5581:
2123:
3600:
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
4511:
3122:
1907:
4213:
3583:
4358:
3933:
3809:
2808:
3468:
4072:
2613:
1881:
108:
2341:
3351:
3214:
1773:
1193:
2925:
3004:
1303:
2697:
1582:
2489:
2206:
4367:
1493:
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
3475:
2702:
2989:
1005:
1488:
4078:
2531:
3253:
2847:
1049:
1353:
4226:
3818:
1610:
1527:
1455:
1417:
3365:
1639:
1095:
3697:
912:
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
2118:{\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 .}
1641:, 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.
3949:
1784:
3657:
2237:
930:
3266:
3129:
1683:
4810:
2859:
3472:
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
38:
5378:
5011:
4549:
3362:
If the
Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as
374:
2389:
3650:
875:
2128:
1103:
5549:
2856:), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation
4860:
126:
5561:
1584:, oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector
582:
5245:
4803:
3643:
1211:
355:
5605:
5179:
4779:
4759:
4735:
4646:
2630:
1530:
538:
1536:
5102:
2932:
461:
4506:{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}
1426:
4833:
4796:
4709:
4678:
4621:
941:
121:
3117:{\displaystyle e^{-iHt/\hbar }=1-{\frac {iHt}{\hbar }}-{\frac {1}{2}}\left({\frac {Ht}{\hbar }}\right)^{2}+\cdots .}
940:
in which both the states and the observables evolve in time. The Schrödinger and
Heisenberg pictures are related as
4943:
210:
955:
picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a
5516:
5225:
5220:
4928:
2620:
1199:
868:
317:
297:
165:
5528:
5200:
557:
287:
5506:
5283:
5205:
5077:
4973:
4843:
597:
335:
235:
5610:
5240:
5174:
5169:
5140:
4838:
4524:
4208:{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}
966:
533:
528:
499:
350:
131:
5308:
5215:
1498:
567:
312:
302:
4353:{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}
5584:
5346:
5154:
5125:
3928:{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }
3221:
2815:
1017:
861:
513:
484:
1322:
5368:
5235:
5159:
5120:
5057:
5036:
4993:
4886:
1460:
518:
479:
432:
407:
330:
190:
4554:
1587:
1504:
1432:
1394:
5435:
5415:
5405:
5395:
5351:
4911:
4770:
602:
5130:
5041:
4539:
3804:{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }
3578:{\displaystyle U(t)=\mathrm {T} \exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right),}
771:
489:
397:
5544:
5092:
4694:(Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
4613:
4606:
2803:{\displaystyle i\hbar {\partial \over \partial t}U(t)|\psi (0)\rangle =HU(t)|\psi (0)\rangle .}
2525:
1615:
1058:
447:
345:
111:
5455:
5230:
5210:
5135:
4988:
4666:
776:
494:
322:
292:
255:
5480:
4978:
4958:
4067:{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}
245:
230:
5511:
5440:
5385:
5115:
4923:
4881:
2608:{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle =H|\psi (t)\rangle ,}
1386:
945:
909:
562:
474:
200:
157:
806:
8:
5496:
5465:
5410:
5390:
5298:
5255:
5110:
5021:
4948:
4938:
4850:
4529:
3682:
956:
937:
661:
469:
387:
215:
195:
147:
5556:
5425:
5323:
5016:
4963:
4855:
4534:
3676:
3636:
3602:
1901:
933:
915:
901:
382:
307:
240:
152:
1497:
be carried by some combination of the state vectors and the operators. For example, a
5566:
5475:
5445:
5373:
5336:
5331:
5313:
5278:
5268:
5072:
4968:
4933:
4916:
4819:
4775:
4755:
4747:
4731:
4705:
4674:
4642:
4617:
3463:{\displaystyle U(t)=\exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right),}
2231:
905:
816:
791:
731:
726:
626:
592:
572:
170:
29:
1425:, a vector space containing all possible states of the system. A quantum-mechanical
5293:
5288:
5145:
5026:
4742:
4697:
3690:
1897:
1778:
1315:
The Schrödinger picture is useful when dealing with a time-independent
Hamiltonian
936:
which keeps the states constant while the observables evolve in time, and from the
821:
811:
801:
701:
681:
666:
636:
504:
392:
5523:
5450:
5430:
5400:
5363:
5358:
5263:
5087:
4719:
2929:
If the
Hamiltonian is independent of time, the solution to the above equation is
846:
716:
696:
442:
282:
1205:
of the system does not vary with time, the time-evolution operator has the form
5501:
5470:
5460:
5082:
5062:
4891:
4727:
4687:
4219:
960:
781:
741:
721:
691:
671:
621:
587:
437:
427:
220:
952:
5599:
5420:
5273:
5164:
5067:
4983:
4953:
4906:
4765:
3594:
3586:
2998:
1876:{\displaystyle \langle \psi (t)|=\langle \psi (t_{0})|U^{\dagger }(t,t_{0}).}
1421:
1364:
1309:
841:
836:
766:
736:
706:
577:
523:
250:
225:
103:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
5303:
4901:
4896:
3590:
1368:
831:
826:
761:
746:
711:
205:
2336:{\displaystyle |\psi (t_{0})\rangle =U(t_{0},t_{0})|\psi (t_{0})\rangle .}
5341:
948:
between operators are preserved in the passage between the two pictures.
796:
751:
686:
641:
4715:
3939:
3256:
2997:
is an operator, this exponential expression is to be evaluated via its
786:
756:
676:
651:
646:
631:
3359:: they only pick up an overall phase factor as they evolve with time.
1385:. More abstractly, the state may be represented as a state vector, or
5195:
4876:
4665:
277:
4788:
2494:
4641:. Schuam's outline series (2nd ed.). McGraw Hill. p. 70.
3346:{\displaystyle |\psi (t)\rangle =e^{-iEt/\hbar }|\psi (0)\rangle .}
3209:{\displaystyle |\psi (t)\rangle =e^{-iHt/\hbar }|\psi (0)\rangle .}
656:
3608:
1768:{\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle .}
1367:
of a quantum-mechanical system is represented by a complex-valued
1188:{\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle .}
4636:
2920:{\displaystyle i\hbar {\frac {\partial }{\partial t}}U(t)=HU(t).}
889:
2506:
index in the time evolution operator with the convention that
4544:
1669:) is defined as the operator which acts on the ket at time
4714:
1298:{\displaystyle U(t,t_{0})=e^{-iH\cdot (t-t_{0})/\hbar },}
3255:
is an arbitrary ket. However, if the initial ket is an
2692:{\displaystyle |\psi (t)\rangle =U(t)|\psi (0)\rangle }
2361:
may be viewed as a two-step time evolution, first from
1577:{\displaystyle \langle \psi |{\hat {p}}|\psi \rangle }
4370:
4229:
4081:
3952:
3821:
3700:
3478:
3368:
3269:
3224:
3132:
3007:
2935:
2862:
2818:
2705:
2633:
2534:
2392:
2240:
2131:
1910:
1904:
of the state ket must not change with time. That is,
1787:
1686:
1618:
1590:
1539:
1507:
1463:
1435:
1397:
1325:
1214:
1106:
1061:
1020:
969:
918:
41:
2484:{\displaystyle U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}).}
4605:
4505:
4352:
4207:
4066:
3927:
3803:
3577:
3462:
3345:
3247:
3208:
3116:
2983:
2919:
2841:
2802:
2691:
2607:
2483:
2335:
2201:{\displaystyle U^{\dagger }(t,t_{0})U(t,t_{0})=I.}
2200:
2117:
1875:
1767:
1633:
1604:
1576:
1521:
1482:
1449:
1411:
1347:
1297:
1187:
1089:
1055:, the time-evolution operator is commonly written
1043:
999:
924:
102:
2495:Differential equation for time evolution operator
5597:
4637:Y. Peleg; R. Pnini; E. Zaarur; E. Hecht (2010).
