How can I calculate the hyperfine structure of a $p$-orbital?

I have a little problem with the calculation of the hyperfine structure of the 3p orbital in the hydrogen atom. The Hamiltonian is:

Were represents the magnetic moment of the proton and the magnetic moment of the electron.

I calculated the spected value for the first term and for the last term, but I don't know how to calculate the second term. A lot of textbooks calculate this only for a s orbital where l=0 but I'm looking this for a p orbital where l=1.

• Are you having a problem with the $\hat{r}$? Try expressing it in terms of $\theta$, $\phi$, and Cartesian unit vectors. – G. Smith Jun 14 at 2:58
• The problem is that for l=1 you have 3 differents spherical harmonics (m=-1,0,1) and I don't know wich I have to choice because for the spected value you have to consider the wave function wich depends on the value of m – Juan Ignacio Jun 14 at 3:17
• In English I think you mean expected value. Wouldn’t you have to calculate three expected values for the three $m$ states? Maybe they will then turn out to be equal. – G. Smith Jun 14 at 3:25
• I was thinking and I concluded that I have to make a matrix 3x3 because it is not diagonal in this base. – Juan Ignacio Jun 14 at 3:36
• If you come up with your own solution, please provide a formal answer. This is allowed on Physics SE and it means the system does not have another unanswered question on it's list. – StephenG Jun 14 at 3:50