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# Born approximation

In scattering theory and, in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small, compared to the incident field, in the scatterer.

For example, the radar scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution.

## Born approximation to the Lippmann-Schwinger equation

The Lippmann-Schwinger equation for the scattering state $\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle$ with a momentum p and out-going (+) or in-going (-) boundary conditions is

$\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i0) V \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle$

where $G^\circ$ is the free particle Green's function, 0 is a positive infinitesimal quantity, and V the interaction potential. $\vert{\Psi_{\mathbf{p}}^{\circ}}\rangle$ is the corresponding free scattering solution sometimes called incident field. The factor $\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle$ on the right hand side is sometimes called driving field.

This equation becomes within Born approximation

$\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i0) V \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle$

which is much easier to solve since the right hand side does not depend on the unknown state $\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle$ anymore.

The obtained solution is the starting point of the Born series.

## Scattering by two potentials

In its simplest form, the incident and scattered waves $\vert{\Psi_{\mathbf{p}}^{\circ}}\rangle$ are plane waves. That is, the scatterer is treated as a perturbation to free space or to a homogeneous medium. They can also be taken as solutions $\vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle$ to a part V1 of the problem V = V1 + V2 that is treated by some other method, either analytical or numerical. The interaction of interest V is treated as a perturbation V2 to some system V1 that can be solved by some other method. In the distorted wave Born approximation (DWBA) to nuclear reactions, numerical optical model waves are used. In scattering of charged particles by charged particles, analytic solutions for coulomb scattering are used. This gives the non-Born preliminary equation

$\vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i0) V^{1} \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle$

and the Born approximation

$\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle + G^1(E_p \pm i0) V^{2} \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle$

Other applications include bremsstrahlung and the Photoelectric effect. For charged particle induced direct nuclear reaction, the procedure is used twice. There are similar methods that do not use Born approximations.