3609:Summary comparison of evolution in all pictures
1644:
4804:
4550:Mathematical formulation of quantum mechanics
3651:
869:
3922:
3848:
3798:
3727:
3337:
3287:
3242:
3200:
3150:
2836:
2794:
2756:
2686:
2651:
2599:
2573:
2327:
2265:
2109:
2063:
2057:
1949:
1943:
1911:
1811:
1788:
1759:
1704:
1599:
1571:
1540:
1516:
1477:
1444:
1406:
1179:
1124:
1038:
994:
97:
71:
4811:
4797:
4724:Quantum Mechanics: Non-Relativistic Theory
3658:
3644:
3589:operator, which is sometimes known as the
876:
862:
4704:(3rd ed., John Wiley 1998) p. 430–1
3555:
3440:
963:. For time evolution from a state vector
2623:. Now using the time-evolution operator
1308:where the exponent is evaluated via its
4588:) must reduce to the identity operator.
3355:The eigenstates of the Hamiltonian are
5598:
4612:(2nd ed.). McGraw Hill. pp.
4603:
1676:to produce the ket at some other time
4818:
4792:
2984:{\displaystyle U(t)=e^{-iHt/\hbar }.}
1363:In elementary quantum mechanics, the
1000:{\displaystyle |\psi (t_{0})\rangle }
4608:McGraw Hill Encyclopaedia of Physics
3259:of the Hamiltonian, with eigenvalue
2849:is a constant ket (the state ket at
1896:The time evolution operator must be
4669:; Bernard Diu; Frank Laloe (1977).
3613:For a time-independent Hamiltonian
13:
4673:. Paris: Wiley. pp. 312–314.
4481:
4443:
4415:
4377:
4328:
4299:
4271:
4236:
4183:
4154:
4126:
4088:
4042:
4019:
3991:
3959:
3907:
3874:
3833:
3783:
3750:
3712:
3495:
2875:
2871:
2718:
2714:
2547:
2543:
1327:
942:active and passive transformations
408:Sum-over-histories (path integral)
94:
68:
24:Part of a series of articles about
14:
5622:
4498:
4432:
4345:
4288:
4200:
4143:
4059:
4008:
3891:
3767:
3518:
3403:
3315:
3248:{\displaystyle |\psi (0)\rangle }
3178:
3090:
3057:
3030:
2973:
2866:
2842:{\displaystyle |\psi (0)\rangle }
2709:
2538:
1429:is a function which takes a ket
1287:
1044:{\displaystyle |\psi (t)\rangle }
45:
5580:
5579:
1348:{\displaystyle \partial _{t}H=0}
932:changes). This differs from the
16:Formulation of quantum mechanics
4752:Principles of Quantum Mechanics
1483:{\displaystyle |\psi '\rangle }
5529:Relativistic quantum mechanics
4726:. Vol. 3 (3rd ed.).
4671:Quantum Mechanics (Volume One)
4630:
4597:
4567:
4455:
4449:
4389:
4383:
4311:
4305:
4248:
4242:
4100:
4094:
3971:
3965:
3919:
3913:
3897:
3845:
3839:
3823:
3795:
3789:
3773:
3724:
3718:
3702:
3552:
3541:
3488:
3482:
3437:
3426:
3378:
3372:
3334:
3328:
3321:
3284:
3278:
3271:
3239:
3233:
3226:
3197:
3191:
3184:
3147:
3141:
3134:
2945:
2939:
2911:
2905:
2893:
2887:
2833:
2827:
2820:
2791:
2785:
2778:
2774:
2768:
2753:
2747:
2740:
2736:
2730:
2683:
2677:
2670:
2666:
2660:
2648:
2642:
2635:
2596:
2590:
2583:
2570:
2564:
2557:
2475:
2449:
2443:
2424:
2415:
2396:
2324:
2311:
2304:
2300:
2274:
2262:
2249:
2242:
2186:
2167:
2161:
2142:
2106:
2093:
2086:
2082:
2069:
2054:
2041:
2034:
2030:
2011:
2005:
1986:
1972:
1968:
1955:
1940:
1934:
1927:
1923:
1917:
1867:
1848:
1834:
1830:
1817:
1804:
1800:
1794:
1756:
1743:
1736:
1732:
1713:
1701:
1695:
1688:
1625:
1605:{\displaystyle |\psi \rangle }
1592:
1564:
1557:
1547:
1522:{\displaystyle |\psi \rangle }
1509:
1465:
1450:{\displaystyle |\psi \rangle }
1437:
1419:. This ket is an element of a
1412:{\displaystyle |\psi \rangle }
1399:
1279:
1260:
1237:
1218:
1176:
1163:
1156:
1152:
1133:
1121:
1115:
1108:
1084:
1065:
1035:
1029:
1022:
991:
978:
971:
558:Relativistic quantum mechanics
90:
83:
64:
1:
5507:Quantum statistical mechanics
5284:Quantum differential calculus
5206:Delayed-choice quantum eraser
4974:Symmetry in quantum mechanics
4659:
1885:
1649:
1358:
598:Quantum statistical mechanics
5606:Foundational quantum physics
1654:The time-evolution operator
1457:and returns some other ket
7:
5309:Quantum stochastic calculus
5299:Quantum measurement problem
5221:Mach–Zehnder interferometer
4518:
1645:The time evolution operator
1499:quantum harmonic oscillator
568:Quantum information science
10:
5627:
1634:{\displaystyle {\hat {p}}}
1090:{\displaystyle U(t,t_{0})}
898:Schrödinger representation
5575:
5537:
5489:
5369:Quantum complexity theory
5347:Quantum cellular automata
5322:
5254:
5188:
5101:
5050:
5037:Path integral formulation
5004:
4869:
4826:
3635:
3632:
3627:is the free Hamiltonian,
5436:Quantum machine learning
5416:Quantum key distribution
5406:Quantum image processing
5396:Quantum error correction
5246:Wheeler's delayed choice
4771:Modern Quantum Mechanics
4560:
4525:Hamilton–Jacobi equation
2368:to an intermediate time
1612:, the momentum operator
603:Quantum machine learning
356:Wheeler's delayed-choice
5352:Quantum finite automata
4667:Cohen-Tannoudji, Claude
4540:Phase space formulation
961:time evolution operator
313:Leggett–Garg inequality
5456:Quantum neural network
4555:Schrödinger functional
4507:
4354:
4209:
4068:
3929:
3805:
3579:
3464:
3347:
3249:
3210:
3118:
2985:
2921:
2843:
2804:
2693:
2609:
2485:
2337:
2202:
2119:
1877:
1769:
1635:
1606:
1578:
1523:
1484:
1451:
1413:
1349:
1299:
1198:In the case where the
1189:
1091:
1045:
1001:
926:
104:
5481:Quantum teleportation
4994:Wave–particle duality
4604:Parker, C.B. (1994).
