Filled shells

The concept of "shell" is not unambiguously defined. In some cases it is referred to as the K, L, M, N etc shells. Here we adopt a more restricted definition. All 2(2l+1) electrons with the same n and l are said to form a closed shell; sometimes this is called a subshell.

All closed shells form symmetric charge distributions.

In a filled shell the (2l+1) states with all possible m_l quantum numbers. The sum of these m_l values gives a total of M_tot = 0. Hence the eigenvalue of the L_z operator must have L_z = 0. For such a closed shell L_z cannot have a different value from 0. Hence L must be zero. The same holds for the S-operator.

Hence L = S = J = 0.