Pauli Exclusion Principle

Discussion in 'Physics & Math' started by Harmony, Dec 27, 2011.

  1. Harmony Harmony Registered Senior Member

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    I watched a TV programme recently that was broadcast on BBC 2 on 18th December "A night with the stars". It was very entertaining and brought to life by Professor Brian Cox and celebrities in the audience. The broad topic was quantum theory and all seemed to make sense except a piece relating to the Pauli exclusion principle.

    The presenter had an uncut diamond and explained that when he warmed it up some of the electrons changed energy levels and as a result all of the electrons in all of the atoms in the universe had to adjust their energy levels because no two could be the same. Now I am familiar with the Pauli exclusion principle applied to individual atoms but this universe wide effect acting instantaneously was a surprise. Did anyone see the programme or can comment on this surprising effect?

    Harmony
     
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  3. prometheus viva voce! Moderator

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    As with all pop science books / programmes, Brian Cox was not being completely truthful. Consider a basic example instead of the uncut diamond - a particle in a box in one dimension. Normally, one says the sides of the box are at \(x = \pm a\). The wavefunctions of the particle in the box are well known and the first few look like this (for a=5) :

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    As you can see, the wavaefunctions extend outside the box, but they are exponentially suppressed meaning they become really insignificant very quickly as you move away from the box. Now suppose you have two boxes that are quite far apart and you give the particle in box 1 a kick so that it goes up to another energy level - since the wavefunction for the particle is very close to zero in the lower state and in the higher energy state at box 2 the wavefunction of the particle in box 2 is basically unaffected. This is the case when you have particles in confining potentials far apart, as is the case for electrons bound in atoms and molecules.
     
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  5. Farsight Valued Senior Member

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    I didn't see the program, but I can tell you that what you were told was absolute woo. The Pauli exclusion principle refers to the electrons in a single atom. No two electrons in a single atom can be in the same quantum state. But an electron in one atom can be in the same quantum state as an electron in the the atom next door.
     
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  7. przyk squishy Valued Senior Member

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    This doesn't sound right. The Pauli exclusion principle implies that electrons can't all be in the same quantum state, but there's more to an electron's quantum state than just its energy. On the other hand, the Pauli exclusion principle follows from the fact that the collective state of any collection of identical fermions (eg. all the electrons in the universe) is always antisymmetric, so there may well have been some truth to what he was saying, even if it was distorted or exaggerated in his presentation.
     
  8. CptBork Robbing the Shalebridge Cradle Valued Senior Member

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    It sounds like either a bunch of woo, or else something got lost in translation. The Pauli Exclusion Principle doesn't only restrict the allowed quantum numbers a system can possess, but their position/momentum wavefunctions as well. Since electrons scattered around the universe don't share the same position/momentum wavefunctions, there's no need to adjust their other quantum numbers in order to satisfy the Exclusion Principle.
     
  9. Trooper Secular Sanity Valued Senior Member

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  10. CptBork Robbing the Shalebridge Cradle Valued Senior Member

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    Looks like a misleading attempt at explaining things to laymen. No two fermions can occupy the same quantum state. That means they can't have the same quantum numbers and the same position/momentum wavefunctions (I say position/momentum wavefunctions because either can be directly calculated from the other, as they form a Fourier transform pair). You can have two hydrogen atoms far apart in space with electrons possessing the exact same energy levels and quantum numbers with respect to the atoms they orbit.

    As far as a particle in one place shifting the energy levels and properties of particles elsewhere, even Newton would have told you that's how it works. Everything in the universe interacts in various ways with everything else, although none of it can (as far as we know to date) have an effect which propagates faster than lightspeed, unless some aspect of that effect (such as entanglement) was already established well in advance in such a way that the initial state of the entangled particles is unknown (hence you can never know what was actually changed until much later).

    I note that Prof. Cox makes mention of the Dirac negative energy electron sea. It's true that electrons rarely/never fall into that sea unless that sea is already missing an electron (such holes in the sea behave in every possible manner as antimatter positrons). But that theoretical Dirac sea is supposed to contain electrons occupying virtually every possible quantum state, including every possible position/momentum and every possible set of internal quantum numbers.
     
