Lippmann–Schwinger equation

The Lippmann–Schwinger equation (named after Bernard Lippmann and Julian Schwinger) is one of the most used equations to describe particle collisions – or, more precisely, scattering – in quantum mechanics. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in atomic, molecular, and optical physics, nuclear physics and particle physics, but also for seismic scattering problems in geophysics. It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and therefore allows calculation of the relevant experimental parameters (scattering amplitude and cross sections).

The most fundamental equation to describe any quantum phenomenon, including scattering, is the Schrödinger equation. In physical problems, this differential equation must be solved with the input of an additional set of initial and/or boundary conditions for the specific physical system studied. The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an integral equation. For scattering problems, the Lippmann–Schwinger equation is often more convenient than the original Schrödinger equation.

The Lippmann–Schwinger equation's general form is (in reality, two equations are shown below, one for the ${\displaystyle +\,}$ sign and other for the ${\displaystyle -\,}$ sign):

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