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Concept
Born approximation
Summary
Generally 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. The Born approximation is named after Max Born who proposed this approximation in early days of quantum theory development.
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 on the scatterer.
For example, the 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.
The Lippmann–Schwinger equation for the scattering state with a momentum p and out-going (+) or in-going (−) boundary conditions is
where is the free particle Green's function, is a positive infinitesimal quantity, and the interaction potential. is the corresponding free scattering solution sometimes called the incident field. The factor on the right hand side is sometimes called the driving field.
Within the Born approximation, the above equation is expressed as
which is much easier to solve since the right hand side no longer depends on the unknown state .
The obtained solution is the starting point of the Born series.
Using the outgoing free Green's function for a particle with mass in coordinate space,
one can extract the Born approximation to the scattering amplitude from the Born approximation to the Lippmann–Schwinger equation above,
where is the transferred momentum.
The Born approximation is used in several different physical contexts.
In neutron scattering, the first-order Born approximation is almost always adequate, except for neutron optical phenomena like internal total reflection in a neutron guide, or grazing-incidence small-angle scattering.
Official source
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