# Measuring total angular momentum of two electrons

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? $$(\vec J)^2=j(j+1)\hbar^2=12\hbar^2,6\hbar^2,2\hbar^2,0$$