Episode 30 - Uncertainty

Tomlin Hawthorne

Imagine the purest musical note you can. A single clean tone, like a tuning fork, at a definite pitch. Let's say it's the A above middle C.

Tomlin Hawthorne

Now I'm going to play you that note for one thousandth of a second. Just the thinnest possible slice of it.

Tomlin Hawthorne

What will you hear?

Tomlin Hawthorne

A click. A tiny tick, like a fingernail on a table. And if I asked you what note it was, you couldn't tell me. Not because your ears are poor. Because a sound that short doesn't have a pitch. It hasn't lasted long enough to have one.

Tomlin Hawthorne

To have a definite pitch, a sound has to go on for a while. And if it goes on for a while, you can't say exactly when it happened. A sound can have a sharp when, or a sharp what note. It can't have both.

Tomlin Hawthorne

That's not quantum mechanics. That's just sound. But it is, almost exactly, the most famous principle in quantum mechanics. By the end of today I hope you'll see that the uncertainty principle isn't a statement about how clumsy our measurements are. It's a statement about what it means to be a wave.

Tomlin Hawthorne

Last time, if you tried it, you held a hair in a laser beam and saw a row of bright dashes on the wall with dark gaps between. Light going round the two sides of the hair, adding in some places and cancelling in others.

Tomlin Hawthorne

And I showed you the double slit. Single electrons, sent one at a time, each landing as a dot, and the dots building up a pattern of stripes that only a wave could make. Find out which slit each one went through, and the stripes vanish.

Tomlin Hawthorne

So an electron on its way to the screen doesn't have a definite path. Today's question follows straight on. What else doesn't it have? And is the fuzziness in our knowledge, or in the world?

Tomlin Hawthorne

I want to begin in a theatre, with the best play I know about science.

Tomlin Hawthorne

Michael Frayn's Copenhagen opened in London in 1998. It has three characters, a bare stage, and no props. The characters are all dead, and they know it. They are Niels Bohr, his wife Margrethe, and Werner Heisenberg.

Tomlin Hawthorne

The play circles one real event. In September 1941, in the middle of the war, Heisenberg travelled from Germany to occupied Copenhagen and went to see Bohr. Heisenberg was by then the leading physicist in Germany and the head of its atomic research. Bohr was his old teacher, practically his second father. He was also half Jewish, and living under German occupation.

Tomlin Hawthorne

They had dinner. Then the two men went for a walk, as they had hundreds of times in the twenties. It lasted perhaps ten minutes. Bohr came back angry. The friendship never recovered.

Tomlin Hawthorne

Nobody knows what was said. The two of them gave different accounts afterwards. Did Heisenberg come to find out what the Allies were doing? To recruit Bohr? To warn him? To ask whether a physicist has the right to work on such a weapon? To suggest that both sides quietly not build it? Was he boasting, or pleading, or fishing?

Tomlin Hawthorne

Frayn's play runs the meeting three times. Each time, the dead try to reconstruct what Heisenberg meant. Each time they get a different answer. And the point, which the play makes with great delicacy, is that it may not be recoverable, even by Heisenberg. That his intentions were not in a definite state. That when you try to pin down a person's motive, you get one clear thing and lose another.

Tomlin Hawthorne

It's a play that takes its shape from a piece of physics. So let's get the physics.

Tomlin Hawthorne

Werner Heisenberg was a prodigy. Born in 1901 in Bavaria, the son of a professor of Greek. Blond, boyish, ferociously competitive, a first-rate pianist, a hiker and skier. He nearly failed his doctorate, because he couldn't answer simple questions about how a microscope works, which is an irony you'll appreciate in a minute.

Tomlin Hawthorne

In June 1925, aged twenty-three, he was struck down by hay fever so severe that his face swelled up, and he fled to a tiny island in the North Sea called Helgoland. It's a lump of red rock with hardly a tree or a flower on it, which was the point. He could breathe.

