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

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Discussion Overview

The discussion centers on the quantum mechanical properties of filled electron shells, specifically addressing why the total orbital angular momentum (L) and total spin angular momentum (S) are both zero for these configurations. The scope includes theoretical aspects of quantum mechanics and angular momentum coupling.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that the text claims the spin momenta of the electrons cancel each other, leading to S=0, and similarly, the orbital momenta cancel to give L=0.
  • Another participant suggests that the filled shell corresponds to a singlet state due to its rotational invariance.
  • Several participants question the conclusion that S=0 is the only state, pointing out that with s1=s2=1/2, the allowed values for S could also include 1, leading to confusion about why S=0 is favored.
  • One participant proposes that the S=0 state is typically lower in energy than the S=1 state, but acknowledges that this depends on the specifics of the Hamiltonian involved.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the conclusion that S=0 is the only valid state for filled shells, with multiple viewpoints on the implications of angular momentum coupling rules. The discussion remains unresolved as to why S=0 is preferred over S=1.

Contextual Notes

There are limitations regarding the assumptions made about the Hamiltonian and the energy states of the system, which are not fully explored in the discussion.

runnerwei
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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!
 
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Because it is rotationally invariant and, hence, must correspond to a singlet state.
 
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?
 
runnerwei said:
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.
 

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