1. The problem statement, all variables and given/known data
Derive the Breit-Rabi equation in the case that the quantum number j is equal to 1/2, specifically the 1S_1/2 state of Hydrogen. This is the equation describing hyper-fine and Zeeman splitting of the energy levels in an applied magnetic field.

2. Relevant equations
We are given the hyperfine Hamiltonian, and told that it will involve diagonalizing a 2x2 matrix. We are told to use perturbation theory with a basis |S,L,m_s,m_l>, denoted as |m_s,m_L>.

What I don't understand is, why will this only involve a 2x2 matrix. Looking at the above Wikipedia page, the way that the Hamiltonian's matrix elements are found are (in |m_s,m_L> notation):
| <+-|H|+-> <+-|H|-+> |
| <-+|H|+-> <-+|H|-+> |

Why are states with m_s = m_L = 1/2 or -1/2 used also; i.e. states like |++> and |- ->? I see the following text that makes me think it has to do with conservation of F number, but I thought that this was not a good quantum number in the high-field regime.

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

The projections, m_F, m_J, m_I are always good quantum number in any regime. The reason for diagonalizing the 2x2 matrix is because this is degenerate perturbation theory.