Tomlin Hawthorne

And there, alone, walking and swimming and reading Goethe, he tried something radical. He'd been struggling, like everyone, with Bohr's atom, with its little orbits that plainly weren't right. And he decided to throw the orbits away. To stop talking about where the electron is and what path it follows inside the atom, since nobody had ever observed any such thing. To build a theory using only what you can observe: the light that comes out.

Tomlin Hawthorne

He worked through one night until about three in the morning, checking that his scheme hung together. It did. He wrote that he was far too excited to sleep, so he walked to the southern tip of the island, climbed a rock that jutted out into the sea, and waited for the sun to come up.

Tomlin Hawthorne

That was the first version of quantum mechanics. Schrödinger's wave equation came a few months later, looking completely different, and for a while there were two rival theories and a good deal of bad feeling, until it was shown that they were the same theory in two languages.

Tomlin Hawthorne

But today's subject is what Heisenberg found two years after the island, in Copenhagen, in February 1927.

Tomlin Hawthorne

Let me get one word in place first. Superposition.

Tomlin Hawthorne

Go back to the electron heading for the two slits. Last time we saw that you can't say it went through the left slit, and you can't say it went through the right. Physicists say it's in a superposition of the two. A blend. Left-and-right, in a way that's neither one, nor the other, nor simply both.

Tomlin Hawthorne

And I need to be careful here, because there's a mild version of this that's easy to accept and wrong. The mild version says: it went through one slit, we just don't know which. That's ignorance. And ignorance can't be the answer, because ignorance gives you two heaps. If each electron really went through one slit and we merely didn't know, the pattern would be the bullet pattern. It isn't. The stripes are there. So it's not that we don't know which. It's that there's no which.

Tomlin Hawthorne

That's superposition. It's the single new idea in quantum mechanics, and everything else follows.

Tomlin Hawthorne

Now, Heisenberg's principle.

Tomlin Hawthorne

It says this. There are certain pairs of properties of a particle that can't both be sharp at once. The best-known pair is position, where it is, and momentum, which is roughly how fast it's going and in what direction. The more precisely a particle has a position, the less precisely it has a momentum. And the other way round. Squeeze one, and the other spreads.

Tomlin Hawthorne

And there's a floor. If you multiply the fuzziness in position by the fuzziness in momentum, the answer can never be smaller than a certain fixed amount. And that amount is set by Planck's constant, that tiny number from episode twenty-seven, the size of the grain.

Tomlin Hawthorne

Now, why? This is where I need the click.

Tomlin Hawthorne

Remember de Broglie. A particle is associated with a wave. And he gave the rule for it: the momentum of the particle corresponds to the wavelength of its wave. A short, tightly spaced wave means high momentum. A long, lazy one means low.

Tomlin Hawthorne

So think about what it would mean for an electron to have one exact momentum. Its wave would have one exact wavelength. Like a pure musical note. A perfect, even ripple.

Tomlin Hawthorne

But a perfect, even ripple goes on for ever. It has no beginning and no end. It's the same everywhere. So where is the electron? Nowhere in particular. Everywhere equally. It has a sharp momentum and no position at all.

Tomlin Hawthorne

Now go the other way. Suppose you want the electron to be somewhere. In one small place. Then its wave has to be bunched up there: a short blip, and nothing elsewhere.

Tomlin Hawthorne

And what's the wavelength of a blip? It doesn't have one. A single bump isn't a ripple. To build a short blip, you have to pile together a whole range of different wavelengths, so that they add up in one spot and cancel everywhere else. That's Fourier, from episode ten. Many wavelengths means many momenta. So an electron in a definite place has a very indefinite momentum.

Tomlin Hawthorne

That's the click. A long note has a pitch and no moment. A click has a moment and no pitch.

Tomlin Hawthorne

So here's what I most want you to take from this episode. The uncertainty principle isn't an extra law, bolted on. It's what you get, automatically, the moment you say a particle is described by a wave. Any wave. It's as true of a sound in a concert hall as of an electron. The only thing quantum mechanics adds is the claim that particles are like that, and a number, Planck's constant, to say how much.

