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A night with the stars (Brian Cox on telly)

by dgwsoft
Tags: brian cox, quantum mechanics
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unusualname
#91
Jan12-12, 05:17 PM
P: 661
Quote Quote by atyy View Post
Yes. It is technically true. In technical terms, this simply reflects the requirement that the wavefunction of a system of fermions must be antisymmetric, and the assumption that there is at any particular time a single wavefunction that contains all fermions in the universe. However, of course when we write a wavefunction for a solid on the earth, we don't expect to have to take account of the fermions on the moon to get a really good approximation. I cannot remember the argument that the fermions on the moon can be neglected for all practical purposes, but it is found in Shankar's QM text http://books.google.com/books?id=2zy...gbs_navlinks_s (around p275, search for "moon"!).
unfortunately the relevant pages 274,275 are not available in my google books preview. But if you have a (free) amazon account you can just search for the word 'moon' in the 'Look Inside!' view


http://www.amazon.co.uk/reader/03064...TF8&query=moon

(The relevant section starts on p 273 called 'When Can We Ignore Symmetrization and AntiSymmetrization?')

The point is that the the type of effect Cox tried to popularize, is in fact completely negligible in practice, even if quantum mechanics, as we currently formulate it, is exactly theoretically correct. But he did link to lecture notes where this point was made explicit to ~50 decimal places in his first post on the thread (several weeks ago)
SecularSanity
#92
Jan12-12, 07:19 PM
P: 27
Quote Quote by James_Sheils View Post
Hello,

I am the author of the review "Double Twit Experiment – What Brian Cox Gets Wrong", as linked by others.

In short, I think presentations like Cox's contributes to a social game that people play, to impress and stupefy. But not to understand.
Bravo, well said. I enjoyed your review. The internet is plagued with so-called wisdom. For those of us who are interested in science, but wish to avoid the pseudo-junk altogether, can you tell us how to find trusted sources?

Wikipedia can be a good starting point, right? From there you can check all the references to see if the authors are from a university, research facility, or published in a reputable journal. Peer reviewed is more reliable and clearly, arXiv is not peer reviewed. It can contain some dubious e-prints but most of the authors care about what they write. If the website ends with .gov or .edu it’s probably a good source, right? Can you think of anything thing else to add?

List of Scientific Journals

How the Scientific Peer Review Process works

Misconceptions about science

What is Science?

P.S. If you’re such a stickler, here’s a suggestion for your next write up.

Why does a photon slow down in a medium?

There are tons of explanations out there. Here is ZapperZ’s explanation from in here and another from yahoo. Is either of these explanations accurate? If not, then perhaps you could provide a better one on your blog.

http://www.physicsforums.com/showpos...93&postcount=4

http://answers.yahoo.com/question/in...8084206AALZBC5
atyy
#93
Jan12-12, 11:42 PM
Sci Advisor
P: 8,658
Let me add that although the topic was first introduced in scenario where the effect is so small as to be practically negligible - the antisymmetry of fermionic wavefunctions that Cox talks about is very important. Matter would not be stable without it, nor neutron stars exist.

http://rmp.aps.org/abstract/RMP/v48/i4/p553_1
http://www.astro.umd.edu/~miller/tea.../lecture17.pdf
James_Sheils
#94
Jan13-12, 09:12 AM
P: 3
Quote Quote by SecularSanity View Post
Bravo, well said. I enjoyed your review. The internet is plagued with so-called wisdom. For those of us who are interested in science, but wish to avoid the pseudo-junk altogether, can you tell us how to find trusted sources?
Thanks for your kind words.

I agree that Wikipedia is a good starting point. Contrary to popular opinion, Wikipedia has a very high fidelity, in physics at least. I hear from specialists in other fields, such as art history, that the pages do not generate enough interest from editors to be reliable. However, in physics there seems to be a good supply of specialist contributors. The only disadvantage I have found is that for a non-specialist, the pages can be difficult to understand. But Wikipedia is a reference source, not an educational program.

