2430:
1981:
5630:
4898:
4297:
5246:
2425:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle A\rangle &={\frac {d}{dt}}\int \Phi ^{*}A\Phi \,d^{3}x\\&=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi \,d^{3}x+\int \Phi ^{*}\left({\frac {\partial A}{\partial t}}\right)\Phi \,d^{3}x+\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)\,d^{3}x\\&=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi \,d^{3}x+\left\langle {\frac {\partial A}{\partial t}}\right\rangle +\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)\,d^{3}x\end{aligned}}}
4452:
3904:
5235:
5625:{\displaystyle {\begin{aligned}\left\langle {\frac {d\Psi }{dt}}{\Big |}{\hat {x}}{\Big |}\Psi \right\rangle +\left\langle \Psi {\Big |}{\hat {x}}{\Big |}{\frac {d\Psi }{dt}}\right\rangle &=\left\langle \Psi {\Big |}{\frac {\hat {p}}{m}}{\Big |}\Psi \right\rangle ,\\\left\langle {\frac {d\Psi }{dt}}{\Big |}{\hat {p}}{\Big |}\Psi \right\rangle +\left\langle \Psi {\Big |}{\hat {p}}{\Big |}{\frac {d\Psi }{dt}}\right\rangle &=\langle \Psi |-V'({\hat {x}})|\Psi \rangle ,\end{aligned}}}
4893:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle x\rangle &={\frac {1}{i\hbar }}\langle \rangle +\left\langle {\frac {\partial x}{\partial t}}\right\rangle \\&={\frac {1}{i\hbar }}\left\langle \left\right\rangle +0\\&={\frac {1}{i\hbar }}\left\langle \left\right\rangle \\&={\frac {1}{i\hbar 2m}}\left\langle {\frac {d}{dp}}p^{2}\right\rangle \\&={\frac {1}{i\hbar 2m}}\langle i\hbar 2p\rangle \\&={\frac {1}{m}}\langle p\rangle \end{aligned}}}
4292:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle p\rangle &=\int \Phi ^{*}V(x,t){\frac {\partial }{\partial x}}\Phi ~dx-\int \Phi ^{*}\left({\frac {\partial }{\partial x}}V(x,t)\right)\Phi ~dx-\int \Phi ^{*}V(x,t){\frac {\partial }{\partial x}}\Phi ~dx\\&=-\int \Phi ^{*}\left({\frac {\partial }{\partial x}}V(x,t)\right)\Phi ~dx\\&=\left\langle -{\frac {\partial }{\partial x}}V(x,t)\right\rangle =\langle F\rangle .\end{aligned}}}
4917:
2858:
6219:. Due to exponential instability of classical trajectories the Ehrenfest time, on which there is a complete correspondence between quantum and classical evolution, is shown to be logarithmically short being proportional to a logarithm of typical quantum number. For the case of integrable dynamics this time scale is much larger being proportional to a certain power of quantum number.
