- #1

DataGG

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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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