    Last edited: Dec 27, 2011
  11. przyk squishy Valued Senior Member

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    Er, that's more or less the point I was making in my post above: the antisymmetry condition for fermions, if you take it and quantum physics seriously, implies that all the electrons in the universe are entangled. There is no such thing as an antisymmetric separable state.

    It also makes the idea of "the" electron in an atom a bit problematic, since again the antisymmetry condition implies that every electron in the universe has exactly the same presence in that atom.
     
  12. Walter L. Wagner Cosmic Truth Seeker Valued Senior Member

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    Exactly. They are different systems if they are different atoms. He seemed somewhat confused on the topic.

    But he is quite entertaining, and generally a very good speaker on the subjects at hand, and gave a good explanation of Hydrogen bonding with Oxygen. I do the same material in my chemistry lectures for High School students. It appears he was doing some form of 'adult education'.
     
  13. CptBork Robbing the Shalebridge Cradle Valued Senior Member

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    But if you look at a reasonably isolated hydrogen atom and see an electron make a transition, it's the local wavefunction of that electron and whatever it emits that gets changed to preserve angular momentum, parity etc. Very little gets changed elsewhere in the universe in order to preserve the total fermion wavefunction antisymmetry.
     
  14. przyk squishy Valued Senior Member

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    Er, if there's any confusion, I'm not saying I agree with what Brain Cox said. His conclusion isn't accurate for the reason you and some others have already given (namely that an electron's energy isn't all there is to its quantum state). I'm just pointing out that all the electrons in the universe are always permanently entangled with one another due to the antisymmetry condition. Basically (and especially if you're explaining what this means in layman's terms) that means that there's a certain nonlocal aspect to the quantum state of all the electrons in the universe.

    So from that perspective there's a faint grain of truth to what Brian Cox was saying, though of course whether that's actually what he was talking about is another matter.
     
  15. przyk squishy Valued Senior Member

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    Actually:
    I'm beginning to have some reservations about even this. If you tried to imagine doing a Bell-type test to measure this entanglement, you'd find that would involve doing non-local measurements - i.e. measuring in bases involving superpositions of different positions - which doesn't really fit the profile of a Bell test anymore.

    Of course what Brian Cox said was wrong either way: the Pauli exclusion principle does not imply that rubbing a piece of diamond will instantaneously affect the energies of all the other electrons in the universe.
     
  16. RJBeery Natural Philosopher Valued Senior Member

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    What I'm hearing in this thread is that Brian Cox is wrong in his description in an effort to make things understandable to the laymen...yet at the same time if you "take quantum physics seriously" then he is technically right.

    Doesn't all of this presume an actual ontological physicality for the wavefunction anyway? This viewpoint suffers from contradictions, as the wavefunction itself of a given system is subject to observer dependence.
     
  17. przyk squishy Valued Senior Member

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    No, you've managed to get a completely distorted impression of this thread. Brian Cox specifically said that if you excited the electrons in a piece of diamond, the Pauli exclusion principle implies that the energies of all the other electrons in the universe would have to readjust themselves. Everyone here is saying that both the reasoning he presents for this and his conclusion are wrong.

    More generally, he is suggesting a link between Pauli exclusion and nonlocality. I (and so far in this thread, only I) am saying that you may be able to consider that there's some truth to this, but not following his reasoning and certainly nothing as extreme as the type of nonlocality he suggested by his example, and I've expressed reservations about saying even as much as I have.
     
    Last edited: Dec 29, 2011
  18. RJBeery Natural Philosopher Valued Senior Member

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    Fair enough, let me restate my point. IF we assign any ontological physicality to the wavefunction as Brian Cox appears to be doing, then doesn't that present a contradiction given the fact that the nonlocal nature of the "universal wavefunction adjustment" of a system would be observer-dependent? Also, we can explicitly heat a diamond (to stick with the given example) and alter the theorized physical wavefunction in a manner of our choosing to cause these nonlocal effects.

