Brian Cox and the Pauli Exclusion Principle

[atyy]

No. It's called the Cluster Decomposition Principle.

I found this article which explains a bit why Bill K's answer is right. The confusing thing for me was that spatially separated entangled states of non-identical particles are nonlocal in the sense of Bell, because they produce distant correlations, and for them entanglement can also be defined by not being a product state. The anti-symmetrization of identical fermions does seem to make them "entangled" by the latter definition, yet is not enough to make them nonlocal.

http://arxiv.org/abs/1009.4147

 

[Ken G]

It means when bound to an atom you can't really consider it as separate particles - you have to consider it as a system because they are entangled.

I agree you can't consider the electrons separately from the rest of the atom, it's a bound system. What I'm doing is further adding that you also can't consider "the electrons in the atom" separately from the electrons in the rest of the universe, because that would be to imply they are distinguishable, and they are not. Of course in practice we do allow ourselves, though formally incorrectly, to use the language "the electrons in the atom", as long as we realize we really mean "whatever electrons show up in our measurement on the atom". They could be any electrons in the universe, and indeed quantum mechanics is quite explicit on this point, that's why the "total wave function" of the universe (if you are the type to believe there is meaning in such a thing, which Brian Cox clearly is) does not say which electrons are in that atom and which electrons aren't. The identity of the electrons in the atom never appears anywhere in the wavefunction of the universe, and importantly so-- that's where the PEP comes from. Granted, the PEP really only cares about electrons with overlapping wavefunctions, but here we are in Brian Cox-land where there is just one wavefunction for the whole universe, which contains all the information in the universe, and since that wavefunction lacks information about the identity of the electrons in that atom, there is, in formal quantum mechanics, no such thing as "the electrons in that atom." Of course this only matters for the tiniest of correlations, but that's what this thread is all about.

The reason electrons can't be in the same state is if electron 1 is in state |u1> and electron 2 in state |u2> the composite system is |u1>|u2>. If |u1> and |u2> are the same state then nothing happens on exchange in contradiction to the fact it must change sign. BUT if bound to an atom its entangled with the nucleus and you can't specify the state of the electron by itself and the argument breaks down. You must consider it as a whole and when that is done it could be a fermion on boson.

All the same, atoms have the structure they have because of the PEP. The entanglements with the nucleus do not change that, though I admit I've never thought about how it might subtly alter the structure of an atom to think about entanglements and not just interaction energies and exchange terms. Has anyone?

Added Later:
I think I may see Kens point. Yes, even in bound atoms you can't tell the difference between electrons. What I am saying is, even though that's still true, the consequences are different - it doesn't necessarily lead to they can't be in the same state.

But it generally does-- we have the level structure of atoms as a result. I presume you are talking about tiny deviations from the simple application of the PEP to electron states that are unentangled with the nucleus, and I've never even thought about those possible repercussions, though we know they have to be small. So I've not disputed your point about entanglements, it was the language about an atom taking its electrons with it that can't be right because "the atom's electrons" is not a formally correct concept. Instead, what the atom "takes with it" are the coordinates of whatever measurement we are doing on the atom, not "its electrons," because the latter require distinguishing that which cannot, by any experiment, be distinguished.

So I'm saying that formally, if we are being more precise than is generally necessary, all we can say is that the total wavefunction of all the electrons in the universe must contain aspects that guarantee we'll usually find some N electrons in that atom if we look. Entanglements with the nucleus, and the interaction energies, are what ensure that, but they are still buried in the evolution of the total wavefunction in ways that does not allow the electrons in that atom to be formally recognized as individual entities, and there's always some probability they will tunnel out of the atom and exchange with other electrons that tunnel in. Not only can no observation rule that out, occasionally we find an observation that requires it. The rest of the time, we'd be nuts to try to take that into account, and a lot of what we do in quantum mechanics is actually a "manual" approximation that we know will work. But even if we are being unnecessarily precise, we agree the universal wavefunction does not propagate signals halfway across the universe. But it certainly could swap electrons halfway across the universe, and not only couldn't we tell the difference, it is one of the central tenets of quantum mechanics that we couldn't tell the difference.

