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In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.[1]

The latter is described by the wavefunction

where is the position vector; ; is the incoming plane wave with the wavenumber k along the z axis; is the outgoing spherical wave; θ is the scattering angle; and is the scattering amplitude. The dimension of the scattering amplitude is length.

The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,


Partial wave expansionEdit

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


where f is the partial scattering amplitude and P are the Legendre polynomials.

The partial amplitude can be expressed via the partial wave S-matrix element S ( ) and the scattering phase shift δ as


Then the differential cross section is given by[3]


and the total elastic cross section becomes


where Im f(0) is the imaginary part of f(0).


The scattering length for X-rays is the Thomson scattering length or classical electron radius, r0.


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

Quantum mechanical formalismEdit

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


The scattering amplitude can be determined by the scattering length in the low-energy regime.

See alsoEdit


  1. ^ Quantum Mechanics: Concepts and Applications Archived 2010-11-10 at the Wayback Machine By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009
  2. ^ Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
  3. ^ Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120.