Breit-Rabi formula ($x=(g_J-g_I){\mu }_{B}B/{\Delta }_{\mathrm{hf}}$)

For electronic $J=1/2$, nuclear spin $I$ and conserved angular momentum component $m$, define ${\Delta }_{\mathrm{hf}}=A(I+1/2)$ and the dimensionless field $x$ shown in the title. The hyperfine-plus-magnetic energies areThe factors $g_I,g_J$ use the same Hamiltonian convention and Bohr-magneton units. The two stretched states require the branch consistent with their unique spin configuration. The formula follows by diagonalizing each two-dimensional fixed-$m$ block; it connects weak-field coupled-spin levels to decoupled high-field states. The name refers to Gregory Breit and Isidor Isaac Rabi.^[ref]

For electronic $J=1/2$, an extremal total projection has only one product state rather than a two-dimensional spin block. With dimensionless angular-momentum operators and nuclear Zeeman coefficient $g_I$ in Bohr-magneton units, its energy is $AI/2\pm {\mu }_{B}B(g_J/2+g_{I}I)$. Extending both signs of a two-state Breit-Rabi formula square root to each stretched projection would invent an extra state.