Born approximation

In the perturbation theory of scattering of waves, especially in quantum mechanics, the lowest approximation in the perturbation series is known as the Born approximation. However, it is used not only in the quantum mechanics, but for example also in the theory of the scattering of electromagnetic waves. It is named after Max Born, who used it in his essay to quantum mechanics of impact processes.

Intuitively, one can imagine the Born approximation using the example of the scattering of radar waves on a plastic stick. In the Bornnäherung it is assumed that the polarized by the external field atoms in the plastic stick swing ( contributing as a small transmitter to the total field ) to the beat of the outer driver field of the incident radar waves. The fact that the atoms thereby producing self - fields, electromagnetic waves, which in turn affect the other atoms ( multiple scattering ), is neglected in this approximation. The Born approximation is valid accordingly, then as a good approximation when the scattering potential is small compared to the incident field small compared to the energy of the incident wave field, and thus the scattered field of a single atom.

Born- approximation of the Lippmann -Schwinger equation

The Lippmann -Schwinger equation for the scattering state with pulse on and off or incoming direction () is

With the Green function of the free particle, a small positive parameter and the interaction potential. The incident field is; one can interpret as a solution of the scattering problem without spreader. The term on the right side of the equation acts as a driver.

This equation can be simplified in terms of the Born approximation to

So that the right side no longer depends on the unknown state. For the explicit form in coordinate representation, see Lippmann -Schwinger equation.

Distorted Wave Born Approximation ( DWBA )

Sometimes, a part A of the scattering process is calculated separately by analytical or numerical method, and the dispersion of a residual potential ( Part B), which is treated as noise in Bornnäherung added. In this case, the " disturbed " (distorted ) waves - taken from Part A as the output wave functions of the disturbance development of Part B - as opposed to those used in the usual application of Bornnäherung flat or spherical waves. One speaks of Distorted Wave Born Approximation or DWBA. If the potential of Part A, Part B and the potential of the solution of the scattering problem from Part A ( with the Green function is calculated ), then the DWBA solution given by:

For example, in some problems of scattering of charged particles on other charged particles (such as bremsstrahlung or the photoelectric effect ) can be chosen as an approach for Part A analytical solutions for Coulomb scattering ( scattering in a Coulomb potential ), which then as incident wave in the Bornnäherung part B feed. In some nuclear reactions, the numerically calculated scattering is, for example, often chosen in a so-called optical potential for Part A.