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] At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction^[2]

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

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 (angle between the incident and scattered direction); 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,

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

The asymptotic form of the wave function in arbitrary external field takes the form^[2]

${\displaystyle \psi =e^{ikr\mathbf {n} \cdot \mathbf {n} '}+f(\mathbf {n} ,\mathbf {n} '){\frac {e^{ikr}}{r}}}$

where ${\displaystyle \mathbf {n} }$ is the direction of incidient particles and ${\displaystyle \mathbf {n} '}$ is the direction of scattered particles.