The Sciences

Everything is Connected

Cosmic VarianceBy Sean CarrollFeb 23, 2012 12:09 PM


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They do things differently over in Britain. For one thing, their idea of a fun and entertaining night out includes going to listen to a lecture/demonstration on quantum mechanics and the laws of physics. Of course, it helps when the lecture is given by someone as charismatic as Brian Cox, and the front row seats are filled with celebrities. (And yes I know, there are people here in the US who would find that entertaining as well — I’m one of them.) In particular, this snippet about harmonics and QM has gotten a lot of well-deserved play on the intertubes.

More recently, though, another excerpt from this lecture has been passed around, this one about ramifications of the Pauli Exclusion Principle. (Headline at io9: “Brian Cox explains the interconnectedness of the universe, explodes your brain.”)

The problem is that, in this video, the proffered mind-bending consequences of quantum mechanics aren’t actually correct. Some people pointed this out, including Tom Swanson in a somewhat intemperately-worded blog post, to which I pointed in a tweet. Which led to some tiresome sniping on Twitter, which you can dig up if you’re really fascinated. Much more interesting to me is getting the physics right.

One thing should be clear: getting the physics right isn’t easy. For one thing, going from simple quantum problems of a single particle in a textbook to the messy real world is often a complicated and confusing process. For another, the measurement process in quantum mechanics is famously confusing and not completely settled, even among professional physicists.

And finally, when one translates from the relative clarity of the equations to a natural-language description in order to reach a broad audience, it’s always possible to quibble about the best way to translate. It’s completely unfair in these situations to declare a certain popular exposition “wrong” just because it isn’t the way you would have done it, or even because it assumes certain technical details that the presenter did not fully footnote. It’s a popular lecture, not a scholarly tome. In this kind of format, there are two relevant questions: (1) is there an interpretation of what’s being said that matches the informal description onto a correct formal statement within the mathematical formulation of the theory?; and (2) has the formalism been translated in such a way that a non-expert listener will come away with an understanding that is reasonably close to reality? We should be charitable interpreters, in other words.

In the video, Cox displays a piece of diamond, in order to illustrate the Pauli Exclusion Principle. The exclusion principle says that no two fermions — “matter” particles in quantum mechanics, as contrasted with the boson “force” particles — can exist in exactly the same quantum state. This principle is why chemistry is interesting, because electrons have to have increasingly baroque-looking orbitals in order to be bound to the same atom. It’s also why matter (like diamond) is solid, because atoms can’t all be squeezed into the same place. So far, so good.

But then he tries to draw a more profound conclusion: that interacting with the diamond right here instantaneously affects every electron in the universe. Here’s the quote:

So here’s the amazing thing: the exclusion principle still applies, so none of the electrons in the universe can sit in precisely the same energy level. But that must mean something very odd. See, let me take this diamond, and let me just heat it up a bit between my hands. Just gently warming it up, and put a bit of energy into it, so I’m shifting the electrons around. Some of the electrons are jumping into different energy levels. But this shift of the electron configuration inside the diamond has consequences, because the sum total of all the electrons in the universe must respect Pauli. Therefore, every electron around every atom in the universe must be shifted as I heat the diamond up to make sure that none of them end up in the same energy level. When I heat this diamond up all the electrons across the universe instantly but imperceptibly change their energy levels.

(Minor quibble: I don’t think that rubbing the diamond causes any “jumping” of electrons; the heating comes from exciting vibrational modes of the atoms in the crystal. But maybe I’m wrong about that? And in any event it’s irrelevant to this particular discussion.)

At face value, there’s no question that what he says here lies somewhere between misleading and wrong. It seems quite plain (that’s the problem with being a clear speaker) that he’s saying that the energy levels of electrons throughout the universe must change because we’ve changed the energy levels of some electrons here in the diamond, and the Pauli exclusion principle says that two electrons can’t be in the same energy level. But the exclusion principle doesn’t say that; it says that no two identical particles can be in the same quantum state. The energy is part of a quantum state, but doesn’t define it completely; we need to include other things like the position, or the spin. (The ground state of a helium atom, for example, has two electrons with precisely the same energy, just different spins.)

