QM Angular momentum addition with known projection (m quantum number)

By | July 12, 2018

Certainly a lot of questions have been asked about this topic, but nowhere I found a concrete answer to my question. I guess there is something very basic about this I don't quite understand yet.

When we add two momenta, say $L$ and $S$, the resulting total angular momentum can range anywhere from $|L-S|$ to $L+S$ in integer steps. I wonder if this is only the case when $m_L$ and $m_S$ are unknown. I think about the vector model where L and S "precess" on a cone around the z direction. When I add them, say $L=S=1$ it makes (in my understanding) an important difference whether eg $m_L=m_S=+1$ or $m_L=-m_S=1$. In the latter case I don't think $J=L+S$ can take on the value $+2$ because L and S point in "different directions", though by the rules of angular momentum coupling $J=2$ is possible as long as $m_J=0=m_L+m_S$. This question is also inspired by one of Hund's rules and the statement that closed subshells of atoms don't contribute any angular momentum. If we have a closed subshell, for example 2D, and add all the projections $m_L$ of electrons in this subshells together, we get zero - good. But why is the total angular momentum zero as well? I mean it could be anything as long as it's projection is zero right? (by general rules for coupling). Basically what I want to know is whether we can determine the value of $J=L+S$ completely if we know $m_L$ and $m_S$ beforhand, for example: What is the total angular momentum of 2 P-electrons (each having l=1) with one of them having $m_l=1$ and the other one having $m_l=-1$?

Thank you for your patience!