Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Closed-Form for commutation relation between powers of raising and lowering operators

  1. Jul 21, 2011 #1
    Hello,

    I am looking to find a closed-form formula for the following commutator
    [itex][J_{-}^{n},J_{+}^{k}][/itex]
    where the operators are raising and lowering operators of the [itex]\mathfrak{su}(2)[/itex] algebra for which [itex][J_{+},J_{-}]=2J_0[/itex] and [itex][J_{0},J_{\pm}]=\pm J_{\pm}[/itex]

    I've already made some progress and I obtained the following relations, which can be proved by induction :

    [itex][J_{-}^{n},J_{+}]=-2nJ_0J_{-}^{n-1}-n(n-1)J_-^{n-1}[/itex]
    [itex][J_{-},J_{+}^{n}]=-2nJ_{+}^{n-1}J_0-n(n-1)J_+^{n-1}[/itex]

    The two last relations can be used to find a recurrence relation for the object I wish to compute, but I am hoping for a closed form formula.

    Any ideas ?

    Thanks
     
  2. jcsd
  3. Jul 22, 2011 #2

    A. Neumaier

    User Avatar
    Science Advisor
    2016 Award

    Re: Closed-Form for commutation relation between powers of raising and lowering opera

    Use your recurrence formula to get the next few terms, guess from these the general form of the result,
    and insert it into your recurrence formula to get recurrences for the unknown coefficients. If a nice closed formula exists, these recurrences should have a simple solution.
     
  4. Jul 23, 2011 #3
    Re: Closed-Form for commutation relation between powers of raising and lowering opera

    Ok ! Thanks.

    I will pursue this...I'll post the result if I get something.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Closed-Form for commutation relation between powers of raising and lowering operators
Loading...