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Show that the expectation value of angular momentum <Lx> is zero

  1. Nov 19, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that the expectation value of angular momentum <Lx> is zero

    2. Relevant equations

    L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩

    L± = Lx ± iLy

    3. The attempt at a solution

    I'm supposed to use ladder operators here to show <Lx> is zero.

    I start with <Lx>=<l,m|Lx|l,m> but don't know where to go from here. I've tried different things but all the methods I've tried seem to lead to a dead end...
     
  2. jcsd
  3. Nov 19, 2011 #2
    Solve your second equation to get [itex]L_x[/itex] in term of [itex]L_+[/itex] and [itex]L_-[/itex].

    Now, substitute this [itex]L_x[/itex] in to [itex]\langle L_x \rangle[/itex] and use the first equation to calculate it.
     
  4. Nov 19, 2011 #3
    I'm not seeing how that would help. Then I just get an equation in terms of L+, L- and Ly.

    Lx=L± minus plus iLy
     
  5. Nov 20, 2011 #4
    Can anyone help me out here?
     
  6. Nov 20, 2011 #5
    Use what mathfeel said and think about orthogonality of |l,m> states.
     
  7. Nov 20, 2011 #6

    vela

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    You have two equations:
    \begin{align*}
    \hat{L}_+ &= \hat{L}_x + i\hat{L}_y \\
    \hat{L}_- &= \hat{L}_x - i\hat{L}_y
    \end{align*}Solve them for Lx in terms of L+ and L-.
     
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