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Mathematics for Physics

141 Scattering Theory

A framework for collisions that is used to relate incoming free-particle states to outgoing amplitudes and cross sections, where a cross section is an effective area that measures the likelihood of a scattering event.

The incident-plus-scattered wave. An incident beam interacting with a short-range target produces an outgoing wave. In the far region the wavefunction is an incident plane wave plus a scattered spherical wave. This principle is used to define the scattering amplitude \(f\).

The asymptotic scattered wave is

\[ \psi(\mathbf{r}) \sim e^{ikz} + f(\theta,\phi)\dfrac{e^{ikr}}{r} \]

where

  • \(f\) is the scattering amplitude.
  • \(\theta\) and \(\phi\) are scattering angles.
  • \(k\) is the wave number.
  • \(r\) is the distance from the target.

The differential cross section. The differential cross section is the squared modulus of the scattering amplitude. This principle is used to convert \(f\) into a measured angular distribution.

The differential cross section is

\[ \dfrac{d\sigma}{d\Omega} = \lvert f(\theta,\phi)\rvert^{2} \]

where

  • \(\sigma\) is the cross section.
  • \(\Omega\) is solid angle.
  • \(f\) is the scattering amplitude.

One-dimensional reflection and transmission. In one dimension the same problem is stated as reflection and transmission amplitudes. This principle is used to analyze barriers and wells on the line.

Note: Also called quantum scattering.

141.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — incident plus scattered wave; amplitude \(f\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — scattering states for short-range potentials.
  3. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(\dfrac{d\sigma}{d\Omega}=|f|^{2}\).