Lippmann–Schwinger equation

The Lippmann -Schwinger equation is used in quantum mechanical perturbation theory, and specifically in the scattering theory. It has the form of an integral equation and is an alternative to the direct solution of the Schrödinger equation with the boundary conditions in the definition of the Green's functions used stuck.

In general, in perturbation theory, the Hamiltonian in the "free Hamiltonian ", to which a solution is known and treated as a small perturbation part ( " potential ") apart. Eigenfunctions of the free Hamiltonian satisfy the equation

With the associated eigenvalue.

As a "free Green's function " refers to an operator that applies to the

This operator is therefore something of a reverse function on the free Hamiltonian. A mathematically correct representation requires the consideration of the distribution.

Now, in a manner analogous to the unknown eigenfunctions of the full Hamiltonian, and its Green's function can be defined. This is now considered the following Lippmann -Schwinger equation

This equation is usually solved iteratively, with the restriction to the first nontrivial order is called the Born approximation.

Scattering theory

The Lippmann -Schwinger equation is accordingly mainly in the scattering theory application. This is calculated as the wave function of a change in the particle dispersion at a potential V being used as the free Hamiltonian kinetic portion on a free particle.

Following is a brief derivation of the equation is carried out in this context ( for a stationary scattering problem, elastic scattering).

The energy of a free particle, the Einschußrichtung of the particle is given by and the direction of the scattering particle is defined as, where, and for all the vectors. It should be noted that it is an elastic scattering, and the magnitude of the momentum vector is not changed.

For the derivation of the Lippmann -Schwinger equation one starts with the Schrödinger equation

This can be solved by the Green function method.

This results in the Lippmann -Schwinger equation of scattering theory.

Here the coordinate representation has been chosen explicitly.

This equation can be solved iteratively by substituting on the right side by the previously obtained solution and as the start value of the iteration as selected.

The first iteration

Is then the mentioned above " Born approximation " in space representation.