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The Lippmann–Schwinger equation 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 and therefore allows calculation of the relevant experimental parameters.
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 :
The potential energy V {\displaystyle V\,} describes the interaction between the two colliding systems. The Hamiltonian H 0 {\displaystyle H_{0}\,} describes the situation in which the two systems are infinitely far apart and do not interact. Its eigenfunctions are | ϕ ⟩ {\displaystyle |\phi \rangle \,} and its eigenvalues are the energies E {\displaystyle E\,}. Finally, i ϵ {\displaystyle i\epsilon \,} is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation. It is a consequence of causality, ensuring that scattered waves consist only of outgoing waves. This is made rigorous by the limiting absorption principle.