4508:
4355:
4210:
4069:
3930:
3806:
3580:
3465:
3348:
3250:
3211:
3119:
2986:
2922:
2844:
2805:
2694:
2610:
2486:
2338:
2203:
2120:
1878:
1770:
1636:
1607:
1579:
1524:
1485:
1452:
1414:
1350:
1300:
1190:
1092:
1046:
1002:
946:commutation relations
927:
298:Elitzur–Vaidman
288:Davisson–Germer
105:
5512:Quantum field theory
5441:Quantum metamaterial
5386:Quantum cryptography
5116:Consistent histories
4774:(Revised Edition),
4368:
4227:
4079:
3950:
3819:
3698:
3476:
3366:
3267:
3222:
3130:
3005:
2933:
2860:
2816:
2703:
2631:
2532:
2526:Schrödinger equation
2390:
2350:Time evolution from
2238:
2129:
1908:
1785:
1684:
1616:
1588:
1537:
1505:
1461:
1433:
1395:
1323:
1212:
1104:
1059:
1018:
967:
916:
563:Quantum field theory
475:Consistent histories
112:Schrödinger equation
39:
5497:Quantum fluctuation
5466:Quantum programming
5426:Quantum logic gates
5411:Quantum information
5391:Quantum electronics
4851:Classical mechanics
4530:Interaction picture
3537:
3422:
1501:may be in a state
938:interaction picture
894:Schrödinger picture
351:Stern–Gerlach
148:Classical mechanics
5550:in popular culture
5332:Quantum algorithms
5180:Von Neumann–Wigner
5160:Objective collapse
4856:Old quantum theory
4535:Heisenberg picture
4503:
4350:
4205:
4064:
3925:
3801:
3603:Heisenberg picture
3575:
3523:
3460:
3408:
3343:
3245:
3206:
3114:
2981:
2917:
2839:
2800:
2689:
2605:
2481:
2382:to the final time
2333:
2198:
2115:
1873:
1765:
1631:
1602:
1574:
1519:
1480:
1447:
1409:
1345:
1295:
1185:
1087:
1041:
1014:to a state vector
997:
934:Heisenberg picture
922:
539:Von Neumann–Wigner
519:Objective-collapse
318:Mach–Zehnder
308:Leggett inequality
303:Franck–Hertz
153:Old quantum theory
100:
5611:Erwin Schrödinger
5593:
5592:
5567:Quantum mysticism
5545:Schrödinger's cat
5476:Quantum simulator
5446:Quantum metrology
5374:Quantum computing
5337:Quantum amplifier
5314:Quantum spacetime
5279:Quantum cosmology
5269:Quantum chemistry
4969:Scattering theory
4917:Zero-point energy
4912:Degenerate levels
4820:Quantum mechanics
4780:978-0-201-53929-5
4760:978-0-306-44790-7
4737:978-0-08-020940-1
4702:Quantum Mechanics
4692:Quantum Mechanics
4648:978-0-07-162358-2
4639:Quantum mechanics
4516:
4515:
4489:
4423:
4336:
4279:
4191:
4134:
4050:
3999:
3882:
3758:
3521:
3406:
3357:stationary states
3093:
3073:
3060:
2882:
2725:
2554:
2232:identity operator
1628:
1560:
1533:of the momentum,
1531:expectation value
925:{\displaystyle V}
906:quantum mechanics
886:
885:
593:Scattering theory
573:Quantum computing
346:Schrödinger's cat
278:Bell's inequality
86:
61:
30:Quantum mechanics
5618:
5583:
5582:
5294:Quantum geometry
5289:Quantum dynamics
5146:Superdeterminism
5078:Rarita–Schwinger
5027:Matrix mechanics
4882:Bra–ket notation
4813:
4806:
4799:
4790:
4789:
4754:, Plenum Press,
4741:
4684:
4653:
4652:
4634:
4628:
4627:
4611:
4601:
4589:
4579:
4571:
4512:
4510:
4509:
4504:
4502:
4501:
4497:
4487:
4486:
4485:
4484:
4448:
4447:
4446:
4436:
4435:
4431:
4421:
4420:
4419:
4418:
4382:
4381:
4380:
4359:
4357:
4356:
4351:
4349:
4348:
4344:
4334:
4333:
4332:
4331:
4304:
4303:
4302:
4292:
4291:
4287:
4277:
4276:
4275:
4274:
4241:
4240:
4239:
4214:
4212:
4211:
4206:
4204:
4203:
4199:
4189:
4188:
4187:
4186:
4159:
4158:
4157:
4147:
4146:
4142:
4132:
4131:
4130:
4129:
4093:
4092:
4091:
4073:
4071:
4070:
4065:
4063:
4062:
4058:
4048:
4047:
4046:
4045:
4024:
4023:
4022:
4012:
4011:
4007:
3997:
3996:
3995:
3994:
3964:
3963:
3962:
3934:
3932:
3931:
3926:
3912:
3911:
3910:
3900:
3895:
3894:
3890:
3880:
3879:
3878:
3877:
3838:
3837:
3836:
3826:
3810:
3808:
3807:
3802:
3788:
3787:
3786:
3776:
3771:
3770:
3766:
3756:
3755:
3754:
3753:
3717:
3716:
3715:
3705:
3660:
3653:
3646:
3630:
3629:
3584:
3582:
3581:
3576:
3571:
3567:
3566:
3551:
3536:
3531:
3522:
3514:
3498:
3469:
3467:
3466:
3461:
3456:
3452:
3451:
3436:
3421:
3416:
3407:
3399:
3352:
3350:
3349:
3344:
3324:
3319:
3318:
3314:
3274:
3254:
3252:
3251:
3246:
3229:
3215:
3213:
3212:
3207:
3187:
3182:
3181:
3177:
3137:
3123:
3121:
3120:
3115:
3104:
3103:
3098:
3094:
3089:
3081:
3074:
3066:
3061:
3056:
3045:
3034:
3033:
3029:
2990:
2988:
2987:
2982:
2977:
2976:
2972:
2926:
2924:
2923:
2918:
2883:
2881:
2870:
2855:
2848:
2846:
2845:
2840:
2823:
2809:
2807:
2806:
2801:
2781:
2743:
2726:
2724:
2713:
2698:
2696:
2695:
2690:
2673:
2638:
2614:
2612:
2611:
2606:
2586:
2560:
2555:
2553:
2542:
2516:and write it as
2515:
2490:
2488:
2487:
2482:
2474:
2473:
2461:
2460:
2442:
2441:
2414:
2413:
2375:, and then from
2342:
2340:
2339:
2334:
2323:
2322:
2307:
2299:
2298:
2286:
2285:
2261:
2260:
2245:
2207:
2205:
2204:
2199:
2185:
2184:
2160:
2159:
2141:
2140:
2124:
2122:
2121:
2116:
2105:
2104:
2089:
2081:
2080:
2053:
2052:
2037:
2029:
2028:
2004:
2003:
1985:
1984:
1975:
1967:
1966:
1930:
1882:
1880:
1879:
1874:
1866:
1865:
1847:
1846:
1837:
1829:
1828:
1807:
1774:
1772:
1771:
1766:
1755:
1754:
1739:
1731:
1730:
1691:
1640:
1638:
1637:
1632:
1630:
1629:
1621:
1611:
1609:
1608:
1603:
1595:
1583:
1581:
1580:
1575:
1567:
1562:
1561:
1553:
1550:
1528:
1526:
1525:
1520:
1512:
1489:
1487:
1486:
1481:
1476:
1468:
1456:
1454:
1453:
1448:
1440:
1418:
1416:
1415:
1410:
1402:
1384:
1354:
1352:
1351:
1346:
1335:
1334:
1318:
1304:
1302:
1301:
1296:
1291:
1290:
1286:
1278:
1277:
1236:
1235:
1204:
1194:
1192:
1191:
1186:
1175:
1174:
1159:
1151:
1150:
1111:
1096:
1094:
1093:
1088:
1083:
1082:
1054:
1050:
1048:
1047:
1042:
1025:
1010:
1006:
1004:
1003:
998:
990:
989:
974:
957:unitary operator
931:
929:
928:
923:
878:
871:
864:
505:Superdeterminism
158:Bra–ket notation
109:
107:
106:
101:
93:
88:
87:
79:
67:
62:
60:
49:
21:
20:
5626:
5625:
5621:
5620:
5619:
5617:
5616:
5615:
5596:
5595:
5594:
5589:
5571:
5557:Wigner's friend
5533:
5524:Quantum gravity
5485:
5471:Quantum sensing
5451:Quantum network
5431:Quantum machine
5401:Quantum imaging
5364:Quantum circuit
