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 latter is described by the wavefunction where r ≡ ( x , y , z ) {displaystyle mathbf {r} equiv (x,y,z)} is the position vector; r ≡ | r | {displaystyle requiv |mathbf {r} |} ; e i k z {displaystyle e^{ikz}} is the incoming plane wave with the wavenumber k along the z axis; e i k r / r {displaystyle e^{ikr}/r} is the outgoing spherical wave; θ is the scattering angle; and f ( θ ) {displaystyle f( heta )} 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, 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 ( = e 2 i δ ℓ {displaystyle =e^{2idelta _{ell }}} ) and the scattering phase shift δℓ as Then the differential cross section is given by