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Johnjbarton (talk | contribs) →Born approximation to the Lippmann–Schwinger equation: as Fourier transform + ref |
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. 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>
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== 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
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