5359:Quantum channel
5318:
5264:Quantum biology
5250:
5226:Elitzur–Vaidman
5201:Davisson–Germer
5184:
5136:Hidden-variable
5126:de Broglie–Bohm
5103:Interpretations
5097:
5046:
5000:
4887:Complementarity
4865:
4822:
4817:
4786:
4738:
4681:
4662:
4657:
4656:
4649:
4635:
4631:
4624:
4602:
4598:
4593:
4592:
4574:
4572:
4568:
4563:
4521:
4493:
4480:
4473:
4469:
4462:
4458:
4442:
4441:
4437:
4427:
4414:
4407:
4403:
4399:
4395:
4376:
4375:
4371:
4369:
4366:
4365:
4340:
4327:
4326:
4322:
4318:
4314:
4298:
4297:
4293:
4283:
4270:
4269:
4265:
4258:
4254:
4235:
4234:
4230:
4228:
4225:
4224:
4195:
4182:
4175:
4171:
4164:
4160:
4153:
4152:
4148:
4138:
4125:
4118:
4114:
4110:
4106:
4087:
4086:
4082:
4080:
4077:
4076:
4054:
4041:
4040:
4036:
4029:
4025:
4018:
4017:
4013:
4003:
3990:
3989:
3985:
3981:
3977:
3958:
3957:
3953:
3951:
3948:
3947:
3906:
3905:
3901:
3896:
3886:
3873:
3866:
3862:
3858:
3854:
3832:
3831:
3827:
3822:
3820:
3817:
3816:
3782:
3781:
3777:
3772:
3762:
3749:
3748:
3744:
3737:
3733:
3711:
3710:
3706:
3701:
3699:
3696:
3695:
3664:
3626:
3619:
3611:
3559:
3544:
3532:
3527:
3513:
3509:
3505:
3494:
3477:
3474:
3473:
3444:
3429:
3417:
3412:
3398:
3394:
3390:
3367:
3364:
3363:
3320:
3310:
3297:
3293:
3270:
3268:
3265:
3264:
3225:
3223:
3220:
3219:
3183:
3173:
3160:
3156:
3133:
3131:
3128:
3127:
3099:
3082:
3080:
3076:
3075:
3065:
3046:
3044:
3025:
3012:
3008:
3006:
3003:
3002:
2968:
2955:
2951:
2934:
2931:
2930:
2874:
2869:
2861:
2858:
2857:
2850:
2819:
2817:
2814:
2813:
2777:
2739:
2717:
2712:
2704:
2701:
2700:
2669:
2634:
2632:
2629:
2628:
2582:
2556:
2546:
2541:
2533:
2530:
2529:
2513:
2507:
2505:
2497:
2469:
2465:
2456:
2452:
2437:
2433:
2409:
2405:
2391:
2388:
2387:
2381:
2374:
2367:
2356:
2318:
2314:
2303:
2294:
2290:
2281:
2277:
2256:
2252:
2241:
2239:
2236:
2235:
2225:
2180:
2176:
2155:
2151:
2136:
2132:
2130:
2127:
2126:
2100:
2096:
2085:
2076:
2072:
2048:
2044:
2033:
2024:
2020:
1999:
1995:
1980:
1976:
1971:
1962:
1958:
1926:
1909:
1906:
1905:
1888:
1861:
1857:
1842:
1838:
1833:
1824:
1820:
1803:
1786:
1783:
1782:
1750:
1746:
1735:
1726:
1722:
1687:
1685:
1682:
1681:
1675:
1668:
1652:
1647:
1620:
1619:
1617:
1614:
1613:
1591:
1589:
1586:
1585:
1563:
1552:
1551:
1546:
1538:
1535:
1534:
1508:
1506:
1503:
1502:
1469:
1464:
1462:
1459:
1458:
1436:
1434:
1431:
1430:
1398:
1396:
1393:
1392:
1371:
1361:
1330:
1326:
1324:
1321:
1320:
1316:
1282:
1273:
1269:
1247:
1243:
1231:
1227:
1213:
1210:
1209:
1202:
1170:
1166:
1155:
1146:
1142:
1107:
1105:
1102:
1101:
1078:
1074:
1060:
1057:
1056:
1052:
1021:
1019:
1016:
1015:
1013:
1008:
985:
981:
970:
968:
965:
964:
917:
914:
913:
882:
853:
852:
851:
616:
608:
607:
553:
552:Advanced topics
545:
544:
543:
495:Hidden-variable
485:de Broglie–Bohm
464:
462:Interpretations
454:
453:
452:
422:
414:
413:
412:
370:
362:
361:
360:
327:
283:CHSH inequality
272:
264:
263:
262:
191:Complementarity
185:
177:
176:
175:
143:
114:
89:
78:
77:
63:
53:
48:
40:
37:
36:
17:
12:
11:
5:
5624:
5614:
5613:
5608:
5591:
5590:
5588:
5587:
5576:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5553:
5552:
5541:
5539:
5535:
5534:
5532:
5531:
5526:
5521:
5520:
5519:
5509:
5504:
5502:Casimir effect
5499:
5493:
5491:
5487:
5486:
5484:
5483:
5478:
5473:
5468:
5463:
5461:Quantum optics
5458:
5453:
5448:
5443:
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5403:
5398:
5393:
5388:
5383:
5382:
5381:
5371:
5366:
5361:
5356:
5355:
5354:
5344:
5339:
5334:
5328:
5326:
5320:
5319:
5317:
5316:
5311:
5306:
5301:
5296:
5291:
5286:
5281:
5276:
5271:
5266:
5260:
5258:
5252:
5251:
5249:
5248:
5243:
5238:
5236:Quantum eraser
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5198:
5192:
5190:
5186:
5185:
5183:
5182:
5177:
5172:
5167:
5162:
5157:
5152:
5151:
5150:
5149:
5148:
5133:
5128:
5123:
5118:
5113:
5107:
5105:
5099:
5098:
5096:
5095:
5090:
5085:
5080:
5075:
5070:
5065:
5060:
5054:
5052:
5048:
5047:
5045:
5044:
5039:
5034:
5029:
5024:
5019:
5014:
5008:
5006:
5002:
5001:
4999:
4998:
4997:
4996:
4991:
4981:
4976:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4920:
4919:
4914:
4909:
4904:
4894:
4892:Density matrix
4889:
4884:
4879:
4873:
4871:
4867:
4866:
4864:
4863:
4858:
4853:
4848:
4847:
4846:
4836:
4830:
4828:
4824:
4823:
4816:
4815:
4808:
4801:
4793:
4784:
4783:
4763:
4745:
4736:
4728:Pergamon Press
4712:
4695:
4688:Albert Messiah
4685:
4679:
4661:
4658:
4655:
4654:
4647:
4629:
4622:
4595:
4594:
4591:
4590:
4565:
4564:
4562:
4559:
4558:
4557:
4552:
4547:
4542:
4537:
4532:
4527:
4520:
4517:
4514:
4513:
4500:
4496:
4492:
4483:
4479:
4476:
4472:
4468:
4465:
4461:
4457:
4454:
4451:
4445:
4440:
4434:
4430:
4426:
4417:
4413:
4410:
4406:
4402:
4398:
4394:
4391:
4388:
4385:
4379:
4374:
4363:
4360:
4347:
4343:
4339:
4330:
4325:
4321:
4317:
4313:
4310:
4307:
4301:
4296:
4290:
4286:
4282:
4273:
4268:
4264:
4261:
4257:
4253:
4250:
4247:
4244:
4238:
4233:
4222:
4220:Density matrix
4216:
4215:
4202:
4198:
4194:
4185:
4181:
4178:
4174:
4170:
4167:
4163:
4156:
4151:
4145:
4141:
4137:
4128:
4124:
4121:
4117:
4113:
4109:
4105:
4102:
4099:
4096:
4090:
4085:
4074:
4061:
4057:
4053:
4044:
4039:
4035:
4032:
4028:
4021:
4016:
4010:
4006:
4002:
3993:
3988:
3984:
3980:
3976:
3973:
3970:
3967:
3961:
3956:
3945:
3942:
3936:
3935:
3924:
3921:
3918:
3915:
3909:
3904:
3899:
3893:
3889:
3885:
3876:
3872:
3869:
3865:
3861:
3857:
3853:
3850:
3847:
3844:
3841:
3835:
3830:
3825:
3814:
3811:
3800:
3797:
3794:
3791:
3785:
3780:
3775:
3769:
3765:
3761:
3752:
3747:
3743:
3740:
3736:
3732:
3729:
3726:
3723:
3720:
3714:
3709:
3704:
3693:
3687:
3686:
3680:
3674:
3667:
3666:
3663:
3662:
3655:
3648:
3640:
3634:
3633:Evolution of:
3624:
3617:
3610:
3607:
3574:
3570:
3565:
3562:
3558:
3554:
3550:
3547:
3543:
3540:
3535:
3530:
3526:
3520:
3517:
3512:
3508:
3504:
3501:
3497:
3493:
3490:
3487:
3484:
3481:
3459:
3455:
3450:
3447:
3443:
3439:
3435:
3432:
3428:
3425:
3420:
3415:
3411:
3405:
3402:
3397:
3393:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3342:
3339:
3336:
3333:
3330:
3327:
3323:
3317:
3313:
3309:
3306:
3303:
3300:
3296:
3292:
3289:
3286:
3283:
3280:
3277:
3273:
3244:
3241:
3238:
3235:
3232:
3228:
3205:
3202:
3199:
3196:
3193:
3190:
3186:
3180:
3176:
3172:
3169:
3166:
3163:
3159:
3155:
3152:
3149:
3146:
3143:
3140:
3136:
3113:
3110:
3107:
3102:
3097:
3092:
3088:
3085:
3079:
3072:
3069:
3064:
3059:
3055:
3052:
3049:
3043:
3040:
3037:
3032:
3028:
3024:
3021:
3018:
3015:
3011:
2980:
2975:
2971:
2967:
2964:
2961:
2958:
2954:
2950:
2947:
2944:
2941:
2938:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2880:
2877:
2873:
2868:
2865:
2838:
2835:
2832:
2829:
2826:
2822:
2799:
2796:
2793:
2790:
2787:
2784:
2780:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2742:
2738:
2735:
2732:
2729:
2723:
2720:
2716:
2711:
2708:
2688:
2685:
2682:
2679:
2676:
2672:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2637:
2604:
2601:
2598:
2595:
2592:
2589:
2585:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2559:
2552:
2549:
2545:
2540:
2537:
2511:
2503:
2496:
2493:
2492:
2491:
2480:
2477:
2472:
2468:
2464:
2459:
2455:
2451:
2448:
2445:
2440:
2436:
2432:
2429:
2426:
2423:
2420:
2417:
2412:
2408:
2404:
2401:
2398:
2395:
2379:
2372:
2365:
2354:
2348:
2343:
2332:
2329:
2326:
2321:
2317:
2313:
2310:
2306:
2302:
2297:
2293:
2289:
2284:
2280:
2276:
2273:
2270:
2267:
2264:
2259:
2255:
2251:
2248:
2244:
2223:
2213:
2208:
2197:
2194:
2191:
2188:
2183:
2179:
2175:
2172:
2169:
2166:
2163:
2158:
2154:
2150:
2147:
2144:
2139:
2135:
2114:
2111:
2108:
2103:
2099:
2095:
2092:
2088:
2084:
2079:
2075:
2071:
2068:
2065:
2062:
2059:
2056:
2051:
2047:
2043:
2040:
2036:
2032:
2027:
2023:
2019:
2016:
2013:
2010:
2007:
2002:
1998:
1994:
1991:
1988:
1983:
1979:
1974:
1970:
1965:
1961:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1929:
1925:
1922:
1919:
1916:
1913:
1894:
1887:
1884:
1872:
1869:
1864:
1860:
1856:
1853:
1850:
1845:
1841:
1836:
1832:
1827:
1823:
1819:
1816:
1813:
1810:
1806:
1802:
1799:
1796:
1793:
1790:
1764:
1761:
1758:
1753:
1749:
1745:
1742:
1738:
1734:
1729:
1725:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1690:
1673:
1666:
1651:
1648:
1646:
1643:
1627:
1624:
1601:
1598:
1594:
1573:
1570:
1566:
1559:
1556:
1549:
1545:
1542:
1529:for which the
1518:
1515:
1511:
1479:
1475:
1472:
1467:
1446:
1443:
1439:
1408:
1405:
1401:
1360:
1357:
1344:
1341:
1338:
1333:
1329:
1306:
1305:
1294:
1289:
1285:
1281:
1276:
1272:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1246:
1242:
1239:
1234:
1230:
1226:
1223:
1220:
1217:
1196:
1195:
1184:
1181:
1178:
1173:
1169:
1165:
1162:
1158:
1154:
1149:
1145:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1110:
1097:, and one has
1086:
1081:
1077:
1073:
1070:
1067:
1064:
1040:
1037:
1034:
1031:
1028:
1024:
1011:
996:
993:
988:
984:
980:
977:
973:
921:
884:
883:
881:
880:
873:
866:
858:
855:
854:
850:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
618:
617:
614:
613:
610:
609:
606:
605:
600:
595:
590:
588:Density matrix
585:
580:
575:
570:
565:
560:
554:
551:
550:
547:
546:
542:
541:
536:
531:
526:
521:
516:
511:
510:
509:
508:
507:
492:
487:
482:
477:
472:
466:
465:
460:
459:
456:
455:
451:
450:
445:
440:
435:
430:
424:
423:
420:
419:
416:
415:
411:
410:
405:
400:
395:
390:
385:
379:
378:
377:
371:
368:
367:
364:
363:
359:
358:
353:
348:
342:
341:
340:
339:
338:
336:Delayed-choice
331:Quantum eraser
326:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
274:
273:
270:
269:
266:
265:
261:
260:
259:
258:
248:
243:
238:
233:
228:
223:
221:Quantum number
218:
213:
208:
203:
198:
193:
187:
186:
183:
182:
179:
178:
174:
173:
168:
162:
161:
160:
155:
150:
144:
141:
140:
137:
136:
135:
134:
129:
124:
116:
115:
110:
99:
96:
92:
85:
82:
76:
73:
70:
66:
59:
56:
52:
47:
44:
33:
32:
26:
25:
15:
9:
6:
4:
3:
2:
5623:
5612:
5609:
5607:
5604:
5603:
5601:
5586:
5578:
5577:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5551:
5548:
5547:
5546:
5543:
5542:
5540:
5536:
5530:
5527:
5525:
5522:
5518:
5515:
5514:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5494:
5492:
5488:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5427:
5424:
5422:
5421:Quantum logic
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5392:
5389:
5387:
5384:
5380:
5377:
5376:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5353:
5350:
5349:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5329:
5327:
5325:
5321:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5275:
5274:Quantum chaos
5272:
5270:
5267:
5265:
5262:
5261:
5259:
5257:
5253:
5247:
5244:
5242:
5241:Stern–Gerlach
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5193:
5191:
5187:
5181:
5178:
5176:
5175:Transactional
5173:
5171:
5168:
5166:
5165:Quantum logic
5163:
5161:
5158:
5156:
5153:
5147:
5144:
5143:
5142:
5139:
5138:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5108:
5106:
5104:
5100:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5055:
5053:
5049:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5009:
5007:
5003:
4995:
4992:
4990:
4987:
4986:
4985:
4984:Wave function
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4959:Superposition
4957:
4955:
4954:Quantum state
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4918:
4915:
4913:
4910:
4908:
4907:Excited state
4905:
4903:
4900:
4899:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4874:
4872:
4868:
4862:
4859:
4857:
4854:
4852:
4849:
4845:
4842:
4841:
4840:
4837:
4835:
4832:
4831:
4829:
4825:
4821:
4814:
4809:
4807:
4802:
4800:
4795:
4794:
4791:
4787:
4781:
4777:
4773:
4772:
4767:
4766:J. J. Sakurai
4764:
4761:
4757:
4753:
4749:
4746:
4744:
4739:
4733:
4729:
4725:
4721:
4720:E.M. Lifshitz
4717:
4713:
4711:
4710:0-471-88702-1
4707:
4703:
4699:
4698:Merzbacher E.