Tomlin Hawthorne

Now I have to undo something, and it's Heisenberg's own fault.

Tomlin Hawthorne

When he first published, he explained the principle with a thought experiment. Suppose you want to see where an electron is. You have to shine light on it. To see it sharply, you need light of very short wavelength. But short wavelength means high-energy photons. And when one of those hits the electron, it gives it a kick. So in finding out where it is, you've disturbed how fast it's going.

Tomlin Hawthorne

That's vivid, and it's what most people still think the principle means. That measuring is clumsy. That you can't look without jostling.

Tomlin Hawthorne

Bohr read the draft and told him it was wrong. They argued about it for days, so fiercely that Heisenberg, by his own account, ended up in tears. And Bohr was right. Notice what the microscope story assumes. It assumes the electron had an exact position and an exact momentum all along, and the only problem is that our measurement messed one of them up. That the fuzziness is in our knowledge.

Tomlin Hawthorne

And that's exactly the mild version again. It's ignorance. And we've just seen that it can't be ignorance, because of the stripes.

Tomlin Hawthorne

The real claim is harder. The electron doesn't have a sharp position and a sharp momentum at the same time. There's nothing there for the measurement to disturb. The fuzziness is in the world.

Tomlin Hawthorne

You may remember that Heisenberg's oral exam nearly failed him because he didn't understand the microscope. He'd got it wrong again.

Tomlin Hawthorne

Now, does this matter outside a laboratory? Enormously. Here's one consequence, and it answers the question from episode twenty-eight more deeply than I managed then.

Tomlin Hawthorne

Why doesn't the electron in an atom fall into the nucleus? I said: because the lowest wave pattern has nothing below it. Here's the same thing in Heisenberg's language. Suppose the electron did sit on the nucleus. Then it'd be in a very small, very definite place. So its momentum would have to be wildly uncertain. Which means it would be moving, violently, in no particular direction. Which means it wouldn't stay put. The more you try to confine it, the harder it fights.

Tomlin Hawthorne

An atom is the size it is because it's a compromise. The electric pull of the nucleus draws the electron in. The uncertainty principle pushes back. They balance at about a ten-billionth of a metre. That's the size of an atom. That's why matter takes up room. That's why you don't sink through your chair.

Tomlin Hawthorne

It also means nothing can ever be completely still. Even at absolute zero, the floor of temperature from episode twelve, atoms keep a little jiggle, because to be perfectly at rest in a perfectly definite place would break the rule.

Tomlin Hawthorne

And there's a larger casualty.

Tomlin Hawthorne

Remember Laplace, in episode nine. Give me the position and the velocity of every particle, and I'll tell you the whole future. Chaos showed that you could never measure them accurately enough. Quantum mechanics says something more drastic. A particle doesn't have an exact position and an exact velocity to be measured. Laplace's starting information doesn't exist.

Tomlin Hawthorne

And the theory, as we saw last time, doesn't tell you what will happen. It tells you the odds. This atom has an even chance of decaying in the next hour. Which hour will it be? There's no answer. Not a hidden one. None.

Tomlin Hawthorne

Einstein couldn't bear it. In December 1926 he wrote to his friend Max Born, the man with the footnote. He granted that the new theory was a great achievement. But something in him, he said, insisted that it wasn't the final word. It produced a great many results, and it left the deepest secret as hidden as before. And then the sentence everyone remembers: he, for one, was sure that God does not play dice.

Tomlin Hawthorne

Bohr's reply, as it's usually told, was: Einstein, stop telling God what to do.

Tomlin Hawthorne

And the cat. You were waiting for the cat.

Tomlin Hawthorne

By 1935, Erwin Schrödinger, who'd written the wave equation, was as unhappy as Einstein about where it had all led. And he devised a thought experiment, not to illustrate the theory, but to make it look ridiculous.

