Resonances

A scattering resonance is a temporary trapping of probability in an interaction region, visible as rapid energy dependence of the scattering amplitude. In a central potential, an isolated elastic resonance often appears as a phase shift that rises rapidly through $\pi/2$ modulo background contributions.

Physically, a resonance behaves like a quasibound state with finite lifetime. Mathematically, it is associated with a pole of the analytically continued scattering matrix at complex energy. The complex-analysis vocabulary behind poles, branch cuts, and analytic continuation is summarized in Complex Analysis Essentials; the choice of analytic sheet is treated in Branch Cuts.

Temporary Trapping and Escape

Imagine an incoming wave encountering an attractive well plus a barrier, or a multichannel system where a closed-channel state is coupled to an open channel. The wave can enter a localized configuration and remain there for a time before leaking back out.

This temporary storage of probability produces:

• enhanced scattering near a resonance energy,
• rapid phase variation,
• a time delay in the outgoing wave,
• a width that measures the decay rate.

The resonance is not a true bound state because it is coupled to the continuum.

Phase-Shift Signature

For one-channel elastic scattering, each partial wave has

$$S_\ell=e^{2i\delta_\ell}.$$

Near an isolated resonance, the phase shift is often written schematically as

$$\delta_\ell(E) = \delta_{\ell,\mathrm{bg}}(E) + \frac{\pi}{2} +\arctan\!\left(\frac{2(E-E_R)}{\Gamma}\right),$$

where $E_R$ is the resonance energy, $\Gamma$ is the width, and $\delta_{\ell,\mathrm{bg}}$ is a slowly varying background phase.

If the background is small, the resonant contribution passes through

$$\delta_\ell(E_R)\approx\frac{\pi}{2}.$$

At that point the partial-wave cross section can approach its elastic unitarity limit.

Width and Lifetime

The width $\Gamma$ has dimensions of energy. The corresponding lifetime scale is

$$\tau \sim \frac{\hbar}{\Gamma}.$$

For a pure pole contribution, $e^{-i(E_R-i\Gamma/2)t/\hbar}$ gives survival probability $e^{-\Gamma t/\hbar}$, so its exponential lifetime is exactly $\hbar/\Gamma$. A physical state follows this law only over the time interval where the isolated-pole approximation applies. Collision delay is a different quantity.

The Wigner time delay for a single elastic partial wave is

$$\tau_\ell(E) = 2\hbar \frac{d\delta_\ell}{dE}.$$

Near a narrow resonance, $d\delta_\ell/dE$ is large and positive. For the constant-width form above and negligible background derivative, the peak delay is $4\hbar/\Gamma$, whereas the survival lifetime is $\hbar/\Gamma$.

Breit–Wigner Form

An isolated narrow resonance admits a local pole factor characterized by $E_R$ and $\Gamma$. It connects the complex pole, rapid phase motion, a Lorentzian limit, and the lifetime scale $\hbar/\Gamma$.

Breit–Wigner Form derives the pole factor and explains background interference, partial widths, threshold corrections, and relativistic parameters.

Quasibound States

A bound state has a normalizable wavefunction and a real energy below threshold. A resonance has outgoing-wave boundary conditions and a complex energy. The imaginary part encodes decay.

In a potential model, a resonance may arise when a centrifugal barrier or shape barrier traps the wave temporarily. In a multichannel system, a resonance may arise when a closed-channel bound state couples to an open scattering channel.

Near threshold, resonances, virtual states, and shallow bound states can all strongly affect the scattering length. Their distinction requires the analytic structure of the amplitude, not just the size of a cross section.

Cross-Section Peak

The partial-wave elastic cross section is

$$\sigma_\ell = \frac{4\pi}{k^2} (2\ell+1) \sin^2\delta_\ell.$$

Partial-Wave Cross Sections develops the general channel sums and unitarity-limited cross sections. Here the formula is used to diagnose the energy-dependent peak produced by resonant phase motion.

If $\delta_\ell$ passes through $\pi/2$, then $\sin^2\delta_\ell=1$ and that partial wave saturates the elastic unitarity limit:

$$\sigma_\ell^{\max} = \frac{4\pi}{k^2} (2\ell+1).$$

A peak in a measured cross section is a useful resonance clue only after backgrounds, thresholds, selection rules, and competing channels have been considered.

Examples

Shape resonances occur when a particle is trapped behind a centrifugal or potential barrier before escaping. Square-Well Scattering supplies the exact matching example. Resonance from a Square Well fits its $p$-wave phase motion and checks the fitted energy and width against the exact complex pole.

Feshbach resonances occur when an open scattering channel is coupled to a closed-channel bound state. They are central in ultracold atom physics because they can tune the scattering length over a wide range.

Nuclear and particle resonances are often described with Breit–Wigner-like forms, but relativistic normalization, spin, phase space, and channel coupling must be handled carefully.

Exercises

1. Suppose a background phase is negligible and a resonance drives $\delta_\ell(E_R)=\pi/2$. What is the partial-wave cross section at $E_R$?

Solution

The elastic partial-wave cross section is

$$\sigma_\ell = \frac{4\pi}{k^2} (2\ell+1) \sin^2\delta_\ell.$$

At resonance, $\sin^2(\pi/2)=1$, so

$$\sigma_\ell(E_R) = \frac{4\pi}{k^2} (2\ell+1).$$

This is the elastic unitarity limit for that partial wave.

2. A resonance has width $\Gamma=10^{-6}\,\mathrm{eV}$. Estimate its lifetime scale.

Solution

Use

$$\tau\sim\frac{\hbar}{\Gamma}.$$

With

$$\hbar\approx6.58\times10^{-16}\,\mathrm{eV\,s},$$

one finds

$$\tau \sim \frac{6.58\times10^{-16}}{10^{-6}}\,\mathrm{s} \approx 6.6\times10^{-10}\,\mathrm{s}.$$

3. Why is phase motion a stronger resonance diagnostic than a cross-section bump alone?

Solution

A cross-section bump can be caused by backgrounds, thresholds, interference, detector acceptance, or channel openings. A resonance is tied to rapid energy variation of the complex amplitude, often visible as phase motion through the resonant region. The phase-shift behavior is therefore closer to the underlying pole structure than the magnitude alone.

References

• G. Breit and E. Wigner, “Capture of slow neutrons,” Physical Review 49, 519–531 (1936), doi:10.1103/PhysRev.49.519.
• C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Reviews of Modern Physics 82, 1225–1286 (2010), doi:10.1103/RevModPhys.82.1225.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.