The Double-Slit Experiment

Richard Feynman, introducing quantum mechanics in his Caltech lectures of the early 1960s, began with one experiment, 'impossible, absolutely impossible, to explain in any classical way', which 'has in it the heart of quantum mechanics. In reality, it contains the only mystery.' He staged it three times with the same apparatus: a source, a wall pierced by two narrow slits, and a backstop where a movable detector counts arrivals. Only the projectile changes. Chapter 2 ran the version with light, whose bright and dark bands, called fringes, convinced Thomas Young that light is a wave.

First, bullets from a wobbly machine gun. Bullets arrive in lumps: whole bullets, one at a time, never half of one. Close slit 2 and arrivals pile into a hump behind slit 1, a probability distribution called P₁; close slit 1 and you get the hump P₂. Open both and the result is simply P₁ + P₂. Every bullet passes through one slit or the other, so the chances of arriving by either route add. No interference.

Second, water waves. Now the detector measures intensity, the energy delivered per second, which takes any value and comes in no lumps. With both slits open the intensity is not the sum of the single-slit intensities: ripples from the two slits reinforce where crest meets crest and cancel where crest meets trough, painting alternating bands. As chapter 2 showed, the wave heights, or amplitudes, add first, and the intensity is the square of the total.

Third, electrons, boiled off a hot tungsten wire. The detector clicks, and every click is a whole electron: they arrive in identical lumps, like bullets. Yet thousands of clicks with both slits open trace the bands of the water waves, and the pattern is not P₁ + P₂. Worse, at some places electrons land steadily when only one slit is open but almost never when both are. Opening a second route lowers the chance of arriving, which no bullet could do. Feynman warned that this experiment had 'never been done in just this way': the slits would have to be impossibly small. He was too pessimistic: at Tübingen in 1961, Claus Jönsson had already photographed electron fringes from two, three, four and five slits.

The decisive test sends electrons one at a time, so that none can jostle another. Pier Giorgio Merli, Gian Franco Missiroli and Giulio Pozzi filmed it in Bologna in 1974, and in 1989 Akira Tonomura and colleagues at Hitachi published the version every textbook now shows. They used an electron biprism, introduced by Gottfried Möllenstedt and Heinrich Düker in 1955: a filament thinner than a micrometre, at a small positive voltage between two grounded plates. Electron waves passing either side are bent together and overlap, just as waves from two slits do. Their 50,000-volt electrons had a wavelength of 5.4 pm.

The beam was so faint, about a thousand electrons per second, that each one crossed the 1.5-metre instrument in about twelve nanoseconds and the next typically trailed it by more than a hundred kilometres. There was almost never more than one electron inside. A detector of nearly 100 percent efficiency recorded each arrival as a single bright dot. After 10 electrons the dots look random; after 100, still random; after 3,000, faint stripes emerge; by 70,000 the fringes are unmistakable. No single dot shows a fringe. The pattern exists only in the statistics of many independent events.

Two conclusions follow. The fringes cannot come from electrons interfering with each other, because they never meet: whatever interferes belongs to each electron alone. And where any one electron lands is unpredictable; only the probabilities are fixed. In 2013 Roger Bach, Sy-Hwang Liou and Herman Batelaan of the University of Nebraska–Lincoln, with Damian Pope of the Perimeter Institute, performed Feynman's version literally: real slits 62 nm wide and 272 nm apart, electrons of 600 eV and wavelength 50 pm arriving about once a second, and a movable mask to close either slit. One slit gave a broad hump; two gave fringes, built dot by dot over about two hours.

Where do the bright fringes fall? Let d be the distance between the slit centres and L the much larger distance to the screen. A point at height y above the centre line is seen from the slits at an angle θ, and the two routes to it differ in length by d sin θ, the path difference. When that is a whole number of wavelengths, mλ with m = 0, ±1, ±2 and so on, the waves arrive in step: a bright fringe. At half-integer numbers of wavelengths they cancel: a dark fringe.

