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Pauli's exclusion principle Intrpretation

  1. Dec 9, 2011 #1
    Hi

    I spent my last couple hours reading books and browsing this forum and web to get answer on my question, obviously in vain:

    I know that wave function for fermions should be antisymmetric, I know fermions spin should be half integer otherwise causality will broken (but I don't why yet), I know spinors can be represented by SU(2) and we can get from there the half integer and exclusion principle...

    But, as I understood from some books and ArXiv papers that basically, and despite the enormous implications of this principle starting from the "space occupied" by atoms to neutron stars formation, there is no clear intuitive explanation why two fermions with the same spin can't be at the same energetic level, and why there will be "repulsive" relations between them despite of that there is no actual "force" between them (contrary to what postulates some time in crystal studies), I agree that even so arguments such as particles has "consciousness" are interesting, they still meta-physical, why fermions are so different of Bosons? are there such a particles which with wave functions not symmetric nor antisymmetric?

    So is there any explanation in modern main-stream physics ? if not it will be really strange for me why such a very important principle didn't get enough attention?

    Thanks in advance.
     
    Last edited: Dec 9, 2011
  2. jcsd
  3. Dec 9, 2011 #2
    The exclusion principle is derived from quantum particles being indistinguishable (I think it may have been postulated first, but later derived). If you have a wave function that describes two particle (say electrons orbiting a nucleus—could be any bound state), switching the positions of the of the particles should not change the expectation values. Since the expectation value is obtained from the square of the wavefunction, both positive and negative solution work. That is an antisymmetric solution. Switching the particle gives the same function, but negative. The symmetric solution doesn't change sign. It sound like you know this already, but I want to be sure.
    The symmetric solution is in the form
    [itex]\Psi(a,b)=(\phi_u(a)\phi_d(b)+\phi_u(b)\phi_d(a))[/itex]
    Now the antisymmetric solution is in the form
    [itex]\Psi(a,b)=(\phi_u(a)\phi_d(b)-\phi_u(b)\phi_d(a))[/itex]
    This could be describing two electrons, one spin up, one spin down, but that doesn't matter, just that the states are different. If they were the same (the [itex]\phi_u=\phi_d[/itex]) the wave function would be 0. Therefore antisymmetric wavefunctions for two particles must have the particles in different states.
    Particles that are described by the top equation have been named bosons while those that follow the bottom are called fermions. Simply put, if two particles with an antisymmetric wave function dropped into the same state, they would cease to exist and violate many conservation laws!
     
  4. Dec 10, 2011 #3
    Yes you were right, why the fermions will repulse each other so that even an enormous gravity of neutron star can't overcome it? haw this repulsion happens if there is no real force? electrical charge basically is a intrinsic property as much as the spin, and both has "plus-negative" charge-orientation of spin), but one of them result in a real force, and the spin result in "virtual" force, how I should understand such a "dual standards" in QM ?

    Hope my question now became clearer.
     
    Last edited: Dec 10, 2011
  5. Dec 10, 2011 #4
    The conservation laws are the most important. If the wave goes to zero, energy would not be conserved and, depending on the particle, charge may not be conserved. Its not a dual standard. Quantum mechanic at the basic level deals with energy before forces.
     
  6. Dec 10, 2011 #5
    yea I'm totaly agree with you that energy/potentials are more fundamental, so why there is interaction between spins while there is no field or energy exchange? and what you mean by :

    charge not be conserved? according to my knowledge it is always conserved, am I wrong?
     
  7. Dec 10, 2011 #6

    Bill_K

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    TMSxPhyFor, In the first place, there is no guarantee that every important principle has a clear, intuitive reason "why" it is so. the universe does not come with an obligation to be easy to understand!

    You say the principle hasn't got enough attention, but the connection between spin and statistics was examined carefully back in the 1930s and 1940s, from many angles. For example one reason "why" is that it prevents states from occurring with negative energy. Also it is necessary to insure that the theory is invariant under time reversal. You'll find a discussion of spin-statistics in any book on quantum field theory.

