12.8 Some important results

This section gives some results that are used frequently in quantum analysis, but usually not explicitly stated.

First a note on notations. It is fairly common to use the letter for orbital angular momentum, for spin, and for combinations of orbital and angular momentum. This subsection will follow these conventions were appropriate.

- 1.
- If all possible angular momentum states are filled
with a fermion, the resulting angular momentum is zero and the wave
function is spherically symmetric. For example, consider the
simplified case that there is one spinless fermion in each spherical
harmonic at a given azimuthal quantum number . Then it is
easy to see from the form of the spherical harmonics that the
combined wave function is independent of the angular position around
the -axis. And all spherical harmonics at that are filled
whatever you take to be the -axis. This makes noble gasses into
the equivalent of billiard balls. More generally, if there is one
fermion for every possible
direction

of the angular momentum, by symmetry the net angular momentum can only be zero. - 2.
- If a spin fermion has orbital angular
momentum quantum number , net (orbital plus spin) angular
momentum quantum number , and net
momentum in the -direction quantum number , its net
state is given in terms of the individual orbital and spin states
as:

If the net spin is , assuming that 0, that becomes

Note that if the net angular momentum is unambiguous, the orbital and spin magnetic quantum numbers and are in general uncertain. - 3.
- For identical particles, an important question is
how the Clebsch-Gordan coefficients change under particle exchange:

For , this verifies that the triplet states 1 are symmetric, and the singlet state 0 is antisymmetric. More generally, states with the maximum net angular momentum and whole multiples of 2 less are symmetric under particle exchange. States that are odd amounts less than the maximum are antisymmetric under particle exchange. - 4.
- When the net angular momentum state is swapped with
one of the component states, the relation is

This is of interest in figuring out what states produce zero net angular momentum, 0. In that case, the right hand side is zero unless and ; and then 1. You can only create zero angular momentum from a pair of particles that have the same square angular momentum; also, only product states with zero net angular momentum in the -direction are involved.