For small angles in radians, sin θ ≈ θ ≈ y/L, so the bright fringes sit at y = mλL/d, evenly spaced by Δy ≈ λL/d. Longer wavelengths and more distant screens spread the stripes; wider slit separations squeeze them. A red laser of 650 nm through slits 0.20 mm apart onto a wall 3.0 metres away gives fringes 9.75 mm apart. Each slit's width, a, adds a broad single-slit diffraction envelope with its first dark points at y ≈ ±λL/a, and roughly 2d/a − 1 fine fringes fit inside its central lobe.

For matter, the wavelength comes from chapter 6: λ = h/p. The Nebraska electrons, at 50 pm, met slits 272 nm apart, so neighbouring fringes were 1.8×10⁻⁴ radians apart, just 0.18 mm on a screen a metre away. Tonomura's fringes were only 700 nm apart where they formed, and electron lenses magnified them 2,000 times to 1.4 mm. That is why electron versions demand a biprism or exquisitely fine slits: tiny wavelengths need tiny slit separations.

Here is the rule that does the work. Quantum mechanics assigns each way an event can happen a probability amplitude: a complex number, pictured as one of chapter 2's rotating arrows, with a length and a direction called its phase. Max Born's 1926 rule, the subject of chapter 9, makes the probability the squared length of the amplitude. Call the amplitudes for reaching a point through slit 1 and through slit 2 ψ₁ and ψ₂. With one slit open, the probability density is |ψ₁|². With both open and nothing distinguishing the routes, the amplitudes add first and the sum is squared: |ψ₁ + ψ₂|².

Squaring a sum is not summing squares. Written out, |ψ₁ + ψ₂|² is |ψ₁|² + |ψ₂|² plus an interference term: twice the product of the arrow lengths times the cosine of φ, the angle between the arrows, which the path difference sets as φ = 2π d sin θ/λ. Where the arrows align, the cosine is 1 and, for equal slits, the probability is four times one slit's value; where they oppose, the cosine is −1 and the probability is zero. Averaged across a fringe the cosine vanishes, leaving twice the one-slit value: probability is redistributed, never created. So |ψ₁ + ψ₂|² ≠ |ψ₁|² + |ψ₂|², and that inequality is the double-slit experiment.

Bullets obey the plain sum because the air and light they scatter on the way record each bullet's route. Feynman reduced the subject to one switch. When an event can happen in several ways that are indistinguishable, meaning nothing in the universe records which way it went, add the amplitudes, then square. When the ways are distinguishable, even in principle, square each and add the probabilities. The stripes' contrast is the fringe visibility V: the difference between the highest and lowest probability across a fringe divided by their sum, 1 for perfect stripes and 0 for none. Unequal slits lower V without destroying it. In an ideal apparatus, only distinguishability destroys it.

Everyone wants to catch the electron in the act. Feynman's version puts a lamp behind the wall: an electron passing a slit scatters a photon, and a flash near slit 1 or slit 2 reveals its route. Every electron is seen at one slit, never both; those seen at slit 1 build P₁, those at slit 2 build P₂, and the total is P₁ + P₂. The fringes are gone. Dim the lamp and only the electrons that slip by unseen still build fringes. Lengthen the light's wavelength beyond about the slit separation, so the flashes blur too much to tell the slits apart, and the fringes return.

Feynman explained this as a kick: light sharp enough to resolve the slits jolts each electron by more than a fringe. As Niels Bohr later recalled their debates at the 1927 Solvay conference, Einstein proposed hanging the slitted screen on springs and reading its recoil to learn each particle's slit. Bohr replied that measuring the recoil so precisely would leave the screen's own position uncertain by more than a fringe, washing the pattern out. Weeks earlier, at Como, Bohr had introduced complementarity: interference and path knowledge are complementary aspects of a quantum object, each revealed by its own experimental arrangement, never both completely in the same one.

