(n=3, l=2, j=5/2) -> (n=2, l=1, j=1/2) hydrogen transition?

[andresordonez]

Hi, I have a doubt about the fine structure of the hydrogenic atoms. In the section 3.2 of the book Physics of Atoms and Molecules by Bransden & Joachain says:

Since the electric dipole operator \mathbf{D} = -e\mathbf{r} does not depend on the spin, the selection rule derived in Chapter 4 for the quantum number l (in the dipole approximation) remains

\Delta l = \pm 1

from which it follows that the selection rule with respect to the quantum number j is

\Delta j = 0, \pm 1

My question is, why \Delta j = \pm 2 is not mentioned here? For example, why the transition
(n=3, l=2, j=5/2) \rightarrow (n=2, l=1, j=1/2) is not considered?

(where n is the energy number, l is the orbital number, and j is the total angular momentum number)

 

[dextercioby]

Why would it be ? How are j and l related ?

 

[andresordonez]

Yeah, sorry I should have posted it before; l and j are related like this:

j = l \pm \frac{1}{2}