advanced35 min studyLesson 397 in Physics

Breit–Wigner resonances and time delay

Connect an isolated scattering pole to phase motion, width and a wave-packet delay without confusing delay with lifetime.

Before you begin

S-matrix unitarity and eigenphases

On this page

The question for today

What you’ll understand

Derive a Breit–Wigner shape from a rapidly varying elastic phase.
Compute the Wigner delay with its factor of two.
Distinguish resonance poles, decay lifetimes and background interference.

A particle can spend an unusually long time near a scatterer before escaping. In a stationary experiment this temporary trapping appears as a rapid phase change and often a sharp cross-section feature. The simplest description is an isolated Breit–Wigner resonance with a slowly varying background. Its energy width controls several related timescales, but the survival lifetime and the peak delay of a scattered wave packet are different quantities and must not be equated without checking their definitions.

Parameterize a single elastic resonance

Focus on one elastic partial wave and an energy interval far enough from a threshold that its wavenumber and background vary slowly across the feature. Split its phase into a smooth background and a resonant contribution that rises continuously through pi over two. The resonance denominator contains an energy displacement and a positive width. It gives a unit-modulus S element for real energy, with a pole below the real energy axis on the appropriate analytically continued scattering sheet.

The resonant squared sine has a Lorentzian profile when the background phase is negligible. Its maximum saturates that partial wave's elastic unitarity limit. Including a background adds coherent phase interference, so a resonance pole can produce an asymmetric peak or even a dip in an observed channel. A visible peak by itself therefore does not establish an isolated Breit–Wigner pole, and fitting a symmetric line across a nearby opening threshold can assign a misleading constant width.

S_\ell(E)=e^{2i\delta_{\rm bg}}\frac{E-E_R-i\Gamma/2}{E-E_R+i\Gamma/2},\qquad \sin^2\delta_{\rm res}=\frac{(\Gamma/2)^2}{(E-E_R)^2+(\Gamma/2)^2}

ER and Γ are energies, with Γ > 0. This isolated elastic model assumes a slowly varying background and width over the fitted interval; thresholds and multiple overlapping poles require a more complete description.↔ Scroll wide equations horizontally.

Differentiate phase to find packet delay

A narrow incoming packet contains nearby energies whose outgoing phases differ. In a partial wave the scattering matrix adds phase $2\delta_\ell(E)$ relative to free propagation. Expand that phase linearly over the packet bandwidth. Stationary phase then shifts the outgoing envelope by reduced Planck constant times the derivative of the added phase. This is the Wigner channel delay, measured relative to a chosen free reference propagation, rather than a trajectory stopwatch inside a sharply defined interaction region.

Differentiate the continuous resonance phase rather than a discontinuous principal arctangent. Its derivative is positive for the ideal isolated pole and largest at the resonant energy. The resulting peak Wigner delay is four times reduced Planck constant over the width. Background derivatives can add positive or negative delays, and packet distortion can invalidate a simple peak-shift interpretation when the bandwidth is not narrow. A negative relative delay does not imply a particle sends information backward in time.

\tau_W=2\hbar\frac{d\delta_\ell}{dE},\qquad \tau_{W,\rm res}=\frac{\hbar\Gamma}{(E-E_R)^2+(\Gamma/2)^2},\qquad \tau_{W,\rm res}(E_R)=\frac{4\hbar}{\Gamma}

The delay has seconds because ℏ has energy times time. The packet must be narrow enough for a local phase-derivative description; the displayed resonance delay omits the background contribution.↔ Scroll wide equations horizontally.

Separate the pole lifetime from scattering delay

A pole energy written as a real resonance energy minus half an imaginary width contributes a temporal amplitude that decreases exponentially with half the probability decay rate. Squaring that amplitude gives survival probability with a lifetime of reduced Planck constant divided by width. This is a metastable-state survival definition. The Wigner delay uses the phase of the full incoming-to-outgoing channel response and includes a different comparison, explaining its different factor at resonance.