Complementarity is not all-or-nothing, and since the 1980s physicists have quantified it. In 1996 Berthold-Georg Englert proved that D² + V² ≤ 1, where V is the visibility and D, the distinguishability, measures how well the best possible reading of the detector could identify the path: 1 for certainty, 0 for a coin toss, with the chance of guessing right equal to half of (1 + D). A detector whose best reading names the correct slit 90 percent of the time has D = 0.8, so V can be at most 0.6. Partial knowledge buys partial stripes. Notice what the inequality never mentions: anyone's awareness of the result.

Is the kick really the culprit? In 1998 Stephan Dürr, Tilman Nonn and Gerhard Rempe split beams of rubidium atoms with standing waves of light and stored which-way information in each atom's internal energy state, a marker that barely touches the atom's motion. The momentum disturbance was far too small to blur the fringes, yet the fringes vanished whenever the marker held path information. The authors traced the loss not to kicks but to correlations between the marker and the atom's motion. A kick is one way to create such a correlation. It is not the essence.

The mechanism, in words: a which-path detector is any physical system that ends in one state, m₁, if the particle took slit 1, and another, m₂, if it took slit 2. Particle and marker then share an entangled state, a superposition of 'slit 1 with m₁' and 'slit 2 with m₂' in which neither part has a description of its own. On the screen, the interference term is multiplied by the overlap γ of the two marker states, whose size runs from 0 to 1. Identical markers, γ = 1, carry no information and the fringes survive; perfectly distinguishable ones, γ = 0, delete the interference term; in between, for equal slits, the visibility equals the size of γ.

None of this needs a scientist or a conscious mind, only a record, anywhere. In 2004 Lucia Hackermüller, Markus Arndt, Anton Zeilinger and colleagues in Vienna laser-heated C₇₀ fullerene molecules flying at 190 m/s into an interferometer whose paths were up to about a micrometre apart. Below about 1,000 kelvin, heat did not disturb their interference. Near 3,000 kelvin each glowing molecule emitted a few visible photons, short enough in wavelength to betray its path, and as the heating rose the visibility fell from its unheated 47 percent to 29, then 7, then 0. The photons vanished into the chamber walls, detected by no one: the environment had made the measurement. This is decoherence, the subject of chapter 22.

If information destroys the fringes, can destroying it restore them? Marlan Scully and Kai Drühl proposed such a quantum eraser in 1982; in 2000 Yoon-Ho Kim, Yanhua Shih, Scully and colleagues built the famous delayed version. Laser light through a double slit struck a crystal at two spots, A and B, either of which could turn one ultraviolet photon into an entangled pair. Each signal photon went to a movable detector, D0, playing the screen. Its twin, the idler, arrived at least 8 nanoseconds later at one of four detectors: D3 or D4 revealed whether the pair came from A or B; D1 or D2, behind a beam splitter mixing the routes, erased that information. Chance made the choice.

The results are subtle and entirely unspooky. The dots at D0 alone never show fringes, whatever the idlers later do. Sorted by their partners, the dots paired with D1 show fringes, those paired with D2 show anti-fringes shifted by half a fringe, and those paired with D3 or D4 show none. Add the D1 and D2 subsets and the stripes cancel back into a featureless hump. The later measurement moved no dot; it only determined how dots already recorded could be sorted to reveal correlations. And since sorting needs the idler records, carried by ordinary means no faster than light, no eraser can send a message, to the past or anywhere else.

John Wheeler's delayed-choice experiment, proposed in 1978, sharpens the point: let a photon enter an interferometer, then decide, while it is inside, whether to recombine the paths, testing for interference, or keep them apart, testing which path it took. In 2007 Vincent Jacques, Alain Aspect, Jean-François Roch and colleagues did it with single photons, a 48-metre interferometer and a quantum random number generator whose choice no light-speed signal could have carried to the photon's entry. Recombined paths gave 94 percent visibility; separate paths were identified with under 1 percent error. The photon did not rewrite its history. It never had a history of the form 'travelled as a wave' or 'travelled as a particle' to rewrite.