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Power Series For Function of Operators

  1. Oct 19, 2005 #1
    Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute.

    My first guess was:
    [tex]
    F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j
    [/tex]

    However, I don't think this is correct as it is possible to have operators between x and p.
    So then I thought that this might be the correct expansion:
    [tex]
    F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i b_{ij} p^j + c_{ij} p^i d_{ij} x^j
    [/tex]

    Obviously, I'm just guessing here and could use some help. Why do I care about this? I am trying to derive a formula for

    [F(x_1,....,x_n), G]

    Suppose I know how G commutes with each of the operators x_i. I want to get an expansion for G commuting with some function F of the x_i operators. I'll work on the details of that, but first I need some help with the power series of F.
     
    Last edited: Oct 19, 2005
  2. jcsd
  3. Oct 19, 2005 #2

    dextercioby

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    Science Advisor
    Homework Helper

    There exists a simple rule, if you'd be applying your discussion to quantum mechanics.

    Daniel.
     
  4. Oct 19, 2005 #3
    So here is what I am actually trying to do. I have:

    [tex]
    [P^j,\phi_r(x)] = -i \hbar \frac{\partial\phi_r(x)}{\partial x_j}
    [/tex]

    and

    [tex]
    [P^j,\pi_r(x)] = -i \hbar \frac{\partial\pi_r(x)}{\partial x_j}
    [/tex]

    For a function [itex]F\left(\phi_r(x),\pi_r(x)\right)[/itex], I need to show the following:

    [tex]
    [P^j,F\left(\phi_r(x),\pi_r(x)\right)] = -i \hbar \frac{\partial}{\partial x_j}F\left(\phi_r(x),\pi_r(x)\right)
    [/tex]

    I was thinking of considering the various commutator relations:

    [tex]
    [P^j,\phi^n_r(x)]
    [/tex]

    but since the operators don't commute, there are (too) many possible combinations to consider. I would be interested in knowing this trick you speak of.

    Thanks.
     
  5. Oct 21, 2005 #4

    dextercioby

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    Science Advisor
    Homework Helper

    Use the Poisson bracket and Dirac's rule giving the canonical quantization.

    Daniel.
     
  6. Oct 25, 2005 #5
    Could someone spell this out for me? I have convinced myself that there is no pretty way to write a power series for a function of operators (that do not necessarily commute). It seems like you'd have a sum of an infinite product...each term in the product with their own index...so you are summing over an infinite number of indices.
     
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