Dressed states E± = ½(δ ± √(δ²+4g²)) — coupling two quantum states
When two quantum states |1⟩ and |2⟩ are coupled with strength g, the eigenstates are dressed states |±⟩ with energies E± = ½(δ ± √(δ²+4g²)), where δ = E₁−E₂ is the detuning. Near resonance (δ≈0), the states cannot cross — they undergo an avoided crossing with minimum splitting 2g (the vacuum Rabi splitting). The mixing angle θ = ½atan(2g/δ) determines the character of the dressed states. This underlies polaritons in microcavities, EIT, and NMR.