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According to Brian Cox in his A night with the Stars lecture$^1$, the Pauli exclusion principle means that no electron in the universe can have the same energy state as any other electron in the universe, and that if he does something to change the energy state of one group of electrons (rubbing a diamond to heat it up in his demo) then that must cause other electrons somewhere in the universe to change their energy states as the states of the electrons in the diamond change.

But when does this change occur? Surely if the electrons are separated by a significant gap then the change cannot be instant because information can only travel at the speed of light. Wouldn't that mean that if you changed the energy state of one electron to be the same as another electron that was some distance away, then surely the two electrons would be in the same state until the information that one other electron is in the same state reaches the other electron.

Or can information be transferred instantly from one place to another? If it can, then doesn't that mean it's not bound by the same laws as the rest of the universe?

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$^1$: The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' A Night with the Stars lecture.

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Could you give a reference? This sound rather dubious the way you state it here. –  leftaroundabout Dec 20 '11 at 11:51
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Like I said, it came from Brian Cox's Evening With The Stars lecture. If you're in the UK you should still be able to see it on iPlayer. –  GordonM Dec 20 '11 at 12:00
    
As it says at blogs.scienceforums.net/swansont/archives/11081: ''Physics, tech and humor. Because science and learning are cool, and life's too short not to laugh.'' - In the present case, the physcis content is minimal, and a superficial misunderstanding is used for making people laugh. –  Arnold Neumaier Mar 5 '12 at 17:04
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If two electrons are distant, then they can't occupying the same state. The state includes the position. –  Peter Shor Apr 17 '13 at 15:58
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10 Answers 10

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The Pauli exclusion principle can be stated as "two electrons cannot occupy the same energy state", but this is really only a rough way of stating it. It's more precise to say that the wavefunction of a system is anti-symmetric with respect to exchange of two electrons. The trouble is that now I have to explain to a non-physicist what "anti-symmetric" means and that's hard without going into the maths. I'll have a go at doing this below.

Anyhow, Brian Cox is being a bit liberal with the truth because I'm not sure it makes sense to say the electrons in his bit of diamond and electrons in far away bits of the universe can be described by a single wavefunction. If this isn't a good description then the Pauli exclusion principle doesn't have any meaning for the system.

Suppose you have two electrons in an atom or some other small system. Then that system is described by some wavefunction $\Psi(e_1, e_2)$ where I've used $e_1$ and $e_2$ to denote the two electrons. The Pauli exclusion principle states:

$$\Psi(e_1, e_2) = -\Psi(e_2, e_1)$$

that is if you swap the two electrons $\Psi$ changes to $-\Psi$. But suppose the two electrons were exactly the same. In that case swapping the electrons cannot change $\Psi$ because they're identical. So we'd have:

$$\Psi(e_1, e_2) = \Psi(e_2, e_1)$$

but the exclusion principle states:

$$\Psi(e_1, e_2) = -\Psi(e_2, e_1)$$

therefore if both are true:

$$\Psi(e_2, e_1) = -\Psi(e_2, e_1)$$ ie $$\Psi = -\Psi$$

The only way you can have $\Psi = -\Psi$ is if $\Psi$ is zero, which means $\Psi$ doesn't exist. This is why if the Pauli exclusion is true, two electrons can't be identical i.e. they can't be in the same energy state.

But this only applies because I could write down a wavefunction $\Psi$ to describe the system. When systems become large, e.g. two footballs in a swimming pool instead of two electrons in an atom, it isn't useful to try and write a wavefunction to describe the system and the exclusion principle doesn't apply. NB this doesn't mean the exclusion principle is wrong, it just means it doesn't apply to that system.

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The answer is nonsense. The wave function of two electrons is not a function of the electrons but a function of their position. Moreover, a sign change in a wave function is irrelevant as this only changes the phase, not the probability amplitude. Thus the argument is spurious and misleading. –  Arnold Neumaier Mar 5 '12 at 17:04
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@ArnoldNeumaier: you'll find exactly this argument in any graduate textbook on quantum mechanics –  John Rennie Mar 5 '12 at 17:18
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...but surely not phrased in this sloppy way. You conclusion that the pauli principle doesn't apply to large systems is also wrong. It still applies, though there are so many states anyway that it is already impossible for practical reasons to create identical states, so that the Pauli principle adds no new information. –  Arnold Neumaier Mar 5 '12 at 17:41
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I agree that this could use an edit. e1 and e2 should be electron positions not electrons, but moreover "Suppose the two electrons were exactly the same" is not very clear... do you mean that the two-particle state is a product of two of the same single-electron state? E.g. Psi(x,y) = psi(x)psi(y)? The exclusion principle indicates that such states are forbidden for fermions, whereas psi1(x)psi2(y) - psi2(x)psi1(y) would be allowed. I assume that's what you're getting at, but it's not very clear. –  Tim Goodman Aug 15 '12 at 19:49
    
This answer does need to be cleaned up...perhaps Arnold could edit it to properly represent the wavefunction's antisymmetry as a function of position with concise definitions about what "exchanging electrons" is. My graduate class on the subject did use position, but in my opinion didn't define very well mathematically what "exchanging electrons" was... –  daaxix Jan 29 '13 at 18:12
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Cox said that when he heats the diamond every electron in the universe shifts energy levels instantly to respect the Pauli Exclusion Principle. The problem here is not that he is talking about energy levels. The set of levels is a set of eigensates of the hamiltonian and the fact that some of them are degenerate does not invalidate what he said. In fact his diagrams showed different spin states at the same energy level so he was making this evident.