I agree with what you say about the other sources, but would always read them with a skeptical mind. As I mentioned in the article, I think the best source for basic physics comes from Walter Lewin's MIT course.

As for the photon question, that's a pretty difficult one to answer, and I can't claim to fully comprehend all the details of modern theory!

I think the explanation you linked was right to avoid single atom explanations, but did not address the faulty assumptions in the question.

As the Double Slit Experiment aims to elucidate, we are not able to measure what happens between a photons emission and its arrival without changing the conditions sufficiently to alter the experiment. And the double slit experiment summarized the very counter-intuitive results concerning detection of photons. They arrive as particles, but do not seem to behave as particles on their journeys.

Encapsulated in the Copenhagen Interpretation of QM is a policy of not trying to speculate about 'where the photon goes' from source to detector.

We might have some mathematical equipment to calculate the probabilities of where the photon might end up, but we don't (or can't) know which path it took. Indeed, QED calculations assumed you need to consider every permissible path to determine the probabilities. So we can't appeal to the mathematical calculations for a satisfactory answer.

Thus, to as 'why' and expect a deterministic 'then the photon does this...' type of narrative asks too much of quantum mechanics.

But, the question could be answered by describing why the extra calculations for the material seems to delay the probability of a photon's arrival, compared with it traveling through empty space. I don't have sufficient quantum mechanical answer for this!
Ken G
#95
Jan13-12, 10:07 AM
PF Gold
P: 3,133
Quote Quote by James_Sheils View Post
I am a maths and physics graduate who has taught physics in secondary schools in the UK for around 6 years. During this time, I've thought quite carefully about which parts of scientific inquiry are worth teaching - which ideas and skills are valuable.
So have many of us. Do you recognize that this practice generates in you a number of opinions, that can be expressed without automatically assuming yours is the complete and final truth of the matter? The most important element of the art of advancing an opinion is the high regard for decorum, civility, and the right to respectfully disagree. Polemic diatribes are both easy, and tempting, but often limit their impact to a relatively small set of die-hard afficionados.
Most important of all, any citizen will benefit from understanding the process of scientific thinking. The role of evidence in falsification, what constitutes a scientific theory, how logic is utilized to determine consequences of a theory, the imaginative guesses that bring about new theories. All of this equips a person with thinking skills and understanding they can apply to enrich their lives, and their understanding of the latest research.
Absolutely, essentially any science educator would agree with that. The issue is, does each person who gets on the internet for a half hour or hour presentation need to feel responsible for all that, or is this more logically the mission of the science educator in the classroom setting?
Values to extract from this include: anti-authoritarianism, fallibilism, logical analysis, philosophical reflection and courageous imaginations.
I agree completely, and indeed from your comments on Dr. Cox I formed the opinion that you are most likely both very capable, and very dedicated, in your science education mission. You probably teach very well, and linked to others who do also. But none of those facts actually justify that vitriolic critique. Not everyone needs to adopt the same mission that you would, in order to be considered of value to science in some objective or demonstrable way. In short, they do not necessarily need to submit to your judgement of their performance. The question is, what audience is your criticism intended for? If you want Brian Cox to pay heed, the tone would rule that out-- I doubt he would read past the first few paragraphs. If your goal is to get people who like to listen to him to boycott him and listen to others that meet with your approval, I doubt you'll have much success-- those inclined to agree with you have probably already formed a similar opinion and don't seek out Dr. Cox's presentations, and those who like them will most likely not be dissuaded, because they simply won't agree with you. If you want to reach that crowd, I think you'd do better with your own presentation-- enlighten and entertain in your own way, achieving those goals that you value, and reach that clientele in that manner. That would accomplish the same goal, but more effectively than a largely unfocused critique-- it's easier to teach than to unteach.
So what to do with a 1 hour presentation? Now, i'm sure there will be much noise about how producers won't agree to programs that present these 'old' ideas. But Cox seems to command a lot of respect - they have already agreed to let him give a one hour lecture with a blackboard.
And this is the fundamental flaw in your position. Here you suggest that your goal is to convince Brian Cox to use his hour differently. Do you really think the way you presented your position is likely to accomplish that? Your comments are not even directed to Dr. Cox, they are directed to people who would listen to him. So your goal is clearly not to get Dr. Cox to use his hour more effectively, which would be a constructive goal (though presumptuous), it is to get those who would listen to his hour to avoid it or join in the Brian-bashing. How is that going to teach people Newton's laws?

