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>. 3. 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.