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Prove some relations but going round in circles

  1. May 18, 2012 #1
    1. The problem statement, all variables and given/known data


    I need to prove some relations but going round in circles.

    ## [\hat{J}_z, \hat{J}_+] = \hbar J_+ ##

    I've got this:

    ##\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)\left(a_+^{\dagger }a_-\right)-\left(a_+^{\dagger }a_-\right)\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

    2. Relevant equations

    ##J_{\pm} = \hbar a_{\pm}^{\dagger} a_{\mp} ##

    ##J_z = \frac{\hbar}{2} \left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

    3. The attempt at a solution


    I have these sort of obvious relations to work from


    ##[a_{\pm}^{\dagger}, a_{\pm}] = 1##
    ##[a_{\pm}^{\dagger}, a_{\mp}] = 0##
    ##[a_{\pm}^{\dagger}, a_{\mp}^{\dagger}] = 0##
    ##[a_{\pm}^{\dagger}, a_{\mp}] = 0##
    ##[a_{\pm}, a_{\mp}] = 0##

    I start expanding and it gets too tough, what am I missing?
     
  2. jcsd
  3. May 18, 2012 #2
    I'd suggest working in commutator notation instead of expanding it out completely. I.e. start from:
    [tex]
    [\hat{J_z},\hat{J_+}] = [\frac{\hbar}{2} \left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right),\hbar a_{+}^{\dagger}a_{-}]
    [/tex]

    Then manipulate that using the general rules of algebra for commutators.
     
  4. May 18, 2012 #3
    Using

    ## [A+B,C] = [A,C] + [B,C] ##
    ## [AB,C] = [A,C]B + A[B,C] ##
    ## [A,BC] = [A,B]C + B[A,C] ##

    It quickly boils down to

    ##\frac{\hbar^2}{2} \hat{a}_{+}^{\dagger} \hat{a}_{-} ##

    I am out by a factor of half but I'm sure it works!
     
  5. May 18, 2012 #4
    Yes it works fine, I factored a half out of the commutator wrongly. Thanks!
     
  6. May 18, 2012 #5
    Sweet! I'm glad it worked out. These types of problems can and will get messy sometimes. Good thing I didn't have to puzzle through it myself. :P
     
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