4684:
4352:
4103:
3528:
4121:
2383:
2042:
2840:
2149:
4516:
4395:
3361:
3264:
1745:
1721:
2297:
1959:
3546:
2803:
2920:
5834:
1114:
1089:
4098:{\displaystyle {\begin{aligned}&\left\langle j'_{n},m'_{n},...,j'_{2},m'_{2},j'_{1},m'_{1}|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\right\rangle \\=&\langle j'_{n},m'_{n}|...\langle j'_{2},m'_{2}|\langle j'_{1},m'_{1}||j_{1},m_{1}\rangle |j_{2},m_{2}\rangle ...|j_{n},m_{n}\rangle \\=&\prod _{k=1}^{n}\left\langle j'_{k},m'_{k}|j_{k},m_{k}\right\rangle \end{aligned}}}
2405:
3259:{\displaystyle {\begin{aligned}\left|j_{1},m_{1},j_{2},m_{2},...j_{n},m_{n}\right\rangle &\equiv \left|j_{1},m_{1}\right\rangle \otimes \left|j_{2},m_{2}\right\rangle \otimes \cdots \otimes \left|j_{n},m_{n}\right\rangle \\&\equiv \left|j_{1},m_{1}\right\rangle \left|j_{2},m_{2}\right\rangle \cdots \left|j_{n},m_{n}\right\rangle \end{aligned}}}
2798:{\displaystyle {\begin{aligned}\sum _{m}|j,m\rangle \langle j,m|&=\sum _{m}|j,-m\rangle \langle j,-m|\\&=\sum _{m}{(-1)}^{2(j-m)}|j,-m\rangle \langle j,-m|\\&=\sum _{m}{(-1)}^{j-m}|j,-m\rangle \langle j,-m|{(-1)}^{j-m}\\&=\sum _{m}T|j,m\rangle \langle j,m|T^{\dagger }\end{aligned}}}
4666:
3351:
Signs are of course not required for just one state, diagrammatically one arrow at a vertex. Sometimes curved arrows with the signs are included to show explicitly the sense of tensor multiplication, but usually just the sign is shown with the arrows left out.
1707:
2827:
takes the set of values given above. There is no difference between the forward-time and reversed-time states for the outer product contraction, so here they share the same diagram, represented as one line without direction, again labelled by
4507:
1945:
2283:
117:
1567:
1485:
1399:
1325:
1251:
2130:
4528:
3551:
2925:
2410:
1590:
3310:
3345:
5063:
4407:
1819:
5631:
5264:
383:
963:). The diagrammatic notation is a considerably large topic in its own right with a number of specialized features â this article introduces the very basics.
884:
5802:
5814:
5113:
2205:
591:
135:
47:
5498:
5056:
1508:
1426:
1340:
1266:
364:
5432:
4683:
1195:
547:
4956:
4935:
4903:
4880:
4857:
4836:
4815:
4794:
4773:
4745:
5355:
470:
4351:
4925:
4870:
4847:
4805:
4784:
5086:
5049:
5012:
4826:
4763:
130:
4891:
924:
of a quantum system allowing calculations to be done symbolically. More specifically, the arrows encode angular momentum states in
4898:. Cambridge Monographs on Atomic, Molecular and Chemical Physics. Vol. 7. University of Chicago: Cambridge University Press.
3276:
Vertices in tensor products have signs (sometimes called "node signs"), to indicate the ordering of the tensor-multiplied states:
5196:
219:
5478:
5473:
4371:
326:
306:
5769:
5181:
988:
877:
174:
5453:
296:
5781:
4661:{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}}
1076:
equals signs "=" are placed between equivalent diagrams, exactly like for multiple algebraic expressions equal to each other.
566:
5330:
2064:
5759:
5536:
5458:
5226:
5096:
606:
344:
244:
5493:
4946:
359:
5427:
5422:
5393:
5091:
948:
542:
537:
508:
140:
5468:
4914:
4735:
321:
311:
5561:
3527:
576:
5378:
1769:
It is important to position the vertex correctly, as forward-time and reversed-time operators would become mixed up.
493:
5863:
5837:
5599:
5407:
4120:
1151:
936:
870:
522:
5310:
5246:
441:
5858:
5621:
5488:
5412:
5373:
5289:
5139:
1405:
527:
488:
416:
339:
199:
5688:
5668:
5658:
5648:
5604:
5164:
611:
2382:
2041:
1702:{\displaystyle T\left|j,m\right\rangle \equiv \left|T(j,m)\right\rangle ={(-1)}^{j-m}\left|j,-m\right\rangle }
1158:
As a general rule, the arrows follow each other in the same sense. In the contrastandard representation, the
5797:
5383:
5345:
5294:
4927:
Operators and
Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics
1144:
995:
960:
780:
498:
456:
406:
354:
120:
2839:
2148:
5708:
5483:
5463:
5388:
5284:
5241:
785:
503:
411:
331:
301:
264:
5733:
5231:
5211:
4515:
4394:
3295:
254:
239:
5134:
4852:. Challenges and Advances in Computational Chemistry and Physics. Vol. 10. Springer. p. 311.
3330:
3269:
and in diagram form, each separate state leaves or enters a common vertex creating a "fan" of arrows -
1018:
925:
166:
5764:
5693:
5638:
5368:
5176:
4992:
815:
571:
483:
209:
5749:
5718:
5663:
5643:
5551:
5508:
5363:
5274:
5201:
5191:
5103:
3360:
1167:
670:
478:
396:
224:
204:
156:
4502:{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}}
5678:
5576:
5269:
5216:
5108:
4378:
2396:
2055:
391:
316:
249:
161:
5809:
1940:{\displaystyle \langle j_{2},m_{2}|j_{1},m_{1}\rangle =\delta _{j_{1}j_{2}}\delta _{m_{1}m_{2}}}
1744:
1720:
5819:
5728:
5698:
5626:
5589:
5584:
5566:
5531:
5521:
5325:
5221:
5186:
5169:
5072:
5018:
5008:
4952:
4931:
4899:
4876:
4853:
4832:
4811:
4790:
4769:
4741:
1257:
1186:
952:
906:
902:
898:
825:
800:
740:
735:
635:
601:
581:
179:
38:
5546:
5541:
5398:
5279:
5000:
4707:
918:
830:
820:
810:
710:
690:
675:
645:
513:
401:
2296:
1958:
5776:
5703:
5683:
5653:
5616:
5611:
5516:
5340:
4697:
940:
855:
725:
705:
451:
291:
2395:
For summations over the outer product, also known in this context as a contraction (c.f.
2054:
For summations over the inner product, also known in this context as a contraction (c.f.
4996:
5754:
5723:
5713:
5335:
5315:
5144:
4721:
4689:
1331:
944:
929:
790:
750:
730:
700:
680:
630:
596:
446:
436:
229:
5004:
4789:. Progress in Theoretical Chemistry and Physics. Vol. 11. springer. p. 183.
1073:
arrowheads are almost always placed at the middle of the line, rather than at the tip,
5852:
5673:
5526:
5417:
5320:
5236:
5206:
5159:
984:
967:
921:
850:
845:
775:
745:
715:
586:
532:
259:
234:
4730:
Wormer and Paldus (2006) provides an in-depth tutorial in angular momentum diagrams.
