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Expectation Values of Angular Momentum Operators

  1. Sep 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that

    < l,m | Lx2 - Ly2 | l,m > = 0

    2. Relevant equations

    L2 = Lx2 + Ly2 + Lz2

    [ Lx, Ly ] = i [STRIKE]h[/STRIKE] Lz

    [ L, Lz ] = i [STRIKE]h[/STRIKE] Lx

    [ Lz, Lx ] = i [STRIKE]h[/STRIKE] Ly



    3. The attempt at a solution

    I tried substituting different commutation values in place of Lx and Ly, but I'm not reducing it any further. I also tried ladder operations, but my professor said they're not needed to solve the problem.
     
  2. jcsd
  3. Sep 4, 2012 #2
    Well I think ladder operators are fine, but there are some "shortcut" ways too:

    You could probably get away with just saying that it's "obvious" that
    [tex]\langle l,m \lvert L_x^2 \lvert l,m \rangle = \langle l,m \lvert L_y^2 \lvert l,m \rangle[/tex]
    because an Lz eigenstate shouldn't know the difference between the x and y directions. If you wanted to make that idea precise, you could find out what happens to the eigenstate and to the angular momentum operators when you do a 90-degree rotation about the z-axis.
     
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