How Do You Derive the Hyperfine Hamiltonian from Magnetic Moments and Fields?

[TFM]

Homework Statement

Derive the hyperfine Hamiltonian starting from $$\hat{H}_H_F = -\hat{\mu}_N \cdot \hat{B_L}$$. Where $$\hat{\mu}_N$$ is the magnetic moment of the nucleus and
$$\hat{B_L}$$ is the magnetic field created by the pion’s motion around the nucleon. Write down the Hamiltonian in the form $$\hat{H}_H_F = ... \vec{I} \cdot \vec{L}$$.

Homework Equations

$$\hat{B_L} = \frac{\mu_0e}{4\pi r^3}\vec{r} \times \vec{v}$$

The Attempt at a Solution

Okay, I have tried putting everything together, and so far I currently have:

$$\hat{H}_{hf} = g_n \mu_n \frac{\vec{I}}{\hbar}\cdot \frac{-\mu_0e}{4\pi r^3} \times V$$

but I am not sure where to go from here. Any suggestions?

TFM

 

[nickjer]

Check wikipedia:

http://en.wikipedia.org/wiki/Hyperfine_structure

Looks like they derive it for you.

 

[TFM]

Thanks for the linkj.

I was koooking through my notes as suggested in the script, and they have a different version, my notes have $$\hat{H}_{HF} = -\hat{\mu}_N\hat{B}_j$$

the notes then go on to say that Bj is parallel to j

is this useful?