How is it that for a closed subshell configuration $L=S=0$? For a closed subshell configuration of a many-electron atom, $M_L=\sum_i m_{\ell_i}=0$ and $M_S=\sum_i m_{s_i}=0$. But I do not understand why does it necessarily mean $L=S=0$. The values $M_L=M_S=0$ are compatible with nonzero values of L and S. Then how does $M_L=M_S=0$ enforce $L=S=0$? I looked at all my books (Bransden, Liboff etc), all of them did a poor job explaining this, in my opinion.
 A: As you say, for a closed subshell $M_L=M_S=0$
And this is true whatever direction you happen to have chosen for the $z$ axis.
If a vector has a $z$ component for any $z$ direction this can only be because it has length zero, as opposed to happening to be in a particular orientation where it lies entirely in the $xy$ plane, which is how the $L>0, M=0$ possibilities arise.
A: The reason $S=0$ is easy. Each electron can be thought as a bar magnetic with its field fully pointed along the $z$ axis. It can either be up or down. The occupancy of the shell goes as $\propto 2\cdot(2\ell+1)$ where $\ell \in \mathbb{N}$ so it's always going to be even. All the spins cancel out.
As for the $L=0$, maths is required for a detailed proof.
But let's consider an example, taking the $p$ shell.
If it's full, then there will be $2$ electrons with the angular momentum fully in the $z$ axis $|\ell=1, m_\ell =1\rangle$ and $2$ electrons with angular momentum fully in the $-z$ axis $|\ell=1, m_\ell =-1\rangle$. Adding these $4$ electrons vectorially, they cancel each other out and hence give zero total angular momentum.
So now let's look at the two (times 2 for spin) electrons with $|\ell=1, m_\ell=0\rangle$.  Their angular momentum is fully in the $xy$ plane, and the question here is: do they add together to make $L_{\text{here}} \neq 0$, or are they pointing the other way to cancel each other out?
Say we look at the $L_x$ basis, which will have a projection quantum number $m_x$. If they were pointing in the same direction hence reinforcing each other, you'd have either the two $m_x$ quantum numbers or the two $m_y$ to be same. But that would violate Pauli's exclusion principle. You need those to be equal and opposite as well, so they all cancel.