 

[Ken G]

I found this article which explains a bit why Bill K's answer is right. The confusing thing for me was that spatially separated entangled states of non-identical particles are nonlocal in the sense of Bell, because they produce distant correlations, and for them entanglement can also be defined by not being a product state. The anti-symmetrization of identical fermions does seem to make them "entangled" by the latter definition, yet is not enough to make them nonlocal.

http://arxiv.org/abs/1009.4147

Sure, but that's not what makes them nonlocal. What makes them nonlocal, formally and if we are being unnecessarily precise and following a probably improper allegiance to the literal details of quantum mechanics, is that their states have evolved within the universal wavefunction from the time of the Big Bang. Put differently, no wavefunctions are completely nonoverlapping, so the cluster decomposition principle is an approximation, whereas formal quantum mechanics is not (if we believe that, anyway). Brian Cox can only be held accountable to formal quantum mechanics, not to approximations like the cluster decomposition principle or other practical matters that allow us to disregard the law of formal quantum mechanics that all electrons are fundamentally and completely indistinguishable.

 

[atyy]

Sure, but that's not what makes them nonlocal. What makes them nonlocal, formally and if we are being unnecessarily precise and following a probably improper allegiance to the literal details of quantum mechanics, is that their states have evolved within the universal wavefunction from the time of the Big Bang. Put differently, no wavefunctions are completely nonoverlapping, so the cluster decomposition principle is an approximation, whereas formal quantum mechanics is not (if we believe that, anyway). Brian Cox can only be held accountable to formal quantum mechanics, not to approximations like the cluster decomposition principle or other practical matters that allow us to disregard the law of formal quantum mechanics that all electrons are fundamentally and completely indistinguishable.

I don't understand this overlapping part - is that something that would hold also for distinguishable particles?

 

[Ken G]

I don't understand this overlapping part - is that something that would hold also for distinguishable particles?

Sure, but the ramifications would be different. For one thing, overlap of indistinguishable fermions produces an "exchange energy" term in the interaction energy. But for the PEP, we don't need any interaction energy, so no exchange energy, we can just look at counting the states. So a quantum statistical mechanical application would be most appropriate. Two overlapping states, once expanded on eigenstates, would normally have a nonzero amplitude of being found in the same eigenstate. But the antisymmetrized wave function would cancel that amplitude. If there is no overlap, there is no such cancellation, because there are no amplitudes for having the same quantum numbers for two completely separated wavefunctions. However, Cox might have been referring to the idea that there is always some tiny amplitude of being anywhere, so there's always overlap, so there's always a need for the antisymmetrized wavefunction to cancel amplitudes of the same quantum numbers, when observations of those quantum numbers are being carried out.

 

[Morberticus]

Here's a decent answer on stack exchange (This time in the context of two election "energy levels" in wells with large separation (ignoring other quantum numbers for the moment))

http://physics.stackexchange.com/qu...-a-two-fermion-double-well-system/22304#22304

"one can't measure the energy "in one well only" with the accuracy needed to distinguish E1 and E2"

"If your measurement apparatus is confined to the vicinity of one well, the error in your energy measurement can't be smaller than E1−E2 so you won't be able to say "which of the two nearby states" the electron is in. The same holds for the vicinity of the other well which is why the measurement in one well can't influence anything detectable near the other well."

So it seems this type of correlation is permitted because of the fundamental uncertainty present in quantum mechanics. It's not just imperceptible, it's formally imperceptible.

 

[Ken G]

I think that post is actually saying something similar to what I said, just in another way, and I can show this. Their key point is that you can't measure the energy "in one well only" with the accuracy needed to distinguish E1 and E2, but why would you want to measure the energy "in one well only" in the first place? That would be a strange interpretation of a quantum mechanical energy measurement, we should actually be looking for energy eigenstates of the joint wavefunction! If we do that, we get no confusion, the two-electron system with two widely separated wells has just a single ground state, and a first excited state that is imperceptibly close. In the limit that the two wells are infinitely separated, that looks like two separate and independent ground states of just one electron each, but remember, that is not the situation if we are not taking that limit-- if we are trying to be absurdly precise, as Brian Cox is essentially doing.

