# Scattering amplitude

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] The latter is described by the wavefunction

${\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}$

where ${\displaystyle \mathbf {r} \equiv (x,y,z)}$ is the position vector; ${\displaystyle r\equiv |\mathbf {r} |}$; ${\displaystyle e^{ikz}}$ is the incoming plane wave with the wavenumber k along the z axis; ${\displaystyle e^{ikr}/r}$ is the outgoing spherical wave; θ is the scattering angle; and ${\displaystyle f(\theta )}$ 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,

${\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\theta )|^{2}\;.}$

## X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r0.

## Neutrons

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

## 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.