Addition of angular momentum - Finding the second tower states

[DataGG]

Homework Statement

I'm supposed to calculate all the states for a system with $l=1$ and $s=1/2$. Let's say $\vec{J} = \vec{L} + \vec{S}$. I want to find the Klebsch-Gordon coefficients.

I know that said system has 2 towers, one with $j=3/2$ and the other with $j=1/2$. I've calculated all the states for $j=3/2$ but now I'm having some problems with $j=1/2$.

So, for the second tower, we've two states: $|j,j_z>=|1/2, 1/2>$ and $|j,j_z>=|1/2, -1/2>$

How am I supposed to find $|j,j_z>=|1/2, 1/2>$? If I do that, finding $|j,j_z>=|1/2, -1/2>$ should be easily done by applying the $J _$ operator.

Homework Equations

$$J _ |j, j_z>=\hbar \sqrt{j(j+1)-j_z(j_z-1)}|j,j_z-1>$$
$$S _ |s, s_z>=\hbar \sqrt{s(s+1)-s_z(s_z-1)}|s,s_z-1>$$
$$L _ |j, j_z>=\hbar \sqrt{l(l+1)-l_z(l_z-1)}|l,l_z-1>$$

The Attempt at a Solution

Well, I've done well for the tower with $j=3/2$. Now with this second tower, I don't know where to begin from. I think this is because for $j = j_z = 3/2$, we know that $j_z = l_z + s_z$ which means $l_z =1$ and $s_z = 1/2$. There's no other way.

For the case with $j=1/2$, we've two options. $l_z=0, s_z=1/2$, which is to say $|l_z,s_z>=|0, 1/2>$ and $l_z=1, s_z=-1/2$, which is to say $|l_z,s_z>=|1, -1/2>$. Should I sum those states somehow?

 

[Orodruin]

You already know that one of the jz = 1/2 states is part of the j=3/2 representation. Since there are only two of those, which is the remaining one?

 

[DataGG]

You already know that one of the jz = 1/2 states is part of the j=3/2 representation. Since there are only two of those, which is the remaining one?

I'm not sure I understand what you're saying.. I know that, for $j_z = 1/2$ there's two states. One being for $j=3/2$ and the other for $j=1/2$. That is:

$|j, j_z> = |3/2, 1/2>$ and $|j, j_z> = |1/2, 1/2>$. Now I need to write this last state using $l_z$ and $s_z$, in order to find the Klebsch-Gordon coefficients.

 

[Orodruin]

Yes, I am fully aware of that. What I am saying is that you know what the state with $j_z = 1/2$ and $j = 3/2$ is, since you have already computed the states with $j = 3/2$. The state you are searching for must be orthogonal to this.

 

[DataGG]

Yes, I am fully aware of that. What I am saying is that you know what the state with $j_z = 1/2$ and $j = 3/2$ is, since you have already computed the states with $j = 3/2$. The state you are searching for must be orthogonal to this.

Oh! I forgot that. I don't know why they need to be orthogonal though, but I guess that's a discussion for another thread. I'll see if I can solve it having that in mind!

Thank you Orodruin!

 

[Orodruin]

I don't know why they need to be orthogonal though

What can you say about the operator $\hat J = \hat L + \hat S$?