4696:
4693:
4689:
4686:
4682:
4680:0-471-16433-X
4676:
4672:
4668:
4664:
4663:
4650:
4644:
4640:
4633:
4625:
4623:0-07-051400-3
4619:
4615:
4610:
4609:
4600:
4596:
4587:
4583:
4577:
4570:
4566:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4522:
4494:
4490:
4477:
4474:
4470:
4466:
4463:
4459:
4452:
4438:
4428:
4424:
4411:
4408:
4404:
4400:
4396:
4392:
4386:
4372:
4364:
4361:
4341:
4337:
4323:
4319:
4315:
4308:
4294:
4284:
4280:
4266:
4262:
4259:
4255:
4251:
4245:
4231:
4223:
4221:
4218:
4217:
4196:
4192:
4179:
4176:
4172:
4168:
4165:
4161:
4149:
4139:
4135:
4122:
4119:
4115:
4111:
4107:
4103:
4097:
4083:
4075:
4055:
4051:
4037:
4033:
4030:
4026:
4014:
4004:
4000:
3986:
3982:
3978:
3974:
3968:
3954:
3946:
3943:
3941:
3938:
3937:
3916:
3902:
3887:
3883:
3870:
3867:
3863:
3859:
3855:
3851:
3842:
3828:
3815:
3812:
3792:
3778:
3763:
3759:
3745:
3741:
3738:
3734:
3730:
3721:
3707:
3694:
3692:
3689:
3688:
3684:
3681:
3678:
3675:
3672:
3669:
3668:
3661:
3656:
3654:
3649:
3647:
3642:
3641:
3638:
3631:
3628:
3623:
3616:
3606:
3604:
3598:
3596:
3595:Freeman Dyson
3592:
3588:
3587:time-ordering
3572:
3568:
3563:
3560:
3556:
3548:
3545:
3538:
3533:
3528:
3524:
3515:
3510:
3506:
3502:
3499:
3491:
3485:
3479:
3470:
3457:
3453:
3448:
3445:
3441:
3433:
3430:
3423:
3418:
3413:
3409:
3400:
3395:
3391:
3387:
3384:
3381:
3375:
3369:
3360:
3358:
3353:
3340:
3331:
3325:
3311:
3307:
3304:
3301:
3298:
3294:
3290:
3281:
3275:
3262:
3258:
3236:
3230:
3216:
3203:
3194:
3188:
3174:
3170:
3167:
3164:
3161:
3157:
3153:
3144:
3138:
3124:
3111:
3108:
3105:
3100:
3095:
3086:
3083:
3077:
3070:
3067:
3062:
3053:
3050:
3047:
3041:
3038:
3035:
3026:
3022:
3019:
3016:
3013:
3009:
3000:
2999:Taylor series
2996:
2991:
2978:
2969:
2965:
2962:
2959:
2956:
2952:
2948:
2942:
2936:
2927:
2914:
2908:
2902:
2899:
2896:
2890:
2884:
2878:
2863:
2853:
2830:
2824:
2810:
2797:
2788:
2782:
2771:
2765:
2762:
2759:
2750:
2744:
2733:
2727:
2721:
2706:
2680:
2674:
2663:
2657:
2654:
2645:
2639:
2626:
2622:
2618:
2602:
2593:
2587:
2579:
2576:
2567:
2561:
2550:
2535:
2527:
2523:
2519:
2510:
2502:
2478:
2470:
2466:
2462:
2457:
2453:
2446:
2438:
2434:
2430:
2427:
2421:
2418:
2410:
2406:
2402:
2399:
2393:
2386:. Therefore,
2385:
2378:
2371:
2364:
2360:
2353:
2349:
2347:
2344:
2330:
2319:
2315:
2308:
2295:
2291:
2287:
2282:
2278:
2271:
2268:
2257:
2253:
2246:
2233:
2229:
2222:
2218:
2214:
2212:
2209:
2195:
2192:
2189:
2181:
2177:
2173:
2170:
2164:
2156:
2152:
2148:
2145:
2137:
2133:
2112:
2101:
2097:
2090:
2077:
2073:
2066:
2060:
2049:
2045:
2038:
2025:
2021:
2017:
2014:
2008:
2000:
1996:
1992:
1989:
1981:
1977:
1963:
1959:
1952:
1946:
1937:
1931:
1920:
1914:
1903:
1899:
1895:
1893:
1890:
1889:
1883:
1870:
1862:
1858:
1854:
1851:
1843:
1839:
1825:
1821:
1814:
1808:
1797:
1791:
1780:
1775:
1762:
1751:
1747:
1740:
1727:
1723:
1719:
1716:
1710:
1707:
1698:
1692:
1679:
1672:
1665:
1661:
1657:
1642:
1622:
1596:
1568:
1554:
1543:
1532:
1513:
1500:
1496:
1491:
1473:
1470:
1441:
1428:
1424:
1423:
1422:Hilbert space
1403:
1390:
1389:
1382:
1378:
1374:
1370:
1366:
1356:
1342:
1339:
1336:
1331:
1313:
1311:
1310:Taylor series
1292:
1283:
1274:
1270:
1266:
1263:
1257:
1254:
1251:
1248:
1244:
1240:
1232:
1228:
1224:
1221:
1215:
1208:
1207:
1206:
1201:
1182:
1171:
1167:
1160:
1147:
1143:
1139:
1136:
1130:
1127:
1118:
1112:
1100:
1099:
1098:
1079:
1075:
1071:
1068:
1062:
1032:
1026:
986:
982:
975:
962:
958:
954:
949:
947:
943:
939:
935:
919:
911:
910:state vectors
908:in which the
907:
903:
899:
895:
891:
879:
874:
872:
867:
865:
860:
859:
857:
856:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
619:
612:
611:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
578:Quantum chaos
576:
574:
571:
569:
566:
564:
561:
559:
556:
555:
549:
548:
540:
537:
535:
534:Transactional
532:
530:
527:
525:
524:Quantum logic
522:
520:
517:
515:
512:
506:
503:
502:
501:
498:
497:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
471:
468:
467:
463:
458:
457:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
425:
418:
417:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
380:
376:
373:
372:
366:
365:
357:
354:
352:
349:
347:
344:
343:
337:
334:
333:
332:
329:
328:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
275:
268:
267:
257:
254:
253:
252:
251:Wave function
249:
247:
244:
242:
239:
237:
234:
232:
231:Superposition
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
188:
181:
180:
172:
169:
167:
164:
163:
159:
156:
154:
151:
149:
146:
145:
139:
138:
133:
130:
128:
125:
123:
120:
119:
118:
117:
113:
80:
74:
57:
54:
50:
42:
35:
34:
31:
28:
27:
23:
22:
19:
5304:Quantum mind
5216:Franck–Hertz
5058:Klein–Gordon
5031:
