Angular momentum and spin

Question

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?

Answer 1

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 $j_z = - 1/2$)