    You seem to feel it's a mysterious thing that two fermions can't occupy the same state. On the contrary, I find it amazing that two bosons can occupy the same state!

    The exclusion principle does not represent a "force", it simply says that certain multiparticle states do not exist. As an analogy, think - why can't the electron in a hydrogen atom sit closer to the nucleus than the ground state? It is not because the nucleus is "repelling" it. It's because there simply is no state that's closer.
     
  8. Dec 10, 2011 #7

    tom.stoer

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    One should have a look at fermionic creation and annihilation operators; b/c of

    [tex](b^\dagger)^2 = 0[/tex]

    one cannot create a state with two identical particles; starting with the vacuum

    [tex]|0\rangle [/tex]

    one can create one particle

    [tex]b^\dagger |0\rangle = |1\rangle[/tex]

    but creating two particles does not work b/c

    [tex](b^\dagger)^2 |0\rangle = 0[/tex]

    [tex](b^\dagger)^2 |0\rangle = b^\dagger|1\rangle = 0[/tex]

    The second equation says that if one tries to "add" a second identical particle to a one-particle-state |1> does not work.
     
  9. Dec 10, 2011 #8
    Even if Pauli exclusion principle exerts the "repulsive force" between particle's spin, it would not be included in the four fundamental forces, I think.
    (This is due to the "mathematical" form of QFT.)

    Because the four fundamental forces need to use bosons such as photon, gluon, W-boson, and graviton.
    In these cases, next Feynman diagrams (interaction energy term) are very important.

    [tex] L_{int} = eA_{\nu} \bar{\psi} \gamma^{\mu} \psi, \quad g W_{\mu}^{+} \bar{u} \gamma^{\mu} d \cdots [/tex]

    We can NOT express Pauli exclusion principle in this form.
    Because Pauli exclusion principle is already included in Dirac's wavefunction such as [itex] \psi[/itex].
    ( As tom.stoer says, creation and annihilation operators of [itex]\psi[/itex] express the "force" of Pauli exclusion princile. )

    So what we call "boson" of Pauli exclusion principle does not exist.
    ( "Mathematical" reason of QFT is related, I think. The upper Feynman diagram is everything in QFT. )
     
    Last edited: Dec 10, 2011
  10. Dec 11, 2011 #9

    tom.stoer

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    Unfortunately there is the notion of "exchange interaction" (or "exchange force") http://en.wikipedia.org/wiki/Exchange_interaction which in a qm context looks like an interaction term in the Hamiltonian. But looking at the QFT concepts (ytuab's post) there is no force, no interaction, no associated field; it's nothing else but a consequence of the algebra of the fermionic operators. Therefore the term "interaction" or "force" is misleading.
     
  11. Dec 11, 2011 #10
    now we getting close to my question, but unfortunately I didn't studied Feynman diagrams yet, can you please explain that in some other way?

    another analogy, dose such a "repulsion" happens between any two particles that has some opposite intrinsic quantum number? (spin, isospin ....)
     
  12. Dec 11, 2011 #11

    tom.stoer

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    I think you don't need any Feynman diagrams.

    An interaction between two fermions is no direct (point-like) interaction, but is mediated via a field A or W (or some others). That means an electron interacts with another electron e.g. via the photon field A.

    The Pauli principle holds as soon as you introcude a free spinor field, even w/o any interaction, i.e. w/o any field A or W.
     
  13. Dec 11, 2011 #12
    But that is the problem it self! there is no interaction at first place, becuase...
    from http://en.wikipedia.org/wiki/Exchange_interaction

    Lets do the flowing mental experiment: if I will try to force an electron of 1/2 spin to be in the first orbit (in atom) that already has electron with the same spin, what will happens?There will be resistance to do that , and this resistance appears as:

    1- one of the electrons will flip it's spin (will change intrinsic property)
    2- one of the electrons will change it's energy level or orbit (will change extrinsic property)

    what will really happen?
     
  14. Dec 11, 2011 #13

    tom.stoer

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    Yes, something like that, but that is not (solely) due to the Pauli principle but due to the additional el.-mag. force which "mixes" with the fermion effects.
     
