# Angular momentum and spin

I am having problems with this excercise.

We look at a system where the total angular momentum is given by an electron with $l=1$ and $s=\frac{1}{2}$.

Now I am supposed to calculate the eigenfunctions $|\frac{3}{2},m_j\rangle, |\frac{1}{2},m_j\rangle$ with respect to $J^2$ and $J_z$, where $J=L^{(1)}+S^{(2)}.$

In part a) I derived the equation $J^2=L^{(1)2}+S^{(2)2}+2L_3^{(1)2}S_3^{(2)2}+L_+^{(1)2}S_-^{(2)2}+L_-^{(1)2}S_+^{(2)2}$. Maybe this one is useful here.

Also a hint was given: I should check that $|1,1\rangle \otimes |\frac{1}{2},\frac{1}{2}\rangle$and $\sqrt{2/3}|1,1\rangle \otimes |1/2,-1/2\rangle-\sqrt{1/3}|1,0\rangle \times |1/2,1/2\rangle$ are eigenfunctions with respect to $J_z$. This was clear to me how to do this and I checked it, but I don't know what this has to do with the actual excercise. Is there anybody who could help me with this excercise?

The states which are given to you are the eigenstates of $J_z$ with maximum possible eigenvalue $j_z=J$, respectively for $J= 3/2$ and $J=1/2$.
Now, to obtain the other states in a particular representation $J$, you just have to successively decrease $j_z$ by one unity and to find the corresponding eigenstate. This can be done, by successively applying $J_- = L_-$ + $S_-$, and, at each step, normalizing the resulting state.
Practically, you may see here, that you have $j_-|j,m\rangle = \sqrt{j(j+1)-m(m-1)}|j,m-1\rangle$.
Remember that $L$ corresponds to $j=1$, and $S$ to $j=1/2$, and that $L_-$ apply only on the $j=1$ states and $S_-$ apply only on the $j=1/2$ states.
As a check, verify that the $2$ eigenstates with $j_z = 1/2$ belonging to the representations $J=3/2$ and $J=1/2$ are orthogonal (idem for the $2$ states with $j_z = - 1/2$)