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.

 

[paranormal]

Yes, your solution is correct. The total angular momentum can take on two possible values: $J=1/2$ or $J=3/2$, corresponding to the two possible outcomes of the measurement. The probabilities for these outcomes are given by the coefficients in front of the state vectors in the given wavefunction, which are $1/5$ and $4/5$ respectively. This solution is in line with the fact that the total angular momentum is the sum of the orbital and spin angular momenta, and can take on values that are the sum or difference of the individual angular momenta.