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. 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, 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

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

where $G^\circ$ is the free particle Green's function, $\epsilon$ 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

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

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.

Applications

The Born approximation is used in quite 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^[2] and to approximate the propagation of long-wavelength waves in elastic media^[3].

The same ideas have also been applied to studying the movements of seismic waves through the Earth^[4].

Distorted wave Born approximation (DWBA)

The Born approximation is simplest when the incident 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.

In the distorted wave Born approximation (DWBA), the incident waves are solutions $\vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle$ to a part $V^1$ of the problem $V=V^1 + V^2$ that is treated by some other method, either analytical or numerical. The interaction of interest $V$ is treated as a perturbation $V^2$ to some system $V^1$ that can be solved by some other method. For nuclear reactions, numerical optical model waves are used. For 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. In condensed-matter research, DWBA is used to analyze grazing-incidence small-angle scattering.

See also

• Born series
• Quantum scattering theory
• Lippmann-Schwinger equation
• Dyson series
• T-matrix
• Green's operator
• Electromagnetic modeling

References

1. Born, Max (1926). "Quantenmechanik der Stossvorgänge". Zeitschrift für Physik 38: 803. Bibcode:1926ZPhy...38..803B. doi:10.1007/BF01397184.
2. Koshino, Mikito; Ando, Tsuneya (2006). "Transport in bilayer graphene: Calculations within a self-consistent Born approximation". Physical Review B 73.
3. Gubernatis, J.E.; Domany, E.; Krumhansl, J.A.; Huberman, M. (1977). "The Born approximation in the theory of the scattering of elastic waves by flaws". Journal of Applied Physics 48.
4. Hudson, J.A.; Heritage, J.R. (1980). "The use of the Born approximation in seismic scattering problems". Geophysical Journal of the Royal Astronomical Society 66: 221–240.

• Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2.
• Newton, Roger G. (2002). Scattering Theory of Waves and Particles. Dover Publications, inc. ISBN 0-486-42535-5.
• Wu and Ohmura, Quantum Theory of Scattering, Prentice Hall, 1962