How to reach the conclusion that L=0 and S=0 if the shell is filled?

[runnerwei]

The text says that the spin momenta of those electrons cancel each other so S=0.
The text also says that the orbital momenta of those electrons cancel each other so L=0.
But, if there are electrons with quantum numbers (l1,s1) and (l2,s2), using S-L coupling, the L=l1+l2,l1+l2-1,...\l1-l2\,
S=s1+s2,s1-s2
How to reach the conclusion that L=0 and S=0 if the shell is filled?


As according to the coupling rule, the allowed values for S is S=s1+s2,s1-s2. As s1=s2=1/2, S can have two values, 0 and 1. So how would they say that S=0, rather than S=1?

Thx

 

[Penn.6-5000]

The Pauli exclusion principle tells us that two electrons can't occupy the same quantum state. To get S=1, both electrons would have to have a spin of +1/2, which is OK only when some other aspect of their state is different, e.g. they're in different orbitals. In this particular problem, those two electrons are filling the same shell. So what we have here is a case of the Pauli exclusion principle saying that two electrons that are in the same shell must have different spins.

For what it's worth, if you manage to excite one of the electrons into a higher energy level, the two electrons can each have the same spin and so the whole atom can have S=1.