5592:
2134:
3611:
The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the
4522:
3133:
1918:
4224:
3594:
4369:
3944:
3820:
2819:
3479:
4083:
2624:
1892:
119:
2352:
3362:
3225:
1784:
1204:
2936:
3015:
1314:
2708:
1593:
2500:
2217:
4378:
1504:
The differences between the Schrödinger and
Heisenberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system
3486:
2713:
3000:
1016:
1499:
4089:
2542:
3264:
2858:
1060:
1364:
4237:
3829:
1621:
1538:
1466:
1428:
3376:
1650:
1106:
3708:
923:
evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the
Hamiltonian which may change if the potential
2129:{\displaystyle \langle \psi (t)|\psi (t)\rangle =\langle \psi (t_{0})|U^{\dagger }(t,t_{0})U(t,t_{0})|\psi (t_{0})\rangle =\langle \psi (t_{0})|\psi (t_{0})\rangle .}
1652:, or both. All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture.
3960:
1795:
3668:
2248:
941:
3277:
3140:
1694:
4821:
2870:
3483:
If the
Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written as
49:
5389:
5022:
4560:
3373:
If the
Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as
385:
2400:
3661:
886:
2139:
1114:
5560:
2867:), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
4871:
137:
17:
5572:
1595:, oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector
593:
5256:
4814:
3654:
1222:
366:
5616:
5190:
4790:
4770:
4746:
4657:
2641:
1541:
549:
1547:
5113:
2943:
472:
4517:{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}
1437:
4844:
4807:
4720:
4689:
4632:
952:
132:
3128:{\displaystyle e^{-iHt/\hbar }=1-{\frac {iHt}{\hbar }}-{\frac {1}{2}}\left({\frac {Ht}{\hbar }}\right)^{2}+\cdots .}
951:
in which both the states and the observables evolve in time. The Schrödinger and
Heisenberg pictures are related as
4954:
221:
966:
picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a
5527:
5236:
5231:
4939:
2631:
1210:
879:
328:
308:
176:
5539:
5211:
568:
298:
5517:
5294:
5216:
5088:
4984:
4854:
608:
346:
246:
5621:
5251:
5185:
5180:
5151:
4849:
4535:
4219:{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}
977:
544:
539:
510:
361:
142:
5319:
5226:
1509:
578:
323:
313:
4364:{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}
5595:
5357:
5165:
5136:
3939:{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }
3232:
2826:
1028:
872:
524:
495:
1333:
5379:
5246:
5170:
5131:
5068:
5047:
5004:
4897:
1471:
529:
490:
443:
418:
341:
201:
4565:
1598:
1515:
1443:
1405:
5446:
5426:
5416:
5406:
5362:
4922:
4781:
613:
5141:
5052:
4550:
3815:{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }
3589:{\displaystyle U(t)=\mathrm {T} \exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right),}
782:
500:
408:
5555:
5103:
4705:(Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
4624:
4617:
2814:{\displaystyle i\hbar {\partial \over \partial t}U(t)|\psi (0)\rangle =HU(t)|\psi (0)\rangle .}
2536:
1626:
1069:
458:
356:
122:
5466:
5241:
5221:
5146:
4999:
4677:
787:
505:
333:
303:
266:
5491:
4989:
4969:
4078:{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}
256:
241:
5522:
5451:
5396:
5126:
4934:
4892:
2619:{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle =H|\psi (t)\rangle ,}
1397:
956:
920:
573:
485:
211:
168:
817:
8:
5507:
5476:
5421:
5401:
5309:
5266:
5121:
5032:
4959:
4949:
4861:
4540:
3693:
967:
948:
672:
480:
398:
226:
206:
158:
5567:
5436:
5334:
5027:
4974:
4866:
4545:
3687:
3647:
3613:
1912:
944:
926:
912:
393:
318:
251:
163:
1508:
be carried by some combination of the state vectors and the operators. For example, a
5577:
5486:
5456:
5384:
5347:
5342:
5324:
5289:
5279:
5083:
4979:
4944:
4927:
4830:
4786:
4766:
4758:
4742:
4716:
4685:
4653:
4628:
3474:{\displaystyle U(t)=\exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right),}
2242:
916:
827:
802:
742:
737:
637:
603:
583:
181:
40:
1436:, a vector space containing all possible states of the system. A quantum-mechanical
5304:
5299:
5156:
5037:
4753:
4708:
3701:
1908:
1789:
1326:
The Schrödinger picture is useful when dealing with a time-independent
Hamiltonian
947:
which keeps the states constant while the observables evolve in time, and from the
832:
822:
812:
712:
692:
677:
647:
515:
403:
5534:
5461:
5441:
5411:
5374:
5369:
5274:
5098:
4730:
2940:
If the
Hamiltonian is independent of time, the solution to the above equation is
857:
727:
707:
453:
293:
1216:
of the system does not vary with time, the time-evolution operator has the form
5512:
5481:
5471:
5093:
5073:
4902:
4738:
4698:
4230:
971:
792:
752:
732:
702:
682:
632:
598:
448:
438:
231:
963:
5610:
5431:
5284:
5175:
5078:
4994:
4964:
4917:
4776:
3605:
3597:
3009:
1887:{\displaystyle \langle \psi (t)|=\langle \psi (t_{0})|U^{\dagger }(t,t_{0}).}
1432:
1375:
1320:
852:
847:
777:
747:
717:
588:
534:
261:
236:
114:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
5314:
4912:
4907:
3601:
1379:
842:
837:
772:
757:
722:
216:
2347:{\displaystyle |\psi (t_{0})\rangle =U(t_{0},t_{0})|\psi (t_{0})\rangle .}
5352:
959:
between operators are preserved in the passage between the two pictures.
807:
762:
697:
652:
4726:
3950:
3267:
3008:
is an operator, this exponential expression is to be evaluated via its
797:
767:
687:
662:
657:
642:
3370:: they only pick up an overall phase factor as they evolve with time.
1396:. More abstractly, the state may be represented as a state vector, or
5206:
4887:
4676:
288:
4799:
2505:
4652:. Schuam's outline series (2nd ed.). McGraw Hill. p. 70.
3357:{\displaystyle |\psi (t)\rangle =e^{-iEt/\hbar }|\psi (0)\rangle .}
3220:{\displaystyle |\psi (t)\rangle =e^{-iHt/\hbar }|\psi (0)\rangle .}
667:
3619:
1779:{\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle .}
1378:
of a quantum-mechanical system is represented by a complex-valued
1199:{\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle .}
4647:
2931:{\displaystyle i\hbar {\frac {\partial }{\partial t}}U(t)=HU(t).}
900:
2517:
index in the time evolution operator with the convention that
4555:
1680:) is defined as the operator which acts on the ket at time
4725:
1309:{\displaystyle U(t,t_{0})=e^{-iH\cdot (t-t_{0})/\hbar },}
3266:
is an arbitrary ket. However, if the initial ket is an
2703:{\displaystyle |\psi (t)\rangle =U(t)|\psi (0)\rangle }
2372:
may be viewed as a two-step time evolution, first from
1588:{\displaystyle \langle \psi |{\hat {p}}|\psi \rangle }
4381:
4240:
4092:
3963:
3832:
3711:
3489:
3379:
3280:
3235:
3143:
3018:
2946:
2873:
2829:
2716:
2644:
2545:
2403:
2251:
2142:
1921:
1915:
of the state ket must not change with time. That is,
1798:
1697:
1629:
1601:
1550:
1518:
1474:
1446:
1408:
1336:
1225:
1117:
1072:
1031:
980:
929:
52:
2495:{\displaystyle U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}).}
4616:
4516:
4363:
4218:
4077:
3938:
3814:
3588:
3473:
3356:
3258:
3219:
3127:
2994:
2930:
2852:
2813:
2702:
2618:
2494:
2346:
2212:{\displaystyle U^{\dagger }(t,t_{0})U(t,t_{0})=I.}
2211:
2128:
1886:
1778:
1644:
1615:
1587:
1532:
1493:
1460:
1422:
1358:
1308:
1198:
1100:
1066:, the time-evolution operator is commonly written
1054:
1010:
935:
113:
2506:Differential equation for time evolution operator
5608:
4648:Y. Peleg; R. Pnini; E. Zaarur; E. Hecht (2010).
