Cross-section peaks at resonance energy E₀ with Lorentzian width Γ — quantum scattering meets complex poles
Breit-Wigner resonance formula (1936) describes the quantum scattering cross-section near a resonance:
σ(E) = σ₀ · (Γ/2)² / [(E−E₀)² + (Γ/2)²]
This is a Lorentzian (Cauchy distribution) peaked at E = E₀ with FWHM = Γ. It arises from a complex pole of the S-matrix at E = E₀ − iΓ/2 in the complex energy plane — a metastable (quasi-bound) state.
Phase shift: The partial-wave phase shift δ passes through π/2 at resonance and increases by π across the resonance.
δ(E) = arctan[Γ/2 / (E₀ − E)]
Wigner time delay τ_delay = 2ℏ dδ/dE peaks sharply at E₀, representing time spent in the resonance state:
τ_delay(E) = ℏΓ / [(E−E₀)² + (Γ/2)²]
The peak delay = 4ℏ/Γ = 4τ. Multiple overlapping resonances interfere — producing Fano profiles when a resonance overlaps a continuum. This formula underlies nuclear physics, atomic physics, and particle physics (Higgs boson: E₀≈125 GeV, Γ≈4 MeV, Q≈3×10⁴).