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Transition from Poisson brackets to commutors?

  1. Sep 9, 2006 #1
    Hi to everyone.

    I am a new member in this forum. I was wondering if there is a rigorous proof
    on to how one passes from Poisson brackets to commutor relations in QM. Any help on that would be appreciated.
  2. jcsd
  3. Sep 9, 2006 #2


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    Staff: Mentor

    As far as I know, the rule that you can make the transition from CM to QM by replacing Poission brackets with commutators can be viewed as a postulate, from which you can derive QM (maybe with additional assumptions/postulates, I'm not very familiar with this). The "proof" is experimental, that is, the physical predictions of QM agree with experiment, at least to date.

    Or are you looking to go the other way, from QM as derived from other postulates to the fact that commutators correspond to Poisson brackets in CM?
  4. Sep 9, 2006 #3

    In principle, it should be possible to proof it in the reverse direction.
    In the limit hbar->0 , the operators & commutators formalism should translate in Poisson brackets.

    It is well known that the Schrödinger equation can lead to the classical least action principle of CM and the Lagragian formulation. This simple fact garantees that the proof will work. However, I have never seen a proof QM->CM focusing on the {} formulation.

  5. Sep 9, 2006 #4


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  6. Sep 10, 2006 #5
    Hi, it is rather well known that there is no unique way to do this: the troubles being collectively named as normal ordering problems and anomalies. Usually one chooses x,p as basic variables, quantizes the corresponding poisson brackets (which comes down to working in the Schrodinger representation) and builds up the representation from there.
  7. Sep 10, 2006 #6
  8. Sep 12, 2006 #7


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    Last edited: Sep 13, 2006
  9. Sep 13, 2006 #8
    Thanks a lot samalkhaiat.
    Quite a hard Latex job!
    Where did you learn about this?

    With my best regards,

  10. Sep 14, 2006 #9


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