3620:Summary comparison of evolution in all pictures
1655:
4815:
4561:Mathematical formulation of quantum mechanics
3662:
880:
3933:
3859:
3809:
3738:
3348:
3298:
3253:
3211:
3161:
2847:
2805:
2767:
2697:
2662:
2610:
2584:
2338:
2276:
2120:
2074:
2068:
1960:
1954:
1922:
1822:
1799:
1770:
1715:
1610:
1582:
1551:
1527:
1488:
1455:
1417:
1190:
1135:
1049:
1005:
108:
82:
4822:
4808:
4735:Quantum Mechanics: Non-Relativistic Theory
3669:
3655:
3600:operator, which is sometimes known as the
887:
873:
4715:(3rd ed., John Wiley 1998) p. 430–1
3566:
3451:
974:. For time evolution from a state vector
2634:. Now using the time-evolution operator
1319:where the exponent is evaluated via its
4599:) must reduce to the identity operator.
3366:The eigenstates of the Hamiltonian are
14:
5609:
4623:(2nd ed.). McGraw Hill. pp.
4614:
1687:to produce the ket at some other time
4829:
4803:
2995:{\displaystyle U(t)=e^{-iHt/\hbar }.}
1374:In elementary quantum mechanics, the
1011:{\displaystyle |\psi (t_{0})\rangle }
4619:McGraw Hill Encyclopaedia of Physics
3270:of the Hamiltonian, with eigenvalue
2860:is a constant ket (the state ket at
1907:The time evolution operator must be
4680:; Bernard Diu; Frank Laloe (1977).
3624:For a time-independent Hamiltonian
24:
4684:. Paris: Wiley. pp. 312–314.
4492:
4454:
4426:
4388:
4339:
4310:
4282:
4247:
4194:
4165:
4137:
4099:
4053:
4030:
4002:
3970:
3918:
3885:
3844:
3794:
3761:
3723:
3506:
2886:
2882:
2729:
2725:
2558:
2554:
1338:
953:active and passive transformations
419:Sum-over-histories (path integral)
105:
79:
35:Part of a series of articles about
25:
5633:
4509:
4443:
4356:
4299:
4211:
4154:
4070:
4019:
3902:
3778:
3529:
3414:
3326:
3259:{\displaystyle |\psi (0)\rangle }
3189:
3101:
3068:
3041:
2984:
2877:
2853:{\displaystyle |\psi (0)\rangle }
2720:
2549:
1440:is a function which takes a ket
1298:
1055:{\displaystyle |\psi (t)\rangle }
56:
5591:
5590:
1359:{\displaystyle \partial _{t}H=0}
943:changes). This differs from the
27:Formulation of quantum mechanics
4763:Principles of Quantum Mechanics
1494:{\displaystyle |\psi '\rangle }
5540:Relativistic quantum mechanics
4737:. Vol. 3 (3rd ed.).
4682:Quantum Mechanics (Volume One)
4641:
4608:
4578:
4466:
4460:
4400:
4394:
4322:
4316:
4259:
4253:
4111:
4105:
3982:
3976:
3930:
3924:
3908:
3856:
3850:
3834:
3806:
3800:
3784:
3735:
3729:
3713:
3563:
3552:
3499:
3493:
3448:
3437:
3389:
3383:
3345:
3339:
3332:
3295:
3289:
3282:
3250:
3244:
3237:
3208:
3202:
3195:
3158:
3152:
3145:
2956:
2950:
2922:
2916:
2904:
2898:
2844:
2838:
2831:
2802:
2796:
2789:
2785:
2779:
2764:
2758:
2751:
2747:
2741:
2694:
2688:
2681:
2677:
2671:
2659:
2653:
2646:
2607:
2601:
2594:
2581:
2575:
2568:
2486:
2460:
2454:
2435:
2426:
2407:
2335:
2322:
2315:
2311:
2285:
2273:
2260:
2253:
2197:
2178:
2172:
2153:
2117:
2104:
2097:
2093:
2080:
2065:
2052:
2045:
2041:
2022:
2016:
1997:
1983:
1979:
1966:
1951:
1945:
1938:
1934:
1928:
1878:
1859:
1845:
1841:
1828:
1815:
1811:
1805:
1767:
1754:
1747:
1743:
1724:
1712:
1706:
1699:
1636:
1616:{\displaystyle |\psi \rangle }
1603:
1575:
1568:
1558:
1533:{\displaystyle |\psi \rangle }
1520:
1476:
1461:{\displaystyle |\psi \rangle }
1448:
1430:. This ket is an element of a
1423:{\displaystyle |\psi \rangle }
1410:
1290:
1271:
1248:
1229:
1187:
1174:
1167:
1163:
1144:
1132:
1126:
1119:
1095:
1076:
1046:
1040:
1033:
1002:
989:
982:
569:Relativistic quantum mechanics
101:
94:
75:
13:
1:
5518:Quantum statistical mechanics
5295:Quantum differential calculus
5217:Delayed-choice quantum eraser
4985:Symmetry in quantum mechanics
4670:
1896:
1660:
1369:
609:Quantum statistical mechanics
5617:Foundational quantum physics
1665:The time-evolution operator
1468:and returns some other ket
7:
5320:Quantum stochastic calculus
5310:Quantum measurement problem
5232:Mach–Zehnder interferometer
4529:
1656:The time evolution operator
1510:quantum harmonic oscillator
579:Quantum information science
10:
5638:
1645:{\displaystyle {\hat {p}}}
1101:{\displaystyle U(t,t_{0})}
909:Schrödinger representation
5586:
5548:
5500:
5380:Quantum complexity theory
5358:Quantum cellular automata
5333:
5265:
5199:
5112:
5061:
5048:Path integral formulation
5015:
4880:
4837:
3646:
3643:
3638:is the free Hamiltonian,
5447:Quantum machine learning
5427:Quantum key distribution
5417:Quantum image processing
5407:Quantum error correction
5257:Wheeler's delayed choice
4782:Modern Quantum Mechanics
4571:
4536:Hamilton–Jacobi equation
2379:to an intermediate time
1623:, the momentum operator
614:Quantum machine learning
367:Wheeler's delayed-choice
5363:Quantum finite automata
4678:Cohen-Tannoudji, Claude
4551:Phase space formulation
972:time evolution operator
324:Leggett–Garg inequality
5467:Quantum neural network
4566:Schrödinger functional
4518:
4365:
4220:
4079:
3940:
3816:
3590:
3475:
3358:
3260:
3221:
3129:
2996:
2932:
2854:
2815:
2704:
2620:
2496:
2348:
2213:
2130:
1888:
1780:
1646:
1617:
1589:
1534:
1495:
1462:
1424:
1360:
1310:
1209:In the case where the
1200:
1102:
1056:
1012:
937:
115:
5492:Quantum teleportation
5005:Wave–particle duality
4615:Parker, C.B. (1994).