If we stick to the absurdly precise non-limit, the presence of two slightly different energy states of a two-electron system, not two separate ground states of two one-electron systems, is a manifestation of the indistinguishability of the electrons, and the need to write a joint wave function that is not only antisymmetric under exchange of the observing coordinates, but also respects that basic indistinguishability. The failure to respect that indistinguishability is what motivates language like "measure the energy of one of the electrons", which translates into "measure the energy of one well only" if you think you can distinguish the electrons. But neither of those statements make formal sense in quantum mechanics, an energy measurement is an energy eigenvalue of the whole system, pure and simple.

So what I'm saying is, Cox's error is in using language that suggests you can talk about "the energy of the electrons in a diamond", in the same breath that he says all the electrons in the universe obey the PEP. If you use the former language, you are treating the electrons as distinguishable, so you get no PEP. If you use the PEP, you don't have "electrons in the diamond", you just have the joint wavefunction of all the electrons, and their eigenstates represent the Hamiltonian of the whole universe, not just the diamond. That is tantamount to saying there is some tiny probability they can tunnel to other places and change places with the electrons that are over there, or if you prefer, the simple fact that you never know which electrons you are really experimenting on, you only know various probabilities and expectation values that you will get when you set up your apparatus and define your measuring coordinates.

Finally, I note that I characterize Cox's language as an "error" because of its inconsistency, but he might hold that the inconsistency is justified if part of the truth is too technical, and the other part isn't. Is inconsistency OK when we "dumb down" one part but not the rest? I don't think so, I think we need to be consistent even when we are simplifying for a nonexpert audience, but that is a different kind of complaint than saying he doesn't understand quantum mechanics.

 

[DevilsAvocado]

I am more worried for the ones who believe they truly understand it all well enough(now i am going into hiding).

:thumbs:

"I think I can safely say that nobody understands quantum mechanics." -- Richard Feynman​

 

[DevilsAvocado]

Then again: The linked-cluster principle does not contradict Bell's findings. There are non-local correlations, but there are no actions at a distance. That's an important difference! Local relativistic QFTs of course admit the Bell correlations (vulgo entanglement), but interactions are local, and the linked-cluster principle holds, i.e., local observables do not depend on interactions/experiments at far-distant places. See Weinberg's Quantum Theory of Fields, Vol. I on this!

Okay, local [Bell] observables do not depend on interactions at far-distant places. So how does the linked-cluster principle explain this?

Real-Time Imaging of Quantum Entanglement

https://www.youtube.com/watch?v=wGkx1MUw2TU
http://www.youtube.com/embed/wGkx1MUw2TU

[URL]http://www.nature.com/srep/2013/130529/srep01914/images_article/srep01914-f4.jpg[/PLAIN]

http://www.nature.com/srep/2013/130529/srep01914/full/srep01914.html

 

[DevilsAvocado]

[...] I think we need to be consistent even when we are simplifying for a nonexpert audience, but that is a different kind of complaint than saying he doesn't understand quantum mechanics.

I think we have to remember that "A Night with the Stars" runs on BBC 2 (watched by 1.6 million viewers), where maybe only a tiny fraction has any deeper understanding of QM, and the majority knows absolutely nothing. I don't want to be rude or something, but if Brian Cox were to talk like you (or me or any other "nerd" on PF), everybody except the already "baptized" will have zapped over to "Dame Edna & Lady Gaga –– Go-go Dancing with the Stars", or something, within a minute.

There's just not any room for Brian to talk about spin, angular momentum, quantum numbers, etc, even if this is what he should have done (to make it technically correct). It just becomes too dense for Average Joe. And still there are comments like this:

"Thank goodness for Simon Pegg. And Jonathan Ross, James May and Sarah Millican. That Brian Cox lecture would have been a struggle without them, that's for sure, writes Sian Brewis. Imagine, a whole programme without a single celebrity in there. How on Earth would we poor viewers be expected to concentrate?"​

Also note that the he did not talk about FTL communication, all he said was - Everything is connected to everything else...

https://www.youtube.com/watch?v=Mn4I-f34cTI
http://www.youtube.com/embed/Mn4I-f34cTI

But this was apparently enough for "Bhagwan-Brahmaputra-Guru", and followers, to light up the "hippie-räucherkegel" and start talking about "interconnected minds" etc, sigh...

Not directly Brian's fault, is it?

