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
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where
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
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.
The partial amplitude can be expressed via the partial wave S-matrix element Sℓ (
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).
X-rays
The scattering length for X-rays is the Thomson scattering length or classical electron radius,
Neutrons
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.
Measurement
The scattering amplitude can be determined by the scattering length in the low-energy regime.