Consider a box with non-interacting fermions, all in distinct quantum states (as they must be). Take just one of them and zap it to move it into a different quantum state, one unoccupied by any other particle. What happens to the other particles in the box? Precisely nothing. Of course if you zap it into a quantum state that is already occupied by another particle, that particle gets bumped somewhere else — but in the real universe there are vastly more unoccupied states than occupied ones, so that can’t be what’s going on. Taken literally as a consequence of the exclusion principle, the statement is wrong.

But it’s possible that there is a more carefully-worded version of the statement that relies on other physics and is correct. And we might learn some physics by thinking about it, so it’s worth a bit of effort. I think it’s possible to come up with interpretations of the statement that make it correct, but in doing so the implications become so completely different from what the audience actually heard that I don’t think we can give it a pass.

The two possibilities for additional physics (over and above the exclusion principle) that could be taken into account to make the statement true are (1) electromagnetic interactions of the electrons, and (2) quantum entanglement and collapse of the wave function. Let’s look at each in turn.

The first possibility, and the one I actually think is lurking behind Cox’s explanation, is that electrons aren’t simply non-interacting fermions; they have an electric field, which means they can interact with other electrons, not to mention protons and other charged particles. If we change the ambient electric field — e.g., by moving the diamond around — it changes the wave function of the electrons, because the energy changes. Physicists would say the we changed the Hamiltonian, the expression for the energy of the system.

There is an interesting and important point to be made here: in quantum mechanics, the wave function for a particle will generically be spread out all over the universe, not confined to a small region. In practice, the overwhelming majority of the wave function might be localized to one particular place, but in principle there’s a very tiny bit of it at almost every point in space. (At some points it might be precisely zero, but those will be relatively rare.) Consequently, when I change the electric field anywhere in the universe, in principle the wave function of every electron changes just a little bit. I suspect that is the physical effect that Cox is relying on in his explanation.

But there are serious problems in accepting this as an interpretation of what he actually said. For one thing, it has nothing to do with the exclusion principle; bosons (who can happily pile on top of each other in the same quantum state) would be affected just as much as fermions. More importantly, it fails as a job of translation, by giving people a completely incorrect idea of what is going on.

The point of this last statement is that when you say “When I heat this diamond up all the electrons across the universe instantly but imperceptibly change their energy levels,” people are naturally going to believe that something has changed about electrons very far away. But that’s not true, in the most accurate meaning we can attach to those words. In particular, imagine there is some physicist located in the Andromeda galaxy, doing experiments on the energy levels of electrons. This is a really good experimenter, with lots of electrons available and the ability to measure energies to arbitrarily good precision. When we rub the diamond here on Earth, is there any change at all in what that experimenter would measure?

Of course the answer is “none whatsoever.” Not just in practice, but in principle. The Hamiltonian of the universe will change when we heat up the diamond, which changes the instantaneous time-independent solutions to the Schoedinger equation throughout space, so in principle the energy levels of all the electrons in the universe do change. But that change is completely invisible to the far-off experimenter; there will be a change, but it won’t happen until the change in the electromagnetic field itself has had time to propagate out to Andromeda, which is at the speed of light. Another way of saying it is that “energy levels” are static, unchanging states, and what really happens is that we poke the electron into a non-static state that gradually evolves. (If it were any other way, we could send signals faster than light using this technique.)

Verdict: if this is what’s going on, there is an interpretation under which Cox’s statement is correct, except that it has nothing to do with the exclusion principle, and more importantly it gives a quite false impression to anyone who might be listening.

The other possibly relevant bit of physics is quantum entanglement and wave function collapse. This is usually the topic where people start talking about instantaneous changes throughout space, and we get mired in interpretive messes. Again, these concepts weren’t mentioned in this part of the lecture, and aren’t directly tied to the exclusion principle, but it’s worth discussing them.

There is something amazing and magical about quantum mechanics that is worth emphasizing over and over again. To wit: unlike in classical mechanics, there are not separate states for every particle in the universe. There is only one state, describing all the particles; modest people call it the “many-particle wave function,” while visionaries call it the “wave function of the universe.” But the point is that you can’t necessarily describe (or measure) what one particle is doing without also having implications for what other particles are doing — even “instantaneously” throughout space (although in ways that have to be carefully parsed).