4519:
4366:
4221:
4080:
3941:
3817:
3591:
3476:
3359:
3261:
3222:
3130:
2997:
2933:
2855:
2816:
2705:
2621:
2497:
2349:
2214:
2131:
1889:
1781:
1647:
1618:
1590:
1535:
1496:
1463:
1425:
1361:
1311:
1201:
1103:
1057:
1013:
957:commutation relations
938:
309:Elitzur–Vaidman
299:Davisson–Germer
116:
18:Schrödinger evolution
5523:Quantum field theory
5452:Quantum metamaterial
5397:Quantum cryptography
5127:Consistent histories
4785:(Revised Edition),
4379:
4238:
4090:
3961:
3830:
3709:
3487:
3377:
3278:
3233:
3141:
3016:
2944:
2871:
2827:
2714:
2642:
2543:
2537:Schrödinger equation
2401:
2361:Time evolution from
2249:
2140:
1919:
1796:
1695:
1627:
1599:
1548:
1516:
1472:
1444:
1406:
1334:
1223:
1115:
1070:
1029:
978:
927:
574:Quantum field theory
486:Consistent histories
123:Schrödinger equation
50:
5508:Quantum fluctuation
5477:Quantum programming
5437:Quantum logic gates
5422:Quantum information
5402:Quantum electronics
4862:Classical mechanics
4541:Interaction picture
3548:
3433:
1512:may be in a state
949:interaction picture
905:Schrödinger picture
362:Stern–Gerlach
159:Classical mechanics
5561:in popular culture
5343:Quantum algorithms
5191:Von Neumann–Wigner
5171:Objective collapse
4867:Old quantum theory
4546:Heisenberg picture
4514:
4361:
4216:
4075:
3936:
3812:
3614:Heisenberg picture
3586:
3534:
3471:
3419:
3354:
3256:
3217:
3125:
2992:
2928:
2850:
2811:
2700:
2616:
2492:
2393:to the final time
2344:
2209:
2126:
1884:
1776:
1642:
1613:
1585:
1530:
1491:
1458:
1420:
1356:
1306:
1196:
1098:
1052:
1025:to a state vector
1008:
945:Heisenberg picture
933:
550:Von Neumann–Wigner
530:Objective-collapse
329:Mach–Zehnder
319:Leggett inequality
314:Franck–Hertz
164:Old quantum theory
111:
5622:Erwin Schrödinger
5604:
5603:
5578:Quantum mysticism
5556:Schrödinger's cat
5487:Quantum simulator
5457:Quantum metrology
5385:Quantum computing
5348:Quantum amplifier
5325:Quantum spacetime
5290:Quantum cosmology
5280:Quantum chemistry
4980:Scattering theory
4928:Zero-point energy
4923:Degenerate levels
4831:Quantum mechanics
4791:978-0-201-53929-5
4771:978-0-306-44790-7
4748:978-0-08-020940-1
4713:Quantum Mechanics
4703:Quantum Mechanics
4659:978-0-07-162358-2
4650:Quantum mechanics
4527:
4526:
4500:
4434:
4347:
4290:
4202:
4145:
4061:
4010:
3893:
3769:
3532:
3417:
3368:stationary states
3104:
3084:
3071:
2893:
2736:
2565:
2243:identity operator
1639:
1571:
1544:of the momentum,
1542:expectation value
936:{\displaystyle V}
917:quantum mechanics
897:
896:
604:Scattering theory
584:Quantum computing
357:Schrödinger's cat
289:Bell's inequality
97:
72:
41:Quantum mechanics
16:(Redirected from
5629:
5594:
5593:
5305:Quantum geometry
5300:Quantum dynamics
5157:Superdeterminism
5089:Rarita–Schwinger
5038:Matrix mechanics
4893:Bra–ket notation
4824:
4817:
4810:
4801:
4800:
4765:, Plenum Press,
4752:
4695:
4664:
4663:
4645:
4639:
4638:
4622:
4612:
4600:
4590:
4582:
4523:
4521:
4520:
4515:
4513:
4512:
4508:
4498:
4497:
4496:
4495:
4459:
4458:
4457:
4447:
4446:
4442:
4432:
4431:
4430:
4429:
4393:
4392:
4391:
4370:
4368:
4367:
4362:
4360:
4359:
4355:
4345:
4344:
4343:
4342:
4315:
4314:
4313:
4303:
4302:
4298:
4288:
4287:
4286:
4285:
4252:
4251:
4250:
4225:
4223:
4222:
4217:
4215:
4214:
4210:
4200:
4199:
4198:
4197:
4170:
4169:
4168:
4158:
4157:
4153:
4143:
4142:
4141:
4140:
4104:
4103:
4102:
4084:
4082:
4081:
4076:
4074:
4073:
4069:
4059:
4058:
4057:
4056:
4035:
4034:
4033:
4023:
4022:
4018:
4008:
4007:
4006:
4005:
3975:
3974:
3973:
3945:
3943:
3942:
3937:
3923:
3922:
3921:
3911:
3906:
3905:
3901:
3891:
3890:
3889:
3888:
3849:
3848:
3847:
3837:
3821:
3819:
3818:
3813:
3799:
3798:
3797:
3787:
3782:
3781:
3777:
3767:
3766:
3765:
3764:
3728:
3727:
3726:
3716:
3671:
3664:
3657:
3641:
3640:
3595:
3593:
3592:
3587:
3582:
3578:
3577:
3562:
3547:
3542:
3533:
3525:
3509:
3480:
3478:
3477:
3472:
3467:
3463:
3462:
3447:
3432:
3427:
3418:
3410:
3363:
3361:
3360:
3355:
3335:
3330:
3329:
3325:
3285:
3265:
3263:
3262:
3257:
3240:
3226:
3224:
3223:
3218:
3198:
3193:
3192:
3188:
3148:
3134:
3132:
3131:
3126:
3115:
3114:
3109:
3105:
3100:
3092:
3085:
3077:
3072:
3067:
3056:
3045:
3044:
3040:
3001:
2999:
2998:
2993:
2988:
2987:
2983:
2937:
2935:
2934:
2929:
2894:
2892:
2881:
2866:
2859:
2857:
2856:
2851:
2834:
2820:
2818:
2817:
2812:
2792:
2754:
2737:
2735:
2724:
2709:
2707:
2706:
2701:
2684:
2649:
2625:
2623:
2622:
2617:
2597:
2571:
2566:
2564:
2553:
2527:and write it as
2526:
2501:
2499:
2498:
2493:
2485:
2484:
2472:
2471:
2453:
2452:
2425:
2424:
2386:, and then from
2353:
2351:
2350:
2345:
2334:
2333:
2318:
2310:
2309:
2297:
2296:
2272:
2271:
2256:
2218:
2216:
2215:
2210:
2196:
2195:
2171:
2170:
2152:
2151:
2135:
2133:
2132:
2127:
2116:
2115:
2100:
2092:
2091:
2064:
2063:
2048:
2040:
2039:
2015:
2014:
1996:
1995:
1986:
1978:
1977:
1941:
1893:
1891:
1890:
1885:
1877:
1876:
1858:
1857:
1848:
1840:
1839:
1818:
1785:
1783:
1782:
1777:
1766:
1765:
1750:
1742:
1741:
1702:
1651:
1649:
1648:
1643:
1641:
1640:
1632:
1622:
1620:
1619:
1614:
1606:
1594:
1592:
1591:
1586:
1578:
1573:
1572:
1564:
1561:
1539:
1537:
1536:
1531:
1523:
1500:
1498:
1497:
1492:
1487:
1479:
1467:
1465:
1464:
1459:
1451:
1429:
1427:
1426:
1421:
1413:
1395:
1365:
1363:
1362:
1357:
1346:
1345:
1329:
1315:
1313:
1312:
1307:
1302:
1301:
1297:
1289:
1288:
1247:
1246:
1215:
1205:
1203:
1202:
1197:
1186:
1185:
1170:
1162:
1161:
1122:
1107:
1105:
1104:
1099:
1094:
1093:
1065:
1061:
1059:
1058:
1053:
1036:
1021:
1017:
1015:
1014:
1009:
1001:
1000:
985:
968:unitary operator
942:
940:
939:
934:
889:
882:
875:
516:Superdeterminism
169:Bra–ket notation
120:
118:
117:
112:
104:
99:
98:
90:
78:
73:
71:
60:
32:
31:
21:
5637:
5636:
5632:
5631:
5630:
5628:
5627:
5626:
5607:
5606:
5605:
5600:
5582:
5568:Wigner's friend
5544:
5535:Quantum gravity
5496:
5482:Quantum sensing
5462:Quantum network
5442:Quantum machine
