How to create good quantum states from good quantum numbers?

[mattlorig]

how to create "good" quantum states from "good" quantum numbers?

I think I am finally understanding what the "good" quantum numbers are in degenerate perturbation theroy. Basically, given a perturbation H', if
[H', L^2] = [H', S^2] = [H', J^2] = [H', Jz] = 0, then
l, s, j, and mj are the "good" quantum numbers.

But, I'm a little confused as to how one goes about creating a "good" quantum state from the "good" quantum numbers.

In particular, I'm wondering about how to create a "good" states for the strong field Zeeman effect ("good" state = | n l ml s ms > ) and the fine structure correction ("good state = | n l s j mj > ).

i.e. how do I go from: | n l s j mj > to SUM( c R_nl Y_lm |s ms> )?

 

[DrClaude]

This is actually a question about the addition of angular momentum. The solution is the Clebsch-Gordan coefficients:
https://en.wikipedia.org/wiki/Clebsch–Gordan_coefficients