    Given your response, it appears I'm agreeing with the thread's sentiment that Brian Cox is mistaken. I'm just emphasizing my objection to the idea of the wavefunction having any ontological significance, while it appears others in the thread are objecting for other reasons. I'd also like to point out that if we heat a diamond (or hell, just measure the electron for locality), it actually WOULD affect the wavefunctions of all other electrons because their probabilities would necessarily be adjusted (i.e. if this electron is measured to be "here" then none others can whereas they might have prior to my measurement). Cox's assertion is correct in the mistaken context of the idea of the wavefunction's physicality; it's the wavefunction's physicality itself that presents the problem.
     
  19. CptBork Robbing the Shalebridge Cradle Valued Senior Member

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    What makes you think it would need to be observer-dependent? Wavefunctions don't need an external observer in order to evolve over time.

    No, the nonlocality of entanglement is only possible because we don't know the exact quantum states of the entangled particles and thus we cannot manipulate these states at will.

    Incorrect. The concept of a multiparticle wavefunction is that you have a particle here, a particle there, but no way of telling which particle is in which location at a specific moment, just that such locations always contain a particle when measured. Yes, if I adjust the properties of just one single particle, the entire universe's wavefunction is altered, but that alteration is localized to the area of the affected particle and makes no measurable difference aside from some miniscule exchange of forces propagating at lightspeed.
     
  20. Trooper Secular Sanity Valued Senior Member

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    CptBork,

    I only watched it because he’s so cute, but he’s not really using this analogy to explain why the atom is mostly empty space, right? He’s only trying to explain the discrete energy levels, isn’t he?

    Brian Cox Demonstrates Why Atoms Are Empty


    “There is no empty space around a nucleus, as in Bohr's superseded model. The picture of an atom being mostly empty stems from the childhood of atomic structure analysis, where most of the atom's extension was found to be transparent for alpha rays and the early models explained that by point like nuclei and electrons.”

    http://www.mat.univie.ac.at/~neum/physfaq/topics/touch
     
  21. RJBeery Natural Philosopher Valued Senior Member

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    OK, when speaking of electron wavefunctions the area of the affected particle is technically infinite, right?...and because we're debating reality rather than practicality, words like "minuscule" may as well be "infinitely large". Either the effect exists or it does not.

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    Secondly, I mentioned that wavefunctions are observer-dependent with EPR in mind. If we're assigning a physicality to wavefunctions we run into contradictions because two relativistically-travelling observers could conclude that the wavefunction representing the entangled system collapsed at different times for each particle. Specifically, they could both claim that the opposite particle had not yet been measured before its wavefunction had collapsed...so we are forced to conclude that the physically real wavefunction had then collapsed for BOTH particles before EITHER of them had been measured. The only alternative is to claim that, even though a particular observer knows the results of a measurement of a particle, its wavefunction does not collapse until it is actually measured...which contradicts the very definition of wavefunction because it no longer represents an accurate probability amplitude. Get it?
     
  22. CptBork Robbing the Shalebridge Cradle Valued Senior Member

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    Pauli's exclusion principle explains why the electrons don't all eventually fall down to the lowest energy ground state, and it does put some restrictions on the sorts of atomic configurations which can exist (i.e. why iron is ferromagnetic), but you don't need to know anything about this principle in order to explain why the hydrogen atom has discrete energy levels- this particular fact simply follows from the basic quantum equations all particles obey (i.e. Schrodinger's eqn, Dirac's eqn...).

    Pauli's principle when combined with Dirac's vacuum particle sea also explains why fermions such as electrons don't just radiate indefinitely and fall to arbitrarily negative energy levels, because the sea is already filled with particles occupying all states with those energies, but again it doesn't explain why these energy levels actually exist, that stuff comes as a separate result in QM.
     
  23. Trooper Secular Sanity Valued Senior Member

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    Last edited: Dec 29, 2011

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