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?