The Breit-Wigner Resonance

Fire particles at a target and most of the time they barely notice it, but tune the beam energy to just the right value and the scattering suddenly erupts. That sharp spike is a resonance: at this special energy the projectile and target briefly bind into a short-lived quasi-state before flying apart, and the cross-section traces out the Breit-Wigner curve $\sigma\propto(\Gamma/2)^2/[(E-E_R)^2+(\Gamma/2)^2]$, a Lorentzian peak centred on the resonance energy $E_R$ with full width $\Gamma$. The width is the inverse lifetime: a narrow resonance is a long-lived state, a broad one a fleeting one. The scene runs the beam and lets you sweep its energy through the resonance, the scattered intensity flaring as you cross $E_R$. What makes a resonance more than just a bump is the phase. The scattering phase shift $\delta$, which encodes how much the outgoing wave is retarded relative to free propagation, sweeps rapidly upward through exactly $\pi/2$ as the energy crosses the resonance, and its steepness there is the Wigner time delay, the extra time the particle lingers near the target. The lower panel shows both at once: the phase climbing through $\pi/2$ and the time delay spiking, perfectly aligned with the cross-section peak above. Narrow the width and watch the phase turn over more sharply and the delay grow, the signature of a long-lived resonance.