Born approximation

This article deals with the Born approximation in scattering theory. For the approximation of the multi-particle Schrödinger equation, also named after Max Born, see Born-Oppenheimer approximation .

In the perturbation theory of the scattering of waves, especially in quantum mechanics , the lowest approximation in the perturbation series is referred to as the Born approximation . It is not only used in quantum mechanics, but z. B. also used in the theory of electromagnetic wave scattering . It is named after Max Born , who used it in his essay On Quantum Mechanics of Impact Processes .

Illustrative example

One can visualize the Born approximation using the example of the scattering of radar waves on a plastic rod. It is assumed that the atoms in the plastic rod polarized by the external field (which contribute to the overall field as small transmitters) oscillate in time with the external driver field of the incident radar waves.

The fact that the atoms themselves generate electromagnetic wave fields, which in turn influence the other atoms (multiple scattering), is neglected in this approximation. Accordingly, the Born approximation is considered a good approximation if the scattering potential is small compared to the energy of the incident wave field and thus the field scattered on a single atom is small compared to the incident field.

Born approximation of the Lippmann-Schwinger equation

The Lippmann-Schwinger equation for the state of scattering with momentum and outgoing or incoming direction ( ) is: ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \pm }$

${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle =\vert {\Psi _{\mathbf {p} }^{0}}\rangle +G^{0}(E_{p}\pm i\varepsilon )V\vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$

With

• the Green function of the free particle ${\displaystyle G^{0}}$
• a small positive parameter ${\displaystyle \varepsilon }$
• the interaction potential ${\displaystyle V}$
• the incident field ; it can be interpreted as a solution to the scattering problem without a scatterer. ${\displaystyle \vert {\Psi _{\mathbf {p} }^{0}}\rangle }$
• the term on the right side of the equation as the driver. ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$

This equation can be simplified in terms of Born's approximation to

${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle =\vert {\Psi _{\mathbf {p} }^{0}}\rangle +G^{0}(E_{p}\pm i\varepsilon )V\vert {\Psi _{\mathbf {p} }^{0}}\rangle }$,

so that the right side no longer depends on the unknown state . ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$

For the explicit form in position representation see Lippmann-Schwinger equation .

Distorted Wave Born Approximation (DWBA)

Sometimes part A of the scattering process is calculated separately in an analytical or numerical way, and the scattering at a residual potential (part B), which is treated as a perturbation in the Born approximation, is added. In this case, the "be disturbed " (distorted) waves - as opposed to those used in the usual application of Bornnäherung flat or spherical waves - from Part A as output wave functions taken B for disturbing development part. One speaks of Distorted Wave Born Approximation or DWBA.

If the potential of part A, the potential of part B and the solution of the scattering problem from part A (with which the Greens function is also calculated), then the DWBA solution results from: ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$ ${\displaystyle \vert {\Psi _{\mathbf {p} }^{1}}^{(\pm )}\rangle }$ ${\displaystyle G^{1}}$

${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle =\vert {\Psi _{\mathbf {p} }^{1}}^{(\pm )}\rangle +G^{1}(E_{p}\pm i\varepsilon )V_{2}\vert {\Psi _{\mathbf {p} }^{1}}^{(\pm )}\rangle .}$

For example, for some problems of the scattering of charged particles on other charged particles (such as bremsstrahlung or the photoelectric effect ), analytical solutions for Coulomb scattering (scattering in a Coulomb potential ) can be chosen as the approach for Part A , which then enter the Include the Born approximation of part B. In some nuclear reactions z. B. often the numerically calculated scattering in an optical potential for part A is chosen.

See also

• Perturbation theory for more general approximation methods

literature

Quantum mechanics textbooks like

• Sakurai, JJ: Modern Quantum Mechanics . Addison-Wesley, 1994, ISBN 0-201-53929-2 .

Individual evidence

1. ^ Journal of Physics. 37, No. 12, 1926, pp. 863-867