An exact bound state below the lowest open threshold is normalizable and has no escape width in a closed self-adjoint problem. A resonance instead represents temporary trapping coupled to continuum states and is not an ordinary normalizable positive-energy eigenvector. A damped mechanical oscillator can illustrate a relation between rapid decay and broad spectral features, but its prepared displacement is not a scattering wave packet and its clock does not directly measure Wigner delay or the survival probability of a quasibound quantum state.

E_{\rm pole}=E_R-i\Gamma/2,\qquad A(t)\propto e^{-iE_Rt/\hbar}e^{-\Gamma t/(2\hbar)},\qquad |A(t)|^2\propto e^{-t/\tau_{\rm life}},\quad \tau_{\rm life}=\frac\hbar\Gamma

The exponential pole approximation describes an intermediate-time isolated resonance, not a universal exact decay law at arbitrarily short or long times. τlife is a survival lifetime, while τW compares scattering packets.↔ Scroll wide equations horizontally.

Work it through

Width, delay and lifetime

An isolated elastic resonance has width $\Gamma=0.20\,\mathrm{eV}$ and negligible background phase and slope. Use $\hbar=0.658\,\mathrm{eV\,fs}$ . Find its survival lifetime, peak Wigner delay, and relative Lorentzian cross section at $E=E_R+0.10\,\mathrm{eV}$ , neglecting the slow kinematic prefactor change.

The probability survival lifetime is $\tau_{\rm life}=\hbar/\Gamma=0.658/0.20=3.29\,\mathrm{fs}$ . The amplitude itself decays on twice that time, which is why squaring matters.
At the resonant center the Wigner delay is $4\hbar/\Gamma=13.16\,\mathrm{fs}$ . It is four times the survival lifetime in this ideal one-channel model, not equal to it.
The energy offset is one half-width. The Lorentzian ratio is $0.10^2/(0.10^2+0.10^2)=1/2$ . Thus this point is one of the half-maximum locations and the full width between them is $0.20\,\mathrm{eV}$ .

The result

The lifetime is 3.29 fs, the peak Wigner delay is 13.2 fs, and the cross section at the specified half-width offset is half its peak in the stated approximation.

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

Decay-rate analogy with a fixed preparation

Reset and set BOTH Damping ratio ζ = 0.10 and Natural frequency ω₀ = 1. Reset for a comparison with ζ = 0.40 and ω₀ = 1. Predict faster decay of the oscillation envelope over several cycles in the second run. Both traces are deterministic prepared releases with x = 1 and v = 0 initially. The lab has no scattering energy scan, resonance width, S matrix or packet-delay measurement.

Displacement x(t)Current time

Displacement x(t): 1
Velocity v(t): 0
Regime: underdamped

Damping ratio ζ0.2Natural frequency ω₀1.4

t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forDecay and spectral width are related, but the measured definition determines the numerical time factor.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The phase derivative gives τW(ER) = 4ℏ/Γ, while probability survival gives τlife = ℏ/Γ. Different definitions account for the factor of four.

Your turn

Try it yourself

An isolated resonance with negligible background phase and slope has its width doubled while its center and slowly varying wavenumber are held fixed. What happens to the peak elastic cross section, the peak Wigner delay and the full width at half maximum in the ideal single-channel model?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

At the center the resonant phase is π/2 for either width. The peak height and the phase slope therefore depend on width differently.

Compare with the solution

The peak partial cross section remains at $4\pi(2\ell+1)/k^2$ , so it is unchanged under the stated fixed-k comparison. The peak delay $4\hbar/\Gamma$ is halved. The Lorentzian full width at half maximum equals Γ and therefore doubles. The survival lifetime also halves, but it remains one quarter of the peak Wigner delay. These conclusions require an isolated elastic resonance with negligible changing background; a channel-dependent branching ratio or threshold-dependent width can alter the observed peak height.

Keep the idea

Take it with you

An isolated resonance links a lower-half-plane pole to rapid continuous phase motion.
The Wigner delay differentiates the full S phase, giving a factor of two times the phase-shift derivative.
Survival lifetime, amplitude decay time and scattering delay are distinct definitions.

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