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I have a problem that asks (verbatim)

Carryout the construction of the eigenvectors of total angular momentum and its z component for $j_1$=3/2 and $j_2$=1/2

I am not completely sure where to start out with this. Our professor has not given the best explanation. The equations I have been trying to use are the following:

$$\textbf{L}^2|l,m\rangle =\hbar^2l(l+1)|l,m\rangle$$ $$\textbf{L}_z|l,m\rangle=\hbar m|l,m\rangle$$

I figure I probably want to use the same equations, but for $\bf J$ instead of $\bf L$. I am also confused what the differences between $j_1$ and $j_2$ would represent. I would greatly appreciate any help!

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I am not sure. I would guess a free particle. – yankeefan11 Mar 17 '14 at 0:36
Some form of system with adding angular momenta – yankeefan11 Mar 17 '14 at 0:38

In order to do solve this, you can do it using the theory of Clebsch-Gordon coefficients. This are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics. So you only need to find out a table with this coefficients for $j_{1} =3/2$ and $j_{2} =1/2$. The theory can be found in this link, and the table of the coefficients in

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