Scattering state

A scattering state describes an incoming preparation and its outgoing components after an interaction. In a stationary calculation, it is typically a generalized energy eigenstate with specified incoming boundary data, rather than a square-integrable bound state.

For a short-range potential, its distant form separates freely propagating incoming and scattered waves. The outgoing boundary condition selects how a localized disturbance radiates away, and the resulting relative coefficient determines a scattering amplitude. A physical normalizable packet is formed by superposing stationary scattering states over an energy or momentum range. This distinction allows finite particle probabilities and localized preparations without treating an infinite plane wave as a normalized state inside the physical Hilbert space.

A Green function for stationary scattering is chosen to contain waves travelling away from its source. In one dimension at positive energy it is proportional to $e^{ik|x-x^{\prime }|}$; the condition selects the $E+i0$ resolvent boundary value.