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

[Jimmy25]

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...

 

[mathfeel]

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 $L_x$ in term of $L_+$ and $L_-$.

Now, substitute this $L_x$ into $\langle L_x \rangle$ and use the first equation to calculate it.

 

[Jimmy25]

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

 

[Jimmy25]

Can anyone help me out here?

 

[kloptok]

Use what mathfeel said and think about orthogonality of |l,m> states.

 

[vela]

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 L_x in terms of L₊ and L_-.