Content deleted Content added
Johnjbarton (talk | contribs) →Born approximation to the Lippmann–Schwinger equation: as Fourier transform + ref |
|||
| (6 intermediate revisions by 3 users not shown) | |||
Line 9:
For example, the scattering of [[radio wave]]s 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.
== Approximate scattering amplitude ==
==Born approximation to the Lippmann–Schwinger equation==▼
Starting with a physical model based on the [[Schrodinger equation|Schrodinger wave equation]] for scattering from a potential {{mvar|V}}
the [[scattering amplitude]], {{mvar|f}}, requires knowing the full scattering [[wavefunction]] <math>\psi</math>,<ref name="Schiff-1987"/>{{rp|325}}
<math display="block">f(\mathbf{k}_f,\mathbf{k}_i) = -\frac{\mu}{2\pi\hbar^2}\int \psi_f^* V(\mathbf{r}) \psi_i d^3r</math>
In the Born approximation, the initial and final wavefunctions approximated plane waves:
<math display="block">f(\mathbf{k}_f,\mathbf{k}_i) = -\frac{\mu}{2\pi\hbar^2}\int e^{-i\mathbf{k}_f\cdot\mathbf{r}} V(\mathbf{r}) e^{i\mathbf{k}_i\cdot\mathbf{r}} d^3r</math>
This is mathematically equivalent to the [[Fourier transform]] of the scattering potential from <math>\mathbf{r}</math> to <math>\mathbf{q}=\mathbf{k}_f - \mathbf{k}_i</math>:<ref name="Schiff-1987"/>{{rp|325}}
<math display="block">f(\mathbf{k}_f,\mathbf{k}_i) = -\frac{\mu}{2\pi\hbar^2}\int V(\mathbf{r}) e^{i\mathbf{q}\cdot\mathbf{r}} d^3r</math>
For a spherically symmetric potential the angular integrations can be performed and the scattering amplitude depends only on the polar angle <math>\theta</math> between the input and output directions:<ref name="Schiff-1987"/>{{rp|325}}
<math display="block">f(\theta) = -\frac{2\mu}{\hbar^2q}\int_0^\infty r \sin(qr) V(r) dr </math>
where <math>q=2k \sin (\theta/2)</math> and <math>k=|\mathbf{k}_f - \mathbf{k}_i|.</math>
The [[Lippmann–Schwinger equation]] for the scattering state <math>\vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle</math> with a momentum '''p''' and out-going (+) or in-going (−) [[boundary condition]]s is
Line 27 ⟶ 39:
The obtained solution is the starting point of a [[perturbation series]] known as the [[Born series]].<ref name="Schiff-1987"/>{{rp|324}}
===
Using the outgoing free Green's function for a particle with mass <math>m</math> in coordinate space,
:<math>
Line 51 ⟶ 63:
|publisher=Cambridge University Press
}}</ref>
. Using this concept, the electronic analogue of Fourier optics has been theoretically studied in [[monolayer]] graphene.<ref>{{cite journal |author=Partha Sarathi Banerjee, Rahul Marathe, Sankalpa Ghosh |title=Electronic analogue of Fourier optics with massless Dirac fermions scattered by quantum dot lattice |journal=Journal of Optics |volume=26 |number=9 |pages=095602 |year=2024 |publisher=IOP Publishing |doi=10.1088/2040-8986/ad645b |url=https://dx.doi.org/10.1088/2040-8986/ad645b|arxiv=2402.11259 }}</ref> The Born approximation has also been used to calculate conductivity in [[bilayer graphene]]<ref>{{cite journal | title= Transport in bilayer graphene: Calculations within a self-consistent Born approximation | last1=Koshino |first1=Mikito | last2=Ando | first2=Tsuneya |journal=Physical Review B |year=2006 |volume=73 | issue=24 | page=245403 | doi=10.1103/physrevb.73.245403|arxiv = cond-mat/0606166 |bibcode = 2006PhRvB..73x5403K | s2cid=119415260 }}</ref> and to approximate the propagation of long-wavelength waves in [[linear elasticity|elastic media]].<ref>{{cite journal | title= The Born approximation in the theory of the scattering of elastic waves by flaws | last1=Gubernatis |first1=J.E. | last2=Domany | first2=E. | last3=Krumhansl |first3=J.A. | last4=Huberman | first4=M. |journal=Journal of Applied Physics |year=1977 |volume=48 | issue=7 | pages=2812–2819 | doi = 10.1063/1.324142|bibcode = 1977JAP....48.2812G }}</ref>
The same ideas have also been applied to studying the movements of [[seismic waves]] through the Earth.<ref>{{cite journal | title= The use of the Born approximation in seismic scattering problems | last1=Hudson |first1=J.A. | last2=Heritage | first2=J.R. |journal=Geophysical Journal of the Royal Astronomical Society |year=1980 |volume=66 | issue=1 | pages=221–240 | doi=10.1111/j.1365-246x.1981.tb05954.x|bibcode = 1981GeoJ...66..221H | doi-access=free }}</ref>
Line 57 ⟶ 69:
== Distorted-wave Born approximation ==
The Born approximation is simplest when the incident waves <math>\vert{\Psi_{\mathbf{p}}^{\circ}}\rangle</math> are [[Plane wave|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 <math>\vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle</math> to a part <math>V^1</math> of the problem <math>V=V^1 + V^2</math> that is treated by some other method, either analytical or numerical. The interaction of interest <math>V</math> is treated as a perturbation <math>V^2</math> to some system <math>V^1</math> 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
Line 86 ⟶ 98:
[[Category:Scattering theory]]
[[Category:Max Born]]
[[Category:Perturbation theory]]
[[Category:1926 introductions]]
| |||