Born approximation

Not to be confused with the Born–Oppenheimer 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.

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

References

  1. ↑ Born, Max (1926). "Quantenmechanik der Stossvorgänge". Zeitschrift für Physik 38: 803. Bibcode:1926ZPhy...38..803B. doi:10.1007/BF01397184.
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