Physics · Module 21
Module 12 built the wavelength of light into two slits and watched the fringes it predicts appear. Module 18 explained what was doing the waving. This module takes that settled, twice-confirmed picture and breaks it: light turns out to arrive in lumps, the way particles do. Then it breaks the opposite picture as well — electrons, which everyone agreed were particles, turn out to interfere with themselves.
Let us be clear about how strong the evidence was, because otherwise what follows is just a story about physicists being surprised.
In Module 12 you sent light through two narrow slits and got a set of evenly spaced bright and dark bands on a screen. The spacing of the bands obeyed
Bands of darkness are the point. Two beams of light arriving at the same place and producing nothing is not something particles do. If light were a stream of bullets, opening a second slit could only ever give you more bullets somewhere; it could never give you fewer. Dark fringes are cancellation, cancellation requires a crest meeting a trough, and only waves have crests and troughs. Module 12 also gave you single-slit diffraction, the Rayleigh resolution limit, thin-film colours and polarisation, all of which need waves and none of which a particle picture can touch.
In Module 18, Maxwell told you what was waving: an electric field and a magnetic field, taking turns, travelling at 1/√(µ0ε0) — µ0 the permeability of free space and ε0 the permittivity, both defined back in Module 18. Not a hypothesis but a derivation — and the speed it predicted agreed, to the few per cent Fizeau's 1849 cogwheel could reach, with the already measured speed of light. That match was the argument.
By 1900 this was as settled as physics ever gets. And then two experiments refused to behave.
The first crack was not the photoelectric effect but something more mundane: the colour of hot things. Heat an iron bar and it glows red, then orange, then white. Classical physics, applied to the light rattling inside a hot closed box — a furnace with a small peephole is the standard version — predicts that the intensity should rise without limit as you go to shorter wavelengths — that every warm object should be blasting out ultraviolet and X-rays. It does not. The prediction was so absurd it earned the name "the ultraviolet catastrophe".
In 1900 Max Planck found a way to fix the formula. He assumed that a vibrating charge cannot hold just any amount of energy. It can hold hf, or twice hf, or three times hf, and nothing in between — whole-number multiples of one small step. Here f is the frequency of the vibration and h is a new constant of nature. With that single assumption the formula matched the measured curve exactly. He regarded it as a mathematical trick and spent years trying to get rid of it. He never could. That constant, h = 6.626 × 10−34 joule-seconds, sets the scale of everything quantum. Wherever h shows up in a formula, the world turns out to be lumpy; every equation in this module and the next has an h in it somewhere. And you are about to measure it yourself.
The experiment is simple enough to describe in a sentence. Shine light on a clean metal plate in a vacuum. Electrons are knocked out. Collect them on a second plate and you have a current you can measure. Now put a voltage across the gap the unhelpful way round, with the collector made negative so that it pushes the arriving electrons back. Only electrons with enough kinetic energy to climb against that voltage still get across. Turn it up until even the fastest fail and the current just stops. Call that the stopping voltage, Vs. An electron of charge e loses energy eVs in climbing that voltage, so the fastest electrons must have set out with exactly that much: eVs = KEmax.
Now, what should happen? Take the wave picture seriously and predict.
You shine bright red light — as bright as you like, a laser if you want — on a zinc plate. No electrons come off at all. Now you switch to a feeble, barely visible ultraviolet lamp, a thousand times dimmer.
What happens?
Immediately — within nanoseconds — and with energies the red light could not produce however bright you made it. Frequency decides whether and how fast. Intensity decides only how many.
Play with that bench before reading on, and specifically do these three things.
One. Set zinc and 500 nm. Nothing. Now push the intensity to maximum. Still nothing — not a reduced current, no current. Now step the wavelength down past about 288 nm and electrons appear the instant you cross it, even at the lowest intensity.
Two. With electrons flowing, change the intensity. The number of electrons changes in proportion — the whole I–V curve scales up and down. But the point where the curve meets zero, the stopping voltage, does not move at all. Brightness has no say in how fast the electrons are.
Three. Now change the wavelength instead. The stopping voltage moves immediately, and it moves in a straight line against frequency.
In 1905 — the same year as relativity, in the same volume of the same journal — Einstein took Planck's mathematical trick and said it was not a trick. Light is not delivered smoothly. It arrives in indivisible lumps, each carrying energy
Then everything falls out at once.
That last point is the one that turned a clever idea into a measurement. Robert Millikan spent ten years trying to disprove Einstein's photon — he thought the idea was reckless — and in 1916 published the most careful photoelectric data anyone had taken. The plot was a perfect straight line. Its slope gave h to within half a percent of Planck's value from an entirely unrelated experiment about hot objects. Millikan reported this while still saying he did not believe the theory behind it.
Do the same experiment yourself. The next bench runs the photoelectric measurement at a series of frequencies, finds the stopping voltage each time by locating where the simulated current dies, plots the points, fits a straight line through them, and reads h off the slope.
Change the metal and run it again. The line shifts up or down — different work function — but the slope stays the same, and the constant you extract from it is the same number every time. That is the real content of the photon idea: h is a property of the universe, not of zinc.
