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

  • Thread starter Jimmy25
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  • #1
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Homework Statement



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

Homework Equations



L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩

L± = Lx ± iLy

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...
 

Answers and Replies

  • #2
181
1

Homework Statement



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

Homework Equations



L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩

L± = Lx ± iLy

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...
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.
 
  • #3
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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
 
  • #4
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Can anyone help me out here?
 
  • #5
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Use what mathfeel said and think about orthogonality of |l,m> states.
 
  • #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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