There are however some other aspects of his statement that are worth objecting to. One problem is that energy states for free electrons are not at discrete levels. It is not clear that what he says makes sense in an open system.

Another problem is that the states are not ordered in such a way that everything can move up one level.

Also the picture he is basing his claims on assumes that electrons are in energy levels of fixed systems, but the other particles are moving too. How does this affect the energy levels of the electrons?

Perhaps the strangest part of his claim was that he said everything would move instantly. Was he invoking the entanglement of the electrons? This does not make sense since he is not merely observing the diamond, he is heating it up. the kind of effect he is describing could not propagate faster than light.

Even taking into account the need to simplify for the public audience I think it is a bit of a stretch to attach any real meaning to what he said or the logic behind it.

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By the way, aside from this it was a great TV lecture. There are not many people who can stand up in front of an audience and make physics that entertaining –  Philip Gibbs Dec 21 '11 at 18:58
    
I think that Jim Parsons would still win in this respect, Phil. He has more physically accurate writers of the script, too. –  Luboš Motl Mar 13 '12 at 20:17
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Such claims as with these electrons should not be taken as strict fact, or used to draw such conclusions about physical reality.

For two identical systems, such as two neutral hydrogen atoms, separated by a great distance, the two electrons have identical energies. When considering both atoms together as one quantum system, even when far apart, we must look at the energy levels of the sum and the difference of wavefunctions of the two electrons. Typically one is slightly larger, the other slightly smaller than the original levels. These shifts in energy are greater with more overlap of wavefunctions, and zero when the atoms are infinitely far apart.

Wavefunctions describing these electrons fall off exponentially with distance away from the system. Think of this as a "spatial half-life" with a distance scale of nanometers, maybe microns. At human-scale distances, for all practical purposes the overlap is zero.

According to ideal math, which gives us ideal mathematical forms for wavefunctions satisfying perfect wave equations in an idealized physical system, the overlap and energy shifts are nonzero, but so absurdly tiny if we're talking about human-scale situations or bigger. Never mind astronomical distances.

These ideal equations are not to be ignored - Nobel Prizes have been won for making precise measurements of electrons, precise to ten digits or so (check wikipedia for the latest) and agreeing with the predictions of quantum electrodynamics. It works well.

Using QED, we can estimate the energy shifts involved in the sorts of situations presented by Dr. Cox. These are so absurdly tiny that they'd be swamped out by much larger tiny effects such as the gravitational effects of a speck of dust anywhere in the vicinity of either atom, Doppler shifts due to even the slightest motion - like trillionths of miles-per-hour.

Heisenberg says that quantum mechanical systems don't even have precise energy levels beyond any particular precision, when you measure the energy over some corresponding time interval. To have a meaningful energy level so precise as to distinguish between the sum and difference wavefunctions, the system must be stationary, untouched, for a very long time - we're talking about cosmic time scales. While the exact math lets calculate such things, the real physical world isn't obligated to be so precise at that level.

In short, Dr. Cox is making a wild and idealized extrapolation.

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I think what Dr. Cox is assuming is that two neutral hydrogen atoms, at great distance, can be viewed as being in an energy eigenstates of the system composed of the two atoms. These energy eigenstates will be have slightly different energies. But this is only true at the time scale of hbar over the difference in the two energies, which is enormous. –  Peter Shor Apr 17 '13 at 16:10
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Brian Cox is wrong. Period. It is not at all about energy states, it is about an antisymmetric wavefunction. For example, every atomic energy level can contain 2 electrons - one with spin up and one with spin down and they can have the same energy. I wouldn't waste any more time on his rubbish...

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-1, it is an antisymmetric wavefunction, but you don't explain how it is wrong very well... –  daaxix Jan 29 '13 at 18:08
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It is a bit hard to say something that captures the essence of your question without some mathematics, but nonetheless, I'll give it a go. :)

In my view, field theory puts all these so called "paradoxes" to rest. To state is simply, there is only one electron field in the universe. What we perceive as particles are excitations in the field due to coupling with other fields. If you take this view, then all you are doing is measuring attributes of the same field at different space-time points.

