Multichannel Scattering Preview

Multichannel scattering occurs when the incoming system can leave in more than one asymptotic channel. A channel may label internal states, spin arrangements, rovibrational levels, reaction products, or different particle partitions. The one-channel elastic formulas remain important, but the $S$-matrix becomes a matrix in channel space.

This is a preview page. It introduces the structure needed to understand inelasticity, thresholds, and Feshbach resonances without developing a full collision-theory formalism.

Inelastic Scattering Preview complements this channel-space map with Q-value kinematics, outgoing-to-incoming flux factors, the golden-rule limit, and detector-inclusive sums.

Channels and Thresholds

A channel $\alpha$ consists of a set of asymptotic quantum numbers and a threshold energy $E_\alpha^{\mathrm{th}}$. At total energy $E$, the channel wave number is

$$k_\alpha = \frac{\sqrt{2\mu_\alpha(E-E_\alpha^{\mathrm{th}})}}{\hbar}$$

when

$$E>E_\alpha^{\mathrm{th}}.$$

Such a channel is open. If $E<E_\alpha^{\mathrm{th}}$, then $k_\alpha$ is imaginary and the channel is closed. Closed channels do not carry outgoing flux to infinity, but they can still affect open-channel scattering virtually.

S-Matrix as a Channel Matrix

S-Matrix defines the full asymptotic operator. Unitarity derives the unit-flux channel sum, projected-subspace deficit, and inelasticity disk. For fixed total angular momentum or partial wave, the scattering matrix has elements

$$S_{\beta\alpha},$$

where $\alpha$ is the incoming channel and $\beta$ is the outgoing channel. Elastic scattering corresponds to $\beta=\alpha$. Inelastic or rearrangement scattering corresponds to $\beta\ne\alpha$.

If all open channels are included and there is no absorption into unobserved sectors, unitarity says

$$S^\dagger S=I$$

on the open-channel space. This is the multichannel version of $|S_\ell|=1$ in one-channel elastic scattering.

Cross Sections

With flux-normalized channel states, a partial-wave transition cross section has the schematic form

$$\sigma_{\beta\leftarrow\alpha}^{(\ell)} = \frac{\pi}{k_\alpha^2} (2\ell+1) \left| S_{\beta\alpha}^{(\ell)} - \delta_{\beta\alpha} \right|^2,$$

for two-body channels with the usual angular-momentum normalization. Precise formulas depend on spin, identical-particle symmetry, degeneracy factors, and channel conventions.

For $\beta\ne\alpha$, the Kronecker delta vanishes and the cross section measures transition probability into another channel.

Elastic Inelasticity Parameter

If one observes only an elastic channel while other open channels exist, the elastic element need not have unit magnitude. A common one-channel parametrization is

$$S_{\alpha\alpha}^{(\ell)} = \eta_\ell e^{2i\delta_\ell}, \qquad 0\le\eta_\ell\le1.$$

Here $\eta_\ell=1$ means purely elastic scattering in that partial wave. Values $\eta_\ell<1$ indicate loss of elastic flux into other channels.

This does not violate unitarity. It means the elastic channel alone is an open subsystem.

Feshbach Resonance Preview

A Feshbach resonance occurs when an open channel is coupled to a closed-channel bound state whose energy is tuned near the open-channel threshold. The closed channel cannot carry flux at infinity, but its bound state can strongly distort open-channel scattering.

Projection language makes this natural. Let $P$ be the open-channel subspace and $Q$ a closed-channel subspace. Eliminating $Q$ produces an energy-dependent term

$$H_{PQ} \frac{1}{E-H_{QQ}} H_{QP},$$

which can become large when $E$ is near an eigenvalue of $H_{QQ}$. In ultracold atoms, magnetic fields can tune this near-threshold condition and thereby tune the scattering length.

Breit–Wigner Form shows how an isolated pole factorizes into entrance and exit partial widths. Near a channel threshold, the present multichannel structure is needed to decide whether that elementary parameterization is adequate.

Threshold Behavior

Channel openings create nonanalytic energy dependence. When a new channel opens, its wave number behaves like

$$k_\beta \propto \sqrt{E-E_\beta^{\mathrm{th}}}.$$

Cross sections and phase shifts can therefore show cusps, rapid variation, or threshold laws. These features are not well captured by a single-channel expansion that assumes one fixed analytic threshold.

Applications

Multichannel scattering appears in:

• molecular collisions with internal excitation,
• spin-changing atomic collisions,
• Feshbach resonances in ultracold gases,
• nuclear reactions with multiple exit channels,
• coupled partial waves in spin-dependent forces,
• reaction and rearrangement scattering.

The common theme is that asymptotic states carry labels beyond just momentum direction.

Common Mistakes

• Applying $|S_\ell|=1$ to an elastic subblock when inelastic channels are open.
• Ignoring closed channels because they do not carry asymptotic flux.
• Forgetting threshold factors when comparing channel cross sections.
• Treating a tunable scattering length as a one-channel parameter when a closed-channel state controls it.
• Using single-channel phase-shift intuition near a channel opening without checking coupled-channel effects.

Exercises

1. What does $\eta_\ell<1$ mean in $S_{\alpha\alpha}^{(\ell)}=\eta_\ell e^{2i\delta_\ell}$?

Solution

It means that elastic flux in channel $\alpha$ is not conserved by itself. Some incoming probability leaves the elastic channel and appears in other open channels. The full multichannel $S$-matrix can still be unitary when those other channels are included.

2. Why can a closed channel affect scattering in an open channel?

Solution

A closed channel has imaginary asymptotic wave number and does not carry flux to infinity. However, the open-channel state can virtually couple into the closed-channel subspace and back. The Feshbach term

$$H_{PQ}(E-H_{QQ})^{-1}H_{QP}$$

encodes this virtual excursion. If $E$ is close to a closed-channel bound state, the effect can be large.

3. Explain why opening a new channel can produce nonanalytic energy dependence.

Solution

Near a threshold,

$$k_\beta \propto \sqrt{E-E_\beta^{\mathrm{th}}}.$$

The square-root dependence is nonanalytic at the threshold energy. Since flux factors and density of states depend on $k_\beta$, scattering observables can show cusps or threshold-law behavior when the channel opens.

References

• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• H. Feshbach, “A unified theory of nuclear reactions. II,” Annals of Physics 19, 287-313, 1962.
• C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Reviews of Modern Physics 82, 1225-1286, 2010.