5412:Quantum imaging
5375:Quantum circuit
5370:Quantum channel
5329:
5275:Quantum biology
5261:
5237:Elitzur–Vaidman
5212:Davisson–Germer
5195:
5147:Hidden-variable
5137:de Broglie–Bohm
5114:Interpretations
5108:
5057:
5011:
4898:Complementarity
4876:
4833:
4828:
4797:
4749:
4692:
4673:
4668:
4667:
4660:
4646:
4642:
4635:
4613:
4609:
4604:
4603:
4585:
4583:
4579:
4574:
4532:
4504:
4491:
4484:
4480:
4473:
4469:
4453:
4452:
4448:
4438:
4425:
4418:
4414:
4410:
4406:
4387:
4386:
4382:
4380:
4377:
4376:
4351:
4338:
4337:
4333:
4329:
4325:
4309:
4308:
4304:
4294:
4281:
4280:
4276:
4269:
4265:
4246:
4245:
4241:
4239:
4236:
4235:
4206:
4193:
4186:
4182:
4175:
4171:
4164:
4163:
4159:
4149:
4136:
4129:
4125:
4121:
4117:
4098:
4097:
4093:
4091:
4088:
4087:
4065:
4052:
4051:
4047:
4040:
4036:
4029:
4028:
4024:
4014:
4001:
4000:
3996:
3992:
3988:
3969:
3968:
3964:
3962:
3959:
3958:
3917:
3916:
3912:
3907:
3897:
3884:
3877:
3873:
3869:
3865:
3843:
3842:
3838:
3833:
3831:
3828:
3827:
3793:
3792:
3788:
3783:
3773:
3760:
3759:
3755:
3748:
3744:
3722:
3721:
3717:
3712:
3710:
3707:
3706:
3675:
3637:
3630:
3622:
3570:
3555:
3543:
3538:
3524:
3520:
3516:
3505:
3488:
3485:
3484:
3455:
3440:
3428:
3423:
3409:
3405:
3401:
3378:
3375:
3374:
3331:
3321:
3308:
3304:
3281:
3279:
3276:
3275:
3236:
3234:
3231:
3230:
3194:
3184:
3171:
3167:
3144:
3142:
3139:
3138:
3110:
3093:
3091:
3087:
3086:
3076:
3057:
3055:
3036:
3023:
3019:
3017:
3014:
3013:
2979:
2966:
2962:
2945:
2942:
2941:
2885:
2880:
2872:
2869:
2868:
2861:
2830:
2828:
2825:
2824:
2788:
2750:
2728:
2723:
2715:
2712:
2711:
2680:
2645:
2643:
2640:
2639:
2593:
2567:
2557:
2552:
2544:
2541:
2540:
2524:
2518:
2516:
2508:
2480:
2476:
2467:
2463:
2448:
2444:
2420:
2416:
2402:
2399:
2398:
2392:
2385:
2378:
2367:
2329:
2325:
2314:
2305:
2301:
2292:
2288:
2267:
2263:
2252:
2250:
2247:
2246:
2236:
2191:
2187:
2166:
2162:
2147:
2143:
2141:
2138:
2137:
2111:
2107:
2096:
2087:
2083:
2059:
2055:
2044:
2035:
2031:
2010:
2006:
1991:
1987:
1982:
1973:
1969:
1937:
1920:
1917:
1916:
1899:
1872:
1868:
1853:
1849:
1844:
1835:
1831:
1814:
1797:
1794:
1793:
1761:
1757:
1746:
1737:
1733:
1698:
1696:
1693:
1692:
1686:
1679:
1663:
1658:
1631:
1630:
1628:
1625:
1624:
1602:
1600:
1597:
1596:
1574:
1563:
1562:
1557:
1549:
1546:
1545:
1519:
1517:
1514:
1513:
1480:
1475:
1473:
1470:
1469:
1447:
1445:
1442:
1441:
1409:
1407:
1404:
1403:
1382:
1372:
1341:
1337:
1335:
1332:
1331:
1327:
1293:
1284:
1280:
1258:
1254:
1242:
1238:
1224:
1221:
1220:
1213:
1181:
1177:
1166:
1157:
1153:
1118:
1116:
1113:
1112:
1089:
1085:
1071:
1068:
1067:
1063:
1032:
1030:
1027:
1026:
1024:
1019:
996:
992:
981:
979:
976:
975:
928:
925:
924:
893:
864:
863:
862:
627:
619:
618:
564:
563:Advanced topics
556:
555:
554:
506:Hidden-variable
496:de Broglie–Bohm
475:
473:Interpretations
465:
464:
463:
433:
425:
424:
423:
381:
373:
372:
371:
338:
294:CHSH inequality
283:
275:
274:
273:
202:Complementarity
196:
188:
187:
186:
154:
125:
100:
89:
88:
74:
64:
59:
51:
48:
47:
28:
23:
22:
15:
12:
11:
5:
5635:
5625:
5624:
5619:
5602:
5601:
5599:
5598:
5587:
5584:
5583:
5581:
5580:
5575:
5570:
5565:
5564:
5563:
5552:
5550:
5546:
5545:
5543:
5542:
5537:
5532:
5531:
5530:
5520:
5515:
5513:Casimir effect
5510:
5504:
5502:
5498:
5497:
5495:
5494:
5489:
5484:
5479:
5474:
5472:Quantum optics
5469:
5464:
5459:
5454:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5393:
5392:
5382:
5377:
5372:
5367:
5366:
5365:
5355:
5350:
5345:
5339:
5337:
5331:
5330:
5328:
5327:
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5282:
5277:
5271:
5269:
5263:
5262:
5260:
5259:
5254:
5249:
5247:Quantum eraser
5244:
5239:
5234:
5229:
5224:
5219:
5214:
5209:
5203:
5201:
5197:
5196:
5194:
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5162:
5161:
5160:
5159:
5144:
5139:
5134:
5129:
5124:
5118:
5116:
5110:
5109:
5107:
5106:
5101:
5096:
5091:
5086:
5081:
5076:
5071:
5065:
5063:
5059:
5058:
5056:
5055:
5050:
5045:
5040:
5035:
5030:
5025:
5019:
5017:
5013:
5012:
5010:
5009:
5008:
5007:
5002:
4992:
4987:
4982:
4977:
4972:
4967:
4962:
4957:
4952:
4947:
4942:
4937:
4932:
4931:
4930:
4925:
4920:
4915:
4905:
4903:Density matrix
4900:
4895:
4890:
4884:
4882:
4878:
4877:
4875:
4874:
4869:
4864:
4859:
4858:
4857:
4847:
4841:
4839:
4835:
4834:
4827:
4826:
4819:
4812:
4804:
4795:
4794:
4774:
4756:
4747:
4739:Pergamon Press
4723:
4706:
4699:Albert Messiah
4696:
4690:
4672:
4669:
4666:
4665:
4658:
4640:
4633:
4606:
4605:
4602:
4601:
4576:
4575:
4573:
4570:
4569:
4568:
4563:
4558:
4553:
4548:
4543:
4538:
4531:
4528:
4525:
4524:
4511:
4507:
4503:
4494:
4490:
4487:
4483:
4479:
4476:
4472:
4468:
4465:
4462:
4456:
4451:
4445:
4441:
4437:
4428:
4424:
4421:
4417:
4413:
4409:
4405:
4402:
4399:
4396:
4390:
4385:
4374:
4371:
4358:
4354:
4350:
4341:
4336:
4332:
4328:
4324:
4321:
4318:
4312:
4307:
4301:
4297:
4293:
4284:
4279:
4275:
4272:
4268:
4264:
4261:
4258:
4255:
4249:
4244:
4233:
4231:Density matrix
4227:
4226:
4213:
4209:
4205:
4196:
4192:
4189:
4185:
4181:
4178:
4174:
4167:
4162:
4156:
4152:
4148:
4139:
4135:
4132:
4128:
4124:
4120:
4116:
4113:
4110:
4107:
4101:
4096:
4085:
4072:
4068:
4064:
4055:
4050:
4046:
4043:
4039:
4032:
4027:
4021:
4017:
4013:
4004:
3999:
3995:
3991:
3987:
3984:
3981:
3978:
3972:
3967:
3956:
3953:
3947:
3946:
3935:
3932:
3929:
3926:
3920:
3915:
3910:
3904:
3900:
3896:
3887:
3883:
3880:
3876:
3872:
3868:
3864:
3861:
3858:
3855:
3852:
3846:
3841:
3836:
3825:
3822:
3811:
3808:
3805:
3802:
3796:
3791:
3786:
3780:
3776:
3772:
3763:
3758:
3754:
3751:
3747:
3743:
3740:
3737:
3734:
3731:
3725:
3720:
3715:
3704:
3698:
3697:
3691:
3685:
3678:
3677:
3674:
3673:
3666:
3659:
3651:
3645:
3644:Evolution of:
3635:
3628:
3621:
3618:
3585:
3581:
3576:
3573:
3569:
3565:
3561:
3558:
3554:
3551:
3546:
3541:
3537:
3531:
3528:
3523:
3519:
3515:
3512:
3508:
3504:
3501:
3498:
3495:
3492:
3470:
3466:
3461:
3458:
3454:
3450:
3446:
3443:
3439:
3436:
3431:
3426:
3422:
3416:
3413:
3408:
3404:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3353:
3350:
3347:
3344:
3341:
3338:
3334:
3328:
3324:
3320:
3317:
3314:
3311:
3307:
3303:
3300:
3297:
3294:
3291:
3288:
3284:
3255:
3252:
3249:
3246:
3243:
3239:
3216:
3213:
3210:
3207:
3204:
3201:
3197:
3191:
3187:
3183:
3180:
3177:
3174:
3170:
3166:
3163:
3160:
3157:
3154:
3151:
3147:
3124:
3121:
3118:
3113:
3108:
3103:
3099:
3096:
3090:
3083:
3080:
3075:
3070:
3066:
3063:
3060:
3054:
3051:
3048:
3043:
3039:
3035:
3032:
3029:
3026:
3022:
2991:
2986:
2982:
2978:
2975:
2972:
2969:
2965:
2961:
2958:
2955:
2952:
2949:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2891:
2888:
2884:
2879:
2876:
2849:
2846:
2843:
2840:
2837:
2833:
2810:
2807:
2804:
2801:
2798:
2795:
2791:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2753:
2749:
2746:
2743:
2740:
2734:
2731:
2727:
2722:
2719:
2699:
2696:
2693:
2690:
2687:
2683:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2648:
2615:
2612:
2609:
2606:
2603:
2600:
2596:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2570:
2563:
2560:
2556:
2551:
2548:
2522:
2514:
2507:
2504:
2503:
2502:
2491:
2488:
2483:
2479:
2475:
2470:
2466:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2437:
2434:
2431:
2428:
2423:
2419:
2415:
2412:
2409:
2406:
2390:
2383:
2376:
2365:
2359:
2354:
2343:
2340:
2337:
2332:
2328:
2324:
2321:
2317:
2313:
2308:
2304:
2300:
2295:
2291:
2287:
2284:
2281:
2278:
2275:
2270:
2266:
2262:
2259:
2255:
2234:
2224:
2219:
2208:
2205:
2202:
2199:
2194:
2190:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2161:
2158:
2155:
2150:
2146:
2125:
2122:
2119:
2114:
2110:
2106:
2103:
2099:
2095:
2090:
2086:
2082:
2079:
2076:
2073:
2070:
2067:
2062:
2058:
2054:
2051:
2047:
2043:
2038:
2034:
2030:
2027:
2024:
2021:
2018:
2013:
2009:
2005:
2002:
1999:
1994:
1990:
1985:
1981:
1976:
1972:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1940:
1936:
1933:
1930:
1927:
1924:
1905:
1898:
1895:
1883:
1880:
1875:
1871:
1867:
1864:
1861:
1856:
1852:
1847:
1843:
1838:
1834:
1830:
1827:
1824:
1821:
1817:
1813:
1810:
1807:
1804:
1801:
1775:
1772:
1769:
1764:
1760:
1756:
1753:
1749:
1745:
1740:
1736:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1701:
1684:
1677:
1662:
1659:
1657:
1654:
1638:
1635:
1612:
1609:
1605:
1584:
1581:
1577:
1570:
1567:
1560:
1556:
1553:
1540:for which the
1529:
1526:
1522:
1490:
1486:
1483:
1478:
1457:
1454:
1450:
1419:
1416:
1412:
1371:
1368:
1355:
1352:
1349:
1344:
1340:
1317:
1316:
1305:
1300:
1296:
1292:
1287:
1283:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1257:
1253:
1250:
1245:
1241:
1237:
1234:
1231:
1228:
1207:
1206:
1195:
1192:
1189:
1184:
1180:
1176:
1173:
1169:
1165:
1160:
1156:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1121:
1108:, and one has
1097:
1092:
1088:
1084:
1081:
1078:
1075:
1051:
1048:
1045:
1042:
1039:
1035:
1022:
1007:
1004:
999:
995:
991:
988:
984:
932:
895:
894:
892:
891:
884:
877:
869:
866:
865:
861:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
629:
628:
625:
624:
621:
620:
617:
616:
611:
606:
601:
599:Density matrix
596:
591:
586:
581:
576:
571:
565:
562:
561:
558:
557:
553:
552:
547:
542:
537:
532:
527:
522:
521:
520:
519:
518:
503:
498:
493:
488:
483:
477:
476:
471:
470:
467:
466:
462:
461:
456:
451:
446:
441:
435:
434:
431:
430:
427:
426:
422:
421:
416:
411:
406:
401:
396:
390:
389:
388:
382:
379:
378:
375:
374:
370:
369:
364:
359:
353:
352:
351:
350:
349:
347:Delayed-choice
342:Quantum eraser
337:
336:
331:
326:
321:
316:
311:
306:
301:
296:
291:
285:
284:
281:
280:
277:
276:
272:
271:
270:
269:
259:
254:
249:
244:
239:
234:
232:Quantum number
229:
224:
219:
214:
209:
204:
198:
197:
194:
193:
190:
189:
185:
184:
179:
173:
172:
171:
166:
161:
155:
152:
151:
148:
147:
146:
145:
140:
135:
127:
126:
121:
110:
107:
103:
96:
93:
87:
84:
81:
77:
70:
67:
63:
58:
55:
44:
43:
37:
36:
26:
9:
6:
4:
3:
2:
5634:
5623:
5620:
5618:
5615:
5614:
5612:
5597:
5589:
5588:
5585:
5579:
5576:
5574:
5571:
5569:
5566:
5562:
5559:
5558:
5557:
5554:
5553:
5551:
5547:
5541:
5538:
5536:
5533:
5529:
5526:
5525:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5505:
5503:
5499:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5432:Quantum logic
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5391:
5388:
5387:
5386:
5383:
5381:
5378:
5376:
5373:
5371:
5368:
5364:
5361:
5360:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5340:
5338:
5336:
5332:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5285:Quantum chaos
5283:
5281:
5278:
5276:
5273:
5272:
5270:
5268:
5264:
5258:
5255:
5253:
5252:Stern–Gerlach
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5204:
5202:
5198:
5192:
5189:
5187:
5186:Transactional
5184:
5182:
5179:
5177:
5176:Quantum logic
5174:
5172:
5169:
5167:
5164:
5158:
5155:
5154:
5153:
5150:
5149:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5119:
5117:
5115:
5111:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5066:
5064:
5060:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5029:
5026:
5024:
5021:
5020:
5018:
5014:
5006:
5003:
5001:
4998:
4997:
4996:
4995:Wave function
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4970:Superposition
4968:
4966:
4965:Quantum state
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4929:
4926:
4924:
4921:
4919:
4918:Excited state
4916:
4914:
4911:
4910:
4909:
4906:
4904:
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4885:
4883:
4879:
4873:
4870:
4868:
4865:
4863:
4860:
4856:
4853:
4852:
4851:
4848:
4846:
4843:
4842:
4840:
4836:
4832:
4825:
4820:
4818:
4813:
4811:
4806:
4805:
4802:
4798:
4792:
4788:
4784:
4783:
4778:
4777:J. J. Sakurai
4775:
4772:
4768:
4764:
4760:
4757:
4755:
4750:
4744:
4740:
4736:
4732:
4731:E.M. Lifshitz
4728:
4724:
4722:
4721:0-471-88702-1
4718:
4714:
4710:
4709:Merzbacher E.
