1986:
617:
291:
2170:
875:
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1822:
1815:
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
477:
145:
1997:
1536:
is the same as the
Fourier transform of the scattering potential . Using this concept, the electronic analogue of Fourier optics has been theoretically studied in monolayer graphene. The Born approximation has also been used to calculate conductivity in
1283:
711:
2183:. For a charged-particle-induced direct nuclear reaction, the procedure is used twice. There are similar methods that do not use the Born approximations. In condensed-matter research, DWBA is used to analyze
1666:
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414:
129:
1124:
1359:
890:
1981:{\displaystyle \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 }
612:{\displaystyle \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 ,}
286:{\displaystyle \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 ,}
2165:{\displaystyle \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 .}
1081:
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321:
348:
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1433:
1379:
1032:
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1759:
703:
372:
870:{\displaystyle G^{(+)}(\mathbf {r} ,\mathbf {r} ')=-{\frac {2m}{\hbar ^{2}}}{\frac {e^{+ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}}
45:
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
71:
that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution.
2411:
Gubernatis, J.E.; Domany, E.; Krumhansl, J.A.; Huberman, M. (1977). "The Born approximation in the theory of the scattering of elastic waves by flaws".
2184:
1502:
56:
method applied to scattering by an extended body. It is accurate if the scattered field is small compared to the incident field on the scatterer.
327:
1616:
20:
2519:
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625:
419:
377:
86:
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Koshino, Mikito; Ando, Tsuneya (2006). "Transport in bilayer graphene: Calculations within a self-consistent Born approximation".
2216:
2201:
80:
1086:
1004:{\displaystyle f_{B}(\theta )=-{\frac {m}{2\pi \hbar ^{2}}}\int d^{3}re^{i\mathbf {q} \cdot \mathbf {r} }V(\mathbf {r} )\;,}
1291:
2545:
1505:. Using the first Born approximation, it has been shown that the scattering amplitude for a scattering potential
2488:
2211:
1602:
are plane waves. That is, the scatterer is treated as a perturbation to free space or to a homogeneous medium.
1361:
In the Born approximation for centrally symmetric field, the scattering amplitude and thus the cross section
2278:
Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.
2338:
1438:
1278:{\displaystyle f_{B}(\theta )=-{\frac {2m}{\hbar ^{2}}}\int _{0}^{\infty }rV(r){\frac {\sin qr}{q}}dr}
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2420:
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2305:"Electronic analogue of Fourier optics with massless Dirac fermions scattered by quantum dot lattice"
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that is treated by some other method, either analytical or numerical. The interaction of interest
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which is much easier to solve since the right hand side no longer depends on the unknown state
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is the corresponding free scattering solution sometimes called the incident field. The factor
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column can be approximated by assuming that each part of the plastic is polarized by the same
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who proposed this approximation in early days of quantum theory development.
64:
1493:, the first-order Born approximation is almost always adequate, except for
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1486:
The Born approximation is used in several different physical contexts.
27:
884:
from the Born approximation to the
Lippmann–Schwinger equation above,
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2321:
1548:
The same ideas have also been applied to studying the movements of
471:
Within the Born approximation, the above equation is expressed as
2448:"The use of the Born approximation in seismic scattering problems"
1126:
is the transferred momentum. In the centrally symmetric potential
685:
Using the outgoing free Green's function for a particle with mass
1661:{\displaystyle \vert {\Psi _{\mathbf {p} }^{1}}^{(\pm )}\rangle }
2303:
Partha
Sarathi Banerjee, Rahul Marathe, Sankalpa Ghosh (2024).
1541:
and to approximate the propagation of long-wavelength waves in
1595:{\displaystyle \vert {\Psi _{\mathbf {p} }^{\circ }}\rangle }
663:{\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }
457:{\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }
409:{\displaystyle \vert {\Psi _{\mathbf {p} }^{\circ }}\rangle }
124:{\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }
26:"DWBA" redirects here. For the Philippine radio station, see
1560:
The Born approximation is simplest when the incident waves
2236:
Born, Max (1926). "Quantenmechanik der
Stossvorgänge".
