5. Resonances
Reading time: ~28 minutes | Pages: 7
Quasi-bound states: particles temporarily trapped in potential, dramatically enhancing scattering.
What is a Resonance?
Physical picture:
- Particle enters potential well
- Bounces around inside for extended time
- Eventually escapes (tunnels through barrier)
- Appears as sharp peak in cross section vs energy
Not a bound state: Energy $E > 0$, particle can escape
Quasi-bound: Long-lived but finite lifetime
Breit-Wigner Formula
Cross section near resonance:
where $E_R$ is resonance energy, $\Gamma$ is width
Peak value (at $E = E_R$):
Maximum cross section for partial wave $\ell$ (unitarity limit)
Resonance Parameters
Resonance energy $E_R$:
- Energy at which phase shift passes through $\pi/2$
- Corresponds to quasi-bound state energy
Width $\Gamma$:
- Full width at half maximum (FWHM) of cross section peak
- Related to lifetime: $\tau = \hbar/\Gamma$
- Narrow resonance: long-lived state
- Broad resonance: short-lived state
Quality factor:
Sharpness of resonance
Phase Shift Behavior
Near resonance:
where $\delta_{bg}$ is slowly varying background
Signature: Rapid change through $\pi/2$
- Below resonance: $\delta_\ell \approx 0$
- At resonance: $\delta_\ell = \pi/2$
- Above resonance: $\delta_\ell \approx \pi$
Origin of Resonances
Type 1: Potential well with barrier
- Attractive well creates quasi-bound levels
- Centrifugal barrier ($\ell > 0$) traps particle temporarily
- Example: Nuclear resonances, molecular potentials
Type 2: Shape resonances
- Geometry of potential creates trapping
- Example: Electron scattering from atoms
Type 3: Feshbach resonances
- Coupling between different channels
- Bound state in closed channel couples to continuum in open channel
- Example: Cold atom collisions, nuclear physics
Time Delay at Resonance
Time particle spends in interaction region:
At resonance: delay equals resonance lifetime
Physical interpretation: Particle makes many bounces before escaping
Example: Square Well with Barrier
Potential: attractive well + centrifugal barrier
Resonances when energy matches quasi-bound levels
Width determined by barrier penetration (WKB approximation)
Argand Diagram for Resonance
Plot $f_\ell$ in complex plane as energy varies:
- Resonance: rapid circular motion
- Circle centered at origin with radius $1/(2k)$
- Speed around circle ∝ $1/\Gamma$
- Visual identification of resonances
Compound Nucleus Resonances
Neutron capture by nucleus:
- Forms excited compound nucleus
- Many overlapping resonances at high excitation
- Statistical description (Hauser-Feshbach theory)
Example: $^{238}$U + n → $^{239}$U* → fission or $\gamma$ emission
Feshbach Resonances in Cold Atoms
Magnetic field tunes scattering length:
where $B_0$ is resonance position, $\Delta B$ is width
Applications:
- Control interactions in BEC
- Create molecular bound states
- BCS-BEC crossover in Fermi gases
Particle Physics Resonances
Many "particles" are actually resonances:
- Δ(1232): $\Gamma \approx 120$ MeV, $\tau \sim 10^{-23}$ s
- ρ meson: $\Gamma \approx 150$ MeV
- Z boson: $\Gamma \approx 2.5$ GeV
Narrow width → longer lived → more "particle-like"
Broad width → short lived → purely resonance
Key Distinctions
| Property | Bound State | Resonance |
|---|---|---|
| Energy | $E < 0$ | $E > 0$ |
| Lifetime | Infinite (stable) | Finite ($\tau = \hbar/\Gamma$) |
| Wave function | Exponential decay | Outgoing wave |
| Observable | Discrete spectrum | Peak in cross section |
| Phase shift | Jump by $\pi$ at threshold | Passes through $\pi/2$ |