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 is:
| ψ ⟩ = | ϕ ⟩ + 1 E − H 0 + i ϵ V | ψ ⟩ . {\displaystyle |\psi \rangle =|\phi \rangle +{\frac {1}{E-H_{0}+i\epsilon }}V|\psi \rangle .\,}
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.
Usage The Lippmann–Schwinger equation is useful in a very large number of situations involving two-body scattering. For three or more colliding bodies it does not work well because of mathematical limitations; Faddeev equations may be used instead. However, there are approximations that can reduce a many-body problem to a set of two-body problems in a variety of cases. For example, in a collision between electrons and molecules, there may be tens or hundreds of particles involved. But the phenomenon may be reduced to a two-body problem by describing all the molecule constituent particle potentials together with a pseudopotential. In these cases, the Lippmann–Schwinger equations may be used. Of course, the main motivations of these approaches are also the possibility of doing the calculations with much lower computational efforts.
Derivation We will assume that the Hamiltonian may be written as
H = H 0 + V {\displaystyle H=H_{0}+V}
where H0 is the free Hamiltonian (or more generally, a Hamiltonian with known eigenvectors). For example, in nonrelativistic quantum mechanics H0 may be
H 0 = p 2 2 m . {\displaystyle H_{0}={\frac {p^{2}}{2m}}.}
Intuitively V is the interaction energy of the system. Let there be an eigenstate of H0:
H 0 | ϕ ⟩ = E | ϕ ⟩ . {\displaystyle H_{0}|\phi \rangle =E|\phi \rangle .}
Now if we add the interaction V {\displaystyle V} into the mix, the Schrödinger equation reads
( H 0 + V ) | ψ ⟩ = E | ψ ⟩ . {\displaystyle \left(H_{0}+V\right)|\psi \rangle =E|\psi \rangle .}
Now consider the Hellmann–Feynman theorem, which requires the energy eigenvalues of the Hamiltonian to change continuously with continuous changes in the Hamiltonian. Therefore, we wish that | ψ ⟩ → | ϕ ⟩ {\displaystyle |\psi \rangle \to |\phi \rangle } as V → 0 {\displaystyle V\to 0} . A naive solution to this equation would be
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