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.[1]
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
(\pm) | |
\vert{\Psi | |
p |
(\pm) | |
\vert{\Psi | |
p |
where
G\circ
\epsilon
V
\circ | |
\vert{\Psi | |
p |
(\pm) | |
\vert{\Psi | |
p |
Within the Born approximation, the above equation is expressed as
(\pm) | |
\vert{\Psi | |
p |
which is much easier to solve since the right hand side no longer depends on the unknown state
(\pm) | |
\vert{\Psi | |
p |
The obtained solution is the starting point of the Born series.
Using the outgoing free Green's function for a particle with mass
m
(+) | |||||
G | (r,
|
e+ik|r-r'| | |
4\pi|r-r'| |
f | ||||
|
\intd3reiq ⋅ V(r) ,
\theta
k
k'
q=k'-k
V=V(r)
f | ||||
|
infty | |
\int | |
0 |
rV(r)
\sinqr | |
q |
dr
q=|q|=2k\sin(\theta/2).
\sigma
p=k/\hbar
\theta
p\sin(\theta/2)
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. The Born approximation has also been used to calculate conductivity in bilayer graphene[3] and to approximate the propagation of long-wavelength waves in elastic media.[4]
The same ideas have also been applied to studying the movements of seismic waves through the Earth.[5]
The Born approximation is simplest when the incident waves
\circ | |
\vert{\Psi | |
p |
In the distorted-wave Born approximation (DWBA), the incident waves are solutions
1 | |
\vert{\Psi | |
p |
V1
V=V1+V2
V
V2
V1
1 | |
\vert{\Psi | |
p |
and the Born approximation
(\pm) | |
\vert{\Psi | |
p |
Other applications include bremsstrahlung and the photoelectric effect. For a charged-particle-induced direct nuclear reaction, the procedure is used twice. There are similar methods that do not use the Born approximations. In condensed-matter research, DWBA is used to analyze grazing-incidence small-angle scattering.
. Roger G. Newton . Newton, Roger G. . Scattering Theory of Waves and Particles . Dover Publications, inc. . 2002 . 0-486-42535-5.