3250:
5230:{\displaystyle {\begin{aligned}m{\frac {d}{dt}}\left\langle \Psi (t)\right|{\hat {x}}\left|\Psi (t)\right\rangle &=\left\langle \Psi (t)\right|{\hat {p}}\left|\Psi (t)\right\rangle ,\\{\frac {d}{dt}}\left\langle \Psi (t)\right|{\hat {p}}\left|\Psi (t)\right\rangle &=\left\langle \Psi (t)\right|-V'({\hat {x}})\left|\Psi (t)\right\rangle .\end{aligned}}}
6579:
2647:
3058:
3700:
3893:
3419:
1248:
1110:
6396:
3537:
2619:
2853:{\displaystyle {\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\int \Phi ^{*}(AH-HA)\Phi ~d^{3}x+\left\langle {\frac {\partial A}{\partial t}}\right\rangle ={\frac {1}{i\hbar }}\langle \rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle .}
3723:
6094:
3279:
3245:{\displaystyle \left\langle \Psi \left|{\frac {d}{dt}}A(t)\right|\Psi \right\rangle =\left\langle \Psi \left|{\frac {\partial A(t)}{\partial t}}\right|\Psi \right\rangle +\left\langle \Psi \left|{\frac {1}{i\hbar }}\right|\Psi \right\rangle ,}
5730:
2997:
4457:
6195:
was derived from the
Ehrenfest theorems by assuming the canonical commutation relation between the coordinate and momentum. If one assumes that the coordinate and momentum commute, the same computational method leads to the
1305:. This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. Otherwise, the evolution equations still may hold
2495:
6614:
is the
Hamiltonian represented in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for
106:
1136:
5891:
1317:
Although, at first glance, it might appear that the
Ehrenfest theorem is saying that the quantum mechanical expectation values obey Newtonâs classical equations of motion, this is not actually the case. If the pair
6187:
2502:
991:
1410:
1864:
1746:
5736:
was introduced as a normalization constant to the balance dimensionality. Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for
6574:{\displaystyle {\frac {\partial }{\partial t}}\langle \phi |x\rangle ={\frac {-1}{i\hbar }}\langle \phi |{\hat {H}}|x\rangle ={\frac {-1}{i\hbar }}\langle \phi |x\rangle H={\frac {-1}{i\hbar }}\Phi ^{*}H,}
5251:
4922:
1460:
4347:
1908:
1790:
1360:
5976:
5971:
3909:
3521:
1986:
6391:
1299:
suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a
Poisson bracket multiplied by
2878:, the derivation is straightforward. The Heisenberg picture moves the time dependence of the system to operators instead of state vectors. Starting with the Heisenberg equation of motion,
3695:{\displaystyle {\frac {d}{dt}}\langle p\rangle ={\frac {1}{i\hbar }}\langle \rangle +\left\langle {\frac {\partial p}{\partial t}}\right\rangle ={\frac {1}{i\hbar }}\langle \rangle ,}
4395:
5640:
to denote the quantum generator of time translation. The next step is to show that this is the same as the
Hamiltonian operator used in quantum mechanics. Stone's theorem implies
4436:
1634:
1601:
5643:
2881:
3053:
3025:
1953:
3888:{\displaystyle {\frac {d}{dt}}\langle p\rangle =\int \Phi ^{*}V(x,t){\frac {\partial }{\partial x}}\Phi ~dx-\int \Phi ^{*}{\frac {\partial }{\partial x}}(V(x,t)\Phi )~dx~.}
954:
5633:
6608:
6317:
1702:
1541:
2439:
1820:
1568:
1516:
3414:{\displaystyle {\frac {d}{dt}}\langle A(t)\rangle =\left\langle {\frac {\partial A(t)}{\partial t}}\right\rangle +{\frac {1}{i\hbar }}\left\langle \right\rangle .}
1489:
983:
1677:
1654:
372:
6209:
6197:
6102:
873:
5744:
1792:
do agree. Thus, for the case of a quantum harmonic oscillator, the expected position and expected momentum do exactly follow the classical trajectories.
1276:
of quantum mechanics, where it amounts to just the expectation value of the
Heisenberg equation of motion. It provides mathematical support to the
580:
124:
1365:
1243:{\displaystyle {\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle \rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle ~,}
36:
6695:
Bondar, D.; Cabrera, R.; Lompay, R.; Ivanov, M.; Rabitz, H. (2012). "Operational
Dynamic Modeling Transcending Quantum and Classical Mechanics".
6826:
353:
536:
899:
1105:{\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\;\;{\frac {d}{dt}}\langle p\rangle =-\left\langle V'(x)\right\rangle ~.}
459:
1284:
4438:. Nevertheless, as explained in the introduction, for states that are highly localized in space, the expected position and momentum will
3439:
1415:
6808:
6300:
119:
1825:
1707:
208:
315:
295:
6097:
2635:
1127:
866:
163:
6215:
The implications of the
Ehrenfest theorem for systems with classically chaotic dynamics are discussed at Scholarpedia article
285:
5896:
2614:{\displaystyle {\frac {\partial \Phi ^{*}}{\partial t}}=-{\frac {1}{i\hbar }}\Phi ^{*}H^{*}=-{\frac {1}{i\hbar }}\Phi ^{*}H.}
555:
4308:
1321:
6831:
6248:
Ehrenfest, P. (1927). "Bemerkung ĂŒber die angenĂ€herte GĂŒltigkeit der klassischen
Mechanik innerhalb der Quantenmechanik".
