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] At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction[2] where
is the incoming plane wave with the wavenumber k along the z axis;
is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and
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, The asymptotic form of the wave function in arbitrary external field takes the form[2] where
is the direction of incidient particles and
is the direction of scattered particles.
When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude.
In the general case, we have[2] Optical theorem follows from here by setting
In the centrally symmetric field, the unitary condition becomes where
This condition puts a constraint on the allowed form for
, i.e., the real and imaginary part of the scattering amplitude are not independent in this case.
is known (say, from the measurement of the cross section), then
[2] In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[3] 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 (
) and the scattering phase shift δℓ as Then the total cross section[4] can be expanded as[2] is the partial cross section.
The total cross section is also equal to
due to optical theorem.
, we can write[2] 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.
A quantum mechanical approach is given by the S matrix formalism.