1119:{\displaystyle \mathbf {q} =\mathbf {k} '-\mathbf {k} }
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Geophysical
Journal of the Royal Astronomical Society
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Born approximation to the
Lippmann–Schwinger equation
1354:{\displaystyle q=|\mathbf {q} |=2k\sin(\theta /2).}
673:The obtained solution is the starting point of the
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464:on the right hand side is sometimes called the
1497:phenomena like internal total reflection in a
880:one can extract the Born approximation to the
681:Born approximation to the scattering amplitude
1034:is the angle between the incident wavevector
8:
2343:: CS1 maint: multiple names: authors list (
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2512:Scattering Theory of Waves and Particles
2185:grazing-incidence small-angle scattering
1503:grazing-incidence small-angle scattering
135:and out-going (+) or in-going (−)
2288:Sakurai, J. J.; Napolitano, J. (2020).
2228:
1402:
1207:
932:
772:
2336:
2446:Hudson, J.A.; Heritage, J.R. (1980).
7:
1613:), the incident waves are solutions
1161:, the scattering amplitude becomes
2473:10.1111/j.1365-246x.1981.tb05954.x
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2041:
2006:
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1228:
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95:
14:
1607:distorted-wave Born approximation
1556:Distorted-wave Born approximation
2132:
2046:
2011:
1951:
1877:
1837:
1631:
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1519:
1471:{\displaystyle p\sin(\theta /2)}
1307:
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100:
1529:{\displaystyle V(\mathbf {r} )}
59:For example, the scattering of
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2017:
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21:Born–Oppenheimer approximation
1:
2315:(9). IOP Publishing: 095602.
2292:. Cambridge University Press.
1761:is treated as a perturbation
1734:{\displaystyle V=V^{1}+V^{2}}
1435:only through the combination
1415:and the scattering amplitude
1076:{\displaystyle \mathbf {k} '}
1056:and the scattered wavevector
2530:Quantum Theory of Scattering
1049:{\displaystyle \mathbf {k} }
19:Not to be confused with the
2514:. Dover Publications, inc.
2217:Rayleigh–Gans approximation
2202:Lippmann–Schwinger equation
2175:Other applications include
1991:and the Born approximation
374:the interaction potential.
81:Lippmann–Schwinger equation
2567:
2413:Journal of Applied Physics
2390:10.1103/physrevb.73.245403
1408:{\displaystyle p=k/\hbar }
316:{\displaystyle G^{\circ }}
25:
18:
343:{\displaystyle \epsilon }
83:for the scattering state
2489:Modern Quantum Mechanics
2331:10.1088/2040-8986/ad645b
2290:Modern Quantum Mechanics
2212:Electromagnetic modeling
1381:depends on the momentum
2486:Sakurai, J. J. (1994).
1428:{\displaystyle \theta }
1374:{\displaystyle \sigma }
1027:{\displaystyle \theta }
2238:Zeitschrift fĂĽr Physik
2166:
1982:
1809:
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1155:
1154:{\displaystyle V=V(r)}
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1028:
1005:
871:
699:
664:
613:
458:
410:
368:
344:
317:
287:
125:
2532:, Prentice Hall, 1962
2167:
1983:
1810:
1808:{\displaystyle V^{1}}
1783:
1781:{\displaystyle V^{2}}
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1736:
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1688:{\displaystyle V^{1}}
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705:in coordinate space,
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126:
37:and in particular in
2181:photoelectric effect
1998:
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882:scattering amplitude
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87:
2464:1981GeoJ...66..221H
2425:1977JAP....48.2812G
2382:2006PhRvB..73x5403K
2250:1926ZPhy...38..803B
2142:
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1552:through the Earth.
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175:
137:boundary conditions
116:
2492:. Addison Wesley.
2258:10.1007/BF01397184
2244:(11–12): 803–827.
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43:Born approximation
2546:Scattering theory
2360:Physical Review B
2309:Journal of Optics
1754:{\displaystyle V}
1267:
1216:
942:
865:
781:
698:{\displaystyle m}
367:{\displaystyle V}
39:quantum mechanics
35:scattering theory
16:Scattering theory
2558:
2525:
2508:Newton, Roger G.
2503:
2478:
2477:
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2433:10.1063/1.324142
2419:(7): 2812–2819.