What's more, you are overlooking the fact that there may be a reason that Dr. Cox is getting this hour (and a blackboard!), and neither you nor I are-- he has proven the ability to entertain and energize his viewers. Personally I think I could put together something that would be entertaining and enlightening also, which you might find less occasion to criticize if we share similar educational values, but I'm not going to get the opportunity to reach such a huge audience. I'm just not, the issue is moot. So I can see value in a certain trade-off there-- yes, perhaps there is an overemphasis on what is titillating rather than what is good basic science, but it's not such a bad exchange to get these ideas out there to people, to help them see that scientists are not just in ivory towers discovering arcane looking equations that somehow helps us build better iPads. Instead, we are getting glimpses deep into the workings of our reality, and getting quite amazed in the process, and we are inclined to want to share some of that experience with a larger audience.
It is disappointing that he has decided to present something so esoteric, yet mostly rely on intellectual intimidation and argument from authority to establish the results. Sure, he tried some underrehearsed explanations and demonstration, but the material was far too broad for even the greatest of educators to do a good job.
OK, so maybe not everything he did worked as well as it could have, and maybe he can learn some lessons for next time. He probably knows that, or if he doesn't, a simple constructive comment might be all that would be needed. What's the purpose behind all the bashing? That's what I really think you should look at more closely, what is really pushing your buttons here? For example, why do you think that his primary motivation is to make himself feel smart? I think it's pretty clear what his primary motivation is, it is to share with others some of the amazing glimpses he feels he has gotten into our reality. Of course it's also fun to feel smart, and of course it's also a rush to be able to entertain, I hardly think we can criticize the comedian for liking to hear a house full of laughter!
Most dangerous of all, it encourages already arrogant students to presume they have understood an idea, when they have merely remembered some impressive words. I have met many students who have tried to explain black holes to me, or something about string theory. I always fell a sympathy that these curious minds have been duped by yet another shallow presentation of scientific inquiry.
But this is unavoidable. Do you really think this never happens to your students? At least the people in question are interested in something that connects with science-- the alternative may be the absence of any of that.
Or, there are the adults I meet who tell me they are 'really interested in science' and then ask me about m-theory, or black-holes.
OK, but the point is, maybe they would not have said they were interested in science and then talked about Newton's laws! That's what you have to include in your calculations. I have had some small success getting people jazzed about Newton's laws, but the fact is, it's just a lot harder-- the number of people who are going to feel that way is just less than it is for the wilder stuff. That I believe is Dr. Cox's primary motivation for his subject selection, not the desire to feel smart.
"Why do some object float in water?" I ask them. Most of them have nothing to say about this. Now I ask you, if a person cannot connect the perceptions of their experience with scientific patterns, what is the possible value in describing the theoretical intricacies of the latest research?
It is simply not an either/or propositon.
In short, I think presentations like Cox's contributes to a social game that people play, to impress and stupefy. But not to understand.
And there's certainly some truth to that. This is a valid criticism that can be raised, but it doesn't make what Dr. Cox is doing worthless or damaging to people's minds, they come to it because it gives them something they like, and it is certainly connected with science. I think it does a lot more good than harm, and if it could be improved in some way, who among us could escape that criticism? None of this justifies that vitriol, even though there are valid aspects to the points you raise.
SecularSanity
#96
Jan13-12, 12:14 PM
P: 27
Quote Quote by James_Sheils View Post
Thanks for your kind words.
And thank you for the reply.

