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Proof of operator relation

  1. Nov 16, 2007 #1
    I'm having a problem proving this operator relation:

    [tex]exp(-i\phi\hat{j_{i}})exp(i\theta\hat{j_{k}})exp(i\phi\hat{j_{i}})=exp(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})[/tex] (1)


    [tex][\hat{j_{i}}, \hat{j_{k}}]=i\epsilon_{ikl}\hat{j_{l}}[/tex]. (2)

    I can prove this for:

    [tex]exp(-i\phi\hat{j_{i}})\hat{j_{k}}exp(i\phi\hat{j_{i}})=cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}[/tex] (3)

    using Baker-Hausdorff lemma.

    Now what I do when I'm trying to prove the first expresion, I expand the middle term in Taylor series, and then trying to use this lemma again, but problem arisses with higher powers of [tex]\hat{j_{k}}[/tex].


    The first term:


    Second term (what I was able to prove (3)):


    And now a problem arisses:


    If (1) is true than it should be:




    but, I can't prove this. Using Baker-Hausdorff lemma for each term becomes too complicated and I get lose in all that mess.
  2. jcsd
  3. Nov 16, 2007 #2


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    What happens when you square equation 3?
  4. Nov 16, 2007 #3
    Nooo, it can't bee :).
    I spent all night trying to solve this in most complicated ways and I didn't saw this...

    Thank you very much!
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