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IMHO this answer would be satisfying if you mentioned how Pauli exclusion principle works in QFT. –  Adam Zalcman Dec 20 '11 at 17:06
    
The antisymmetry of the wave function is a consequence of the anticommutation rule for fermions in QFT. And the Pauli principle is expressed by the fact that applying the same creation operator twice annihilates a state. –  Arnold Neumaier Mar 5 '12 at 18:16
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I wrote to Professor Cox the day after his TV lecture raising basically the issues discussed here. In particular the idea of instantaneous action seems particularly hard to swallow, and it seems the answer, which seems no answer at all, is that only information cannot be transmitted faster than light velocity. But telling another electron a billion km away of its energy state sounds like information to me. As to the other points, he invited me to read his book which I am doing but am no closer to accepting his arguments yet. It all has a slightly mystical feel to it which I know he would hate to be accused of, and indeed much of quantum mechanics has been held in such belief in the past. It is only the 'well it works approach' which has justified the seemingly outrageous propositions, there being no other serious contenders. We are on a road to understanding something fantastic but nobody has any idea of what it is.

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The Pauli exclusion principle does not explicitly reference energy. It states that no two identical fermions (in this case electrons) may occupy the same quantum state. In a system where each quantum state corresponds to a unique energy level, then you could infer that no two identical fermions may occupy the same energy level, otherwise no such constrain exists. An example already given is where two electrons may occupy the same energy level in an atom. This is because the energy level corresponds to the orbital angular momentum of the particle, but there is an additional degree of freedom, the spin of the electron. Of course the Pauli exclusion principle operates here by ensuring NO MORE than 2 electrons can be at this energy level, because there are only two possible spin states.

If I understand you correctly, the way you've interpreted what Brian Cox is saying is that, given some electrons in a diamond, which are at some energy, no other electrons in the universe may also be at that energy. So if you change the energy of the diamond by heating it up, then any electrons in the universe which happen to be at this new energy will have to adjust in order to accommodate this change. I haven't seen what Brian Cox said, so I don't know whether what you've paraphrased is accurate, but if this is what he was implying, then he's dead wrong. This is simply not true.

One could extend this reasoning to consider an electron in an atom. Given this rule that no other electrons in the universe may also be at this energy, one must conclude that no other atom may contain an electron at this energy, thus making consistent chemical behaviour impossible. The point is that being in one atom or another corresponds to a different quantum state, so no violation occurs when one electron in one atom happens to be in the same quantum state as an electron in another atom, since there's the added degree of freedom of being in one atom or the other. The same kind of thing holds for crystals and all the other electrons in the universe.

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My take on what he said was that the energy he imparted to the diamond will pass as heat to the atoms and electrons of air immediately surrounding the diamond, then on to others next to them, eventually it will be radiated into space and, as energy cannot be destroyed will continue it's journey across the universe long after the planet has been consumed by our expanding sun , minutely raising the energy levels of the electrons it meets. In other words entropy increasing from the chemical energy in his food to mechanical energy in his hand to thermal energy in the diamond/air/interstellar space.

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@GordonM One has to be careful when making arguments about energy (energy leves). There is a concept which has been overlooked in this interesting discussion. Let us accept for a moment professor Cox's argument. Now, let us assume that an electron somewhere inside the diamond gets a sufficient amout of energy, 0.5eV say, to jump from energry level E_1 to another energy level or virtual state E_2. There are about 10^59 Kg of matter in the universe, and therefore it is very likely that there are an extremely large number of electrons in the universe, which occupy that particular energy level E_2, and they should move to some other energy level, and let us assume there is no domino effect by doing this. The amount of energy needed for all those electrons in the universe to do so can be immense! It is clear that we have a problem here. Where is that extraordinarily large amount of energy going to come from? One can argue that, some electrons will move up and some will move down one level so it will all balance out in the end! If that is the case then, we should be able to observe this effect happening with atoms in our laboratories, as atoms should suddenly emit light just because a piece of diamond at some other part of the universe was warmed up a bit!!??

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Any discussion on the subject of particle physics is bound to be fatuous unless it can conducted in explicit mathmatical terms and what I understand on the matter for example would be unlikely to fill the back of a postcard so I hesitate to comment. Having seen the program however, my take on what Cox was trying to explain was the inter-relatedness of all matter. IE photons exist only in the sense of 2 events that occur in 2 different places. So Somewhere in the universe an electron jumps to a lower energy state and somewhere else another electron jumps to a higher state. We interpret this as a photon passing between the 2 locations. Before the event the path of the photon (ie the location of the 2 events) can only be predicted in terms of probability and mapped as a field. Intuitively it seems reasonable to me that the Pauli exclusion principle could apply to any system one might care to define even though different parts of the system may be physically very remote from each other. I do not believe therefore that Cox was suggesting that because an electron on one atom (say the hydrogen atom at the back of my eye) was in one energy state then no electrons attached to other hydrogen atoms in the universe could have an electron in a similar state. What he did seem to be suggesting however was that the Pauli exclusion principle applies across the system defined by the 2 events and not simply constrained to a single atom. IE the principle is a universal law.

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If this is actually what Prof. Cox said (I expect that it isn't what he intended to say, but maybe what he actually said), it's misleading. The Pauli exclusion principle just says that two electrons can't occupy the same state, but the description of the state includes position as well as energy. –  Peter Shor Apr 17 '13 at 16:04
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protected by Qmechanic Sep 14 '13 at 18:56

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