Addition of angular momentum - Finding the second tower states

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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?

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Orodruin
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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?

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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
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Homework Helper
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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.

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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
Staff Emeritus