Quantum statistics of angular momentum

Discussion in 'Physics & Math' started by Bruinthor, Jan 26, 2016.

  1. Bruinthor Registered Member

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    I am looking for assistance with regard to the problem the quantum statistics of angular momentum for indistinguishable fermions and bosons alike. To set the stage assume a set of tops spinning on a table: there may be a non-zero total angular momentum even though the system as a whole is not rotating. Generalize to three dimension and allow the tops to move then throw quantum theory.
    I am well versed in spherical harmonics and clebsch-gordan coefficients so simple references to such will not be helpful. If anyone has insights or knows of relevant pedagogic material please let me know.
     
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  3. James R Just this guy, you know? Staff Member

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    Can you please explain how this is possible?
     
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  5. Bruinthor Registered Member

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    In a real world test friction would apply torque to the table which would might start it to spinning if it were not effectively anchored to the floor. If you are willing to ignore friction and the fact that the earth is moving the tops and the table would not move nor would the orientation of any of them change.
     
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  7. James R Just this guy, you know? Staff Member

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    I think I understand you, apart from the part about the tops not moving. You said they were spinning (and I'll assume for the sake of argument that they are all spinning in the same direction). That means that the total angular momentum of "the system" - by which is presumably meant the tops and the table - is non-zero.

    Now, the table is not rotating, we assume, but the tops are rotating. The total angular momentum of the system is the sum of the angular momenta of its parts, which is clearly non-zero.

    I don't see the problem. Is there a problem?
     
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  8. Q-reeus Banned Valued Senior Member

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    I'm not versed in the maths details but seems you are referring to: https://en.wikipedia.org/wiki/Spin–statistics_theorem
    Or maybe spin waves as in magnetically polarized ferrimagnetic media is more it. Your OP is a bit vague.
    If by pedagogical you mean pictorial representations then an appropriate search on YouTube will probably turn up something to your liking.
     
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  9. expletives deleted Registered Senior Member

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    Your example reminds me of the natural migration of imbalances to the edges or boundaries of a system. Like the imbalance in charges move to the surfaces of a solid sphere to present the net charge of the whole system on its surface even though the solid sphere interior may be in a neutral charge state from just below that surface all the way to its centre. May I suggest you google all kinds of edge effects and boundary effects including the other terms you are most interested in, like quantum statistics and angular momentum and so on, and see what turns up? If anything interesting does turn up, please let me know! I haven't studied your precise scenario before, so I too would be very interested to hear what you turn up, or what other more learned members advise you.
     
  10. Bruinthor Registered Member

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    I have googled, dogpiled etc. without finding anything beyond basic discussions of spherical harmonics, differences between fermions/bosons and clebsch-gordan coefficients.
     
  11. hansda Valued Senior Member

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    I think the tops might precess due to Earth's Gravity. http://hyperphysics.phy-astr.gsu.edu/hbase/top.html
     
    Last edited: Jan 26, 2016
  12. Bruinthor Registered Member

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    adjective: pedagogic
    of or relating to teaching.
     
  13. Bruinthor Registered Member

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  14. hansda Valued Senior Member

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  15. Fednis48 Registered Senior Member

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    I have quite a bit of experience with quantum angular momentum, so I may be able to help, but I don't understand the question in the OP. Are you asking what will happen if the quantum spins are allowed to interact with each other? Or what the spin statistics are in a thermal distribution?
     
  16. quantum_wave Contemplating the "as yet" unknown Valued Senior Member

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    http://chemwiki.ucdavis.edu/Physica..._Spectroscopy/Electronic_Spectroscopy:_Theory

    Every little bit may help. Not a lot here that directly applies, but note the difference between the singlet state and the triplet state. In the singlet state all spin is the same, and in the triplet state, at least one election is not in sync. Don't know if there is anything in that to give you anything, but the concept my lead you somewhere :shrug:
     
  17. Bruinthor Registered Member

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    A more concrete example would be calculating the partition function of a mass of steam that has a none zero total angular momentum. Assuming an unknown but fixed triplet of principle moments of inertia I have the solution for an individual molecule.
     
  18. Bruinthor Registered Member

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    This article is kind of where I am now. I am hoping to learn more about how systems with many more particles are handled (assuming such methods exist). Spin glass for example but with elements able move about in three dimensions. I am also as interested in bosons as fermions.
     
  19. Bruinthor Registered Member

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    I am going ask a more concrete question:

    https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

    In this linked article the Microcanonical ensemble section there is a derivation of the Fermi-Dirac distribution. How can this be extended to include angular momentum. Summarizing what I presume to apply.
    For a quantum particles the angular momentum is usually parameterized by two variables J and Jz. J(J+1)hbar^2 is the square magnitude of the angular momentum vector and Jz*hbar is one component of the angular momentum vector. For fermions J ranges from ½ to infinity in integer steps while Jz goes from -J to J in integer steps. In going over to multiple particles we can calculate the total Jz by adding the weighted Jzi for each component state just as the example does for the energy.
    How can the the component state Ji be included. Summing like energy or Jzi will not work.

    Note 1:
    The method referenced in the link does not work unless there a large degeneracy in the indentical states.

    Note 2:
    https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics
    Contains a similar derivation for the Bose-Einstein distribution hidden below Derivation in the canonical approach.
    The single particle J in this boson case goes from 0 to infinity in integer steps.

    Note 3:
    I know one way to solve this, find the eigenstates of square magnitude of the sum of the angular momentum. I have not been able to make much progress in towards a general solution.
     
    Last edited: Jan 28, 2016
  20. Fednis48 Registered Senior Member

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    Ok, I think I understand a little better now. One thing to keep in mind is that spin doesn't usually show up in treatments of Fermi-Dirac statistics and the like because spin has no intrinsic effect on thermodynamics beyond its contribution to the energy. If you have a cloud of non-interacting particles with angular momentum, you can handle the spin just by noting that J contributes to energy, and applying the appropriate thermodynamic distribution to the full {momentum+spin} set of degrees of freedom. Absent an external field, Jz does not affect energy at all, so the spins' orientations will be completely random and independent of J or momentum. If the spinning particles do interact, you get situations like the spin glasses you mentioned, which can be much more complicated (as far as I know, spin glasses still aren't really "solved"). In such cases, though, it's important to specify the details of your spin-spin interaction.
     
  21. Bruinthor Registered Member

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    fednis48: That mostly fits with what I have learned.
    The total Jz of a rotating body would be non-zero and I presume it would be zero for a non-rotating body. I would dearly like to prove it.
     
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  22. danshawen Valued Senior Member

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    The vacuum is not Lorentz invariant, because of spin and invariant c. This makes all rotations relativistically dependent on r, expressed as light travel time.

    Forcing quantum rotations into a EUCLIDEAN space by means of complex rotation is an insane manner of mitigating a misunderstanding of spin statistics, derived of a bias for 3D geometry mathematicians already know how to deal with. There is only light travel time perceived as space, in every direction. And as far as rotations go, r is everything.

    I got a lot out of this discussion from all participants. Thanks. It explained much more than you know, including the derivation of some critical physics vocabulary which was added in the 1990's while I was most active in another field.
     
  23. exchemist Valued Senior Member

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    Dan, my dear chap, I have to say I think you are talking out of your arse.

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    This is about statistical thermodynamics and how spin states are treated within it. Ballocking on about relativity seems to be utterly beside the point.
     
    Last edited: Feb 6, 2016
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