Why filled shell of electrons has L=0 and S=0?

[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?
Thanks a lot!

 

[Dickfore]

Because it is rotationally invariant and, hence, must correspond to a singlet state.

 

[runnerwei]

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?

 

[element4]

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?

Probably because the S = 0 state usually has lower energy than S = 1, but this depends on the details of the Hamiltonian.