Imagine we have a situation with two electrons, each in a separate atom, with different energy levels in each atom. Quantum mechanics tells us that it’s possible for the system to be in the following kind of state: each electron is either in energy level 1 or energy level 2, and we don’t know which one (more carefully, they are in a superposition), but we do know that they are in different energy levels. So if we measure the first electron and find it in level 1, we know for sure that the other electron is in level 2, and vice-versa. This is true even if the two electrons are a jillion miles away from each other.

As far as I can tell, this isn’t at all what Brian Cox was talking about; he discusses heating up the electrons in a diamond by rubbing on it, not measuring their energies by observing them and then drawing conclusions about entangled electrons very far away. (In a real-world context it’s very unlikely that distant electrons are entangled in any noticeable way, although strictly speaking you could argue that everything is slightly entangled with everything else.) But there is some underlying moral similarity — this is, as mentioned, the context in which people traditionally talk about instantaneous changed in quantum mechanics.

So let’s go back to our observer in Andromeda. Imagine that we have such a situation with two electrons in two atoms, in a mutually entangled state. We measure our electron to be in energy level 1. Is it true that we instantly know that our far-away friend will measure their electron to be in energy level 2? Yes, absolutely true.

But consider the same experiment from the point of view of our far-away friend. They know what the state of the electrons is, so they know that when they observe their electron it will be either in level 1 or level 2, and ours will be in the other one. And let’s say they even know that we are going to make a measurement at some particular moment in time. What changes about any measurement they could make on their electron, before and after we measure ours?

Absolutely nothing. Before we made our measurement, they didn’t know the energy level of their electron, and would give 50/50 chances for finding it in level 1 or 2. After we made our measurement, it’s in some particular state, but they don’t know what that state is. So again they would give a 50/50 chance for getting either result. From their point of view, nothing has changed.

It has to work out this way, of course. Otherwise we could indeed use quantum entanglement to send signals faster than light (which we can’t). Indeed, note that we had to refer to “time” in some particular reference frame, stretching across millions of light-years. In some other frame, relativity teaches us that the order of measurements could be completely different. So it can’t actually matter. It’s possible to say that the wave function of the universe changes instantaneously throughout space when we make a measurement; but that statement has no consequences. It’s just one of an infinite number of legitimate descriptions of the situation, corresponding to different choices of how we define “time.”

Verdict: I don’t think this is what Cox was talking about. He doesn’t mention entanglement, or collapse of the wave function, or anything like that. But even if he had, I would personally judge it extremely misleading to tell people that the energy of very far-away electrons suddenly changed because I was rubbing a diamond here in this room.

Just to complicate things a bit more, Brian in a tweet refers to this discussion of the double-well potential as some quantitative justification for what he’s getting at in the lecture. These notes are a bit confusing, but I’ve had a go at them.

The reason they are confusing is because they start off talking about the exclusion principle and indistinguishable particles, but when it comes time to look at equations they only consider single-particle quantum mechanics. They have a situation with two “potential wells” — think of two atoms, perhaps quite far away, in which an electron might find itself. They then consider the wave function for a single electron, ψ(x). And they show, perfectly correctly, that the lowest energy states of this system have nearly identical energies, and have the feature that the electron has an equal probability of being in either of the two atoms.

Which, as far as it goes, is completely fine. It illustrates an interesting example where the lowest-energy state of the electron can be really spread out in space, rather than being localized on a single atom. In particular, the very existence of the other atom far away has a tiny but (in principle) perceptible effect on the shape of the wave function in the vicinity of the nearby atom.

But this says very little about what we purportedly care about, which is the Pauli exclusion principle, something that only makes sense when we have more than one electron. (It says that no two electrons can be in the same state; it has nothing interesting to say about what one electron can do.) It’s almost as if the notes cut off before they could be finished. If we wanted to think about the exclusion principle, we would need to think about two electrons, with positions let’s say x1 and x2, and a joint quantum wave function ψ(x1, x2). Then we would note that fermions have the property that such a wave function must be “odd” in its arguments: ψ(x1, x2) = -ψ(x2, x1). Physically, we’re saying that the wave function goes to minus itself when we exchange the two particles. But if the two particles were in exactly the same state, the wave function would necessarily be unchanged when we exchanged the particles. And a function that is both equal to another function and equal to minus that function is necessarily zero. So that’s the exclusion principle: given that minus sign under exchange, two particles can never be in precisely the same quantum state.

