Zeeman effect and defining the g_F Factor

[TFM]

Homework Statement

A hydrogen atom is interacting with an external magnetic field.
1. Derive the equation for the $$g_F$$-factor of the hyperfine states.

Homework Equations

The Attempt at a Solution

Okay, so the question asks to define the gF factor, however, I am not quite sure where to start.

I know firstly that it is based on the diagram of the vector arrows, as (crudely drawn) attached:

I also know the answer I need to get (it is mentioned in the notes):

$$g_F = g_J\frac{F(F + 1) + j(j + 1) - I(I +1)}{2F(F + 1)} + \frac{\mu_N}{\mu_B}g_I \frac{F(F + 1) + I(I + 1) - j(j +1)}{2F(F + 1)}$$

Also, for the gJ, it starts with:

$$H_J = \frac{\mu_N}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} = \frac{\mu_N}{\hbar}(\hat{J} + \vec{S})\cdot \vec{B}$$

and I know for gF, we have:

$$H_J = \frac{\mu_B}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} - g_I ({\frac{\mu_N}{\hbar} \vec{I}\cdot \vec{B})$$

Anyone got any suggestions about what I should do first?

TFM

 

[Hao]

I suspect the approach would be to treat the external magnetic field as a perturbation to the internal magnetic field of the Hydrogen atom.

In other words, $$\vec{B} = \vec{B}_{Internal} + \vec{B}_{External}$$.

Considering $$\vec{B}_{Internal}$$ first, we can find the orbital angular momentum (l), spin angular momentum (s),ml,ms (or J,m?) eigenstate of the Hamiltonian (possibly after some simplifying assumptions?).

We then apply first order perturbation theory for $$\vec{B}_{External}$$.

Since you know the energy, you should be able to get the g factor.

Hopefully, this is the way to go.