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Group QM question

  1. Sep 4, 2008 #1
    In a book about group theory in physics (Wu-Ki Tung) he is using the Euler angle representation of a rotation I'm unsure how he gets the following results...

    [tex] R(\alpha,\beta,\gamma) = e^{-i \alpha J_z}e^{-i \beta J_y}e^{-i \gamma J_z}[/tex]

    he writes

    [tex] R^{-1} J_3 R = -sin \beta (J_+ e^{i \gamma} + J_ e^{-i \gamma})/2 + J_3 cos \beta [/tex]
    and
    [tex] e^{i \gamma J_3}J_2 e^{-i \gamma J_3} = i[-J_+ e^{i \gamma} + J_- e^{-i \gamma}]/2 [/tex]

    how in the world is he getting this!? [tex] J_2 = (J_+ - J_-)/2i [/tex] i know that must be used but how is he getting this?! I tried doing a long expansion of all the exponentials in a taylor series but that didnt get me anywhere. Any help would be aweosme (you can also find this book in google books the page this is on is p.141 Group theory in Physics Wu-ki Tung) Thanks to anyone
     
  2. jcsd
  3. Sep 5, 2008 #2
    You can use the following identity.

    [itex]e^{A}Be^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...[/itex]
     
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