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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. The latter is described by the wavefunction
where is the position vector; ; [clarification needed] is the incoming plane wave with the wavenumber along the axis; is the outgoing spherical wave; is the scattering angle; and is the scattering amplitude. The dimension of the scattering amplitude is length.
Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
where fℓ is the partial scattering amplitude and Pℓ are the Legendre polynomials.
Then the differential cross section is given by
and the total elastic cross section becomes
where Im f(0) is the imaginary part of f(0).
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by .
Quantum mechanical formalism
A quantum mechanical approach is given by the S matrix formalism.
- Quantum Mechanics: Concepts and Applications By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009
- Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
- Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120.