I feel ones time might be more fruitfully spent perusing Herzberg, Atomic Spectra and Atomic Structure (1937)
On this election night, I’m feeling really cranked on about QM. I would just like to comment on the singlet and triplet spectroscopic states as per Herzberg, pg.75.
Traditionally, these are explained as spin 1/2 electrons being aligned or anti-aligned, to form spin 1 or spin 0 states, but this heuristic doesn’t really wash.
There are 2 X 2 = 4 states of the two electrons, represented as |++> , |+-> , |-+> , |—> in Dirac’s “bra - ket” notation, named for the inner product “bracket” notation, e.g. <+-|++> = 0 .
The “bra” is the dual of the “ket”, and their inner product represents a probability of finding the system in both states simultaneously. ( The ket is, I believe, the punning basis for Schroedinger’s cat. )
But there are two anti-aligned states, and they resolve into the triplet and the singlet thus:
triplet: |++> , |+-> + |+-> , |—>
singlet: |+-> - |-+>
So, there are TWO anti-aligned states: the spin 1 state with z-component 0, and the spin 0 state.
This defies heuristic explanation, except that the singlet is antisymmetric under exchange, and the Lz=0 triplet state is symmetric. It all works out beautifully. Learn to love it.
This is almost literally, the ABCs or 1+1=2, of atomic QM. It stands as rock.