Pauli Exclusion Principle and Brian Cox

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Pauli Exclusion Principle and Brian Cox

Postby BeerBottle » Wed Dec 21, 2011 11:32 am UTC

Just watched Prof. Brian Cox's BBC TV special 'A Night with the Stars' - (think Royal Institution Christmas lecture with celebrities being set on fire). Most of it was more than enjoyable but he mentioned something very starge about the Pauli Exclusion Principle which I'd never heard before. After describing how the PEP results in the electronic structure of atoms which then leads to chemical reactivity (which I'm fine with) he then mentioned that the PEP means that all electrons in all atoms everywhere are in slightly different energy levels. He even went on to say that if he changed the energy of electrons in a diamond he was holding (by warming it up) then this led to instantaneous changes in all electron energy levels in the universe. At this point I was all W T F ???

Now Brian Cox is genrally a pretty solid guy and does some great tv programs. I'm don't think I'm misuderstanding him as he was very explicit (see link below to watch for yourself) - but to me this just sounded crazy. Surely two isolated H atoms have the same electron energy levels? Don't they? Otherwise couldn't you use PEP to send FTL messages by changing the energy level of one H atom and measuring the other one?

So the question is - is Brian Cox talking rubbish?

For the geographically privilaged you can watch the whole program on BBC iplayer:
http://www.bbc.co.uk/iplayer/episode/b0 ... the_Stars/
Less fortunate souls can see it on youtube:
http://www.youtube.com/watch?v=4f9wcSLs ... re=related

The bit I'm talking about is about 35 minutes in.

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Re: Pauli Exclusion Principle and Brian Cox

Postby eSOANEM » Wed Dec 21, 2011 12:42 pm UTC

I thought the exact same thing when I saw it. As far as I understand it, he's right in that PEP means that there cannot be any other electron in the universe in exactly the same energy level, however that would not require that changing one electron's energy affects all electrons in the universe (each change could only directly affect at most one other electron (that in the energy level you wish to jump to) which is very different from "all of them") or that it happens instantaneously (it would be conceivable that such information would be carried at c away from the changed energy level and that PEP would only apply locally).

As I said, I am not a physicist, but that statement certainly seems at odds to everything I've read elsewhere.
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Re: Pauli Exclusion Principle and Brian Cox

Postby SlyReaper » Wed Dec 21, 2011 1:29 pm UTC

I thought the PEP simply forbade any two particles being in the same quantum state. Surely if two electrons are in measurably different positions, they could have the same energy level without violating the PEP?
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Re: Pauli Exclusion Principle and Brian Cox

Postby Rococo » Wed Dec 21, 2011 2:02 pm UTC

He's absolutely, incontrovertibly right. Here is a good example of the concept, using a double-well potential: http://www.hep.man.ac.uk/u/forshaw/Bose ... 0Well.html

The key point is that we think of an electron in an atom as being 'isolated', but that's not how quantum mechanics actually works: there is no such thing as a localized energy eigenstate and no such thing as a truly isolated system.

Imagine you have a universe that is n well-separated hydrogen ions, milling about. There are n energy eigenstates that are nearly the same, but slightly separated. Now, the key thing to realize is that the rules of fermions are such that they must be entangled. Put in two electrons and they cannot be in the states /0> and /1>, but must be in /01>-/10>. Once you keep adding until you have n electrons, you'll end up with an entangled mess such that measuring one electron near a particular atom excludes the rest from being there, and the same applies to energy.

This is all assuming that we're in stationary states. If you suppose that in the real world things are being localized pretty often by measurement-like processes, it gets a bit messier. As the simulation at the bottom of the linked page shows, you'll start with a localized packet, which is a sum of energy eigenstates, and the probability will slowly tunnel out to all the other potentials in the universe. And of course all this influences every other particle's wavefunction, since they are all entangled.

