Total Angular Momentum Measurements

[andre220]

Homework Statement

Consider a particle with orbital momentum $l=1$ and spin $s = 1/2$ to be in the state described by
$$\Psi = \frac{1}{\sqrt{5}}| 1,1\rangle|\downarrow\rangle+\frac{2}{\sqrt{5}}|1,0\rangle|\uparrow\rangle$$

If the total angular momentum is measured what would be the possible outcomes? What are the corresponding probabilities?

Homework Equations

$\mathbf{J} = \mathbf{L}+\mathbf{S}$

The Attempt at a Solution

Okay, so on the surface this seems pretty simple, but I want to make sure that I am not thinking about this wrong.

For the first state: $J=1-1/2 =1/2$ with probability $1/5$ and the second state $J=1+1/2=3/2$ with probability $4/5$. Is this correct?

 

[vela]

No, it's not correct. You need to express the state in terms of eigenstates of $J^2$.

 

[andre220]

On second looks, it's not as easy as I thought. For some reason I thought it was $|j,m\rangle$ and instead it is $|l,m\rangle|s_z\rangle$

What would be a good way to approach this problem? In thinking about it again, I need to determine $j$, but I am not sure how to go about doing so.

 

[vela]

Look up the appropriate Clebsch-Gordon coefficients.