What Is the Quantum State |l,m> After Angular Momentum Measurement?

[atomicpedals]

Homework Statement

The square angular momentum L² and the z-component L_z of a free particle are measured. They are found to be L²=6$\hbar^2$ and L_z=$\hbar$

What is the state |l,m> of the system after measurement?

Homework Equations

L²|l,m>=$\hbar^2$l|l,m>
L_z|l,m>=$\hbar$m|l,m>

The Attempt at a Solution

Is this simply:

L²|lm>=$\hbar^2$l|l,m>=6$\hbar^2$|6,m>
L_z|l,m>=$\hbar$m|l,m>=$\hbar$|l,1>

|l,m> = |6,1> ?

 

[atomicpedals]

I'm getting some contradiction from my notes, the other way I see it the eigenstates could be |2,1>.

 

[vela]

It's $\vert 2 ~ 1 \rangle$. Your first "relevant equation" is incorrect.

 

[atomicpedals]

Cool, means page two of my notes needs a quick correction. Thanks!

 

[atomicpedals]

As a follow up to see if I'm slowly catching on; would the expectation value for L_x then be given by <lm|L_x|lm>?

 

[vela]

Yes.