Avoided Level Crossing: Two coupled oscillators with frequencies ω₁, ω₂ and coupling g
are described by the 2×2 Hamiltonian H = [[ω₁, g],[g, ω₂]]. Eigenvalues: ω± = (ω₁+ω₂)/2 ± √((Δ/2)² + g²)
where Δ=ω₁−ω₂ is the detuning. As Δ→0, the modes avoid crossing — minimum splitting is 2g (the "normal mode splitting").
This occurs universally: cavity QED (photon-atom), polaritons, magnon-phonon hybridization, molecular orbital theory.
At zero detuning, eigenstates are symmetric/antisymmetric superpositions — neither mode is "pure." At large detuning,
eigenmodes return to bare oscillator character. The energy gap 2g is the signature of strong coupling.