Sorry, but I couldn’t resist. However, I’ll refrain from linking the video.

You’re young, handsome, and your accent makes you sound intelligent, but here’s some womanly advice. Critics should cover their own butt and stick to the bare necessities, don’t cha think? What’s up with the banana?

Thanks again.

Cheers!
unusualname
#97
Jan13-12, 02:02 PM
P: 661
There are several science programs on bbc tv and radio, some more populist than others. Brian Cox's are more at the entertainment end of the scale, but I for one quite enjoyed the four episodes in The Wonders of The Universe series, for example (even with the ott music in the first series of broadcasts).

The target audience is certainly not elitist types, and you should probably avoid these programs if you have 'a stick up your bottom' attitude to such populist science.

There're always the online lectures of Susskind for example if you want a dry Diracesque introduction to QM. Feynman's style can be seen in the Messenger Lectures http://www.microsoft.com/education/f...&c1=en-us&c2=0 (requires silverlight - microsoft compatible only) , I personally doubt his double-slit lecture (lecture 6) will enlighten the uninitiated any more than Cox's attempts.
cpsietc
#98
Jan30-12, 04:35 PM
P: 1
I am still surprised by what was said about the consequences for electrons throughout the Universe of warming a diamond in one's hand. For a start, diamond is an electrical insulator with a large energy gap of more than 5 electron volts whereas the average thermal energy of an electron at room temperature (3/2 kT) is only 0.04 eV. Increasing this by at most 5% falls far short of the minimum needed to cause any electrons to jump into higher energy levels (assuming the "box of carbon atoms" contains no impurities); it will just cause the atomic lattice to vibrate a bit more.

Ignoring anomalies (if any?) caused by relativistic effects such as electron creation and annihilation or the lack of any FTL signals, the Pauli Exclusion Principle does of course hold for all electrons everywhere, regardless of whether they are pictured as bound to nuclei, zipping along on their own at almost the speed of light or just drifting about in a plasma.

The double-well example is fine as far as it goes, but only bound states corresponding to fixed separations of the wells are considered. In a gas, unless two nuclei are part of the same molecule, they will not usually remain a fixed distance apart and therefore will not give rise to a set of stationary states with exact electron energy levels.

I think I'm right in saying that at present, the conventional view of astronomers is that a good 90% of ordinary (baryonic) matter (nearly all H) is in the plasma state. If this is correct, then around 90% of all electrons are not bound to any nuclei at all!

When two of these "free" electrons are in relative motion, there could always be some inertial observers for whom their energies are equal alongside others for whom they are unequal. Therefore, I do not see how it is possible in general to substitute rules about electron energies for the basic requirement of antisymmetry of the electron component of the total wave-function, a property which is both observer-independent and permanent.

I agree of course that quantum mechanics does imply that "everything is connected to everything else" through entanglement, but I don't think the scenarios chosen to illustrate this amazing idea were at all convincing.
pds
#99
Feb2-12, 08:30 AM
P: 1
Quote Quote by becox View Post
Seems to be some confusion here about the Pauli Principle. Jeff Forshaw and myself write about it in detail in our book The Quantum Universe, chapter 8. The essential point is that two widely separated hydrogen atoms should not be treated as isolated systems. If you'd like to see how we teach this to undergraduates in Manchester, have a read of this:

http://www.hep.manchester.ac.uk/u/fo...le%20Well.html

But I do also recommend our book, because the argument is extended to explain semiconductors.

doodyone - in particular, I suggest you pay close attention, especially if you're an undergraduate. You might up your degree classification!

Brian
If it is the case that electrons occupy slightly different energy levels, then wouldn't it follow then the spectra would show similar subtle variations? In Chapter 11 of the Quantum Universe, it mentions the "Lamb Shift" and this is accounted for by factoring in particle interactions within the atom. Wouldn't this Lamb Shift be undetectable if there is also a certain "arbitrariness" about the actual energy levels? Or is it a question of scale? Or maybe, I haven't understood!
mc^2
#100
Mar1-12, 08:06 AM
P: 4
Quote Quote by dgwsoft View Post
http://www.bbc.co.uk/programmes/b018nn7l

I did enjoy Brian Cox's program on quantum mechanics last night, but one bit left me thinking "no, that's not right!".