1070:
are often labelled next to the arrows to refer to a specific angular momentum state,
5556:
5154:
5149:
840:
835:
770:
755:
720:
214:
4807:
Fundamental World of
Quantum Chemistry: A Tribute to the Memory of Per-Olov Löwdin
5594:
805:
760:
695:
650:
4831:. Theoretical and Computational Chemistry. Vol. 14. Elsevier. p. 97.
4726:. Translated by A. Sen; R. N. Sen. Israel Program for Scientific Translations.
4702:
4679:
4522:
4401:
2278:{\displaystyle \left|j_{2},m_{2}\right\rangle \left\langle j_{1},m_{1}\right|}
1159:
1113:
1088:
956:
795:
765:
685:
660:
655:
640:
5022:
112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
17:
5448:
5129:
2135:
it is conventional to denote the result as a closed circle labelled only by
1139:
Arrows are directed to or from vertices, a state transforming according to:
286:
4382:
1562:{\displaystyle T{\hat {\mathbf {J} }}T^{\dagger }=-{\hat {\mathbf {J} }}}
1480:{\displaystyle T{\hat {\mathbf {L} }}T^{\dagger }=-{\hat {\mathbf {L} }}}
1394:{\displaystyle T{\hat {\mathbf {S} }}T^{\dagger }=-{\hat {\mathbf {S} }}}
1320:{\displaystyle T{\hat {\mathbf {p} }}T^{\dagger }=-{\hat {\mathbf {p} }}}
665:
1246:{\displaystyle T{\hat {\mathbf {x} }}T^{\dagger }={\hat {\mathbf {x} }}}
5041:
4786:
Theoretical
Chemistry and Physics of Heavy and Superheavy Elements
4740:. Chemical Physics. Vol. 13 (2nd ed.). Springer-Verlag.
2838:
2147:
5045:
4983:
P.E.S. Wormer; J. Paldus (2006). "Angular
Momentum Diagrams".
4828:
Relativistic
Electronic Structure Theory: Part 2. Applications
4804:
E.J. BrÀndas; P.O. Löwdin; E. BrÀndas; E.S. Kryachko (2004).
1055:
headed arrow, pointing in the opposite direction to the ket.
4720:
Yutsis, Adolfas P.; Levinson, I. B.; Vanagas, V. V. (1962).
4765:
Springer
Handbook of Atomic, Molecular, and Optical Physics
970:(sometimes translated as Yutsis) in the twentieth century.
4345:. Similarly for more than three pairs of angular momenta.
4948:
Group Theory - Birdtracks, Lie's, and
Exceptional Groups
4723:
Mathematical
Apparatus of the Theory of Angular Momentum
4537:
4416:
928:
and include the abstract nature of the state, such as
27:
Pictorial computational technique in quantum chemistry
4531:
4410:
3549:
3333:
3298:
2923:
2408:
2208:
2125:{\displaystyle \sum _{m}\langle j,m|j,m\rangle =2j+1}
2067:
1822:
1593:
1511:
1429:
1343:
1269:
1198:
974:
Equivalence between Dirac notation and Jucys diagrams
50:
3540:
For the inner product of two tensor product states:
5790:
5742:
5575:
5507:
5441:
5354:
5303:
5257:
5122:
5079:
1147:
is designated by an oriented line leaving a vertex,
943:. The diagrams consist of arrows and vertices with
913:, or more accurately from a mathematical viewpoint
4978:
4976:
4875:. NATO Science Series C. Vol. 113. Springer.
4660:
4501:
4097:
3339:
3304:
3258:
2797:
2277:
2124:
1939:
1712:The time-reversed diagrams for kets and bras are:
1701:
1561:
1490:and therefore the total angular momentum operator
1479:
1393:
1319:
1245:
111:
4919:. Vol. 44. University of Chicago: Draugija.
4377:Calculations with real quantum systems, such as
3521:. Similarly for more than three angular momenta.
1080:The most basic diagrams are for kets and bras:
5057:
917:, are a diagrammatic method for representing
878:
8:
5031:for the time reversal operator, here we use
3989:
3949:
3918:
3853:
3816:
3770:
2764:
2761:
2677:
2674:
2590:
2587:
2494:
2491:
2440:
2437:
2104:
2078:
1877:
1823:
1572:Acting on an eigenstate of angular momentum
106:
80:
5064:
5050:
5042:
4872:Methods in Computational Molecular Physics
1166:, is used. It is unitary, which means the
885:
871:
29:
4644:
4632:
4620:
4606:
4594:
4582:
4568:
4556:
4544:
4532:
4530:
4485:
4473:
4461:
4447:
4435:
4423:
4411:
4409:
4080:
4067:
4058:
4049:
4033:
4018:
4007:
3983:
3970:
3961:
3943:
3930:
3921:
3912:
3899:
3890:
3885:
3876:
3860:
3848:
3839:
3823:
3802:
3793:
3777:
3748:
3735:
3713:
3700:
3687:
3674:
3665:
3656:
3640:
3624:
3608:
3580:
3564:
3550:
3548:
3332:
3297:
3241:
3228:
3205:
3192:
3172:
3159:
3129:
3116:
3087:
3074:
3051:
3038:
3011:
2998:
2976:
2963:
2950:
2937:
2924:
2922:
2785:
2776:
2747:
2738:
2712:
2698:
2692:
2657:
2645:
2631:
2624:
2605:
2570:
2549:
2535:
2528:
2509:
2474:
2468:
2452:
2423:
2417:
2409:
2407:
2264:
2251:
2231:
2218:
2207:
2090:
2072:
2066:
1929:
1919:
1914:
1902:
1892:
1887:
1871:
1858:
1849:
1843:
1830:
1821:
1665:
1651:
1592:
1548:
1546:
1545:
1533:
1518:
1516:
1515:
1510:
1466:
1464:
1463:
1451:
1436:
1434:
1433:
1428:
1380:
1378:
1377:
1365:
1350:
1348:
1347:
1342:
1306:
1304:
1303:
1291:
1276:
1274:
1273:
1268:
1232:
1230:
1229:
1220:
1205:
1203:
1202:
1197:
951:". The sense of each arrow is related to
98:
87:
86:
72:
57:
49:
4951:. Princeton, NJ: Princeton Univ. Press.
1154:is depicted as a line entering a vertex.
1039:Symmetrically, the corresponding bra is
4972:
987:vector of a single particle with total
947:as labels, hence the alternative term "
54:
37:
4810:. Vol. 3. Springer. p. 385.
4768:(2nd ed.). springer. p. 60.
959:of the angular momentum states (c.f.
7:
5027:These authors use the theta variant
4869:G.H.F. Diercksen; S. Wilson (1983).
4930:. University of Chicago: Elsevier.
3273:lines attached to a single vertex.
935:The notation parallels the idea of
3305:{\displaystyle \circlearrowright }
1420:, this must also become negative:
417:Sum-over-histories (path integral)
103:
77:
33:Part of a series of articles about
25:
4913:Lietuvos FizikĆł draugija (2004).
4849:Relativistic Methods for Chemists
4734:I. Lindgren; J. Morrison (1986).
4370:The diagrams are well-suited for
3340:{\displaystyle \circlearrowleft }
966:They were developed primarily by
5833:
5832:
4682:
4514:
4393:
4350:
4119:
3526:
3359:
2381:
2295:
2163:The outer product of two states
2040:
1957:
1777:The inner product of two states
1743:
1719:
1549:
1519:
1467:
1437:
1381:
1351:
1307:
1277:
1233:
1206:
1112:
1087:
4896:Theoretical Atomic Spectroscopy
4846:M. Barysz; Y. Ishikawa (2010).