  15. Dec 11, 2011 #14
    tom.stoer, What TmsxphyFor says is due to Pauli exclusion principle, I think.
    Due to the first electron's spin, the second electron changes its orbit (to the higher energy level).
    It is very difficult to explain from the viewpoint of the energy.
    (For example, the energy level of the 2s orbit is much higher than 1s orbit. )

    And the magnetic force of the electron spin is not strong enough to fix their states.
     
  16. Dec 11, 2011 #15

    tom.stoer

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    No, the el.-mag force is required, too.

    Write down the Hamiltonian of two free (non-interacting) fermions and try to describe something like that; it will not work.
     
  17. Dec 11, 2011 #16
    Sorry. I don't understand what you mean well.
    Pauli exclusion principle is related to wavefunction itself (= determinant ) not Hamiltonian.

    From the viewpoint of the energy, for example, the fine structure (2P3/2 and 2P1/2) is actually observed.
    If only the energy of the magnetic force determines the spin state, 2P3/2 state would NOT be observed.
    ( As a result, the fine structure is not observed.)

    The fine structure is caused by the spin and orbital magnetic interaction of one electron ( about "Bohr magneton").
    The two electrons are apart from each other, so the spin-spin interaction is weaker than the spin-orbital interaction from the viewpoint of the energy.
    First, the fine structure (= 0.00005 eV ) itself is very small, so the spin-spin interaction is weaker than that.

    So the Pauli exclusion principle can not be explained by the magnetic force (energy).
     
  18. Dec 11, 2011 #17
    Yes, maybe the example not very accurate, but think about it in abstract way, no electromagnetism, or we have neutrons instead of electrons.
     
  19. Dec 12, 2011 #18
    Bolding mine.

    Could you perhaps expand a bit on this answer. I mean, you're a basically saying that the Pauli principle is caused by a theoretical construct (the algebra), but I'm thinking that it must be possible to go deeper and state what part of nature cause a formulation of the algebra to be such that it does not allow two fermions in the same state.

    We are basically deriving our knowledge of physics from a set of basic rules, such as energy/momentum/spin conservation, and descriptions of the 4 forces. So, the question is: is the Pauli exclusion principle directly derived from a more fundamental postulate, or is it a new postulate in itself?
     
  20. Dec 12, 2011 #19

    tom.stoer

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    First what I am saying is that Pauli's exclusion principle is NOT a force or an interaction. Suppose you have an Hamiltonian for free, non-interacting fermions. Then already at that level the exclusion principle does apply. There is no "force" between two fermions generated by this Hamiltonian; it's not that one electron "pushes away" the other one. Pauli's exclusion principle guarantuees that two identical fermions will NEVER EXIST.

    Now when you try to change the quantum state of one fermion slightly such that it becomes equal to the state of a second electron, then you are talking about a dynamical system where the dynamics (e.g. the el.-mag. force between two electrons) is affected by the spins. The force itself becomes spin dependent (which is interpretetd as an exchange force or something like that), but even w/o any such force the Pauli principle does apply.

    Regarding a deeper reason for the Pauli exclusion principle you may want to have a look at the Spin-statistics theorem
    http://en.wikipedia.org/wiki/Pauli_exclusion_principle
    http://en.wikipedia.org/wiki/Spin-statistics_theorem
     
  21. Dec 12, 2011 #20
    Yes right! on big distances, that also what historically happen, they tried at first to explain Hund first rule by "magnetic" nature of spin-to-spin interaction , but later they found that this is not enough, and Pauli exclusion principle is the lord in this, and this why it's considered as the one who makes atoms "occupy volume", and this way I'm saying if there is any explanation to it!

    yes I'm aware of that, and i want to understand the physical reasons of such operators algebra!
     
  22. Dec 12, 2011 #21

    tom.stoer

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  23. Dec 12, 2011 #22
    I think The Story of Spin (by S.Tomonaga) explains about this "unnaturally" strong interaction between spins in detail.

