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Hamilton-Jacobi equation for interacting spins ?

  1. Oct 8, 2004 #1
    The Schrodinder equation leads to the classical Hamilton-Jacobi
    equation with an additional term called "quantum potential". This is
    the start of the Bohmian interpretation of quantum mechanics. The
    derivation is straigthforward by assuming psi = R exp(iS/hb). The
    vanishing of the additional term for the classical limit hb->0
    illustrates nicely the link between QM and CM.

    I would like to know if a similar find could be done for a system of
    interacting spins (or qbits !). I take interacting spins as an
    example of 'discrete system' (no space coordinate). Is there also an
    Hamilton-Jacobi to be derived ? If yes, is there also a classical
    limit to be seen ?

    I would be interrested in ideas, suggestions and eventually
    references related to this topic.
  2. jcsd
  3. Oct 8, 2004 #2
    Beware that taking simply hbar=0 is simple in H-J formulation of QM, but it is far from giving you the link between QM and CM when hbar is different from 0 (you only look at a space point with such approximation).
    Have a look at the WBK and optical approximations for a better understanding.

    And, yes you have equivalent formulations of QM in the H-J formalism with the spin space.
    There's a lot of papers in arxiv concerning this topic. I've found one, but beware as it is not the most accurate one. At least, it will give you additional paper pointers (Bogan, quant-ph0212110) :wink: .

  4. Oct 8, 2004 #3
    The Bogan paper is not a "spin-only" formualtion. I know of several attempts to incorporate spin into Bohmian mechanics, but they all add it onto the usual position approach. It would be interesting to see a spin-only formulation, not necessarily for any foundational reason, but because it might provide a method of simulating some quantum computations on a classical computer. I would suggest that one could approach this by picking a privelliged spin direction to have a definite value, playing the role that position plays in the usual Bohmian formalism.
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