(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:

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