The different elements of a rank k tensor operator are related to one another by geometry (clebsch-gordon coefficients).

where is all other quantum numbers, and is the reduced matrix element.

proof:

so or braket=0.

implies coefficients vary as clebsch-gordon coefficients. This means we only need one term to get the other terms via clebsch-gordon coefficients.

For spin-spin interaction, the interaction is like a rank 2 tensor. for .

source psfile jl@crush.caltech.edu index anomalous_Zeeman_effect