Scattering amplitude

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In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process.[1] The latter is described by the wavefunction


\psi(\mathbf{r}) = e^{ikz} + f(\theta)\frac{e^{ikr}}{r} \;,

where \mathbf{r}\equiv(x,y,z) is the position vector; r\equiv|\mathbf{r}|; e^{ikz} is the incoming plane wave with the wavenumber k along the z axis; e^{ikr}/r is the outgoing spherical wave; \theta is the scattering angle; and f(\theta) is the scattering amplitude. The dimension of the scattering amplitude is length.

The differential cross-section is given as


\frac{d\sigma}{d\Omega} = |f(\theta)|^2 \;.

In the low-energy regime the scattering amplitude is determined by the scattering length.

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Partial wave expansion [edit]

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[2]

f(\theta)=\sum_{\ell=0}^\infty (2\ell+1) f_\ell(k) P_\ell(\cos(\theta)) \;,

where f_\ell(k) is the partial amplitude and P_\ell(\cos(\theta)) is the Legendre polynomial.

The partial amplitude can be expressed via the S-matrix element S_\ell=e^{2i\delta_\ell} and the scattering phase \delta_\ell as

f_\ell = \frac{S_\ell-1}{2ik} = \frac{e^{2i\delta_\ell}-1}{2ik} = \frac{e^{i\delta_\ell} \sin\delta_\ell}{k} = \frac{1}{k\cot\delta_\ell-ik} \;.

X-rays [edit]

The scattering length for X-rays is the Thompson scattering length or classical electron radius, r_0.

Neutrons [edit]

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

Quantum mechanical formalism [edit]

A quantum mechanical approach is given by the S matrix formalism.

References [edit]

  1. ^ Quantum Mechanics: Concepts and Applications By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009, ©2008
  2. ^ Michael Fowler/ 1/17/08 Plane Waves and Partial Waves