# Homework Help: Total Angular Momentum Measurements

1. Oct 23, 2014

### andre220

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
$\mathbf{J} = \mathbf{L}+\mathbf{S}$

3. 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?

2. Oct 23, 2014

### vela

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

3. Oct 23, 2014

### 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.

4. Oct 23, 2014

### vela

Staff Emeritus
Look up the appropriate Clebsch-Gordon coefficients.