The Lippmann–Schwinger equation (named after Bernard A. 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. 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 has been proved to be mathematically and intuitively more convenient than the original Schrödinger equation.The Lippmann–Schwinger equation general shape is (in reality, two equations are shown below, one for the sign and other for the sign):In the equations above, is the wave function of the whole system (the two colliding systems considered as a whole) at an infinite time before the interaction; and , at an infinite time after the interaction (the "scattered wave fuction"). The potential energy describes the interaction between the two colliding systems. The hamiltonian describes the situation in which the two systems are infinitely far apart and do not interact. Its eigenfunctions are and its eigenvalues are the energies . Finally, is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation and has no physical meaning.
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