Why can't this be used to send messages? It might be easier to take a proposed way of sending messages and shooting it down, but I'll try to give a useful example. Let's take the simplest possible system: the universe has one electron and one proton, and all of a sudden another proton pops into existence far away. As a result, the electron's bound state goes from 100% near one proton to equally shared between the two- and this happens instantly. So at first glance, maybe you'd think that you could immediately look for the electron near the first proton, and you'd have a 50% chance of knowing that the second proton was created immediately no matter how far away it was. But here's the trick: when the second proton appears, the stationary states of the electron instantly shift, but its actual wavefunction does not immediately change. Again, as the simulation shows it takes time for the wavefunction to adjust by tunnelling through to the new potential, and that time is longer when the new potential is farther away. One could propose that this tunnelling time is such that you must wait at least distance/c for any measurable disturbance at the original location. That said, there is a tension between tunnelling and causality: http://en.wikipedia.org/wiki/Hartman_effect , and I'd be indebted to anyone who knows more about the issue and can clear it up a bit. But a key point to take away is that instantaneous shifting of the eigenstates of a system does not translate to instantaneous shifting of the actual state itself.

edit: corrected a key mistake about the entanglement of the state.

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Re: Pauli Exclusion Principle and Brian Cox

Postby Minerva » Thu Dec 22, 2011 1:46 pm UTC

It's one of those quantum-mechanics phenomena that can happen and will happen, but it's so improbable, so subtle or so insignificant that it is interesting and fun and beautiful from a theoretical physics point of view but utterly meaningless in terms of ever having any practical effect on anything in the macroscopic universe.

In that regard, for example, it's just like Brian's example of the diamond spontaneously disappearing from its box and moving somewhere elsewhere within the universe.
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Re: Pauli Exclusion Principle and Brian Cox

Postby Minerva » Thu Dec 22, 2011 1:50 pm UTC

I am sure he checked that this was solid before he said it, because we all know that everybody would be calling him a twat or a "nobber" or (insert name here that he calls crackpots) if he slipped up, and he would never hear the end of it.
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Re: Pauli Exclusion Principle and Brian Cox

Postby Gigano » Thu Dec 22, 2011 8:39 pm UTC

I was a bit baffled by this myself, but after reading Rococo's explanation it does seem to make sense to me. At least in the context of quantum mechanics; nothing truly makes sense in quantum mechanics.
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Re: Pauli Exclusion Principle and Brian Cox

Postby Charlie! » Thu Dec 22, 2011 9:12 pm UTC

Why accept something that's probably wrong, jut because someone's smart, or the subject is complicated? The claim seems pretty nobberish, anyhow. The exclusion principle keeps electrons from being in the same state, but many different states can have the same energy. For example, in a system with left-right symmetry, an electron moving to the left had better be able to have the same energy as one moving to the right, or tiny Erwin Schroedinger will hunt it down and punch it in the face. On the other hand, you could say that the pauli exclusion princple plays a role in reducing degeneracy, since to get electrons with the same energy you have to find symmetrical states, or tune a parameter so two states have the same energy, you can't just put all electrons in the same state.

And no, antisymmetrizing states with the same energy doesn't somehow change their energy. (edit: Oh wait, it actually can in the case of two particles that repel or attract each other. But it doesn't introduce any asymmetry into previously symmetric situations, so the key point remains.)
Last edited by Charlie! on Sat Dec 24, 2011 1:46 am UTC, edited 2 times in total.
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Re: Pauli Exclusion Principle and Brian Cox

Postby some_dude » Fri Dec 23, 2011 1:16 am UTC

I've always trusted Griffith's explanation that the PEP in practice only apply to identical fermions with overlapping wavefunctions. This is correct, right?

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Re: Pauli Exclusion Principle and Brian Cox

Postby mfb » Fri Dec 23, 2011 5:30 pm UTC

It applys to all fermions of the same type. And all eigenstates are non-zero everywhere, so they always overlap. However, for large distances you do not have to care as the overlap is so small that it does not matter.

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Re: Pauli Exclusion Principle and Brian Cox

Postby Rococo » Sat Dec 24, 2011 10:50 pm UTC

Charlie! wrote:Why accept something that's probably wrong, jut because someone's smart, or the subject is complicated?

You shouldn't, of course.