A 100 W bulb emits roughly 1020 photons every second. Your eye, on a dark night, can register a handful. At any ordinary brightness the lumpiness of light is as invisible as the graininess of sand is from an aeroplane: the beach looks perfectly smooth. The comparison is not exact, and the difference is worth having. Sand looks smooth from up there because your eye cannot resolve one grain. Light looks smooth because so many photons arrive every second that the number landing anywhere hardly wobbles at all — which is why you catch the lumpiness by starving a camera of light, not by looking harder. Young could not have seen photons in 1801 with the equipment he had; his fringes were made by billions of them arriving together, and billions of anything look continuous.
You can, however, get very close to seeing them. A modern camera sensor at high ISO in near-darkness produces a speckled, grainy image, and a good part of that grain is genuine photon shot noise — the statistical scatter in how many photons happened to land on each pixel. When your phone's night mode looks noisy, you are looking at the lumpiness of light.
Now the situation is uncomfortable. Light interferes, so it is a wave. Light knocks electrons out one lump at a time, so it is a particle. Both experiments are correct and neither is going away.
In 1924 a French graduate student named Louis de Broglie wrote a thesis containing an idea so bold that his examiners did not know what to do with it and sent it to Einstein for an opinion. His reasoning went roughly like this.
Notice what the third ledger box is really saying. Nothing is "a wave" or "a particle" by nature. What matters is the ratio of its de Broglie wavelength to the size of whatever it is being sent through. Electrons in a crystal: ratio about 1, so wave behaviour is unmissable. A cricket ball going through a doorway: ratio 10−34, so it goes straight through and lands where Module 1 said it would. Newtonian mechanics is not wrong; it is what quantum mechanics looks like when λ is negligible, exactly as Newton's dynamics is what relativity looks like when v/c is negligible.
An electron microscope can resolve individual atoms; the best optical microscope cannot resolve anything smaller than about 200 nanometres, no matter how good its lenses are.
Why?
The Rayleigh criterion from Module 12 — roughly, you cannot resolve detail much finer than the wavelength you are working with — applies to electron waves exactly as it does to light. Visible light is stuck at 400–700 nm, and no lens engineering can beat that. But an electron accelerated through 100 000 volts has a de Broglie wavelength of about 0.004 nm — a hundred thousand times shorter than green light — so the same criterion allows a hundred thousand times finer detail in principle. In practice, magnetic lenses are far cruder than glass ones and cannot be corrected for spherical aberration the way glass lenses can, so the best electron microscopes reach about 0.05 nm rather than 0.004 nm. That is still smaller than an atom, which is why the pictures work. Every atomic-resolution image you have ever seen was taken with a wave whose existence nobody suspected before 1924.
Notice that the electron microscope is not a workaround for the wave limit. It uses the wave limit. The electron is being treated as a wave from beginning to end — focused by magnetic lenses, diffracted by the specimen, interfering on the detector. Which raises the obvious question: if it is a wave, in what sense is it also a particle?
Module 12, in the box beside the two-slit bench, told you that this experiment has been done with a source so faint that only one photon was in the apparatus at a time — and that the fringes still built up, dot by dot. Here is that experiment done with electrons, which is easier to arrange and stranger to watch.
Take the double slit. Replace the lamp with an electron gun, turned down until electrons leave one at a time — so far apart that the previous one has hit the screen and been recorded before the next one is emitted. Put a detector screen behind the slits that registers each arrival as a single dot at a single place.
Each electron arrives as a point. Never half an electron; never a smeared-out patch. One dot, one place, every time. So far, entirely particle-like.
Now let it run for a few hours, so that tens of thousands of individual electrons have gone through one at a time, and look at the accumulated pattern of dots.
What is on the screen?
Fringes. The same fringes, with the same spacing λD/d, built up one dot at a time by electrons that never met each other.
Sit with what that requires. Each electron arrives at a definite point, so it is not a spread-out wave hitting the whole screen at once. But where it is allowed to arrive depends on both slits being open. Here is how you know. Cover one slit and let the dots pile up; pick a spot that has collected plenty of them. Now uncover the second slit, clear the screen, and fire the same number again. That same spot collects almost none. You gave the electrons an extra way in, and it made that place harder to reach. Whatever an electron is, each one went through in a way that knew about both slits.
Watch the order in which the picture appears. The first fifty dots look like noise — you could not tell them from a random scatter. At five hundred you begin to suspect stripes. At ten thousand the fringes are unmistakable, and the dark bands are genuinely dark: places where thousands of electrons could have landed and essentially none did.
No individual electron ever "made a fringe". Each one landed somewhere, unpredictably. What was predictable — precisely, to as many decimals as you care to measure — was the probability of landing at each place. And that probability distribution is exactly the intensity pattern of a wave of wavelength h/p.
That is the actual content of quantum mechanics, and it is worth stating plainly:
What travels is a wave of possibility. What arrives is a particle. The wave tells you the odds; it does not tell you the outcome.The whole thing, more or less
By now you must want to ask which slit the electron actually went through. It is the obvious question. It is a perfectly sensible question about any object you have ever met. And the experiment has an answer for you, which is worse than a refusal.