4914:. However, the converse is also true: the Schrödinger equation can be inferred from the Ehrenfest theorems. We begin from
595:
333:
233:
5908:
348:
531:
526:
497:
129:
6351:
6212:, shows that the essential difference between quantum and classical mechanics reduces to the value of the commutator .
310:
300:
1869:
1751:
565:
6757:
482:
859:
511:
3276:
out of the first term, since the state vectors are no longer time dependent in the
Heisenberg Picture. Therefore,
430:
516:
477:
405:
328:
188:
4443:
1277:
600:
4356:
6192:
6089:{\displaystyle m{\frac {\partial H(x,p)}{\partial p}}=p,\qquad {\frac {\partial H(x,p)}{\partial x}}=V'(x),}
4911:
4400:
2433:
1959:
follow the classical trajectories, at least for as long as the wave function remains localized in position.
769:
487:
445:
395:
343:
109:
1606:
1573:
4350:
774:
492:
400:
320:
290:
253:
1288:
243:
228:
6346:
155:
6769:
6714:
6257:
804:
560:
472:
198:
3030:
3002:
6205:
3429:
659:
467:
385:
213:
193:
145:
1115:
The Ehrenfest theorem is a special case of a more general relation between the expectation of any
6738:
6704:
6273:
2875:
2639:
1362:
were to satisfy Newton's second law, the right-hand side of the second equation would have to be
1273:
1119:
903:
380:
305:
238:
150:
6584:
6216:
1917:
6804:
6730:
6296:
1266:
1116:
916:
814:
789:
729:
724:
624:
590:
570:
168:
27:
5725:{\displaystyle i\hbar \left|{\frac {d\Psi }{dt}}\right\rangle ={\hat {H}}|\Psi (t)\rangle ~,}
3708:
commutes with itself and has no time dependence. By expanding the right-hand-side, replacing
2992:{\displaystyle {\frac {d}{dt}}A(t)={\frac {\partial A(t)}{\partial t}}+{\frac {1}{i\hbar }},}
1659:
An exception occurs in case when the classical equations of motion are linear, that is, when
6777:
6722:
6656:
of the system. The time dependence of the expectation value, in this picture, is due to the
6265:
819:
809:
799:
699:
679:
664:
634:
502:
390:
1798:
1546:
1494:
6649:
1570:). This means, in the case of Newton's second law, the right side would be in the form of
1465:
1292:
959:
844:
714:
694:
440:
280:
6773:
6718:
6261:
1682:
1521:
6657:
1662:
1639:
891:
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
3530:
Suppose we wanted to know the instantaneous change in the expectation of the momentum
6820:
6653:
6277:
6201:
4449:
Similarly, we can obtain the instantaneous change in the position expectation value.
1968:
1306:
1296:
839:
834:
764:
734:
704:
575:
521:
248:
223:
6742:
2866:
is time-independent so that its derivative is zero and we can ignore the last term.
