Measuring total angular momentum of two electrons

Question

I'm trying to work out the total angular momentum for a system consisting of two electrons, both with spin $s_{1,2} = 1/2$ and $l_{1,2} = 1$.

I first calculate $$(\vec L)^2=l(l+1)\hbar^2 = 6\hbar^2,2\hbar^2,0 $$ for $l=l_1+l_2,l_1+l_2-1,...,|l_1-l_2|=2,1,0$ respectively.

Then I calculate $$(\vec S)^2=s(s+1)\hbar^2 = 2\hbar^2,0 $$ for $s=s_1+s_2,s_1+s_2-1,...,|s_1-s_2|=1,0$ respectively.

Finally, calculating $(\vec J)^2$, I first conclude that $$j=j_1+j_2=(l_1+s_1)+(l_2+s_2)=(s_1+s_2)+(l_1+l_2)=3,2,1,0$$Is this correct for the value of $j$? Does it really have 4 possibilities? Also, where do I go from here? Can I use the formula below to calculate the total angular momentum here? $$(\vec J)^2=j(j+1)\hbar^2=12\hbar^2,6\hbar^2,2\hbar^2,0$$

Answer 1

Yes, that is all correct. You're applying the formalism correctly and you're getting the correct results. As to where to go from there, that depends on what it is you want to do.