Put a detector at the slits — a small device that registers, without stopping the electron, which one it passed. Now you know the path of every electron, and you can still watch the dots accumulate.
You switch on the which-slit detector and run the experiment again, exactly as before, one electron at a time.
What appears on the screen?
The fringes vanish. Completely. What is left is one broad hump, not two. Feynman draws two because his slits are far apart compared with the spread from each one; here the slits are 400 nm apart while each slit alone spreads the beam over about a millimetre, so the two humps land on top of each other and add to a single smooth band. Turn the detector off and they come back. The reason is not that the detector disturbs the electron — it is that any detector creates a record of which slit was used, and fringes exist only when the two routes are indistinguishable. Make one route visible and the interference goes. How precisely? If V measures the fringe contrast (1 for full stripes, 0 for none) and D measures how reliably the detector names the slit (1 for certain, 0 for useless), then V2 + D2 can never exceed 1. Perfect path knowledge forces V = 0; perfect fringes force D = 0. No design beats this bound.
The partial setting is the interesting one. Detect a quarter of the electrons and you do not get a quarter of the screen ruined; you get fringes of reduced contrast everywhere, because the detected quarter fills in the dark bands. Knowledge and interference trade off continuously. You can have a bit of each — that is exactly what the middle of the slider shows you — but more of one always means less of the other.
Feynman was blunt about what this means:
We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. Richard Feynman, Lectures on Physics, Volume III, Chapter 1
He meant it literally. Everything else in quantum mechanics — atoms, chemistry, lasers, semiconductors, all of Modules 22 and 23, and Module 25 — is worked out from this one behaviour. There is a rule that tells you exactly what to expect, and the Deeper box below gives it in four lines. What there is not is any account of why that rule and not some other. Nobody has one. This is the floor. Asking which slit the electron went through, when nothing was there to see, is like asking what is north of the North Pole: every word is ordinary, and there is still nothing for the question to point at. Put a detector at the slits and the question does get an answer — but then, as you have just watched, the fringes are gone. That is the trap. The question only has an answer in an experiment that has already destroyed the thing you were asking about. (This is where the North Pole comparison breaks down: that question can never be answered, while this one can, at a price you may not want to pay.)
Here is the rule, and it is short. For each way an event could happen, quantum mechanics hands you an arrow: it has a length and a direction, like an arrow drawn on paper. That arrow is called the amplitude for that way. To find the probability of the event, add the arrows for all the ways you cannot tell apart — tip to tail, the way you add two forces — and then square the length of the arrow you are left with.
Two slits, no detector: the two routes cannot be told apart, so lay the two arrows tip to tail and square the length of the total. Where the arrows point the same way the total is long, and you get a bright fringe. Where they point opposite ways they nearly cancel, and that is your dark fringe. With a detector, the routes can be told apart, so square each arrow's length first and add the two numbers afterwards. Two squares are both positive — they can never cancel — so no dark fringes — and here, with the slits this close together, the two humps overlap into a single plain band.
The entire difference between the two experiments is whether you add before or after squaring. That is the whole of it, and it is what every quantum calculation in the world is doing underneath.
Take a torch or a distant streetlight, and hold two fingers almost touching in front of one eye so that you are squinting through a narrow gap. You will see fine dark lines running along the gap: that is diffraction, Module 12, and you can do it right now. For proper fringes, cut two fine parallel slits a fraction of a millimetre apart in a piece of foil with a scalpel and look at a distant white LED through them.
Then do the arithmetic that makes it strange. Estimate how many photons per second from that LED get through your slits — a few times 1012 is a fair guess. Now imagine turning it down by a factor of 1013, so that one photon crosses the apparatus at a time. Every experiment ever done says the same fringes build up dot by dot. The pattern you are looking at right now, in an ordinary bright room, is not made by photons bumping into each other. It is made by each photon separately, going through both slits.
| Idea | What it says |
|---|---|
| The problem | Module 12 proved light is a wave. The photoelectric effect proves it arrives in lumps. Both stand. |
| Photons | E = hf, with h = 6.626 × 10−34 J s. Frequency sets the energy of each lump; intensity sets how many arrive. |
| Photoelectric equation | KEmax = hf − φ. Explains the threshold, the independence from brightness, and the absence of any delay. Plot Vs against f: a straight line of slope h/e. |
| de Broglie | λ = h/p, for everything. Negligible for a cricket ball; the size of an atom for an electron at a few tens of volts. Confirmed by Davisson and Germer. |
| One at a time | Electrons sent singly still build interference fringes. Each arrives as a point; the pattern lives in the probabilities. |
| Which slit | Detecting the path reduces fringe contrast. Path information V and visibility D trade off under V² + D² ≤ 1: no design reaches full fringes and full path knowledge at the same time. |
| The rule | Add amplitudes over indistinguishable routes, then square. Distinguishable routes: square first, then add. Every quantum prediction is that sentence. |
Next: what happens when you trap one of these waves inside an atom, where — exactly as with the string in Module 7 — only certain wavelengths fit.