4707:
4704:
4700:
4697:
4693:
4691:0-471-16433-X
4687:
4683:
4679:
4675:
4674:
4661:
4655:
4651:
4644:
4636:
4634:0-07-051400-3
4630:
4626:
4621:
4620:
4611:
4607:
4598:
4594:
4588:
4581:
4577:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4533:
4505:
4501:
4488:
4485:
4481:
4477:
4474:
4470:
4463:
4449:
4439:
4435:
4422:
4419:
4415:
4411:
4407:
4403:
4397:
4383:
4375:
4372:
4352:
4348:
4334:
4330:
4326:
4319:
4305:
4295:
4291:
4277:
4273:
4270:
4266:
4262:
4256:
4242:
4234:
4232:
4229:
4228:
4207:
4203:
4190:
4187:
4183:
4179:
4176:
4172:
4160:
4150:
4146:
4133:
4130:
4126:
4122:
4118:
4114:
4108:
4094:
4086:
4066:
4062:
4048:
4044:
4041:
4037:
4025:
4015:
4011:
3997:
3993:
3989:
3985:
3979:
3965:
3957:
3954:
3952:
3949:
3948:
3927:
3913:
3898:
3894:
3881:
3878:
3874:
3870:
3866:
3862:
3853:
3839:
3826:
3823:
3803:
3789:
3774:
3770:
3756:
3752:
3749:
3745:
3741:
3732:
3718:
3705:
3703:
3700:
3699:
3695:
3692:
3689:
3686:
3683:
3680:
3679:
3672:
3667:
3665:
3660:
3658:
3653:
3652:
3649:
3642:
3639:
3634:
3627:
3617:
3615:
3609:
3607:
3606:Freeman Dyson
3603:
3599:
3598:time-ordering
3583:
3579:
3574:
3571:
3567:
3559:
3556:
3549:
3544:
3539:
3535:
3526:
3521:
3517:
3513:
3510:
3502:
3496:
3490:
3481:
3468:
3464:
3459:
3456:
3452:
3444:
3441:
3434:
3429:
3424:
3420:
3411:
3406:
3402:
3398:
3395:
3392:
3386:
3380:
3371:
3369:
3364:
3351:
3342:
3336:
3322:
3318:
3315:
3312:
3309:
3305:
3301:
3292:
3286:
3273:
3269:
3247:
3241:
3227:
3214:
3205:
3199:
3185:
3181:
3178:
3175:
3172:
3168:
3164:
3155:
3149:
3135:
3122:
3119:
3116:
3111:
3106:
3097:
3094:
3088:
3081:
3078:
3073:
3064:
3061:
3058:
3052:
3049:
3046:
3037:
3033:
3030:
3027:
3024:
3020:
3011:
3010:Taylor series
3007:
3002:
2989:
2980:
2976:
2973:
2970:
2967:
2963:
2959:
2953:
2947:
2938:
2925:
2919:
2913:
2910:
2907:
2901:
2895:
2889:
2874:
2864:
2841:
2835:
2821:
2808:
2799:
2793:
2782:
2776:
2773:
2770:
2761:
2755:
2744:
2738:
2732:
2717:
2691:
2685:
2674:
2668:
2665:
2656:
2650:
2637:
2633:
2629:
2613:
2604:
2598:
2590:
2587:
2578:
2572:
2561:
2546:
2538:
2534:
2530:
2521:
2513:
2489:
2481:
2477:
2473:
2468:
2464:
2457:
2449:
2445:
2441:
2438:
2432:
2429:
2421:
2417:
2413:
2410:
2404:
2397:. Therefore,
2396:
2389:
2382:
2375:
2371:
2364:
2360:
2358:
2355:
2341:
2330:
2326:
2319:
2306:
2302:
2298:
2293:
2289:
2282:
2279:
2268:
2264:
2257:
2244:
2240:
2233:
2229:
2225:
2223:
2220:
2206:
2203:
2200:
2192:
2188:
2184:
2181:
2175:
2167:
2163:
2159:
2156:
2148:
2144:
2123:
2112:
2108:
2101:
2088:
2084:
2077:
2071:
2060:
2056:
2049:
2036:
2032:
2028:
2025:
2019:
2011:
2007:
2003:
2000:
1992:
1988:
1974:
1970:
1963:
1957:
1948:
1942:
1931:
1925:
1914:
1910:
1906:
1904:
1901:
1900:
1894:
1881:
1873:
1869:
1865:
1862:
1854:
1850:
1836:
1832:
1825:
1819:
1808:
1802:
1791:
1786:
1773:
1762:
1758:
1751:
1738:
1734:
1730:
1727:
1721:
1718:
1709:
1703:
1690:
1683:
1676:
1672:
1668:
1653:
1633:
1607:
1579:
1565:
1554:
1543:
1524:
1511:
1507:
1502:
1484:
1481:
1452:
1439:
1435:
1434:
1433:Hilbert space
1414:
1401:
1400:
1393:
1389:
1385:
1381:
1377:
1367:
1353:
1350:
1347:
1342:
1324:
1322:
1321:Taylor series
1303:
1294:
1285:
1281:
1277:
1274:
1268:
1265:
1262:
1259:
1255:
1251:
1243:
1239:
1235:
1232:
1226:
1219:
1218:
1217:
1212:
1193:
1182:
1178:
1171:
1158:
1154:
1150:
1147:
1141:
1138:
1129:
1123:
1111:
1110:
1109:
1090:
1086:
1082:
1079:
1073:
1043:
1037:
997:
993:
986:
973:
969:
965:
960:
958:
954:
950:
946:
930:
922:
921:state vectors
919:in which the
918:
914:
910:
906:
902:
890:
885:
883:
878:
876:
871:
870:
868:
867:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
630:
623:
622:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
589:Quantum chaos
587:
585:
582:
580:
577:
575:
572:
570:
567:
566:
560:
559:
551:
548:
546:
545:Transactional
543:
541:
538:
536:
535:Quantum logic
533:
531:
528:
526:
523:
517:
514:
513:
512:
509:
508:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
478:
474:
469:
468:
460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
436:
429:
428:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
391:
387:
384:
383:
377:
376:
368:
365:
363:
360:
358:
355:
354:
348:
345:
344:
343:
340:
339:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
286:
279:
278:
268:
265:
264:
263:
262:Wave function
260:
258:
255:
253:
250:
248:
245:
243:
242:Superposition
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
199:
192:
191:
183:
180:
178:
175:
174:
170:
167:
165:
162:
160:
157:
156:
150:
149:
144:
141:
139:
136:
134:
131:
130:
129:
128:
124:
91:
85:
68:
65:
61:
53:
46:
45:
42:
39:
38:
34:
33:
30:
19:
5315:Quantum mind
5227:Franck–Hertz
5069:Klein–Gordon
5042:
5023:Formulations
5016:Formulations
4945:Interference
4935:Entanglement
4913:Ground state
4908:Energy level
4881:Fundamentals
4845:Introduction
4796:
4780:
4762:
4734:
4712:
4702:
4681:
4649:
4643:
4618:
4610:
4596:
4592:
4586:
4580:
3681:
3632:
3625:
3623:
3610:
3602:Dyson series
3482:
3372:
3367:
3365:
3271:
3228:
3136:
3005:
3003:
2939:
2862:
2822:
2635:
2627:
2532:
2528:
2519:
2511:
2510:We drop the
2509:
2394:
2387:
2380:
2373:
2369:
2362:
2356:
2238:
2231:
2227:
2221:
1902:
1787:
1688:
1681:
1674:
1670:
1666:
1664:
1505:
1503:
1431:
1398:
1391:
1387:
1383:
1380:wavefunction
1373:
1325:
1318:
1208:
961:
908:
904:
898:
444:Klein–Gordon
413:
380:Formulations
217:Energy level
212:Entanglement
195:Fundamentals
182:Interference
133:Introduction
29:
5573:EPR paradox
5353:Quantum bus
5222:Double-slit
5200:Experiments
5166:Many-worlds
5104:Schrödinger
5053:Phase space
5043:Schrödinger
5033:Interaction
4990:Uncertainty
4960:Nonlocality
4955:Measurement
4950:Decoherence
4940:Hamiltonian
4754:Online copy
4727:L.D. Landau
3694:Interaction
3682:Schrödinger
3596:where T is
3137:Therefore,
2632:Hamiltonian
2136:Therefore,
1330:; that is,
1211:Hamiltonian
964:Schrödinger
913:formulation
833:von Neumann
818:Schrödinger
594:EPR paradox
525:Many-worlds
459:Schrödinger
414:Schrödinger
409:Phase-space
399:Interaction
304:Double-slit
282:Experiments
257:Uncertainty
227:Nonlocality
222:Measurement
207:Decoherence
177:Hamiltonian
5611:Categories
5501:Extensions
5335:Technology
5181:Relational
5132:Copenhagen
5028:Heisenberg
4975:Tunnelling
4838:Background
4759:R. Shankar
4671:References
3951:Observable
3688:Heisenberg
3268:eigenstate
3229:Note that
1911:. For the
1897:Properties
1661:Definition
1370:Background
828:Sommerfeld
743:Heisenberg
738:Gutzwiller
678:de Broglie
626:Scientists
540:Relational
491:Copenhagen
394:Heisenberg
252:Tunnelling
153:Background
5207:Bell test
5062:Equations
4888:Born rule
4625:786, 1261
4510:ℏ
4475:−
4450:ρ
4444:ℏ
4384:ρ
4373:constant
4357:ℏ
4306:ρ
4300:ℏ
4271:−
4243:ρ
4212:ℏ
4177:−
4155:ℏ
4071:ℏ
4042:−
4020:ℏ
3955:constant
3934:⟩
3914:ψ
3903:ℏ
3860:⟩
3840:ψ
3824:constant
3810:⟩
3790:ψ
3779:ℏ
3750:−
3739:⟩
3719:ψ
3702:Ket state
3536:∫
3530:ℏ
3522:−
3514:
3421:∫
3415:ℏ
3407:−
3399:
3349:⟩
3337:ψ
3327:ℏ
3310:−
3299:⟩
3287:ψ
3254:⟩
3242:ψ
3212:⟩
3200:ψ
3190:ℏ
3173:−
3162:⟩
3150:ψ
3120:⋯
3102:ℏ
3074:−
3069:ℏ
3053:−
3042:ℏ
3025:−
2985:ℏ
2968:−
2887:∂
2883:∂
2878:ℏ
2848:⟩
2836:ψ
2806:⟩
2794:ψ
2768:⟩
2756:ψ
2730:∂
2726:∂
2721:ℏ
2698:⟩
2686:ψ
2663:⟩
2651:ψ
2638:to write
2611:⟩
2599:ψ
2585:⟩
2573:ψ
2559:∂
2555:∂
2550:ℏ
2339:⟩
2320:ψ
2277:⟩
2258:ψ
2149:†
2121:⟩
2102:ψ
2078:ψ
2075:⟨
2069:⟩
2050:ψ
1993:†
1964:ψ
1961:⟨
1955:⟩
1943:ψ
1926:ψ
1923:⟨
1903:Unitarity
1855:†
1826:ψ
1823:⟨
1803:ψ
1800:⟨
1771:⟩
1752:ψ
1716:⟩
1704:ψ
1637:^
1611:⟩
1608:ψ
1583:⟩
1580:ψ
1569:^
1555:ψ
1552:⟨
1528:⟩
1525:ψ
1489:⟩
1482:ψ
1456:⟩
1453:ψ
1418:⟩
1415:ψ
1339:∂
1299:ℏ
1278:−
1269:⋅
1260:−
1191:⟩
1172:ψ
1136:⟩
1124:ψ
1050:⟩
1038:ψ
1006:⟩
987:ψ
858:Zeilinger
703:Ehrenfest
432:Equations
109:⟩
106:Ψ
95:^
83:⟩
80:Ψ
57:ℏ
5596:Category
5390:Timeline
5142:Ensemble
5122:Bayesian
5084:Majorana
5000:Collapse
4872:Glossary
4855:Timeline
4779:(1993);
4761:(1994);
4733:(1977).
4701:, 1966.
4530:See also
3631:, where
3604:, after
3575:′
3560:′
3460:′
3445:′
2245:, since
2230: =
2222:Identity
1485:′
1438:operator
1062:at time
1018:at time
783:Millikan
708:Einstein
693:Davisson
648:Blackett
633:Aharonov
501:Ensemble
481:Bayesian
386:Overview
267:Collapse
247:Symmetry
138:Glossary
5549:Related
5528:History
5267:Science
5099:Rydberg
4850:History
3648:Picture
2630:is the
2535:). The
2357:Closure
2241:is the
1909:unitary
962:In the
901:physics
823:Simmons
813:Rydberg
778:Moseley
758:Kramers
748:Hilbert
733:Glauber
728:Feynman
713:Everett
683:Compton
454:Rydberg
143:History
5242:Popper
4789:
4769:
4745:
4719:
4688:
4656:
4631:
4499:
4433:
4346:
4289:
4201:
4144:
4060:
4009:
3892:
3768:
3004:Since
2823:Since
2626:where
970:, the
903:, the
853:Zeeman
848:Wigner
798:Planck
768:Landau
753:Jordan
404:Matrix
334:Popper
5152:Local
5094:Pauli
5074:Dirac
4572:Notes
2226:When
1376:state
911:is a
808:Raman
793:Pauli
788:Onnes
723:Fermi
698:Debye
688:Dirac
653:Bloch
643:Bethe
511:Local
449:Pauli
439:Dirac
237:State
5079:Weyl
4787:ISBN
4767:ISBN
4743:ISBN
4717:ISBN
4686:ISBN
4654:ISBN
4629:ISBN
4556:POVM
3696:(I)
3690:(H)
3684:(S)
1913:norm
1790:bras
1788:For
1506:must
955:and
843:Wien
838:Weyl
803:Rabi
773:Laue
763:Lamb
718:Fock
673:Bose
668:Born
663:Bohr
658:Bohm
638:Bell
4591:,
4589:= 0
4584:At
3636:0,S
3511:exp
3396:exp
2865:= 0
2539:is
2525:= 0
2368:to
1792:,
1399:ket
915:of
907:or
899:In
5613::
4741:.
