709:. The second iteration substitutes the first Born approximation in the right hand side and the result is called the second Born approximation. In general the n-th Born approximation takes n-terms of the series into account. The second Born approximation is sometimes used, when the first Born approximation vanishes, but the higher terms are rarely used. The Born series can formally be summed as
401:
1551:
897:
491:
1071:
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224:
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703:
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45:
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749:
1615:
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27:
is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential
190:. In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction
925:
916:
1179:
396:{\displaystyle |\psi \rangle =|\phi \rangle +G_{0}(E)V|\phi \rangle +^{2}|\phi \rangle +^{3}|\phi \rangle +\dots }
1564:
1273:
611:
570:
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17:
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108:
1664:
706:
112:
1693:
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1589:
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120:
115:, which is the first order term of the Born series. The series can formally be understood as
50:
1656:
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710:
163:
1079:
502:
83:
1603:
908:
1652:
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30:
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526:
for a free particle can be retarded/advanced or standing wave operator for retarded
116:
1270:
Its solution by iteration leads to the Born series for the full Green's operator
705:
on the right hand side of the
Lippmann-Schwinger equation and it gives the first
1546:{\displaystyle G(E)=G_{0}(E)+G_{0}(E)VG_{0}(E)+^{2}G_{0}(E)+^{3}G_{0}(E)+\dots }
157:
907:
The Born series can also be written for other scattering quantities like the
649:. The first iteration is obtained by replacing the full scattering solution
892:{\displaystyle |\psi \rangle =^{-1}|\phi \rangle =^{-1}V|\phi \rangle .}
743:, giving the formal solution to Lippmann-Schwinger equation in the form
1660:
486:{\displaystyle |\psi \rangle =|\phi \rangle +G_{0}(E)V|\psi \rangle .}
1586:
Scattering Theory: The
Quantum Theory on Nonrelativistic Collisions
1639:
Born, Max (1926). "Quantenmechanik der Stoßvorgänge".
1344:
1276:
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1112:
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86:
53:
33:
1153:. The standing wave Green's operator would give the
1066:{\displaystyle T(E)=V+VG_{0}(E)V+V^{2}+V^{3}+\dots }
1545:
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1145:
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669:
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72:
39:
218:The Born series for the scattering states reads
8:
1260:{\displaystyle G(E)=G_{0}(E)+G_{0}(E)VG(E).}
883:
819:
761:
713:with the common ratio equal to the operator
692:
664:
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554:
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427:
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335:
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236:
1329:{\displaystyle G(E)=(E-H+i\epsilon )^{-1}}
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195:
171:
165:
128:
91:
85:
58:
52:
32:
1608:Scattering Theory of Waves and Particles
1631:
1161:Born series for full Green's operator
7:
1165:The Lippmann-Schwinger equation for
642:{\displaystyle |\psi ^{(P)}\rangle }
601:{\displaystyle |\psi ^{(-)}\rangle }
560:{\displaystyle |\psi ^{(+)}\rangle }
608:or standing wave scattering states
406:It can be derived by iterating the
156:of the Born series are related to
14:
677:with free particle wave function
214:Born series for scattering states
911:which is closely related to the
152:. The speed of convergence and
1534:
1528:
1509:
1502:
1496:
1483:
1477:
1471:
1452:
1445:
1439:
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1420:
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1314:
1292:
1286:
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1251:
1245:
1236:
1230:
1214:
1208:
1192:
1186:
1146:{\displaystyle G_{0}^{(+)}(E)}
1140:
1134:
1129:
1123:
1048:
1041:
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1022:
1007:
1000:
994:
981:
969:
963:
938:
932:
876:
860:
853:
847:
825:
812:
799:
792:
786:
767:
754:
698:{\displaystyle |\phi \rangle }
685:
670:{\displaystyle |\psi \rangle }
657:
631:
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616:
590:
584:
575:
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534:
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279:
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243:
229:
145:{\displaystyle V\to \lambda V}
133:
1:
1563:Joachain, Charles J. (1983).
47:(more precisely in powers of
210:and large collision energy.
111:). It is closely related to
1610:. Dover Publications, inc.
917:Lippmann-Schwinger equation
408:Lippmann–Schwinger equation
1710:
15:
1103:stands only for retarded
919:for the T-matrix we get
1584:Taylor, John R. (1972).
1566:Quantum collision theory
1160:
903:Born series for T-matrix
16:Not to be confused with
73:{\displaystyle G_{0}V,}
1641:Zeitschrift fĂĽr Physik
1547:
1330:
1261:
1147:
1097:
1067:
893:
737:
736:{\displaystyle G_{0}V}
699:
671:
643:
602:
561:
520:
487:
397:
204:
184:
183:{\displaystyle G_{0}V}
146:
101:
74:
41:
1548:
1331:
1262:
1148:
1098:
1096:{\displaystyle G_{0}}
1068:
894:
738:
700:
672:
644:
603:
562:
521:
519:{\displaystyle G_{0}}
488:
398:
205:
185:
154:radius of convergence
147:
107:is the free particle
102:
100:{\displaystyle G_{0}}
75:
42:
1342:
1274:
1180:
1110:
1080:
926:
913:scattering amplitude
750:
717:
681:
653:
612:
571:
530:
503:
416:
225:
194:
164:
127:
84:
51:
31:
18:Bourne (film series)
1653:1926ZPhy...38..803B
1133:
1661:10.1007/bf01397184
1647:(11–12): 803–827.