The claim seems pretty nobberish, anyhow. The exclusion principle keeps electrons from being in the same state, but many different states can have the same energy. For example, in a system with left-right symmetry, an electron moving to the left had better be able to have the same energy as one moving to the right, or tiny Erwin Schroedinger will hunt it down and punch it in the face. On the other hand, you could say that the pauli exclusion princple plays a role in reducing degeneracy, since to get electrons with the same energy you have to find symmetrical states, or tune a parameter so two states have the same energy, you can't just put all electrons in the same state.

And no, antisymmetrizing states with the same energy doesn't somehow change their energy. (edit: Oh wait, it actually can in the case of two particles that repel or attract each other. But it doesn't introduce any asymmetry into previously symmetric situations, so the key point remains.)

This is a very interesting point that deserves to be addressed. I think we can cover it adequately by just looking at a two-electron, double-well potential system. Let's call the two energy eigenfunctions /a> and /b>, which are slightly nondegenerate. Then the system state is /ab>-/ba>. On the other hand, we can change bases instead to a location basis, with the eigenstates /L> and /R> for an electron in the left or right well. Looking at the stationary states, the obvious definition for this is /L>=/a>+/b> and /R>=/a>-/b>, and if you do the substitution the position representation for the system is just /LR>-/RL>.

So here's the issue: you are right, in the sense that /L> and /R> have the same distribution over the energy eigenstates- or, if you prefer, the same average energy. That should make Schroedinger happy. On the other hand, if you actually measure the energy of one or the other, you force yourself into the energy basis, and in that case you will always find them unequal, with one electron in /a> and one in /b>. In a nutshell, the issue is that the position and energy operators, in this case, do not commute, so it is physically meaningless to talk about a electron that is both 'moving to the left or right' and 'has a defined energy value.' Whenever you measure energy, you will never get both electrons to have the same value, and you can't really talk about the energy they have when they are in some other state, so the only reasonable thing to say is that the electrons can never have the same energy.

Something I do agree with you and SlyReaper about: it actually doesn't seem to me like it is Pauli exclusion that is enforcing this. We could imagine two isolated systems which each have a simultaneous position and energy eigenstates. Then, if you could join these two systems together and put two electrons inside, you'd again have a state that looked like /LR>-/RL>, but now since /R> and /L> have the same energy you would really have two electrons with the same, well defined energy and no Pauli Exclusion problem. So what really keeps this from happening, I think, is the fact that the energy eigenstates for an electron are always going to be spread around all the possible attractive potentials in the system, so it is impossible to add another potential well to your system without your electrons trying to spread into it to lower their energies.

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Re: Pauli Exclusion Principle and Brian Cox

Postby Charlie! » Sun Dec 25, 2011 3:54 am UTC

Rococo wrote:So here's the issue: you are right, in the sense that /L> and /R> have the same distribution over the energy eigenstates- or, if you prefer, the same average energy. That should make Schroedinger happy. On the other hand, if you actually measure the energy of one or the other, you force yourself into the energy basis, and in that case you will always find them unequal, with one electron in /a> and one in /b>.

In the double well potential, there are no two states (with the same spin) that have the same energy, so it's impossible to replicate the main point of my post.

A situation that does have degeneracy is two free electrons, one moving left and one moving right, with momentum +1 and -1 along the x axis. So now you measure the energies of these two electrons, but since we already wrote down their momentum they're already in energy eigenstates, so nothing changes. Were the energies of these two electrons made different by the Pauli exclusion principle? No.

Other good systems with degeneracy would be a particle in a 3d box, or the hydrogen atom, or electrons in perfect crystals.
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Re: Pauli Exclusion Principle and Brian Cox

Postby Ormpskel » Mon Dec 26, 2011 2:43 am UTC

Charlie! wrote: Other good systems with degeneracy would be a particle in a 3d box, or the hydrogen atom, or electrons in perfect crystals.



The electron is a hydrogen atom has no inherent degeneracy, as there is only electron. You need multiple objects for there to be degeneracy, by its very definition. The electrons in a perfect, infinite crystalline lattice (which I assume you are describing) vary in their thermal energy, orbital quantum numbers and spin quantum numbers.

