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. At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction
where is the position vector; ; 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 is the scattering amplitude. 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
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
Optical theorem follows from here by setting
In the centrally symmetric field, the unitary condition becomes
where and are the angles between and and some direction . 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. For example, if in is known (say, from the measurement of the cross section), then can be determined such that is uniquely determined within the alternative .
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 ( ) and the scattering phase shift δℓ as
Then the total cross section
can be expanded as
is the partial cross section. The total cross section is also equal to due to optical theorem.
For , we can write
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.
The scattering amplitude can be determined by the scattering length in the low-energy regime.