1543:
1326:
1257:
1171:resolvent identity
1143:
1113:
1093:
1063:
889:
733:
707:Born approximation
695:
667:
639:
598:
557:
516:
483:
393:
200:
180:
142:
113:Born approximation
97:
70:
37:
1617:978-0-486-42535-1
1595:978-0-471-84900-1
1576:978-0-7204-0294-0
1569:. North Holland.
1076:For the T-matrix
203:{\displaystyle V}
121:coupling constant
40:{\displaystyle V}
1701:
1673:
1672:
1636:
1621:
1604:Newton, Roger G.
1599:
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1167:Green's operator
1152:
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1105:Green's operator
1102:
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711:geometric series
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498:Green's operator
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160:of the operator
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149:
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123:by substitution
119:introducing the
109:Green's operator
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1594:
1588:. John Wiley.
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1188:
1185:
1169:is called the
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1120:
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1090:
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496:Note that the
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1189:
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1168:
1158:
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1057:
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1044:
1038:
1030:
1026:
1019:
1016:
1011:
1003:
997:
989:
985:
978:
975:
972:
966:
958:
954:
950:
947:
944:
941:
935:
929:
922:
921:
920:
918:
915:. Iterating
914:
910:
902:
886:
880:
872:
867:
864:
856:
850:
842:
838:
834:
831:
828:
822:
816:
806:
803:
795:
789:
781:
777:
773:
770:
764:
758:
746:
745:
744:
730:
725:
721:
712:
708:
689:
661:
628:
621:
587:
580:
546:
539:
511:
507:
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480:
474:
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67:
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59:
55:
34:
26:
19:
1644:
1640:
1634:
1607:
1585:
1565:
1557:Bibliography
1269:
1164:
1075:
906:
495:
405:
217:
117:power series
24:
22:
158:eigenvalues
25:Born series
1689:Scattering
1683:Categories
1626:References
1669:126244962
1541:…
1319:−
1311:ϵ
1299:−
1157:instead.
1061:…
884:⟩
881:ϕ
865:−
832:−
820:⟩
817:ϕ
804:−
774:−
762:⟩
759:ψ
693:⟩
690:ϕ
665:⟩
662:ψ
637:⟩
622:ψ
596:⟩
588:−
581:ψ
567:advanced
555:⟩
540:ψ
478:⟩
475:ψ
442:⟩
439:ϕ
428:⟩
425:ψ
391:…
385:⟩
382:ϕ
336:⟩
333:ϕ
287:⟩
284:ϕ
251:⟩
248:ϕ
237:⟩
234:ψ
137:λ
134:→
1694:Max Born
1606:(2002).
1155:K-matrix
909:T-matrix
1649:Bibcode
1667:
1614:
1592:
1573:
80:where
1665:S2CID
1612:ISBN
1590:ISBN
1571:ISBN
23:The
1657:doi
1685::
1663:.
1655:.
1645:38
1643:.
1173:,
1671:.
1659::
1651::
1620:.
1598:.
1579:.
1538:+
1535:)
1532:E
1529:(
1524:0
1520:G
1514:3
1510:]
1506:V
1503:)
1500:E
1497:(
1492:0
1488:G
1484:[
1481:+
1478:)
1475:E
1472:(
1467:0
1463:G
1457:2
1453:]
1449:V
1446:)
1443:E
1440:(
1435:0
1431:G
1427:[
1424:+
1421:)
1418:E
1415:(
1410:0
1406:G
1402:V
1399:)
1396:E
1393:(
1388:0
1384:G
1380:+
1377:)
1374:E
1371:(
1366:0
1362:G
1358:=
1355:)
1352:E
1349:(
1346:G
1322:1
1315:)
1308:i
1305:+
1302:H
1296:E
1293:(
1290:=
1287:)
1284:E
1281:(
1278:G
1255:.
1252:)
1249:E
1246:(
1243:G
1240:V
1237:)
1234:E
1231:(
1226:0
1222:G
1218:+
1215:)
1212:E
1209:(
1204:0
1200:G
1196:=
1193:)
1190:E
1187:(
1184:G
1141:)
1138:E
1135:(
1130:)
1127:+
1124:(
1119:0
1115:G
1089:0
1085:G
1058:+
1053:3
1049:]
1045:V
1042:)
1039:E
1036:(
1031:0
1027:G
1023:[
1020:V
1017:+
1012:2
1008:]
1004:V
1001:)
998:E
995:(
990:0
986:G
982:[
979:V
976:+
973:V
970:)
967:E
964:(
959:0
955:G
951:V
948:+
945:V
942:=
939:)
936:E
933:(
930:T
887:.
877:|
873:V
868:1
861:]
857:V
854:)
851:E
848:(
843:0
839:G
835:V
829:V
826:[
823:=
813:|
807:1
800:]
796:V
793:)
790:E
787:(
782:0
778:G
771:I
768:[
765:=
755:|
731:V
726:0
722:G
686:|
658:|
632:)
629:P
626:(
617:|
591:)
585:(
576:|
550:)
547:+
544:(
535:|
512:0
508:G
481:.
471:|
467:V
464:)
461:E
458:(
453:0
449:G
445:+
435:|
431:=
421:|
388:+
378:|
372:3
368:]
364:V
361:)
358:E
355:(
350:0
346:G
342:[
339:+
329:|
323:2
319:]
315:V
312:)
309:E
306:(
301:0
297:G
293:[
290:+
280:|
276:V
273:)
270:E
267:(
262:0
258:G
254:+
244:|
240:=
230:|
198:V
178:V
173:0
169:G
140:V
131:V
93:0
89:G
68:,
65:V
60:0
56:G
35:V
20:.
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