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Addition of Angular Momenta

  1. Jan 14, 2013 #1
    Hey!

    While I was reading some book in Quantum Mechanics, I ran across the following, and couldn't
    know how can this be true or actually how was it assumed.

    How by adding equation (7.91)and (7.92), we get (7.110), see attachment.
     

    Attached Files:

  2. jcsd
  3. Jan 14, 2013 #2
    Well isn't
    [tex]
    \vec{J}=\vec{J}_1+\vec{J}_2
    [/tex]?
    Then you can work component by component and obtain the result.
     
  4. Jan 14, 2013 #3
    Yes, but this is not the 'real' addition, each of the operators you've listed belong to different spaces..
     
  5. Jan 14, 2013 #4
    But [itex]\vec{J}[/itex] may be defined in this way on the space defined as the direct sum of the spaces where 1 and 2 act, or not?
     
  6. Jan 14, 2013 #5
    Please read carefully what's written in the attachment.
     

    Attached Files:

  7. Jan 14, 2013 #6

    Bill_K

    User Avatar
    Science Advisor

    They do act on different subspaces. But actually it's not the direct sum, it's the direct product. To be technical about it, J1 is really J1 ⊗ I, and J2 is really I ⊗ J2, and J = J1 + J2 = J1 ⊗ J2.

    Now if you focus on two of the components, say x and y components, and look at their commutator,

    [Jx, Jy] = [J1x, J1y] ⊗ [J2x, J2y] = i J1z ⊗ J2z = i Jz
     
  8. Jan 15, 2013 #7
    Yes, the direct product, I messed up my operations.
     
  9. Jan 15, 2013 #8
    Thanks! This was helpful.
     
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