4729:,
4711:,
4627:.
3676:)
3616:.
3608:.
3274::
3012::
2710:,
2237:,
1691::
1673:,
1501:.
1402:,
1390:,
1366:.
1323:.
4823:e
4816:t
4809:v
4793:.
4773:.
4751:.
4694:.
4662:.
4637:.
4597:t
4595:(
4593:U
4587:t
4506:/
4502:t
4493:S
4489:,
4486:0
4482:H
4478:i
4471:e
4467:)
4464:t
4461:(
4455:S
4440:/
4436:t
4427:S
4423:,
4420:0
4416:H
4412:i
4408:e
4404:=
4401:)
4398:t
4395:(
4389:I
4353:/
4349:t
4340:S
4335:H
4331:i
4327:e
4323:)
4320:0
4317:(
4311:S
4296:/
4292:t
4283:S
4278:H
4274:i
4267:e
4263:=
4260:)
4257:t
4254:(
4248:S
4208:/
4204:t
4195:S
4191:,
4188:0
4184:H
4180:i
4173:e
4166:S
4161:A
4151:/
4147:t
4138:S
4134:,
4131:0
4127:H
4123:i
4119:e
4115:=
4112:)
4109:t
4106:(
4100:I
4095:A
4067:/
4063:t
4054:S
4049:H
4045:i
4038:e
4031:S
4026:A
4016:/
4012:t
4003:S
3998:H
3994:i
3990:e
3986:=
3983:)
3980:t
3977:(
3971:H
3966:A
3931:)
3928:t
3925:(
3919:S
3909:|
3899:/
3895:t
3886:S
3882:,
3879:0
3875:H
3871:i
3867:e
3863:=
3857:)
3854:t
3851:(
3845:I
3835:|
3807:)
3804:0
3801:(
3795:S
3785:|
3775:/
3771:t
3762:S
3757:H
3753:i
3746:e
3742:=
3736:)
3733:t
3730:(
3724:S
3714:|
3670:e
3663:t
3656:v
3650:(
3633:H
3629:S
3626:H
3584:,
3580:)
3572:t
3568:d
3564:)
3557:t
3553:(
3550:H
3545:t
3540:0
3527:i
3518:(
3507:T
3503:=
3500:)
3497:t
3494:(
3491:U
3469:,
3465:)
3457:t
3453:d
3449:)
3442:t
3438:(
3435:H
3430:t
3425:0
3412:i
3403:(
3393:=
3390:)
3387:t
3384:(
3381:U
3352:.
3346:)
3343:0
3340:(
3333:|
3323:/
3319:t
3316:E
3313:i
3306:e
3302:=
3296:)
3293:t
3290:(
3283:|
3272:E
3251:)
3248:0
3245:(
3238:|
3215:.
3209:)
3206:0
3203:(
3196:|
3186:/
3182:t
3179:H
3176:i
3169:e
3165:=
3159:)
3156:t
3153:(
3146:|
3123:.
3117:+
3112:2
3107:)
3098:t
3095:H
3089:(
3082:2
3079:1
3065:t
3062:H
3059:i
3050:1
3047:=
3038:/
3034:t
3031:H
3028:i
3021:e
3006:H
2990:.
2981:/
2977:t
2974:H
2971:i
2964:e
2960:=
2957:)
2954:t
2951:(
2948:U
2926:.
2923:)
2920:t
2917:(
2914:U
2911:H
2908:=
2905:)
2902:t
2899:(
2896:U
2890:t
2875:i
2863:t
2845:)
2842:0
2839:(
2832:|
2809:.
2803:)
2800:0
2797:(
2790:|
2786:)
2783:t
2780:(
2777:U
2774:H
2771:=
2765:)
2762:0
2759:(
2752:|
2748:)
2745:t
2742:(
2739:U
2733:t
2718:i
2695:)
2692:0
2689:(
2682:|
2678:)
2675:t
2672:(
2669:U
2666:=
2660:)
2657:t
2654:(
2647:|
2636:U
2628:H
2614:,
2608:)
2605:t
2602:(
2595:|
2591:H
2588:=
2582:)
2579:t
2576:(
2569:|
2562:t
2547:i
2533:t
2531:(
2529:U
2523:0
2520:t
2515:0
2512:t
2490:.
2487:)
2482:0
2478:t
2474:,
2469:1
2465:t
2461:(
2458:U
2455:)
2450:1
2446:t
2442:,
2439:t
2436:(
2433:U
2430:=
2427:)
2422:0
2418:t
2414:,
2411:t
2408:(
2405:U
2395:t
2391:1
2388:t
2384:1
2381:t
2377:0
2374:t
2370:t
2366:0
2363:t
2342:.
2336:)
2331:0
2327:t
2323:(
2316:|
2312:)
2307:0
2303:t
2299:,
2294:0
2290:t
2286:(
2283:U
2280:=
2274:)
2269:0
2265:t
2261:(
2254:|
2239:U
2235:0
2232:t
2228:t
2207:.
2204:I
2201:=
2198:)
2193:0
2189:t
2185:,
2182:t
2179:(
2176:U
2173:)
2168:0
2164:t
2160:,
2157:t
2154:(
2145:U
2124:.
2118:)
2113:0
2109:t
2105:(
2098:|
2094:)
2089:0
2085:t
2081:(
2072:=
2066:)
2061:0
2057:t
2053:(
2046:|
2042:)
2037:0
2033:t
2029:,
2026:t
2023:(
2020:U
2017:)
2012:0
2008:t
2004:,
2001:t
1998:(
1989:U
1984:|
1980:)
1975:0
1971:t
1967:(
1958:=
1952:)
1949:t
1946:(
1939:|
1935:)
1932:t
1929:(
1882:.
1879:)
1874:0
1870:t
1866:,
1863:t
1860:(
1851:U
1846:|
1842:)
1837:0
1833:t
1829:(
1820:=
1816:|
1812:)
1809:t
1806:(
1774:.
1768:)
1763:0
1759:t
1755:(
1748:|
1744:)
1739:0
1735:t
1731:,
1728:t
1725:(
1722:U
1719:=
1713:)
1710:t
1707:(
1700:|
1689:t
1685:0
1682:t
1678:0
1675:t
1671:t
1669:(
1667:U
1634:p
1604:|
1576:|
1566:p
1559:|
1521:|
1477:|
1449:|
1411:|
1394:)
1392:t
1388:x
1386:(
1384:ψ
1354:0
1351:=
1348:H
1343:t
1328:H
1304:,
1295:/
1291:)
1286:0
1282:t
1275:t
1272:(
1266:H
1263:i
1256:e
1252:=
1249:)
1244:0
1240:t
1236:,
1233:t
1230:(
1227:U
1214:H
1194:.
1188:)
1183:0
1179:t
1175:(
1168:|
1164:)
1159:0
1155:t
1151:,
1148:t
1145:(
1142:U
1139:=
1133:)
1130:t
1127:(
1120:|
1096:)
1091:0
1087:t
1083:,
1080:t
1077:(
1074:U
1064:t
1047:)
1044:t
1041:(
1034:|
1023:0
1020:t
1003:)
998:0
994:t
990:(
983:|
931:V
888:e
881:t
874:v
102:|
92:H
86:=
76:|
69:t
66:d
62:d
54:i
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.