989:angular momentum quantum number
955:, which roughly corresponds to
5782:Relativistic quantum mechanics
4112:lots of inner product arrows:
4059:
3962:
3922:
3891:
3886:
3849:
3803:
3666:
2852:The tensor product ⊗ of
2777:
2748:
2708:
2699:
2693:
2658:
2641:
2632:
2606:
2571:
2565:
2553:
2545:
2536:
2510:
2475:
2453:
2424:
2091:
1850:
1661:
1652:
1639:
1627:
1553:
1523:
1471:
1441:
1385:
1355:
1311:
1281:
1237:
1210:
567:Relativistic quantum mechanics
99:
92:
73:
1:
5760:Quantum statistical mechanics
5537:Quantum differential calculus
5459:Delayed-choice quantum eraser
5227:Symmetry in quantum mechanics
5005:10.1016/S0065-3276(06)51002-0
4985:Advances in Quantum Chemistry
4783:U. Kaldor; S. Wilson (2003).
1152:contrastandard representation
903:quantum many-particle systems
607:Quantum statistical mechanics
2823:was used, and the fact that
1173:equals the inverse operator
1051:. In diagram form this is a
5562:Quantum stochastic calculus
5552:Quantum measurement problem
5474:MachâZehnder interferometer
4372:ClebschâGordan coefficients
1334:operator becomes negative:
577:Quantum information science
5880:
3288:indicates the ordering is
2843:Outer product contraction.
2152:Inner product contraction.
1162:operator, denoted here by
937:Penrose graphical notation
932:and transformation rules.
5828:
5622:Quantum complexity theory
5600:Quantum cellular automata
5290:Path integral formulation
4924:P.E.T. Jorgensen (1987).
4916:Lietuvos fizikos ĆŸurnalas
4890:Zenonas Rudzikas (2007).
4825:P. Schwerdtfeger (2004).
4365:Examples and applications
4357:Time reversed equivalent.
3533:Time reversed equivalent.
2388:Time reversed equivalent.
2047:Time reversed equivalent.
1406:angular momentum operator
1032:. As a diagram this is a
911:angular momentum diagrams
5689:Quantum machine learning
5669:Quantum key distribution
5659:Quantum image processing
5649:Quantum error correction
5499:Wheeler's delayed choice
1584:, it can be shown that:
901:and its applications to
612:Quantum machine learning
365:Wheeler's delayed-choice
5605:Quantum finite automata
4737:Atomic Many-Body Theory
1145:standard representation
996:magnetic quantum number
979:Angular momentum states
915:angular momentum graphs
322:LeggettâGarg inequality
5709:Quantum neural network
4945:P. CvitanoviÄ (2008).
4662:
4503:
4099:
4023:
3341:
3306:
3260:
2844:
2799:
2288:and the diagrams are:
2279:
2153:
2126:
1950:and the diagrams are:
1941:
1703:
1563:
1481:
1395:
1321:
1247:
113:
5734:Quantum teleportation
5247:Waveâparticle duality
4762:G.W.F. Drake (2006).
4663:
4504:
4100:
4003:
3342:
3307:
3261:
2842:
2808:where the result for
2800:
2280:
2151:
2127:
1942:
1704:
1564:
1482:
1396:
1322:
1248:
1189:leaves it invariant:
953:Hermitian conjugation
307:Elitzur–Vaidman
297:Davisson–Germer
114:
5765:Quantum field theory
5694:Quantum metamaterial
5639:Quantum cryptography
5369:Consistent histories
4991:. Elsevier: 59â124.
4529:
4408:
3547:
3331:
3296:
2921:
2406:
2206:
2065:
1820:
1591:
1509:
1427:
1341:
1267:
1196:
1185:. Its action on the
1062:the quantum numbers
961:Schrödinger equation
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
48:
5750:Quantum fluctuation
5719:Quantum programming
5679:Quantum logic gates
5664:Quantum information
5644:Quantum electronics
5104:Classical mechanics
4997:2006AdQC...51...59W
4379:multielectron atoms
4057:
4041:
3884:
3868:
3847:
3831:
3801:
3785:
3664:
3648:
3632:
3616:
3588:
3572:
1168:Hermitian conjugate
360:Stern–Gerlach
157:Classical mechanics
5803:in popular culture
5585:Quantum algorithms
5433:Von NeumannâWigner
5413:Objective collapse
5109:Old quantum theory
4658:
4652:
4499:
4493:
4095:
4093:
4045:
4029:
3872:
3856:
3835:
3819:
3789:
3773:
3652:
3636:
3620:
3604:
3576:
3560:
3337:
3302:
3256:
3254:
2845:
2795:
2793:
2743:
2629:
2533:
2473:
2422:
2397:tensor contraction
2275:
2154:
2122:
2077:
2056:tensor contraction
1937:
1750:Time reversed bra
1726:Time reversed ket
1699:
1559:
1502:becomes negative:
1477:
1404:Since the orbital
1391:
1317:
1260:becomes negative:
1243:
1017:, is denoted as a
548:Von NeumannâWigner
528:Objective-collapse
327:Mach–Zehnder
317:Leggett inequality
312:Franck–Hertz
162:Old quantum theory
109:
5864:Quantum mechanics
5846:
5845:
5820:Quantum mysticism
5798:Schrödinger's cat
5729:Quantum simulator
5699:Quantum metrology
5627:Quantum computing
5590:Quantum amplifier
5567:Quantum spacetime
5532:Quantum cosmology
5522:Quantum chemistry
5222:Scattering theory
5170:Zero-point energy
5165:Degenerate levels
5073:Quantum mechanics
4958:978-0-691-11836-9
4937:978-0-08-087258-2
4905:978-0-521-02622-2
4882:978-90-277-1638-5
4859:978-1-4020-9975-5
4838:978-0-08-054047-4
4817:978-1-4020-2583-9
4796:978-1-4020-1371-3
4775:978-0-387-26308-3
4747:978-3-540-16649-8
2734:
2620:
2524:
2464:
2413:
2068:
1556:
1526:
1474:
1444:
1388:
1358:
1314:
1284:
1258:momentum operator
1240:
1213:
1187:position operator
907:quantum chemistry
899:quantum mechanics
895:
894:
602:Scattering theory
582:Quantum computing
355:Schrödinger's cat
287:Bell's inequality
95:
70:
39:Quantum mechanics
16:(Redirected from
5871:
5859:Angular momentum
5836:
5835:
5547:Quantum geometry
5542:Quantum dynamics
5399:Superdeterminism
5331:RaritaâSchwinger
5280:Matrix mechanics
5135:Braâket notation
5066:
5059:
5052:
5043:
5036:
5026:
4980:
4962:
4941:
4920:
4909:
4886:
4863:
4842:
4821:
4800:
4779:
4751:
4727:
4708:Feynman diagrams
4692:
4687:
4686:
4667:
4665:
4664:
4659:
4657:
4656:
4649:
4648:
4637:
4636:
4625:
4624:
4611:
4610:
4599:
4598:
4587:
4586:
4573:
4572:
4561:
4560:
4549:
4548:
4518:
4508:
4506:
4505:
4500:
4498:
4497:
4490:
4489:
4478:
4477:
4466:
4465:
4452:
4451:
4440:
4439:
4428:
4427:
4397:
4354:
4344:
4297:
4287:
4277:
4267:
4257:
4247:
4238:
4192:
4187:
4177:
4167:
4157:
4147:
4137:
4123:
4104:
4102:
4101:
4096:
4094:
4090:
4086:
4085:
4084:
4072:
4071:
4062:
4053:
4037:
4022:
4017:
3988:
3987:
3975:
3974:
3965:
3948:
3947:
3935:
3934:
3925:
3917:
3916:
3904:
3903:
3894:
3889:
3880:
3864:
3852:
3843:
3827:
3806:
3797:
3781:
3758:
3754:
3753:
3752:
3740:
3739:
3718:
3717:
3705:
3704:
3692:
3691:
3679:
3678:
3669:
3660:
3644:
3628:
3612:
3584:
3568:
3553:
3530:
3520:
3474:
3457:
3440:
3422:
3404:
3386:
3363:
3346:
3344:
3343:
3338:
3311:
3309:
3308:
3303:
3265:
3263:
3262:
3257:
3255:
3251:
3247:
3246:
3245:
3233:
3232:
3215:
3211:
3210:
3209:
3197:
3196:
3182:
3178:
3177:
3176:
3164:
3163:
3143:
3139:
3135:
3134:
3133:
3121:
3120:
3097:
3093:
3092:
3091:
3079:
3078:
3061:
3057:
3056:
3055:
3043:
3042:
3021:
3017:
3016:
3015:
3003:
3002:
2981:
2980:
2968:
2967:
2955:
2954:
2942:
2941:
2913:
2891:
2873:
2822:
2804:
2802:
2801:
2796:
2794:
2790:
2789:
2780:
2751:
2742:
2727:
2723:
2722:
2711:
2696:
2661:
2656:
2655:
2644:
2628:
2613:
2609:
2574:
2569:
2568:
2548:
2532:
2517:
2513:
2478:
2472:
2456:
2427:
2421:
2385:
2375:
2358:
2340:
2322:
2299:
2284:
2282:
2281:
2276:
2274:
2270:
2269:
2268:
2256:
2255:
2241:
2237:
2236:
2235:
2223:
2222:
2199:is an operator:
2198:
2180:
2131:
2129:
2128:
2123:
2094:
2076:
2044:
2034:
2002:
1984:
1961:
1946:
1944:
1943:
1938:
1936:
1935:
1934:
1933:
1924:
1923:
1909:
1908:
1907:
1906:
1897:
1896:
1876:
1875:
1863:
1862:
1853:
1848:
1847:
1835:
1834:
1812:
1794:
1761:
1747:
1737:
1723:
1708:
1706:
1705:
1700:
1698:
1694:
1676:
1675:
1664:
1646:
1642:
1615:
1611:
1583:
1568:
1566:
1565:
1560:
1558:
1557:
1552:
1547:
1538:
1537:
1528:
1527:
1522:
1517:
1486:
1484:
1483:
1478:
1476:
1475:
1470:
1465:
1456:
1455:
1446:
1445:
1440:
1435:
1400:
1398:
1397:
1392:
1390:
1389:
1384:
1379:
1370:
1369:
1360:
1359:
1354:
1349:
1326:
1324:
1323:
1318:
1316:
1315:
1310:
1305:
1296:
1295:
1286:
1285:
1280:
1275:
1252:
1250:
1249:
1244:
1242:
1241:
1236:
1231:
1225:
1224:
1215:
1214:
1209:
1204:
1132:
1116:
1107:
1091:
1050:
1031:
941:Feynman diagrams
926:braâket notation
919:angular momentum
887:
880:
873:
514:Superdeterminism
167:Braâket notation
118:
116:
115:
110:
102:
97:
96:
88:
76:
71:
69:
58:
30:
21:
5879:
5878:
5874:
5873:
5872:
5870:
5869:
5868:
5849:
5848:
5847:
5842:
5824:
5810:Wigner's friend
5786:
5777:Quantum gravity
5738:
5724:Quantum sensing
5704:Quantum network
5684:Quantum machine
5654:Quantum imaging
5617:Quantum circuit
5612:Quantum channel
5571:
5517:Quantum biology
5503:
5479:ElitzurâVaidman
5454:DavissonâGermer
5437:
5389:Hidden-variable
5379:de BroglieâBohm
5356:Interpretations
5350:
5299:
5253:
5140:Complementarity
5118:
5075:
5070:
5040:
5039:
5015:
4982:
4981:
4974:
4969:
4959:
4944:
4938:
4923:
4912:
4906:
4889:
4883:
4868:
4860:
4845:
4839:
4824:
4818:
4803:
4797:
4782:
4776:
4761:
4758:
4756:Further reading
4748:
4733:
4719:
4716:
4698:Ladder operator
4688:
4681:
4678:
4673:
4672:
4671:
4670:
4669:
4651:
4650:
4640:
4638:
4628:
4626:
4616:
4613:
4612:
4602:
4600:
4590:
4588:
4578:
4575:
4574:
4564:
4562:
4552:
4550:
4540:
4533:
4527:
4526:
4519:
4511:
4510:
4492:
4491:
4481:
4479:
4469:
4467:
4457:
4454:
4453:
4443:
4441:
4431:
4429:
4419:
4412:
4406:
4405:
4398:
4367:
4362:
4361:
4360:
4359:
4358:
4355:
4347:
4346:
4342:
4335:
4328:
4321:
4314:
4307:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4236:
4229:
4222:
4215:
4208:
4201:
4194:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4130:
4124:
4092:
4091:
4076:
4063:
4028:
4024:
3999:
3993:
3992:
3979:
3966:
3939:
3926:
3908:
3895:
3766:
3760:
3759:
3744:
3731:
3709:
3696:
3683:
3670:
3559:
3555:
3545:
3544:
3538:
3537:
3536:
3535:
3534:
3531:
3523:
3522:
3518:
3511:
3504:
3497:
3490:
3483:
3476:
3472:
3465:
3458:
3455:
3448:
3441:
3438:
3431:
3424:
3420:
3413:
3406:
3402:
3395:
3388:
3384:
3377:
3370:
3364:
3329:
3328:
3294:
3293:
3253:
3252:
3237:
3224:
3223:
3219:
3201:
3188:
3187:
3183:
3168:
3155:
3154:
3150:
3141:
3140:
3125:
3112:
3111:
3107:
3083:
3070:
3069:
3065:
3047:
3034:
3033:
3029:
3022:
3007:
2994:
2972:
2959:
2946:
2933:
2932:
2928:
2919:
2918:
2911:
2902:
2893:
2889:
2882:
2875:
2871:
2864:
2857:
2850:
2848:Tensor products
2812:
2792:
2791:
2781:
2725:
2724:
2697:
2630:
2611:
2610:
2534:
2515:
2514:
2457:
2404:
2403:
2393:
2392:
2391:
2390:
2389:
2386:
2378:
2377:
2373:
2366:
2359:
2356:
2349:
2342:
2338:
2331:
2324:
2320:
2313:
2306:
2300:
2260:
2247:
2246:
2242:
2227:
2214:
2213:
2209:
2204:
2203:
2196:
2189:
2182:
2178:
2171:
2164:
2161:
2063:
2062:
2052:
2051:
2050:
2049:
2048:
2045:
2037:
2036:
2032:
2025:
2018:
2011:
2004:
2000:
1993:
1986:
1982:
1975:
1968:
1962:
1925:
1915:
1910:
1898:
1888:
1883:
1867:
1854:
1839:
1826:
1818:
1817:
1810:
1803:
1796:
1792:
1785:
1778:
1775:
1767:
1766:
1765:
1764:
1763:
1751:
1748:
1740:
1739:
1727:
1724:
1681:
1677:
1650:
1623:
1619:
1601:
1597:
1589:
1588:
1573:
1529:
1507:
1506:
1447:
1425:
1424:
1361:
1339:
1338:
1287:
1265:
1264:
1256:but the linear
1216:
1194:
1193:
1137:
1136:
1135:
1134:
1133:
1122:
1117:
1109:
1108:
1097:
1092:
1040:
1021:
981:
976:
945:quantum numbers
930:tensor products
891:
862:
861:
860:
625:
617:
616:
562:
561:Advanced topics
554:
553:
552:
504:Hidden-variable
494:de BroglieâBohm
473:
471:Interpretations
463:
462:
461:
431:
423:
422:
421:
379:
371:
370:
369:
336:
292:CHSH inequality
281:
273:
272:
271:
200:Complementarity
194:
186:
185:
184:
152:
123:
62:
46:
45:
28:
23:
22:
15:
12:
11:
5:
5877:
5875:
5867:
5866:
5861:
5851:
5850:
5844:
5843:
5841:
5840:
5829:
5826:
5825:
5823:
5822:
5817:
5812:
5807:
5806:
5805:
5794:
5792:
5788:
5787:
5785:
5784:
5779:
5774:
5773:
5772:
5762:
5757:
5755:Casimir effect
5752:
5746:
5744:
5740:
5739:
5737:
5736:
5731:
5726:
5721:
5716:
5714:Quantum optics
5711:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5666:
5661:
5656:
5651:
5646:
5641:
5636:
5635:
5634:
5624:
5619:
5614:
5609:
5608:
5607:
5597:
5592:
5587:
5581:
5579:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5524:
5519:
5513:
5511:
5505:
5504:
5502:
5501:
5496:
5491:
5489:Quantum eraser
5486:
5481:
5476:
5471:
5466:
5461:
5456:
5451:
5445:
5443:
5439:
5438:
5436:
5435:
5430:
5425:
5420:
5415:
5410:
5405:
5404:
5403:
5402:
5401:
5386:
5381:
5376:
5371:
5366:
5360:
5358:
5352:
5351:
5349:
5348:
5343:
5338:
5333:
5328:
5323:
5318:
5313:
5307:
5305:
5301:
5300:
5298:
5297:
5292:
5287:
5282:
5277:
5272:
5267:
5261:
5259:
5255:
5254:
5252:
5251:
5250:
5249:
5244:
5234:
5229:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5173:
5172:
5167:
5162:
5157:
5147:
5145:Density matrix
5142:
5137:
5132:
5126:
5124:
5120:
5119:
5117:
5116:
5111:
5106:
5101:
5100:
5099:
5089:
5083:
5081:
5077:
5076:
5071:
5069:
5068:
5061:
5054:
5046:
5038:
5037:
5013:
4971:
4970:
4968:
4965:
4964:
4963:
4957:
4942:
4936:
4921:
4910:
4904:
4887:
4881:
4865:
4864:
4858:
4843:
4837:
4822:
4816:
4801:
4795:
4780:
4774:
4757:
4754:
4753:
4752:
4746:
4731:
4728:
4715:
4712:
4711:
4710:
4705:
4700:
4694:
4693:
4690:Physics portal
4677:
4674:
4655:
4647:
4643:
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4614:
4609:
4605:
4601:
4597:
4593:
4589:
4585:
4581:
4577:
4576:
4571:
4567:
4563:
4559:
4555:
4551:
4547:
4543:
4539:
4538:
4536:
4521:Diagram for a
4520:
4513:
4512:
4496:
4488:
4484:
4480:
4476:
4472:
4468:
4464:
4460:
4456:
4455:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4422:
4418:
4417:
4415:
4400:Diagram for a
4399:
4392:
4391:
4390:
4389:
4388:
4387:
4386:
4375:
4366:
4363:
4356:
4349:
4348:
4340:
4333:
4326:
4319:
4312:
4305:
4298:
4288:
4278:
4268:
4258:
4248:
4234:
4227:
4220:
4213:
4206:
4199:
4188:
4178:
4168:
4158:
4148:
4138:
4125:
4118:
4117:
4116:
4115:
4114:
4106:
4105:
4089:
4083:
4079:
4075:
4070:
4066:
4061:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4027:
4021:
4016:
4013:
4010:
4006:
4002:
4000:
3998:
3995:
3994:
3991:
3986:
3982:
3978:
3973:
3969:
3964:
3960:
3957:
3954:
3951:
3946:
3942:
3938:
3933:
3929:
3924:
3920:
3915:
3911:
3907:
3902:
3898:
3893:
3888:
3883:
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3851:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3815:
3812:
3809:
3805:
3800:
3796:
3792:
3788:
3784:
3780:
3776:
3772:
3769:
3767:
3765:
3762:
3761:
3757:
3751:
3747:
3743:
3738:
3734:
3730:
3727:
3724:
3721:
3716:
3712:
3708:
3703:
3699:
3695:
3690:
3686:
3682:
3677:
3673:
3668:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3615:
3611:
3607:
3603:
3600:
3597:
3594:
3591:
3587:
3583:
3579:
3575:
3571:
3567:
3563:
3558:
3554:
3552:
3532:
3525:
3524:
3516:
3509:
3502:
3495:
3488:
3481:
3470:
3463:
3453:
3446:
3436:
3429:
3418:
3411:
3400:
3393:
3382:
3375:
3367:Tensor product
3365:
3358:
3357:
3356:
3355:
3354:
3349:
3348:
3336:
3313:
3301:
3267:
3266:
3250:
3244:
3240:
3236:
3231:
3227:
3222:
3218:
3214:
3208:
3204:
3200:
3195:
3191:
3186:
3181:
3175:
3171:
3167:
3162:
3158:
3153:
3149:
3146:
3144:
3142:
3138:
3132:
3128:
3124:
3119:
3115:
3110:
3106:
3103:
3100:
3096:
3090:
3086:
3082:
3077:
3073:
3068:
3064:
3060:
3054:
3050:
3046:
3041:
3037:
3032:
3028:
3025:
3023:
3020:
3014:
3010:
3006:
3001:
2997:
2993:
2990:
2987:
2984:
2979:
2975:
2971:
2966:
2962:
2958:
2953:
2949:
2945:
2940:
2936:
2931:
2927:
2926:
2907:
2898:
2887:
2880:
2869:
2862:
2849:
2846:
2806:
2805:
2788:
2784:
2779:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2750:
2746:
2741:
2737:
2733:
2730:
2728:
2726:
2721:
2718:
2715:
2710:
2707:
2704:
2701:
2695:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2660:
2654:
2651:
2648:
2643:
2640:
2637:
2634:
2627:
2623:
2619:
2616:
2614:
2612:
2608:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2573:
2567:
2564:
2561:
2558:
2555:
2552:
2547:
2544:
2541:
2538:
2531:
2527:
2523:
2520:
2518:
2516:
2512:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2477:
2471:
2467:
2463:
2460:
2458:
2455:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2426:
2420:
2416:
2412:
2411:
2387:
2380:
2379:
2371:
2364:
2354:
2347:
2336:
2329:
2318:
2311:
2301:
2294:
2293:
2292:
2291:
2290:
2286:
2285:
2273:
2267:
2263:
2259:
2254:
2250:
2245:
2240:
2234:
2230:
2226:
2221:
2217:
2212:
2194:
2187:
2176:
2169:
2160:
2159:Outer products
2157:
2156:
2155:
2133:
2132:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2093:
2089:
2086:
2083:
2080:
2075:
2071:
2046:
2039:
2038:
2030:
2023:
2016:
2009:
1998:
1991:
1980:
1973:
1963:
1956:
1955:
1954:
1953:
1952:
1948:
1947:
1932:
1928:
1922:
1918:
1913:
1905:
1901:
1895:
1891:
1886:
1882:
1879:
1874:
1870:
1866:
