How can I calculate the expectation of L(y) using the commutation relation?

[wam_mi]

Homework Statement

I am trying to calculate the expectation of the y-component of the angular momentum L.
$<L_{y}>$. How should I approach this?

Homework Equations

I try to write it in terms of the following commutator

$L(y) = \frac{2*pi}{ih} [L_{x}, L_{z}]$

The Attempt at a Solution

 

[Avodyne]

What do you know about the state in which you are to take the expectation value?

 

[wam_mi]

What do you know about the state in which you are to take the expectation value?

Hi there, the state is given as |l, m>, where l is the orbital angular momentum quantum number, and m is the magnetic moment quantum number.

I want to prove that to compute <l,m| L(y) |l,m> = <L(y)> =0, how should I approach this?

Many thanks!

 

[gabbagabbahey]

If I were you, I'd write $L_y$ in terms of the raising and lowering operators $L_{\pm}$...

 

[wam_mi]

If I were you, I'd write $L_y$ in terms of the raising and lowering operators $L_{\pm}$...

Hi there, thank you for your reply.
I was told that I have to use the commutation relation between the L(x) and L^2 to get the expectation value of L(y). How can I do that though?

Thanks

 

[gabbagabbahey]

Hi there, thank you for your reply.
I was told that I have to use the commutation relation between the L(x) and L^2 to get the expectation value of L(y). How can I do that though?

Thanks

I'm not sure...the only way I know of showing it is to use the raising and lowering operators.

You seem to be working on basically the exact same problem as jazznaz in this thread, so maybe you two should work together and see what you can come up with.