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1789:
1788:to some system
1768:
1763:
1762:
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1721:
1708:
1697:
1696:
1695:of the problem
1675:
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1669:
1623:
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1561:
1558:
1507:
1506:
1495:neutron optical
1484:
1437:
1436:
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1382:
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2479:
2458:(1): 221–240.
2438:
2403:
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2350:
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2226:
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2220:
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2204:
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2192:
2189:
2177:bremsstrahlung
2173:
2172:
2161:
2158:
2153:
2150:
2147:
2140:
2134:
2129:
2123:
2118:
2114:
2110:
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2025:
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2019:
2013:
2008:
2003:
1989:
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1977:
1972:
1969:
1966:
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1948:
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1933:
1929:
1926:
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1911:
1907:
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1396:
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901:
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694:
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447:
444:
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430:
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405:
399:
393:
388:
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363:
354:quantity, and
350:is a positive
339:
310:
306:
294:
293:
282:
279:
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256:
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114:
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69:electric field
15:
13:
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4:
3:
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2195:
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2099:
2094:
2090:
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2077:
2073:
2062:
2052:
2032:
2020:
1994:
1993:
1992:
1967:
1957:
1935:
1931:
1924:
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1918:
1913:
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1900:
1896:
1892:
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1800:
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1637:
1612:
1608:
1603:
1583:
1555:
1553:
1551:
1550:seismic waves
1546:
1544:
1543:elastic media
1540:
1512:
1504:
1500:
1499:neutron guide
1496:
1492:
1487:
1481:
1479:
1462:
1458:
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1272:
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1260:
1257:
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885:
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796:
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776:
767:
764:
758:
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716:
708:
707:
706:
692:
680:
678:
676:
671:
648:
606:
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578:
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569:
566:
561:
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548:
544:
540:
531:
512:
500:
474:
473:
472:
469:
467:
466:driving field
442:
397:
361:
353:
352:infinitesimal
337:
329:
326:
325:free particle
308:
304:
280:
268:
246:
240:
237:
234:
229:
225:
216:
212:
208:
199:
180:
168:
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141:
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29:
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2312:
2308:
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2207:Dyson series
2174:
1990:
1610:
1606:
1604:
1559:
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1488:
1485:
1482:Applications
1287:
1013:
879:
684:
672:
621:
470:
465:
295:
132:
78:
58:
54:perturbation
51:
42:
32:
2197:Born series
675:Born series
63:by a light
61:radio waves
2540:Categories
2322:2402.11259
2223:References
1668:to a part
52:It is the
2398:119415260
2266:126244962
2157:⟩
2149:±
2128:Ψ
2100:±
2071:⟩
2063:±
2042:Ψ
2030:⟩
2021:±
2007:Ψ
1976:⟩
1968:±
1947:Ψ
1919:±
1901:∘
1890:⟩
1884:∘
1873:Ψ
1862:⟩
1854:±
1833:Ψ
1656:⟩
1648:±
1627:Ψ
1590:⟩
1584:∘
1573:Ψ
1455:θ
1449:
1423:θ
1403:ℏ
1369:σ
1335:θ
1329:
1255:
1229:∞
1220:∫
1208:ℏ
1194:−
1185:θ
1109:−
1022:θ
974:⋅
945:∫
933:ℏ
929:π
917:−
908:θ
847:−
834:π
810:−
773:ℏ
759:−
658:⟩
649:±
635:Ψ
604:⟩
598:∘
587:Ψ
573:ϵ
567:±
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538:⟩
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510:⟩
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452:⟩
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429:Ψ
404:⟩
398:∘
387:Ψ
338:ϵ
309:∘
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206:⟩
200:∘
189:Ψ
178:⟩
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155:Ψ
119:⟩
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96:Ψ
65:styrofoam
2551:Max Born
2510:(2002).
2191:See also
2179:and the
1105:′
1070:′
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2460:Bibcode
2421:Bibcode
2378:Bibcode
2246:Bibcode
1605:In the
323:is the
28:DWBA-FM
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2264:
1288:where
1014:where
296:where
41:, the
2394:S2CID
2368:arXiv
2317:arXiv
2262:S2CID
1501:, or
2516:ISBN
2494:ISBN
2345:link
1611:DWBA
79:The
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2429:doi
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2327:doi
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