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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. The latter is described by the wavefunction
where is the position vector; ; 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 differential cross-section is given as
In the low-energy regime the scattering amplitude is determined by the scattering length.
Partial wave expansion 
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
where is the partial amplitude and is the Legendre polynomial.
The partial amplitude can be expressed via the S-matrix element and the scattering phase as
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.