Scattering States and Observables

A scattering experiment connects a prepared beam with an observed pattern of outgoing particles. This chapter develops that connection in five steps: preparation, boundary conditions, probability current, amplitude, and cross section. Keep the distinction between a complex amplitude and a measured rate in view throughout.

Helpful background. Free-particle waves describe incoming motion; probability current explains transport; normalization conventions distinguish wave packets from continuum eigenstates.

Begin with the experiment

What is a scattering experiment? introduces beam flux, targets, detectors, and angular distributions. A detector counts particles in a finite acceptance region, with finite efficiency and resolution. The cross section is extracted by accounting for those factors and the incident flux.

For a uniform beam incident on a thin sample of independent targets, a small angular bin has the approximate count rate

$$R_{\mathrm{bin}}\approx j_{\mathrm{inc}}N_{\mathrm{targets}}\eta \frac{d\sigma}{d\Omega}\Delta\Omega.$$

Here $\eta$ is the detection efficiency, assumed nearly constant across the bin, and the differential cross section is also taken nearly constant there. Multiple scattering and appreciable beam attenuation would require a more detailed model. The formula makes one point immediate: twice the incident flux gives twice the count rate, with the same cross section.

Specify the wave that describes the preparation

Scattering states and boundary conditions explains the stationary solutions used to resolve a collision by energy. An outgoing scattering solution contains a specified incident wave plus the outgoing wave generated by the interaction. The boundary condition selects the solution; solving the differential equation without it is incomplete.

A physical wave packet can be normalized to total probability one. An ideal stationary plane wave extends throughout space and instead uses continuum or flux normalization. These descriptions serve different purposes: a packet describes a preparation and its evolution, while stationary solutions provide the components from which that packet can be assembled.

Follow the current to the detector

Probability current and flux connects the wavefunction to transport. Flux through a surface is current dotted with its oriented normal. In one dimension, a reflected current points opposite to the incident current; the nonnegative reflection probability uses the magnitude of their ratio.

The scattering amplitude is the complex coefficient of the outgoing spherical wave in the usual short-range, three-dimensional setting. Its phase is essential when several contributions reach the same final state. Squaring each contribution separately would discard their interference.

Differential and total cross sections then turns outgoing flux into an angular distribution and integrates over the relevant final states. In a single elastic channel with unit incident-wave amplitude, $d\sigma/d\Omega=|f|^2$. Different incoming and outgoing channel speeds introduce a velocity ratio; identical particles require exchange symmetry and consistent final-state counting.

Try the connection yourself

Flux and efficiency. A detector’s count rate falls by half after its efficiency falls from $0.8$ to $0.4$. The beam and target are unchanged. Has the physical cross section changed?

Solution

No. Correcting each measurement for its known efficiency gives the same cross section. An uncorrected count rate combines the interaction’s response with the preparation and detector properties.

Reflected current. For a one-dimensional wave with incident current $j_{\mathrm{in}}>0$ and reflected current $j_{\mathrm{ref}}<0$, what is the reflection probability?

Solution

It is $R=-j_{\mathrm{ref}}/j_{\mathrm{in}}=|j_{\mathrm{ref}}|/j_{\mathrm{in}}$. The sign of the current describes direction; $R$ is nonnegative. For a lossless single-channel problem, $R+T=1$ after the transmitted velocity factor is included.

Continue to operators and unitarity to organize these amplitudes, or return to the volume guide to choose another route.

References

• Joachain, C. J. Quantum Collision Theory. 3rd ed., North-Holland (1983). Collision observables, normalization, and flux.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Scattering states, packets, and cross sections.