The gist of it was that all the electrons in the universe have to be in constant communication to ensure that no two of them are ever in the same state. If he changed the energies of electrons in a diamond, by heating it in his hand, all the other electrons in the world would have to adjust their energies too.

I think this may have been an attempt to show that entanglement follows from the Pauli exclusion principle, but was it a simplification too far?

The Pauli principle confused me when I first heard it at school: did it mean that no two hydrogen atoms in the universe could be in their ground states simultaneously? I have always understood, since then, that it doesn't mean that, because which proton the electron is bound to is part of its state. So "in the first energy level around this proton" is a different state from "in the first energy level around that proton".

The exclusion principle states that no two electrons can be in the same *state* not, as Cox seemed to be implying, that they may not have numerically the same energies. That is not forbidden as far as I know. We would not see nice spectral lines from billions of hydrogen atoms all making the same state transition at the same time, if it was.

I now know there is a deeper explanation of the exclusion principle, namely that the multi-particle wave-function of a half-integral spin particle is antisymmetric, and that means the probability of finding two of them in the same place is zero. So OK, Pauli and entanglement are connected. But I always like a simple explanation if one is available. What does the panel think? Did what Cox said amount to a good explanation for a general audience, or does it risk perpetuating a misunderstanding?
If all the electrons in the universe have to be in constant communication to ensure that no two of them are ever in the same state, then this may contradict the principle of conservation of energy. If we control a material in such a way that it's electrons would occupy most of the lowest possible energy states - this would indicate according to Cox explanation that all the other electrons in the universe would have a lower probability to occupy these lowest energy states and a higher probability to exist in higher energy states. This cannot be correct.
Ken G
#101
Mar1-12, 08:54 AM
PF Gold
P: 3,133
I'm sure Dr. Cox understands conservation of energy. His viewpoint is simply that if there is a probability that an electron will be in an energy state, this affects the accessibility of the state, so if I remove energy from an electron such that it would have a higher probability of moving into some state, and there is already some probability of an electron being in that state, the fact that all electrons are entangled (by their indistinguishability) implies that they are all "affected" in some sense. I think the real problem here is that Dr. Cox's words are being overinterpreted-- the key point is that electrons are identical, and thus entangled. Hence, any counterargument that first pretends the electrons have separate identities is already missing the point. Perhaps he was not careful to make this distinction-- it is crucial that all language like "this electron" or "that electron" be avoided when one is discussing Pauli exclusion.
mc^2
#102
Mar1-12, 10:34 AM
P: 4
"I think the real problem here is that Dr. Cox's words are being overinterpreted."

I agree; the real problem is to try to find the right words to describe the situation in terms of a layman's frame of reference while minimizing the possibility of misleading them.
Ken G
#103
Mar1-12, 10:36 AM
PF Gold
P: 3,133
Exactly. I'm sympathetic of that problem-- we might not all agree with how Dr. Cox negotiates it, but we're all in glass houses on that score. If one person thinks Cox is doing more harm than good by stressing the more mystical elements, another can say he is doing more good than harm by simply getting people interested in some of the more fascinating new elements of what we have discovered. The fact is it might take centuries before we really understand what all this means, remember Feynman's wonderful words about quantum mechanics:
"We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it.... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem."
unusualname
#104
Mar1-12, 12:22 PM
P: 661
Quote Quote by mc^2 View Post
If all the electrons in the universe have to be in constant communication to ensure that no two of them are ever in the same state, then this may contradict the principle of conservation of energy. If we control a material in such a way that it's electrons would occupy most of the lowest possible energy states - this would indicate according to Cox explanation that all the other electrons in the universe would have a lower probability to occupy these lowest energy states and a higher probability to exist in higher energy states. This cannot be correct.
It's only a problem if our ability to "control" the material is inconsistent with global unitary evolution. ie is Brian's Cox's choice to rub the diamond any different from a diamond being shifted around underground by a natural process such as an earthquake?

