Breit-Rabi Formula Derivation for j=1/2

[teroenza]

Homework Statement

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.

Homework 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>.

The Attempt at a Solution

I see that much of the information is outlined in this link:
https://en.wikipedia.org/wiki/Zeeman_effect

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.

To solve this system, we note that at all times, the total angular momentum projection

will be conserved. Furthermore, since

between states

will change between only

. Therefore, we can define a good basis as:

 

[bombermanFX]

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.