none of them have spread out across the screen. And yet collectively their landings draw the picture of a wave.

This is the experimental fact that Feynman, in the third chapter of the first volume of his Lectures on Physics, said “contains the only mystery” of quantum mechanics. “It cannot be explained in any classical way,” he wrote, “and has in it the heart of quantum mechanics. In reality, it contains the only mystery.” Every other so-called paradox in the subject (Schrödinger’s cat, Bell inequalities, the EPR pair, the delayed-choice eraser) is really this same fact dressed up in fancier clothes. A particle event is registered at one place. A pattern of wave events is recorded in the ensemble. The two pictures are both correct, and neither is the whole story.

Why does the pattern emerge? The clean answer, written in the wavepacket language of Phase vii, is that each electron is described by a wavefunction ψ(r, t) that, when it arrives at the two slits, splits into two cylindrical waves emanating from each slit. The two waves propagate together to the screen, where their amplitudes add. The probability of detection at any point on the screen is the modulus squared of the summed amplitude, |ψ₁ + ψ₂|², not the sum of the separate probabilities |ψ₁|² + |ψ₂|². The cross term ψ₁*ψ₂ + ψ₂*ψ₁ is the interference. It oscillates with position because the two amplitudes pick up different phases on their way to different points on the screen, and that oscillation is the fringe pattern. The arrival is then sampled randomly from this probability distribution. One sample is one dot; ten thousand samples paint the picture.

The wavepacket picture, however, leaves something dangling. The wave passes through both slits, but the dot lands at one place. What chooses the place? The Born rule, which we met in Chapter 09 of Phase iii, says only that the place is sampled from the probability distribution |ψ|². The rule does not say what physical process pulls a particular sample out of the distribution on any given trial. This is the famous measurement problem, and the double slit dramatizes it sharper than any other experiment because the wave and the particle aspects of the same single event can be seen with the eye, in the same image, at the same time.

Now turn a knob and ruin the magic. Suppose you place a small detector at one of the slits, something that will tell you whether the electron went through slit A or slit B on any given pass. Maybe a very faint light, photons that scatter off the passing electron and reveal its location. Run the experiment again, this time recording which-path information on every electron. The dots accumulate as before, but the bands do not form. The pattern reverts to two smeared blobs, the sum of the patterns you would get from each slit on its own, with no interference whatsoever.

This is not a mechanical failure. The detector does not bump the electron off course in any classical sense. It is a fact about quantum amplitudes. If a measurement somewhere in the apparatus reliably distinguishes path A from path B, then the two amplitudes cannot interfere because the system as a whole (electron plus detector plus environment) ends up in two orthogonal states, one tagged “A” and one tagged “B.” When you compute |ψ_A|² + |ψ_B|² as separate contributions there is no cross term. The interference vanishes the instant the which-path information leaks into the world, even if no human ever looks at the detector. Reading the record is irrelevant. The mere fact of the record existing, somewhere physical, is what kills the pattern. This is why even an inadvertent scattered photon can erase the fringes, and why Tonomura’s team had to work in ultra-high vacuum: any stray gas molecule colliding with the electron during transit would have written a faint which-path entry into the universe and degraded the visibility.

I(θ) ∝ cos²( π d sin θ / λ )

E(θ) = sinc²( π b sin θ / λ )    where sinc(x) = sin(x)/x

I(θ) ∝ cos²( π d sin θ / λ ) · sinc²( π b sin θ / λ )

I(θ) ∝ ⟨ψ₁|ψ₁⟩ + ⟨ψ₂|ψ₂⟩ + 2 Re⟨A|B⟩ Re(ψ₁* ψ₂)

There is one more thing to notice. The double-slit experiment, as Tonomura and his successors have run it, does not depend on the particle being an electron. The same fringes appear when you fire single neutrons through wide slits. They appear with helium atoms (Carnal and Mlynek, 1991). They appear with C60 buckyballs, the 720-atom soccer-ball molecule (Markus Arndt and Anton Zeilinger’s group in Vienna, 1999). They have been demonstrated with molecules of 2000 atoms and tens of thousands of atomic mass units, well into the territory where most chemists would call the object a small protein. In every case, the fringe spacing is λL/d with the de Broglie wavelength of the object, and in every case the fringes vanish if the apparatus contains anything (a hot gas, a stray photon, a thermal bath warm enough to scatter infrared off the molecule’s vibrations) that could in principle record which slit the object went through. The mystery does not get smaller as the object gets bigger. It just gets harder to see.

This is the experimental reason that quantum mechanics is universal. There is no scale at which it stops working. There is only the engineering threshold below which the which-path information stays bottled up in the wavefunction and above which it leaks out. Build a clean enough apparatus, and the fringes appear for anything that has a wavelength. The world we walk around in feels classical because it is full of things that radiate, breathe, and shed photons. Take those things into a vacuum and chill them down and they remember they are waves.

Feynman’s remark about the only mystery is sometimes read as a closing of the case (the mystery is here, here it stays, let’s move on). It is more useful as an invitation. The two-slit pattern is the simplest test bed for every interpretive move in the subject. The Copenhagen view sees the dot as the collapse of the wavefunction onto an eigenstate of position; the Many-Worlds view sees it as the splitting of the universal wavefunction into a branch where the dot is there and another where it is somewhere else; the de Broglie-Bohm pilot wave view sees a real particle steered by a real wave; quantum Bayesianism sees the dot as the answer to a question and the wavefunction as the questioner’s prior. None of these stories changes the prediction for the pattern; they only change the language we use when we explain it to each other after dinner. The double slit gives us the same fringes regardless of what we believe about them. That is its enduring power.

We will return to it. In the next chapter we put a particle in a parabolic well and watch the energy ladder of the harmonic oscillator emerge, the same ladder Planck guessed in his interpolation and the same one that quantizes the electromagnetic field itself. The two-slit lessons travel with us. Every superposition in this book, every interference effect we will see in molecules and condensed matter, is a relative of the experiment Young did with a card and a pinhole in 1801. Keep the picture in your head: a single dot, then thousands of dots, organizing themselves into a wave they each individually never had.