My guess is that he wanted to talk about the non-local nature of QM, and he preferred something that could be 'explained' in a few minutes before the audience had lost interest. It's probably impossible to be 100% technically correct all the time in a show like this, and if Brian can make some viewers interested enough to learn more about the subject, that can't be a bad thing, can it? Who knows, maybe the next "Feynman/Schrödinger/Einstein" becomes interested in QM (instead of Lady Gaga) because of a show like this. That is just great, IMHO!

So, how 'wrong' was Brian Cox?

Well, as in all cases where people talk about QM – it's a matter of interpretations. Matt Leifer sums it up pretty well:

My interpretation of what Brian Cox was trying to say is slightly different, but not necessarily more likely. Since the Pauli exclusion principle comes from the requirement that the global wavefunction of a bunch of fermions has to be antisymmetric under exchange, you can argue that there is a sense in which all the electrons in the universe are entangled, e.g. in a universe of only 2 electrons with two possible positions “here” and “far away”, the wavefunction would have to be of the form:

psi(1,here) psi(2,far) – psi(1,far)psi(2,here)

This looks like it is entangled, but there is nothing you could do to prove it by local measurements, since you can’t do a measurement that distinguishes 1 from 2 in any way.

Of course, anyone who has read Sakurai knows that, if we only have access to observables in the “here” region, then this state will be indistinguishable from a single particle state psi(here), which is manifestly local. If you are one of those people who think of wavefunctions as being physically real, then there is one description in which Brian is correct and one in which he is incorrect and since they are operationally equivalent it is impossible to adjudicate. Of course, if you insist on only regarding measurement outcomes as physically real, then you will never observe any nonlocality this way, and I don’t think anyone would argue with that.

There are obviously always different views:

https://www.youtube.com/watch?v=ASZWediSfTU
http://www.youtube.com/embed/ASZWediSfTU

 

[Ken G]

My guess is that he wanted to talk about the non-local nature of QM, and he preferred something that could be 'explained' in a few minutes before the audience had lost interest. It's probably impossible to be 100% technically correct all the time in a show like this, and if Brian can make some viewers interested enough to learn more about the subject, that can't be a bad thing, can it?

I basically agree, and I think he provides more of a service by getting people jazzed about it than he does a disservice by feeding the lunatic fringe. I'm just saying that if we are to critique the formal correctness of his language, the main problem would be failing to enforce the indistinguishability within a set of electrons that he is also invoking the PEP to talk about. I think you can make the same points he is making, and get the same people jazzed about them, without feeding the misconception that we have one set of distinguishable electrons within a diamond, and a different set halfway across the universe, and they affect each other via the PEP-- when the PEP is demonstrably a principle relating to particles that you cannot possibly distinguish. How important is it to get that right? I don't know, I think it's important when it's not any harder than getting it wrong, but everyone has to kind of draw their own line there.

Well, as in all cases where people talk about QM – it's a matter of interpretations. Matt Leifer sums it up pretty well:

Indeed I said something very similar earlier-- we have never tested the kinds of effects that Cox is talking about, so we can't really say "the universe is like this", because some don't accept that there "really is" any such thing as a universal wave function. But Cox does believe in that concept, and he could use language that is consistent with that concept, but he is only partly doing that, and partly not doing that. It is the inconsistency that is my issue.

ETA: let me put it this way. Sometimes when we say something that sounds a little too shocking, it is because we said it wrong. This is probably one of those cases. If one wants to invoke the PEP to explain energy shifts of electrons halfway across the universe, one needs to recognize that in the context of formal QM, and in particular the PEP, there is no such thing as electrons halfway across the universe, because they have to be indistinguishable to obey the PEP. Hence, to see the kinds of energy shifts he is talking about, one would need to do energy measurements on the entire universe of electrons. If one is doing energy measurements on the entire universe of electrons, it is a whole lot less surprising, but perhaps just as interesting, that what you do to a diamond can effect the energies of the electrons of the universe, expressly because the electrons of the universe are all indistinguishable from those in the diamond.

 

[Morberticus]

some don't accept that there "really is" any such thing as a universal wave function. But Cox does believe in that concept

I think Cox's position is one I sympathise with, but even non-local realism is under attack.

http://www.nature.com/nature/journal/v446/n7138/full/nature05677.html

 

[Dale]

Closed for moderation