    Alkali metal such as sodium shows doublet spectrum.
    This is caused by the quantization (up and down) of the valence electron spin 1/2 along the orbital magnetic momment l.

    And the alkaline earth metal such as Mg, shows the singlet or triplet spectrum.
    This shows that each electron spin is interacted to each other more strongly than its orbital magnetic moment.
    ( up + up = 1, down +down = -1 up + down = 0, So half integer "1/2" disappears. )
    But considering the two electron's distance and their spin magnetic moment (=Bohr magneton), this spin-spin interaction is very weak.
    This was a mystery for a long time according to this book.
    Actually the energy difference among the singlet and triplet states were as big as Coulomb energy, which can not be explained by weak spin-spin magnetic interaction.

    As a result, we have to consider the antisymmetric property of wavefunction to explain this strong interaction.
    But this book does not say "force" but "interaction" about it. So it is a little complicated, I think.
     
  24. Dec 12, 2011 #23
    I mentioned this explanation from the beginning, but, Personally, I can't understand what this antisymmetric wave function means physically, if you can visualize for me difference between "waves" described by antisymmetric and usual functions, and how it "pushes" very close other neutrons , I will be quite impressive!

    As I said at the beginning, I searched a lot for an answer, and wikipedia is an obvious option, sorry but it seems I'm ignorant enough to not see the "answer" I'm looking for between those lines, I will appreciate if you will rephrase it for me as you understand it.
     
  25. Dec 12, 2011 #24
    The talk `Exchange, antisymmetry and Pauli repulsion' linked at the bottom of this page appears to provide what you want.
     
  26. Dec 12, 2011 #25

    jambaugh

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    TMSxPhysFor,

    I would share with you my thoughts on the matter though I'm not sure they're an "explanation" or even helpful to anyone but me.

    Firstly we must understand that this issue comes up when we move from the single particle to the many particle theory. I've looked at this from the abstract mathematical point of view. Remember that spin reflects the rotation group representation and thus to of its underlying Lie algebra, su(2), and that of the more broad group's Lie algebra, i.e. Poincare and ultimately Poincare plus SM gauge.

    To my mind we should stop thinking from the classical background of how quantum versions of many particles should behave but rather think in terms of how a given representation or sequence of representations should manifest as a number of quanta, possessing the various gauge charges.

    There is clearly some as yet not fully understood connection between the rank of the spin representation with the ranks of the gauge groups. Thus e.g. unit charge particle have odd half integer spin. We can see by stabilizing deformations (equivalent to adding just a bit of curvature to space/space-time) that translational degrees of freedom too connect with spin degrees of freedom. (That is we can replace the local Poincare group with a deSitter or anti-deSitter and consider the constraint this puts on representations).

    Ok. Then one considers how the many multi-spinor or multivector representations can manifest as a graded set of irreducible representations where the grading would correspond to particle count. Two specific sequences of irreps stand out. The totally symmetrized and totally antisymmetrized cases of a base representation.

    Of course there are other irreps but they can be interpreted as entangled admixtures of distinct types of particles. Only the bosonic and fermionic cases have a nice interpretation as a composite of many identical types of particles.

    The antisymmetry in the fermionic case then forbids our interpreting the aggregate as a number of identical particles wherein any two of them have simultaneously identical values for their observables. If we want to describe a mish-mash system as "5 electrons" then the nature of that mish-mash is that the "5 electron" description only works if they obey the PEP. I see it as a logical constraint, not so much a physical one....(of course it's physical but not in the conventional "physical constraining force" way of thinking).

    Finally as to the why of the Spin Statistics theorem and its implied correlation, I've more to study on this point but I believe it has a great dealt to do with whether in a very large irreducible representation of the grand gauge+space-time group the specific representations can manifest in the classical limit as localized particles.

    (What I would love to do is sit down with an impossibly large super-computer and crank out some computations with extreme irreps of candidate GUT gauge+space-time groups and see how they behave I.T.O correspondence to classical limits. Do they look like recognizable particles or what?)

    Anyway I hope this makes some sort of sense.
     
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