When a supernova explodes it undoubtedly has a significant effect on the state vector of the universe, but it ought to be consistent with unitary evolution according to the Schrödinger Eqn.

Of course, this isn't an issue if you don't believe in macroscopic wavefunctions, especially one describing the entire universe, but in that case you need corrections to the current standard formulation of QM.

The no-communication theorem says a measurement in one place cannot change the probability distribution of any observable outside the future light-cone of the first measurement.

But science has no consensus on the nature of free-will, and such theorems may not apply. However, if free-will does break unitarity in a deterministic way then we may also need a reformulation of relativity since we would otherwise have the possibility of causal paradoxes.
Ken G
#105
Mar1-12, 12:35 PM
PF Gold
P: 3,133
Quote Quote by unusualname View Post
Of course, this isn't an issue if you don't believe in macroscopic wavefunctions, especially one describing the entire universe, but in that case you need corrections to the current standard formulation of QM.
Yet that's a pretty small "but". It is a "but" that is more or less the defining quality of science!
The no-communication theorem says a measurement in one place cannot change the probability distribution of any observable outside the future light-cone of the first measurement.
Actually, I don't think the theorem can quite say that. A probability distribution is always contingent upon what you already regard as known, versus what unknowns you are simply averaging over. So changes in knowledge, here, can change probability distributions about distant events, reckoned here, without any causality violations (as in EPR type experiments). Hence, you can reckon that the probability distribution somewhere else, outside your light cone, can be changed by your measurement-- it is just the physicists outside your light cone that cannot know that. It's a question of what any probability distribution is contingent on.
unusualname
#106
Mar1-12, 01:21 PM
P: 661
Quote Quote by Ken G View Post
The no-communication theorem says a measurement in one place cannot change the probability distribution of any observable outside the future light-cone of the first measurement.
Actually, I don't think the theorem can quite say that. A probability distribution is always contingent upon what you already regard as known, versus what unknowns you are simply averaging over. So changes in knowledge, here, can change probability distributions about distant events, reckoned here, without any causality violations (as in EPR type experiments). Hence, you can reckon that the probability distribution somewhere else, outside your light cone, can be changed by your measurement-- it is just the physicists outside your light cone that cannot know that. It's a question of what any probability distribution is contingent on.
Yes, obviously I meant the probability distribution wrt to the observer observing the observable.
SecularSanity
#107
Mar7-12, 10:11 AM
P: 27
Disclaimer: Pre-coffee

I thought that quantum entanglement had to be created by direct interactions between subatomic particles, but this guy says that the entire universe is in this entangled state. I don’t know but I don't like it.

Was Brian Cox Wrong?
This entangled state, which is the whole universe. Essentially, that will choose a particular state for the electron here, which corresponds for a particular state in the electrons on Andromeda.
However, I did find a poor quality video of John Bell stating, “You cannot get away with saying that there is no action at a distance. You cannot separate off from what happens in one place from what happens in another. They have to be described and explained jointly.”

Bell Himself Explaining the Implications of his Inequality

Does it prove that the entire universe is in an entangled state simply because there are methods of creating entanglement? Is quantum nonlocality equivalent to entanglement? Aren’t there limits to quantum nonlocality, e.g. Tsirelson's bound?

BTW, doesn’t he look like Johnny Depp as Willy Wonka?

“Oh, you should never, never doubt what nobody is sure about.”~ Willy Wonka
sheaf
#108
Mar7-12, 03:45 PM
P: 203
As far as I know, the discussions on this issue are still ongoing. I thought I'd describe the situation from the viewpoint of my armchair.