1861:
1857:
1852:
1846:
1842:
1838:
1833:
1829:
1825:
1808:
1801:
1790:
1783:
1774:
1771:
1749:
1742:
1741:
1725:
1718:
1717:
1716:
1715:
1714:
1710:
1709:
1697:
1693:
1690:
1687:
1684:
1680:
1674:
1671:
1668:
1663:
1660:
1657:
1654:
1649:
1645:
1641:
1638:
1635:
1632:
1629:
1626:
1622:
1618:
1614:
1610:
1607:
1604:
1600:
1596:
1570:
1569:
1555:
1551:
1544:
1541:
1536:
1532:
1525:
1521:
1514:
1488:
1487:
1473:
1469:
1462:
1459:
1454:
1450:
1443:
1439:
1432:
1402:
1401:
1387:
1383:
1376:
1373:
1368:
1364:
1357:
1353:
1346:
1328:
1327:
1313:
1309:
1302:
1299:
1294:
1290:
1283:
1279:
1272:
1254:
1253:
1239:
1235:
1228:
1223:
1219:
1212:
1208:
1201:
1156:
1155:
1148:
1118:
1111:
1110:
1093:
1086:
1085:
1084:
1083:
1082:
1078:
1077:
1074:
1071:
1058:In each case;
1036:headed arrow.
980:
977:
975:
972:
922:quantum states
893:
892:
890:
889:
882:
875:
867:
864:
863:
859:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
627:
626:
623:
622:
619:
618:
615:
614:
609:
604:
599:
597:Density matrix
594:
589:
584:
579:
574:
569:
563:
560:
559:
556:
555:
551:
550:
545:
540:
535:
530:
525:
520:
519:
518:
517:
516:
501:
496:
491:
486:
481:
475:
474:
469:
468:
465:
464:
460:
459:
454:
449:
444:
439:
433:
432:
429:
428:
425:
424:
420:
419:
414:
409:
404:
399:
394:
388:
387:
386:
380:
377:
376:
373:
372:
368:
367:
362:
357:
351:
350:
349:
348:
347:
345:Delayed-choice
340:Quantum eraser
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
282:
279:
278:
275:
274:
270:
269:
268:
267:
257:
252:
247:
242:
237:
232:
230:Quantum number
227:
222:
217:
212:
207:
202:
196:
195:
192:
191:
188:
187:
183:
182:
177:
171:
170:
169:
164:
159:
153:
150:
149:
146:
145:
144:
143:
138:
133:
125:
124:
119:
108:
105:
101:
94:
91:
85:
82:
79:
75:
68:
65:
61:
56:
53:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5876:
5865:
5862:
5860:
5857:
5856:
5854:
5839:
5831:
5830:
5827:
5821:
5818:
5816:
5813:
5811:
5808:
5804:
5801:
5800:
5799:
5796:
5795:
5793:
5789:
5783:
5780:
5778:
5775:
5771:
5768:
5767:
5766:
5763:
5761:
5758:
5756:
5753:
5751:
5748:
5747:
5745:
5741:
5735:
5732:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5674:Quantum logic
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5633:
5630:
5629:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5606:
5603:
5602:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5582:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5527:Quantum chaos
5525:
5523:
5520:
5518:
5515:
5514:
5512:
5510:
5506:
5500:
5497:
5495:
5494:SternâGerlach
5492:
5490:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5446:
5444:
5440:
5434:
5431:
5429:
5428:Transactional
5426:
5424:
5421:
5419:
5418:Quantum logic
5416:
5414:
5411:
5409:
5406:
5400:
5397:
5396:
5395:
5392:
5391:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5367:
5365:
5362:
5361:
5359:
5357:
5353:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5308:
5306:
5302:
5296:
5293:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5262:
5260:
5256:
5248:
5245:
5243:
5240:
5239:
5238:
5237:Wave function
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5212:Superposition
5210:
5208:
5207:Quantum state
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5171:
5168:
5166:
5163:
5161:
5160:Excited state
5158:
5156:
5153:
5152:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5127:
5125:
5121:
5115:
5112:
5110:
5107:
5105:
5102:
5098:
5095:
5094:
5093:
5090:
5088:
5085:
5084:
5082:
5078:
5074:
5067:
5062:
5060:
5055:
5053:
5048:
5047:
5044:
5034:
5030:
5024:
5020:
5016:
5014:9780120348510
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4979:
4977:
4973:
4966:
4960:
4954:
4950:
4949:
4943:
4939:
4933:
4929:
4928:
4922:
4918:
4917:
4911:
4907:
4901:
4897:
4893:
4888:
4884:
4878:
4874:
4873:
4867:
4866:
4861:
4855:
4851:
4850:
4844:
4840:
4834:
4830:
4829:
4823:
4819:
4813:
4809:
4808:
4802:
4798:
4792:
4788:
4787:
4781:
4777:
4771:
4767:
4766:
4760:
4759:
4755:
4749:
4743:
4739:
4738:
4732:
4729:
4725:
4724:
4718:
4717:
4713:
4709:
4706:
4704:
4701:
4699:
4696:
4695:
4691:
4685:
4680:
4675:
4653:
4645:
4641:
4633:
4629:
4621:
4617:
4607:
4603:
4595:
4591:
4583:
4579:
4569:
4565:
4557:
4553:
4545:
4541:
4534:
4524:
4517:
4494:
4486:
4482:
4474:
4470:
4462:
4458:
4448:
4444:
4436:
4432:
4424:
4420:
4413:
4403:
4396:
4384:
4380:
4376:
4373:
4369:
4368:
4364:
4353:
4339:
4332:
4325:
4318:
4311:
4304:
4294:
4284:
4274:
4264:
4254:
4244:
4233:
4226:
4219:
4212:
4205:
4198:
4184:
4174:
4164:
4154:
4144:
4134:
4128:
4127:Inner product
4122:
4113:
4111:
4087:
4081:
4077:
4073:
4068:
4064:
4054:
4050:
4046:
4042:
4038:
4034:
4030:
4025:
4019:
4014:
4011:
4008:
4004:
4001:
3996:
3984:
3980:
3976:
3971:
3967:
3958:
3955:
3952:
3944:
3940:
3936:
3931:
3927:
3913:
3909:
3905:
3900:
3896:
3881:
3877:
3873:
3869:
3865:
3861:
3857:
3844:
3840:
3836:
3832:
3828:
3824:
3820:
3813:
3810:
3807:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3768:
3763:
3755:
3749:
3745:
3741:
3736:
3732:
3728:
3725:
3722:
3719:
3714:
3710:
3706:
3701:
3697:
3693:
3688:
3684:
3680:
3675:
3671:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3598:
3595:
3592:
3589:
3585:
3581:
3577:
3573:
3569:
3565:
3561:
3556:
3543:
3542:
3541:
3529:
3515:
3508:
3501:
3494:
3487:
3480:
3469:
3462:
3452:
3445:
3435:
3428:
3417:
3410:
3399:
3392:
3381:
3374:
3368:
3362:
3353:
3334:
3326:
3325:anticlockwise
3322:
3318:
3314:
3299:
3291:
3287:
3283:
3279:
3278:
3277:
3274:
3272:
3248:
3242:
3238:
3234:
3229:
3225:
3220:
3216:
3212:
3206:
3202:
3198:
3193:
3189:
3184:
3179:
3173:
3169:
3165:
3160:
3156:
3151:
3147:
3145:
3136:
3130:
3126:
3122:
3117:
3113:
3108:
3104:
3101:
3098:
3094:
3088:
3084:
3080:
3075:
3071:
3066:
3062:
3058:
3052:
3048:
3044:
3039:
3035:
3030:
3026:
3024:
3018:
3012:
3008:
3004:
2999:
2995:
2991:
2988:
2985:
2982:
2977:
2973:
2969:
2964:
2960:
2956:
2951:
2947:
2943:
2938:
2934:
2929:
2917:
2916:
2915:
2910:
2906:
2901:
2897:
2886:
2879:
2868:
2861:
2855:
2847:
2841:
2837:
2835:
2832:only and not
2831:
2826:
2820:
2816:
2811:
2786:
2782:
2773:
2770:
2767:
2758:
2755:
2752:
2744:
2739:
2735:
2731:
2729:
2719:
2716:
2713:
2705:
2702:
2689:
2686:
2683:
2680:
2671:
2668:
2665:
2662:
2652:
2649:
2646:
2638:
2635:
2625:
2621:
2617:
2615:
2602:
2599:
2596:
2593:
2584:
2581:
2578:
2575:
2562:
2559:
2556:
2550:
2542:
2539:
2529:
2525:
2521:
2519:
2506:
2503:
2500:
2497:
2488:
2485:
2482:
2479:
2469:
2465:
2461:
2459:
2449:
2446:
2443:
2434:
2431:
2428:
2418:
2414:
2402:
2401:
2400:
2398:
2384:
2370:
2363:
2353:
2346:
2335:
2328:
2317:
2310:
2304:
2303:Outer product
2298:
2289:
2271:
2265:
2261:
2257:
2252:
2248:
2243:
2238:
2232:
2228:
2224:
2219:
2215:
2210:
2202:
2201:
2200:
2193:
2186:
2175:
2168:
2158:
2150:
2146:
2145:
2144:
2142:
2138:
2119:
2116:
2113:
2110:
2107:
2101:
2098:
2095:
2087:
2084:
2081:
2073:
2069:
2061:
2060:
2059:
2057:
2043:
2029:
2022:
2015:
2008:
1997:
1990:
1979:
1972:
1966:
1965:Inner product
1960:
1951:
1930:
1926:
1920:
1916:
1911:
1903:
1899:
1893:
1889:
1884:
1880:
1872:
1868:
1864:
1859:
1855:
1844:
1840:
1836:
1831:
1827:
1816:
1815:
1814:
1807:
1800:
1789:
1782:
1773:Inner product
1772:
1770:
1759:
1755:
1746:
1735:
1731:
1722:
1713:
1695:
1691:
1688:
1685:
1682:
1678:
1672:
1669:
1666:
1658:
1655:
1647:
1643:
1636:
1633:
1630:
1624:
1620:
1616:
1612:
1608:
1605:
1602:
1598:
1594:
1587:
1586:
1585:
1581:
1577:
1542:
1539:
1534:
1530:
1512:
1505:
1504:
1503:
1501:
1497:
1493:
1460:
1457:
1452:
1448:
1430:
1423:
1422:
1421:
1419:
1415:
1411:
1407:
1374:
1371:
1366:
1362:
1344:
1337:
1336:
1335:
1333:
1300:
1297:
1292:
1288:
1270:
1263:
1262:
1261:
1259:
1226:
1221:
1217:
1199:
1192:
1191:
1190:
1188:
1184:
1180:
1176:
1172:
1169:
1165:
1161:
1160:time reversal
1153:
1149:
1146:
1142:
1141:
1140:
1130:
1126:
1121:
1115:
1105:
1101:
1096:
1090:
1081:
1075:
1072:
1069:
1065:
1061:
1060:
1059:
1056:
1054:
1048:
1044:
1037:
1035:
1029:
1025:
1020:
1016:
1012:
1008:
1004:
1000:
997:
993:
990:
986:
985:quantum state
978:
973:
971:
969:
968:Adolfas Jucys
964:
962:
958:
957:time reversal
954:
950:
946:
942:
938:
933:
931:
927:
923:
920:
916:
912:
908:
904:
900:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
628:
621:
620:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
587:Quantum chaos
585:
583:
580:
578:
575:
573:
570:
568:
565:
564:
558:
557:
549:
546:
544:
543:Transactional
541:
539:
536:
534:
533:Quantum logic
531:
529:
526:
524:
521:
515:
512:
511:
510:
507:
506:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
476:
472:
467:
466:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
434:
427:
426:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
385:
382:
381:
375:
374:
366:
363:
361:
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353:
352:
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343:
342:
341:
338:
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333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
277:
276:
266:
263:
262:
261:
260:Wave function
258:
256:
253:
251:
248:
246:
243:
241:
240:Superposition
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
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190:
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178:
176:
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172:
168:
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163:
160:
158:
155:
154:
148:
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142:
139:
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134:
132:
129:
128:
127:
126:
122:
89:
83:
66:
63:
59:
51:
44:
43:
40:
36:
32:
31:
19:
18:Jucys diagram
5557:Quantum mind
5469:FranckâHertz
5311:KleinâGordon
5265:Formulations
5258:Formulations
5187:Interference
5177:Entanglement
5155:Ground state
5150:Energy level
5123:Fundamentals
5087:Introduction
5032:
5028:
4988:
4984:
4947:
4926:
4915:
4895:
4871:
4848:
4827:
4806:
4785:
4764:
4736:
4722:
4337:
4330:
4323:
4316:
4309:
4302:
4292:
4282:
4272:
4262:
4252:
4242:
4231:
4224:
4217:
4210:
4203:
4196:
4182:
4172:
4162:
4152:
4142:
4132:
4126:
4109:
4107:
3539:
3513:
3506:
3499:
3492:
3485:
3478:
3467:
3460:
3450:
3443:
3433:
3426:
3415:
3408:
3397:
3390:
3379:
3372:
3366:
3350:
3324:
3320:
3316:
3289:
3285:
3281:
3275:
3270:
3268:
2908:
2904:
2899:
2895:
2884:
2877:
2866:
2859:
2853:
2851:
2833:
2829:
2824:
2818:
2814:
2809:
2807:
2394:
2368:
2361:
2351:
2344:
2333:
2326:
2315:
2308:
2302:
2287:
2191:
2184:
2173:
2166:
2162:
2140:
2136:
2134:
2053:
2027:
2020:
2013:
2006:
1995:
1988:
1977:
1970:
1964:
1949:
1805:
1798:
1787:
1780:
1776:
1768:
1757:
1753:
1733:
1729:
1711:
1579:
1575:
1571:
1499:
1495:
1491:
1489:
1417:
1413:
1409:
1403:
1329:
1255:
1182:
1178:
1174:
1170:
1163:
1157:
1138:
1128:
1124:
1119:
1103:
1099:
1094:
1079:
1067:
1063:
1057:
1052:
1046:
1042:
1038:
1033:
1027:
1023:
1014:
1010:
1006:
1002:
998:
991:
982:
965:
934:
914:
910:
896:
442:KleinâGordon
378:Formulations
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
5815:EPR paradox
5595:Quantum bus
5464:Double-slit
5442:Experiments
5408:Many-worlds
5346:Schrödinger
5295:Phase space
5285:Schrödinger
5275:Interaction
5232:Uncertainty
5202:Nonlocality
5197:Measurement
5192:Decoherence
5182:Hamiltonian
2914:is written
1009:â 1, ..., â
831:von Neumann
816:Schrödinger
592:EPR paradox
523:Many-worlds
457:Schrödinger
412:Schrödinger
407:Phase-space
397:Interaction
302:Double-slit
280:Experiments
255:Uncertainty
225:Nonlocality
220:Measurement
205:Decoherence
175:Hamiltonian
5853:Categories
5743:Extensions
5577:Technology
5423:Relational
5374:Copenhagen
5270:Heisenberg
5217:Tunnelling
5080:Background
4714:References
4703:Fock space
4523:9-j symbol
4402:6-j symbol
4239:, that is
4108:there are
3423:, that is
2341:, that is
2003:, that is
1177:, that is
994:and total
905:, notably
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
5449:Bell test
5304:Equations
5130:Born rule
5023:0065-3276
4383:molecular
4005:∏
3990:⟩
3950:⟩
3919:⟩
3854:⟨
3817:⟨
3771:⟨
3335:↺
3300:↻
3290:clockwise
3217:⋯
3148:≡
3105:⊗
3102:⋯
3099:⊗
3063:⊗
3027:≡
2787:†
2765:⟨
2762:⟩
2736:∑
2717:−
2703:−
2687:−
2678:⟨
2675:⟩
2669:−
2650:−
2636:−
2622:∑
2600:−
2591:⟨
2588:⟩
2582:−
2560:−
2540:−
2526:∑
2504:−
2495:⟨
2492:⟩
2486:−
2466:∑
2441:⟨
2438:⟩
2415:∑
2105:⟩
2079:⟨
2070:∑
1912:δ
1885:δ
1878:⟩
1824:⟨
1689:−
1670:−
1656:−
1617:≡
1554:^
1543:−
1535:†
1524:^
1472:^
1461:−
1453:†
1442:^
1386:^
1375:−
1367:†
1356:^
1312:^
1301:−
1293:†
1282:^
1238:^
1222:†
1211:^
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
5838:Category
5632:Timeline
5384:Ensemble
5364:Bayesian
5326:Majorana
5242:Collapse
5114:Glossary
5097:Timeline
4676:See also
4385:systems.
4343:⟩
4241:⟨
4237:⟩
4191:⟩
4088:⟩
4055:′
4039:′
4026:⟨
3882:′
3866:′
3845:′
3829:′
3799:′
3783:′
3756:⟩
3662:′
3646:′
3630:′
3614:′
3586:′
3570:′
3557:⟨
3519:⟩
3473:⟩
3456:⟩
3439:⟩
3421:⟩
3403:⟩
3385:⟩
3249:⟩
3213:⟩
3180:⟩
3137:⟩
3095:⟩
3059:⟩
3019:⟩
2912:⟩
2890:⟩
2872:⟩
2821:⟩
2360:⟨
2357:⟩
2339:⟩
2321:⟩
2244:⟨
2239:⟩
2197:⟩
2179:⟩
2033:⟩
2005:⟨
2001:⟩
1983:⟩
1811:⟩
1793:⟩
1752:⟨
1736:⟩
1696:⟩
1644:⟩
1613:⟩
1582:⟩
1330:and the
1123:⟨
1106:⟩
1041:⟨
1030:⟩
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
5791:Related
5770:History
5509:Science
5341:Rydberg
5092:History
4993:Bibcode
2856:states
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
5484:Popper
5021:
5011:
4955:
4934:
4902:
4879:
4856:
4835:
4814:
4793:
4772:
4744:
4301:|
4195:|
4131:|
3477:|
3459:|
3442:|
3425:|
3407:|
3389:|
3371:|
2894:|
2892:, ...
2876:|
2858:|
2813:|
2374:|
2343:|
2325:|
2307:|
2183:|
2165:|
2139:, not
2019:|
1987:|
1969:|
1797:|
1779:|
1760:|
1728:|
1574:|
1131:|
1098:|
1053:double
1049:|
1034:single
1022:|
1013:+ 1, â
949:graphs
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
5394:Local
5336:Pauli
5316:Dirac
4967:Notes
3319:sign
3312:, and
3284:sign
3282:minus
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
5321:Weyl
5019:ISSN
5009:ISBN
4953:ISBN
4932:ISBN
4900:ISBN
4877:ISBN
4854:ISBN
4833:ISBN
4812:ISBN
4791:ISBN
4770:ISBN
4742:ISBN
4381:and
4193:and
3323:for
3317:plus
2323:and
2181:and
1985:and
1813:is:
1795:and
1332:spin
983:The
939:and
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
5001:doi
4892:"8"
4129:of
3369:of
3321:(+)
3286:(â)
2399:):
2305:of
2058:):
1967:of
1408:is
1120:Bra
1095:Ket
1019:ket
897:In
5855::
5017:.
5007:.
4999:.
4989:51
4987:.
4975:^
4894:.
4525:,
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4322:,
4315:,
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4271:,
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4230:,
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4216:,
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4181:,
4171:,
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4151:,
4141:,
3512:,
3505:,
3498:,
3491:,
3484:,
3475:=
3466:,
3449:,
3432:,
3414:,
3405:,
3396:,
3387:,
3378:,
3327:,
3315:a
3292:,
3280:a
2903:,
2883:,
2874:,
2865:,
2836::
2817:,
2367:,
2350:,
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2190:,
2172:,
2143::
2026:,
2012:,
1994:,
1976:,
1804:,
1786:,
1756:,
1732:,
1578:,
1498:+
1494:=
1416:Ă
1412:=
1181:=
1150:a
1143:a
1127:,
1102:,
1066:,
1045:,
1026:,
1005:,
1001:=
909:,
5065:e
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2031:1
2028:m
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2014:m
2010:2
2007:j
1999:2
1996:m
1992:2
1989:j
1981:1
1978:m
1974:1
1971:j
1931:2
1927:m
1921:1
1917:m
1904:2
1900:j
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1890:j
1881:=
1873:1
1869:m
1865:,
1860:1
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1851:|
1845:2
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1540:=
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1458:=
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1372:=
1363:T
1352:S
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1308:p
1298:=
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1234:x
1227:=
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1200:T
1183:T
1179:T
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1129:m
1125:j
1104:m
1100:j
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886:e
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100:|
90:H
84:=
74:|
67:t
64:d
60:d
52:i
20:)
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