Tomlin Hawthorne

Take a single radioactive atom. Over the next hour, it has an even chance of decaying. Quantum mechanics says that, until someone looks, the atom is in a superposition: decayed and not decayed, blended.

Tomlin Hawthorne

Fine, said Schrödinger. Now put that atom in a sealed steel box. With a detector. Connected to a hammer. Poised over a flask of poison. And put a cat in the box. If the atom decays, the hammer falls, and the cat dies.

Tomlin Hawthorne

If the atom is in a blend of decayed and not decayed, then the hammer is in a blend of fallen and not fallen. And the cat is in a blend of alive and dead. Until you open the box.

Tomlin Hawthorne

He called this a ridiculous case. He meant it as an objection. Surely, he was saying, nobody believes this about a cat. So the theory must be missing something. Where does the blending stop?

Tomlin Hawthorne

It's one of the small ironies of the subject that his protest became the mascot.

Tomlin Hawthorne

Is there an answer? There's a partial one, worked out mostly since the 1970s. The blend is delicate. The stripes in the double slit vanish if anything records the path. Now, an electron in a vacuum can stay unrecorded. A cat can't. A cat is warm, it's breathing, it's being hit by air molecules and giving off heat, trillions of times a second. Every one of those interactions is, in effect, the environment taking a note of whether the cat is alive. So the blend of alive and dead is destroyed, by leakage, almost before it's begun. The process is called decoherence. It's why you never see big things in two states at once, and it's the main obstacle to building a quantum computer.

Tomlin Hawthorne

Does that fully solve the puzzle? Physicists disagree, strongly. That's the episode after next.

Tomlin Hawthorne

The myth of the week is that the uncertainty principle means we can never really know anything. That it's a licence for vagueness. That it shows science has admitted everything is uncertain.

Tomlin Hawthorne

It's one of the most exact statements in physics. It tells you precisely how much fuzziness there is, and its predictions have been confirmed to many decimal places. A theory that says exactly how uncertain something is, is not being uncertain.

Tomlin Hawthorne

And the question to take away. It's about that play.

Tomlin Hawthorne

Frayn uses the uncertainty principle as a metaphor for the human heart. You can't fix a person's motive without changing it. It's a beautiful idea and it makes a great play.

Tomlin Hawthorne

But is it legitimate? Is it an insight, or just a pun? In 1996 a physicist called Alan Sokal got so irritated with the way literary theorists were borrowing words like uncertainty and relativity that he wrote a spoof article, pure nonsense dressed in physics jargon, and submitted it to a respected cultural studies journal. They published it. It caused a great scandal, and a good deal of hurt feeling on both sides.

Tomlin Hawthorne

So where's the line? When Frayn says a motive is uncertain in Heisenberg's sense, is he illuminating something or misusing a word? When a critic says a text has no meaning until it's read, is that quantum mechanics, or is it something that was perfectly sayable before 1927? You know the humanities side of this better than I do. I'd be curious where you'd draw it.

Tomlin Hawthorne

Your challenge is to hear the uncertainty principle. Find a tone generator, an app or a website. Set it to 440. Play the tone for two full seconds. A clear note.

Tomlin Hawthorne

Now play it for a tenth of a second. Still a note, just about.

Tomlin Hawthorne

Now, if the app allows, a hundredth of a second, or as short as it'll go. It's a click. The pitch has gone.

Tomlin Hawthorne

Nothing changed but the duration. You haven't measured it badly. The short sound hasn't got a pitch to measure. That's Heisenberg, in your ears.

Tomlin Hawthorne

And read Copenhagen, or better, see it. Act One is enough to get the idea.

Tomlin Hawthorne

Next time. Einstein doesn't give up. In 1935 he finds what he's sure is the fatal flaw: quantum mechanics seems to allow two particles to affect each other instantly across any distance. He calls it spooky. For thirty years everyone thinks the argument is philosophy. Then a man from Belfast shows that you can settle it with a pencil. And we'll do it.