Regardless of the discussions regarding whether Brian Cox should perhaps have said “quantum state”, rather than “energy level” in the TV show, this whole discussion has made me try to understand the applicability of the concept of entanglement to a situation such as this. Certainly Cox and Forshaw in their book did have entanglement in mind in connection with this issue, since they state:

There is only ever one set of energy levels and when anything changes (e.g. an electron changes from one energy level to another) then everything else must instantaneously adjust itself so that no two fermions are ever in the same energy level.
The idea that electrons ‘know’ about each other instantaneously sounds like it has the potential to violate Einstein’s Theory of Relativity. Perhaps we can build some sort of signalling apparatus that exploits this instantaneous communication to transmit information at faster-than-light speeds. This apparently paradoxical feature of quantum theory was first appreciated in 1935 by Einstein in collaboration with Boris Podolsky and Nathan Rosen; Einstein called it ‘spooky action at a distance’ and did not like it. It took some time before people realized that, despite its spookiness, it is impossible to exploit these long range correlations to transfer information faster than light and that means the law of cause and effect can rest safe.
Entanglement does indeed allow quantum measurements to display “instantaneous” influences, but no information can be transmitted using this mechanism. But, how would you go about applying entanglement to the scenarios they're discussing?

The model Cox and Forshaw are using is the double rectangular potential well. This model is described here. The energy eigenstates of a single rectangular well are split into pairs of energy eigenstates with very closely spaced energy eigenvalues. One member of a pair is a wavefunction with odd reflection symmetry about the origin and the other has even reflection symmetry.

We now populate the double well system with a pair of fermions. For simplicity, they could be spinless electrons, which would have to be in different states to respect their fermionic nature. As an example, they could be in each of the two lowest energy eigenstates, so the system state would be
[tex]|\Psi \rangle={1\over{\sqrt{2}}}(|E_1 \rangle |E_2 \rangle-|E_2 \rangle |E_1 \rangle) \ \ \ (0)[/tex]

The sort of question we would like to ask is whether or not there is entanglement between quantities measured in the left hand well and quantities measured in the right hand well?

Conventionally, entanglement questions would be treated by decomposing the full Hilbert space in the form
[tex]{\mathcal{H}=\mathcal{H_{L}}\otimes\mathcal{H_{R}}}[/tex]
For example, in the "classic" EPR entanglement scenario, this sort of decomposition is clear - [itex]\mathcal{H_{L}}[/itex] is the two dimensional Hilbert space of spin states of a LH-travelling spin 1/2 decay product of a spin 0 singlet state, and [itex]\mathcal{H_{R}}[/itex] the RH-travelling equivalent.
For any pure state [itex]|\Psi\rangle\in\mathcal{H}[/itex]I can choose an orthonormal basis [itex]\{|\Psi^L_{i}\rangle\}[/itex]for [itex]\mathcal{H_{L}}[/itex] and [itex]\{|\Psi^R_{i}\rangle\}[/itex]for [itex]\mathcal{H_{R}}[/itex] such that
[tex]|\Psi\rangle=\sum\limits_{i}\alpha_{i}|\Psi^L_{i} \rangle \otimes|\Psi^R_{i}\rangle \ \ \ (1)[/tex]
here [itex]\alpha_{i}[/itex] are a bunch of coefficients (which can be chosen to be real and positive). This is the Schmidt decomposition. Given this, a good measure of entanglement - namely the entanglement entropy - can be defined as
[tex]S_{A}=-\sum\limits_{i} \alpha_{i}^2log \alpha_{i}^2[/tex]
The higher the entropy of a state, the more entangled it is.

Now trying to apply this to the double well scenario, we immediately run into trouble, because it is not clear how to perform the decomposition [itex]\mathcal{H}=\mathcal{H_{L}}\otimes\mathcal{H_{R}}[/itex].