The notes don’t say any of that, however; they just talk about the two lowest energy levels in a double-well potential for a single electron. They don’t demonstrate anything interesting about the exclusion principle. The analysis does imply, correctly, that changing the Hamiltonian of a particle somewhere far away (e.g. by altering the shape of one of the wells) changes, even if by just a little bit, the energy of the wave function defined over all space. That’s connected to the first possible interpretation of Cox’s lecture above, that heating up the diamond changes the Hamiltonian of the universe and therefore affects the wave function of every electron. Which also has nothing to do with the exclusion principle, so at least it’s consistent.

In terms of explaining the mysteries of quantum mechanics to a wide audience, which is the point here, I think the bottom line is this: rubbing a diamond here in this room does not have any instantaneous effect whatsoever on experiments being done on electrons very far away. There are two very interesting and conceptually central points worth making: that the Pauli exclusion principle helps explain the stability of matter, and that quantum mechanics says there is a single state for the whole universe rather than separate states for each individual particle. But in this case these became mixed up a bit, and I suspect that this part of the lecture wasn’t the most edifying for the audience. (The rest of the lecture still remains pretty awesome.)

Update: I added this as a comment, but I’m promoting it to the body of the post because hopefully it makes things clearer for people who like a bit more technical precision in their quantum mechanics. [Note the mid-update extra update.]

Consider the double-well potential talked about in the notes I linked to near the end of the post. Think of this as representing two hydrogen nuclei, very far away. And imagine two electrons in this background, close to their ground states.

To start, think of the electrons as free particles, not interacting with each other. (That’s a very bad approximation in this case, contrary to what is said in the notes, but we can fix it later.) As the notes correctly state, for any singleelectron there will be two low-lying states, one that is even E(x) and one that is odd O(x). When we now add the other electron in, they can’t both be in the same lowest-lying state (the even one), because that would violate Pauli. So you are tempted to put one in E(x1) and the other in O(x2).

But that’s not right, because they’re indistinguishable fermions. The two-particle wave function needs to obey ψ(x1, x2) = -ψ(x2, x1). So the correct state is the antisymmetric product: ψ(x1, x2) = E(x1) O(x2) – O(x1) E(x2).

That means that neither electron is really in an energy level; they are both part of an entangled superposition. If you zap one of them into a completely different energy, nothing whatsoever happens to the other one. It would now be possible for the other one to decay to be purely in the ground state, rather than a superposition of E and O, but that would require some interaction to allow the decay. (All this is ignoring spins. If we allow for spin, they could both be in the ground-state energy level, just with opposite spins. When we zapped one, what happens to the other is again precisely nothing. That’s what you get for considering non-interacting particles.)

[Second update: the below two italicized paragraphs are wrong, my bad. It’s actually quite a good approximation (although still an approximation) to ignore the electromagnetic interactions of the electrons, because after antisymmetrization you will almost always find precisely one electron in each well. If electrons were bosons, you’d get a similar quantum state because the interactions would be important, but for fermions the exclusion principle does the job. Final paragraph is still okay.]

But of course it’s a very bad approximation to ignore the interaction between the two electrons, precisely because of the above analysis; it’s not true that one is here and one is far away, they both are equally distributed between being here and being far away, and can interact noticeably.

Since electrons repel, the true ground state is one in which the wave function for one is strongly concentrated one one hydrogen atom, and the wave function for the other is strongly concentrated on the other. Of course it’s the antisymmetrized product of those two possibilities, because they are identical fermions. The energies of both are identical.

Now when you zap one electron to change its energy, you do change the energy of the other one, in principle. But it has nothing to do with the exclusion principle; it’s just because you’ve changed the amount of electrostatic repulsion by changing the spatial wave function of one of the electrons.

Furthermore, while you instantaneously change “the energy levels” available to the far-away electron by jiggling the one nearby, you don’t actually change the position-space wave function in the far-away region at all. As I said in the post, you’ve poked the other electron into a superposition rather than being in an energy eigenstate. Its wave function (to the extent that we can talk about it, e.g. by integrating out the other particles) is now a function of time. And the place where it’s actually evolving is completely inside your light cone, not infinitely far away. So there is literally nothing someone could do, in principle as well as practice, to detect any change as a far-away observer.

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