A mathematical function describing the probability amplitude for a scattering process.The scattering amplitude is a complex-valued function in quantum mechanics that encodes the probability amplitude for a scattering process — that is, the likelihood that a particle (or wave) will be deflected by a target into a particular direction. Its absolute square gives the differential cross-section, which is what experiments actually measure: how many particles scatter into a given solid angle. The scattering amplitude is central to understanding everything from particle collisions in accelerators to the way light scatters off a glass sphere or how neutrons interact with atomic nuclei.
The modern formulation of scattering amplitude dates to the late 1920s and 1930s, with Max Born's approximation method (1926) and the development of partial wave analysis by Gregory Breit and Eugene Wigner (1936). The S-matrix concept, which generalizes the scattering amplitude, was introduced by John Archibald Wheeler in 1937 and developed by Werner Heisenberg in 1943.
In quantum scattering theory, the scattering amplitude f(θ) appears in the asymptotic form of the wavefunction far from the scatterer: ψ(r) ≈ e^{ikz} + f(θ)e^{ikr}/r. The first term is the incoming plane wave, the second is the outgoing spherical wave modulated by f(θ). The amplitude is computed by solving the Schrödinger equation with the scattering potential, often using perturbation theory (Born approximation) or partial wave analysis. It contains all the physical information about the interaction — resonances, phase shifts, and the strength of the force.
Scattering amplitudes are used in particle physics to predict collision outcomes at the LHC, in condensed matter physics to interpret neutron and X-ray diffraction experiments, and in medical imaging techniques like computed tomography where X-ray scattering must be modeled.
The scattering amplitude for identical particles can be zero at certain angles due to quantum interference — a phenomenon known as 'forbidden scattering' that has no classical analog.