If we want to ask the question "is there any entanglement in the double well model?" a key problem is that the two electrons in the system are indistinguishable fermions, so when one tries to construct a two particle state, it must be antisymmetric in the two electron identities. For example, ignoring spins, the position wavefunction representation of a two particle state might be constructed from single particle wavefunctions as:
[tex]\Psi(x_1,x_2)={1\over{\sqrt{2}}}(\psi(x_1)\phi(x_2)-\psi(x_2)\phi(x_1)) \ \ \ (2)[/tex]
An n-particle state would be the same, except it would be a normalised sum over all the even permutations of [itex]x_1,x_2,...x_n[/itex] minus all the odd permutations. Such states/wavefuctions are sometimes called Slater determinants.

Now, there is a fairly large body of literature around which discusses entanglement in multi-fermion systems. However, much of it is concerned with treating entanglement in systems appropriate to quantum computing - for example entanglement between quantum dots. In these cases, the mere fact that you cannot express a two particle state as a product state, but rather a difference of such, like in (2), is deemed *not* to constitute entanglement. For example Shi defines entanglement in a multi-fermion system to be the inability to express the state (by choosing a suitable single particle basis) as a single Slater determinant (like (2) for the case of 2 particles). In other words, a state is *not* entangled if you *can* express it as a single Slater determinant.

Adopting this definition would immediately rule out the double well energy eigenstate (0) as being entangled – it's a single Slater determinant. But is this criterion really appropriate for the double well discussions? As far as I can tell, the reasoning behind considering (2) as unentangled has immediately made an assumption regarding remote exchange correlations, namely that they can be ignored due to the large separation. Schliemann, whilst arguing the case for using Slater rank as the entanglement criterion states ( where I've substitued the wavefunctions in (2) for his notation) states:

However, if the moduli of [itex] \psi(x_1), \phi(x_2)[/itex] have only vanishingly small overlap, these exchange correlations will also tend to zero for any physically meaningful operator. This situation is generically realized if the supports of the single-particle wavefunctions are essentially centered around locations being sufficiently apart from each other, or the particles are separated by a sufficiently large energy barrier.
So by construction the double-well electrons will be unentangled if we use Slater rank as the entanglement criterion, so this doesn't really help.

There are other approaches to entanglement of fermions, such as the one discussed by Zanardi et al(http://arxiv.org/abs/quant-ph/0308043). They state that it is meaningless to discuss entanglement of a state

without specifying the manner in which one can manipulate and probe its constituent physical degrees of freedom. In this sense entanglement is always relative to a particular set of experimental capabilities.

This approach avoids the need to perform the decomposition (1) and instead focuses on the properties of various observables on the state being checked for entanglement. The criterion of Zanardi et al seems quite complex, but its essence is captured in a simpler formulation described in a reference by Kaplan, to which I was referred by PF user Morberticus. Basically the question of whether or not a state is entangled is asked *with respect to a pair of observables* [itex]A[/itex], and [itex]B[/itex]. A state [itex]\Psi[/itex] is deemed entangled with respect to [itex]A[/itex], and [itex]B[/itex] if the covariance function
[tex]C_{AB}\equiv \langle\Psi|AB|\Psi\rangle-\langle\Psi|A|\Psi\rangle\langle\Psi|B|\Psi\rangle \ \ \ (3)[/tex]
is non zero.

However, to apply this to our double well system, we need to be able to define the operators A and B appropriate to "making an energy measurement in the LH well" and "making an energy measurement in the RH well".

The only energy operator I can think of that would be consistent in the two-fermion system would be the total energy operator [itex]E_1+E_2[/itex]. This is symmetric in permutation of the electron identities 1 and 2 as it should be. However, to evaluate (3) to check for entanglement, I'm still left with the job of defining a "left hand well energy operator" [itex]E^{A}_1+E^{A}_2[/itex] and a "right hand well energy operator" [itex]E^{B}_1+E^{B}_2[/itex].

I've no idea how to do such a thing, and I'm inclined to agree with the conclusion of Arnold Neumaier in his answer to my question on physics stackexchange (http://physics.stackexchange.com/que...le-well-system), namely that there is no simple way to progress this discussion !


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