5012:Formulations
5005:Formulations
4934:Interference
4924:Entanglement
4902:Ground state
4897:Energy level
4870:Fundamentals
4834:Introduction
4785:
4769:
4751:
4723:
4701:
4691:
4670:
4638:
4632:
4607:
4599:
4585:
4581:
4575:
4569:
3670:
3621:
3614:
3612:
3599:
3591:Dyson series
3471:
3361:
3356:
3354:
3260:
3217:
3125:
2994:
2992:
2928:
2851:
2811:
2624:
2616:
2521:
2517:
2508:
2500:
2499:We drop the
2498:
2383:
2376:
2369:
2362:
2358:
2351:
2345:
2227:
2220:
2216:
2210:
1891:
1776:
1677:
1670:
1663:
1659:
1655:
1653:
1494:
1492:
1420:
1387:
1380:
1376:
1372:
1369:wavefunction
1362:
1314:
1307:
1197:
950:
897:
893:
887:
433:Klein–Gordon
402:
369:Formulations
206:Energy level
201:Entanglement
184:Fundamentals
171:Interference
122:Introduction
18:
5562:EPR paradox
5342:Quantum bus
5211:Double-slit
5189:Experiments
5155:Many-worlds
5093:Schrödinger
5042:Phase space
5032:Schrödinger
5022:Interaction
4979:Uncertainty
4949:Nonlocality
4944:Measurement
4939:Decoherence
4929:Hamiltonian
4743:Online copy
4716:L.D. Landau
3683:Interaction
3671:Schrödinger
3585:where T is
3126:Therefore,
2621:Hamiltonian
2125:Therefore,
1319:; that is,
1200:Hamiltonian
953:Schrödinger
902:formulation
822:von Neumann
807:Schrödinger
583:EPR paradox
514:Many-worlds
448:Schrödinger
403:Schrödinger
398:Phase-space
388:Interaction
293:Double-slit
271:Experiments
246:Uncertainty
216:Nonlocality
211:Measurement
196:Decoherence
166:Hamiltonian
5600:Categories
5490:Extensions
5324:Technology
5170:Relational
5121:Copenhagen
5017:Heisenberg
4964:Tunnelling
4827:Background
4748:R. Shankar
4660:References
3940:Observable
3677:Heisenberg
3257:eigenstate
3218:Note that
1900:. For the
1886:Properties
1650:Definition
1359:Background
817:Sommerfeld
732:Heisenberg
727:Gutzwiller
667:de Broglie
615:Scientists
529:Relational
480:Copenhagen
383:Heisenberg
241:Tunnelling
142:Background
5196:Bell test
5051:Equations
4877:Born rule
4614:786, 1261
4499:ℏ
4464:−
4439:ρ
4433:ℏ
4373:ρ
4362:constant
4346:ℏ
4295:ρ
4289:ℏ
4260:−
4232:ρ
4201:ℏ
4166:−
4144:ℏ
4060:ℏ
4031:−
4009:ℏ
3944:constant
3923:⟩
3903:ψ
3892:ℏ
3849:⟩
3829:ψ
3813:constant
3799:⟩
3779:ψ
3768:ℏ
3739:−
3728:⟩
3708:ψ
3691:Ket state
3525:∫
3519:ℏ
3511:−
3503:
3410:∫
3404:ℏ
3396:−
3388:
3338:⟩
3326:ψ
3316:ℏ
3299:−
3288:⟩
3276:ψ
3243:⟩
3231:ψ
3201:⟩
3189:ψ
3179:ℏ
3162:−
3151:⟩
3139:ψ
3109:⋯
3091:ℏ
3063:−
3058:ℏ
3042:−
3031:ℏ
3014:−
2974:ℏ
2957:−
2876:∂
2872:∂
2867:ℏ
2837:⟩
2825:ψ
2795:⟩
2783:ψ
2757:⟩
2745:ψ
2719:∂
2715:∂
2710:ℏ
2687:⟩
2675:ψ
2652:⟩
2640:ψ
2627:to write
2600:⟩
2588:ψ
2574:⟩
2562:ψ
2548:∂
2544:∂
2539:ℏ
2328:⟩
2309:ψ
2266:⟩
2247:ψ
2138:†
2110:⟩
2091:ψ
2067:ψ
2064:⟨
2058:⟩
2039:ψ
1982:†
1953:ψ
1950:⟨
1944:⟩
1932:ψ
1915:ψ
1912:⟨
1892:Unitarity
1844:†
1815:ψ
1812:⟨
1792:ψ
1789:⟨
1760:⟩
1741:ψ
1705:⟩
1693:ψ
1626:^
1600:⟩
1597:ψ
1572:⟩
1569:ψ
1558:^
1544:ψ
1541:⟨
1517:⟩
1514:ψ
1478:⟩
1471:ψ
1445:⟩
1442:ψ
1407:⟩
1404:ψ
1328:∂
1288:ℏ
1267:−
1258:⋅
1249:−
1180:⟩
1161:ψ
1125:⟩
1113:ψ
1039:⟩
1027:ψ
995:⟩
976:ψ
847:Zeilinger
692:Ehrenfest
421:Equations
98:⟩
95:Ψ
84:^
72:⟩
69:Ψ
46:ℏ
5585:Category
5379:Timeline
5131:Ensemble
5111:Bayesian
5073:Majorana
4989:Collapse
4861:Glossary
4844:Timeline
4768:(1993);
4750:(1994);
4722:(1977).
4690:, 1966.
4519:See also
3620:, where
3593:, after
3564:′
3549:′
3449:′
3434:′
2234:, since
2219: =
2211:Identity
1474:′
1427:operator
1051:at time
1007:at time
772:Millikan
697:Einstein
682:Davisson
637:Blackett
622:Aharonov
490:Ensemble
470:Bayesian
375:Overview
256:Collapse
236:Symmetry
127:Glossary
5538:Related
5517:History
5256:Science
5088:Rydberg
4839:History
3637:Picture
2619:is the
2524:). The
2346:Closure
2230:is the
1898:unitary
951:In the
890:physics
812:Simmons
802:Rydberg
767:Moseley
747:Kramers
737:Hilbert
722:Glauber
717:Feynman
702:Everett
672:Compton
443:Rydberg
132:History
5231:Popper
4778:
4758:
4734:
4708:
4677:
4645:
4620:
4488:
4422:
4335:
4278:
4190:
4133:
4049:
3998:
3881:
3757:
2993:Since
2812:Since
2615:where
959:, the
892:, the
842:Zeeman
837:Wigner
787:Planck
757:Landau
742:Jordan
393:Matrix
323:Popper
5141:Local
5083:Pauli
5063:Dirac
4561:Notes
2215:When
1365:state
900:is a
797:Raman
782:Pauli
777:Onnes
712:Fermi
687:Debye
677:Dirac
642:Bloch
632:Bethe
500:Local
438:Pauli
428:Dirac
226:State
5068:Weyl
4776:ISBN
4756:ISBN
4732:ISBN
4706:ISBN
4675:ISBN
4643:ISBN
4618:ISBN
4545:POVM
3685:(I)
3679:(H)
3673:(S)
1902:norm
1779:bras
1777:For
1495:must
944:and
832:Wien
827:Weyl
792:Rabi
762:Laue
752:Lamb
707:Fock
662:Bose
657:Born
652:Bohr
647:Bohm
627:Bell
4580:,
4578:= 0
4573:At
3625:0,S
3500:exp
3385:exp
2854:= 0
2528:is
2514:= 0
2357:to
1781:,
1388:ket
904:of
896:or
888:In
5602::
4730:.
