Consider a completely filled d subshell, i.e., one containing the ten electrons allowed by the exclusion principle. Ignore the interactions between the electrons, so that the Hartree approximation quantum numbers n, l, ml, ms can be used to describe each electron.
(a) Show that there is only one possible quantum state for the system that satisfies the exclusion principle.
(b) Show that in this state the z components of the total spin angular momentum, the total orbital angular momentum, and the total angular momentum, are all zero.
(c) Give an argument showing that these conclusions imply that the magnitudes of the total spin angular momentum, the total orbital angular momentum, and the total angular momentum, are also all zero. (Hint: If an angular momentum vector is not of zero magnitude, but has zero z component in one quantum state, then there are other quantum states in which it has a nonzero z component.)
(d) Now consider the interactions between the electrons that are actually present. Can they change the conclusion about the total angular momentum of the subshell? What about the total spin angular momentum and total orbital angular momentum?