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 a stationary-state scattering process.[1] 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.

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

Partial wave expansion[edit]

Main article: Partial wave analysis

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, .


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

Quantum mechanical formalism[edit]

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[clarification needed].