The boltzmann distribution curve shows that for a group of particles, there is a range of energies, with 0 have 0 energy, and a non zero number having energies way beyond the mean. Even if 2 electrons have the same energies (i.e they are in the same, degenerate 2p set of orbitals), they can have a different oribtal quantum number, thusly filling the 2px, 2py or 2pz orbitals. Lastly if they filled the same 2pz orbitals, they have differing quantum spin values, either spin up, or spin down. To do otherwise would create 2 negative charges in the exact region of space, which is logically impossible - they must repel each other. As energy can neither be created nor destroyed, it follows that the 2 particles must differ in 1 value - either their energy, or their description within the system.

The majority of this is better explained by cox, and the rest is explained better by Rococo, quantum entanglement not really being my cup of tea.

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Re: Pauli Exclusion Principle and Brian Cox

Postby Dopefish » Mon Dec 26, 2011 6:59 am UTC

The impression I'm getting is that we're discussing real systems where all of everything in the universe comes into play, and so you simply don't have an idealised lone hydrogen atom or 3d box or double well or other 'nice' solvable systems, and I'm not sure if theres any true symmetries that can be appealed to in real life.

You can have very good approximations of all those nice systems, but there would be tiny differences from reality due to having everything around that would perturb the energy levels in different ways (although both perturbations may be rediculously small and unmeasurable in practice).

In that light, I might be able to buy that every position gives rise to a slightly different environment (and hence energy), and by PEP no two fermions can be in the same position, and so the statement would be true if you could convince me that all those enviroments gave rise to a unique energy. I'm not convinced that you couldn't have two different states with the same energy in different places simply by chance though. Intentionally preparing things as such would probably be impossible since you can't control everything in the universe, but in the all of everything I feel like there would be some true degeneracy.

Even if it is technically right though, I don't like it since it's apt to muddle people who are learning QM and actually do get to deal with idealised situations (or even 'real' situations where the approximations are perfectly good), since the world of hypothetical systems you can definitely get degeneracy.

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Re: Pauli Exclusion Principle and Brian Cox

Postby eSOANEM » Mon Dec 26, 2011 8:58 am UTC

Dopefish wrote:In that light, I might be able to buy that every position gives rise to a slightly different environment (and hence energy), and by PEP no two fermions can be in the same position, and so the statement would be true if you could convince me that all those enviroments gave rise to a unique energy. I'm not convinced that you couldn't have two different states with the same energy in different places simply by chance though. Intentionally preparing things as such would probably be impossible since you can't control everything in the universe, but in the all of everything I feel like there would be some true degeneracy.


The other thing to think about is that, as you sit by your computer, the vast majority of the electrons around you are not in specific eigenstates but rather a superposition of such states. Once you raise the energy level of one electron, this eigenstate can no longer appear in the other superpositions. The thing then of course, is that that electron's state some time after its energy is raised again becomes a superposition of eigenstates and, until it's measured, many electrons' wave functions can include that eigenstate provided the overall wave function is such that only one can actually be in that eigenstate when measured.

It's then worth noting, that if you look at a note on a spectrum analysis chart, cut out one of the higher harmonics and then run the new wave through the speakers, it won't sound very different at all. Higher energy levels are like the higher harmonics (almost identical for infinite wells for example) and so, cutting out a higher one will not make much difference to the superposition of states the electron occupies as a whole.
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Re: Pauli Exclusion Principle and Brian Cox

Postby Charlie! » Mon Dec 26, 2011 8:34 pm UTC

Dopefish wrote:The impression I'm getting is that we're discussing real systems where all of everything in the universe comes into play, and so you simply don't have an idealised lone hydrogen atom or 3d box or double well or other 'nice' solvable systems, and I'm not sure if theres any true symmetries that can be appealed to in real life.

Even for real-life systems, there is absolutely no problem with having a nonzero probability density at identical energies - that is, identical energies are not treated any different than slightly different energies, if you have some other difference (like direction of momentum). And measuring the energy of an electron going left wouldn't change the energy of an electron going right. Even the messiness of real life does not salvage this idea.
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Re: Pauli Exclusion Principle and Brian Cox

Postby doogly » Wed Dec 28, 2011 6:57 am UTC

Seriously, degenerate eigenvalues are a thing, even for Hamiltonian operators.
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