4718:,
4700:,
4616:.
3665:)
3605:.
3597:.
3263::
3001::
2699:,
2226:,
1680::
1662:,
1490:.
1391:,
1379:,
1355:.
1312:.
4812:e
4805:t
4798:v
4782:.
4762:.
4740:.
4683:.
4651:.
4626:.
4586:t
4584:(
4582:U
4576:t
4495:/
4491:t
4482:S
4478:,
4475:0
4471:H
4467:i
4460:e
4456:)
4453:t
4450:(
4444:S
4429:/
4425:t
4416:S
4412:,
4409:0
4405:H
4401:i
4397:e
4393:=
4390:)
4387:t
4384:(
4378:I
4342:/
4338:t
4329:S
4324:H
4320:i
4316:e
4312:)
4309:0
4306:(
4300:S
4285:/
4281:t
4272:S
4267:H
4263:i
4256:e
4252:=
4249:)
4246:t
4243:(
4237:S
4197:/
4193:t
4184:S
4180:,
4177:0
4173:H
4169:i
4162:e
4155:S
4150:A
4140:/
4136:t
4127:S
4123:,
4120:0
4116:H
4112:i
4108:e
4104:=
4101:)
4098:t
4095:(
4089:I
4084:A
4056:/
4052:t
4043:S
4038:H
4034:i
4027:e
4020:S
4015:A
4005:/
4001:t
3992:S
3987:H
3983:i
3979:e
3975:=
3972:)
3969:t
3966:(
3960:H
3955:A
3920:)
3917:t
3914:(
3908:S
3898:|
3888:/
3884:t
3875:S
3871:,
3868:0
3864:H
3860:i
3856:e
3852:=
3846:)
3843:t
3840:(
3834:I
3824:|
3796:)
3793:0
3790:(
3784:S
3774:|
3764:/
3760:t
3751:S
3746:H
3742:i
3735:e
3731:=
3725:)
3722:t
3719:(
3713:S
3703:|
3659:e
3652:t
3645:v
3639:(
3622:H
3618:S
3615:H
3573:,
3569:)
3561:t
3557:d
3553:)
3546:t
3542:(
3539:H
3534:t
3529:0
3516:i
3507:(
3496:T
3492:=
3489:)
3486:t
3483:(
3480:U
3458:,
3454:)
3446:t
3442:d
3438:)
3431:t
3427:(
3424:H
3419:t
3414:0
3401:i
3392:(
3382:=
3379:)
3376:t
3373:(
3370:U
3341:.
3335:)
3332:0
3329:(
3322:|
3312:/
3308:t
3305:E
3302:i
3295:e
3291:=
3285:)
3282:t
3279:(
3272:|
3261:E
3240:)
3237:0
3234:(
3227:|
3204:.
3198:)
3195:0
3192:(
3185:|
3175:/
3171:t
3168:H
3165:i
3158:e
3154:=
3148:)
3145:t
3142:(
3135:|
3112:.
3106:+
3101:2
3096:)
3087:t
3084:H
3078:(
3071:2
3068:1
3054:t
3051:H
3048:i
3039:1
3036:=
3027:/
3023:t
3020:H
3017:i
3010:e
2995:H
2979:.
2970:/
2966:t
2963:H
2960:i
2953:e
2949:=
2946:)
2943:t
2940:(
2937:U
2915:.
2912:)
2909:t
2906:(
2903:U
2900:H
2897:=
2894:)
2891:t
2888:(
2885:U
2879:t
2864:i
2852:t
2834:)
2831:0
2828:(
2821:|
2798:.
2792:)
2789:0
2786:(
2779:|
2775:)
2772:t
2769:(
2766:U
2763:H
2760:=
2754:)
2751:0
2748:(
2741:|
2737:)
2734:t
2731:(
2728:U
2722:t
2707:i
2684:)
2681:0
2678:(
2671:|
2667:)
2664:t
2661:(
2658:U
2655:=
2649:)
2646:t
2643:(
2636:|
2625:U
2617:H
2603:,
2597:)
2594:t
2591:(
2584:|
2580:H
2577:=
2571:)
2568:t
2565:(
2558:|
2551:t
2536:i
2522:t
2520:(
2518:U
2512:0
2509:t
2504:0
2501:t
2479:.
2476:)
2471:0
2467:t
2463:,
2458:1
2454:t
2450:(
2447:U
2444:)
2439:1
2435:t
2431:,
2428:t
2425:(
2422:U
2419:=
2416:)
2411:0
2407:t
2403:,
2400:t
2397:(
2394:U
2384:t
2380:1
2377:t
2373:1
2370:t
2366:0
2363:t
2359:t
2355:0
2352:t
2331:.
2325:)
2320:0
2316:t
2312:(
2305:|
2301:)
2296:0
2292:t
2288:,
2283:0
2279:t
2275:(
2272:U
2269:=
2263:)
2258:0
2254:t
2250:(
2243:|
2228:U
2224:0
2221:t
2217:t
2196:.
2193:I
2190:=
2187:)
2182:0
2178:t
2174:,
2171:t
2168:(
2165:U
2162:)
2157:0
2153:t
2149:,
2146:t
2143:(
2134:U
2113:.
2107:)
2102:0
2098:t
2094:(
2087:|
2083:)
2078:0
2074:t
2070:(
2061:=
2055:)
2050:0
2046:t
2042:(
2035:|
2031:)
2026:0
2022:t
2018:,
2015:t
2012:(
2009:U
2006:)
2001:0
1997:t
1993:,
1990:t
1987:(
1978:U
1973:|
1969:)
1964:0
1960:t
1956:(
1947:=
1941:)
1938:t
1935:(
1928:|
1924:)
1921:t
1918:(
1871:.
1868:)
1863:0
1859:t
1855:,
1852:t
1849:(
1840:U
1835:|
1831:)
1826:0
1822:t
1818:(
1809:=
1805:|
1801:)
1798:t
1795:(
1763:.
1757:)
1752:0
1748:t
1744:(
1737:|
1733:)
1728:0
1724:t
1720:,
1717:t
1714:(
1711:U
1708:=
1702:)
1699:t
1696:(
1689:|
1678:t
1674:0
1671:t
1667:0
1664:t
1660:t
1658:(
1656:U
1623:p
1593:|
1565:|
1555:p
1548:|
1510:|
1466:|
1438:|
1400:|
1383:)
1381:t
1377:x
1375:(
1373:ψ
1343:0
1340:=
1337:H
1332:t
1317:H
1293:,
1284:/
1280:)
1275:0
1271:t
1264:t
1261:(
1255:H
1252:i
1245:e
1241:=
1238:)
1233:0
1229:t
1225:,
1222:t
1219:(
1216:U
1203:H
1183:.
1177:)
1172:0
1168:t
1164:(
1157:|
1153:)
1148:0
1144:t
1140:,
1137:t
1134:(
1131:U
1128:=
1122:)
1119:t
1116:(
1109:|
1085:)
1080:0
1076:t
1072:,
1069:t
1066:(
1063:U
1053:t
1036:)
1033:t
1030:(
1023:|
1012:0
1009:t
992:)
987:0
983:t
979:(
972:|
920:V
877:e
870:t
863:v
91:|
81:H
75:=
65:|
58:t
55:d
51:d
43:i
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