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. The plane wave is described by the wavefunction

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

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

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

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.

See also

• Veneziano amplitude
• Plane wave expansion