Calculating transition rate between nljm resolved states

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I know how to calculate transition rates between nl resolved states in a hydrogen-like atom, but I don't know how to calculate transition rates between nljm states.

I know that dipole transition rate is
[itex]\frac{32}{3}\frac{\pi^3 \alpha c}{\lambda^3}\left<\psi_1|\mathbf{d}|\psi_2\right>[/itex]
The matrix elements on the right can be separated into a radial integral and an angular part. The radial part is
[itex]\int R_{nl} R_{n'l'} r^3\,\mathrm{d}r[/itex]
If we pretend that spin doesn't exit, the angular part goes something like
[itex]\iint {Y_l^m}^{*} Y_1^{0,\pm 1} Y_{l'}^{m'} \sin(\theta) \,d\theta\,d\phi[/itex]
I can solve this using Clebsch Gordan coefficients, and it seems to give the right answer. But, I have no idea what to do when I add in spin angular momentum. It seems this integral doesn't make sense if I replace l with with 1/2 integer j.

I tried some various things, and I ended up with something that gave the right answer in some cases but not in others. I'm always off by some multiple of some rational. I know it has to do with degeneracy and angular momentum addition, but I can't figure it out. Any resources?


When including spin, ##j## replaces ##l## in the angular part.

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