6726:
6644:-valued function of time, will have time dependence, the momentum operator itself,
5240:
3898:
1974:. If we want to know the instantaneous time derivative of the expectation value of
829:
824:
759:
744:
709:
203:
1636:. The difference between these two quantities is the square of the uncertainty in
2999:
Ehrenfest's theorem follows simply upon projecting the Heisenberg equation onto
6641:
1795:
For general systems, if the wave function is highly concentrated around a point
794:
749:
684:
639:
6318:"Remarks concerning the status & some ramifications of Ehrenfest's theorem"
2490:{\displaystyle {\frac {\partial \Phi }{\partial t}}={\frac {1}{i\hbar }}H\Phi }
1123:
895:
784:
754:
674:
649:
644:
629:
4910:
It was established above that the Ehrenfest theorems are consequences of the
4442:
follow classical trajectories, which may be understood as an instance of the
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
5973:, the commutator equations can be converted into the differential equations
3433:
275:
6734:
654:
6648:
does not, in this picture: Rather, the momentum operator is a constant
6269:
6781:
6660:
of the wavefunction for which the expectation value is calculated. An
6182:{\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\hat {x}}).}
6661:
4902:
This result is actually in exact accord with the classical equation.
5886:{\displaystyle im=\hbar {\hat {p}},\qquad i=-\hbar V'({\hat {x}}).}
6709:
5895:
Assuming that observables of the coordinate and momentum obey the
4906:
Derivation of the Schrödinger equation from the Ehrenfest theorems
1955:. In that case, the expected position and expected momentum will
1405:{\displaystyle -V'\left(\left\langle x\right\rangle \right),}
1283:
The reason is that Ehrenfest's theorem is closely related to
2432:
where we are integrating over all of space. If we apply the
1859:{\displaystyle V'\left(\left\langle x\right\rangle \right)}
1741:{\displaystyle V'\left(\left\langle x\right\rangle \right)}
6803:, Graduate Texts in Mathematics, vol. 267, Springer,
6664:
example of an operator which does have time dependence is
1603:, while in the Ehrenfest theorem it is in the form of
6587:
6399:
6354:
6105:
5979:
5911:
5747:
5646:
5249:
4920:
4455:
4403:
4359:
4342:{\displaystyle (\langle X\rangle ,\langle P\rangle )}
4311:
3907:
3726:
3540:
3442:
3282:
3061:
3033:
3005:
2884:
2650:
2505:
2442:
1984:
1920:
1872:
1828:
1801:
1754:
1710:
1685:
1665:
1642:
1609:
1576:
1549:
1524:
1497:
1468:
1418:
1368:
1355:{\displaystyle (\langle x\rangle ,\langle p\rangle )}
1324:
1139:
994:
962:
919:
39:
1914:
the same, since both will be approximately equal to
5966:{\displaystyle {\hat {H}}=H({\hat {x}},{\hat {p}})}
4301:As explained in the introduction, this result does
3516:{\displaystyle H(x,p,t)={\frac {p^{2}}{2m}}+V(x,t)}
3055:from the left, or taking the expectation value, so
956:on a massive particle moving in a scalar potential
6758:"Commutation relations for functions of operators"
6602:
6573:
6385:
6181:
6088:
5965:
5885:
5724:
5624:
5229:
4892:
4430:
4389:
4341:
4291:
3887:
3694:
3515:
3413:
3244:
3047:
3019:
2991:
2852:
2613:
2489:
2424:
1947:
1902:
1858:
1814:
1784:
1740:
1696:
1671:
1648:
1628:
1595:
1562:
1535:
1510:
1483:
1455:{\displaystyle -\left\langle V'(x)\right\rangle .}
1454:
1404:
1354:
1242:
1104:
977:
948:
100:
6756:Transtrum, M. K.; Van Huele, J. F. O. S. (2005).
6386:{\displaystyle \phi ^{*}=\langle \phi ,x\rangle }
6295:. World Scientific Pub Co Inc. pp. 108â109.
5527:
5508:
5482:
5463:
5416:
5392:
5345:
5326:
5300:
5281:
4353:, because the right-hand side of the formula is
2642:. Placing this into the above equation we have
1903:{\displaystyle \left\langle V'(x)\right\rangle }
1785:{\displaystyle \left\langle V'(x)\right\rangle }
6632:Although the expectation value of the momentum
6210:derivation of the Koopmanâvon Neumann mechanics
6680:is just the (non-operator) time, a parameter.
890:, named after Austrian theoretical physicist
867:
8:
6690:
6688:
6686:
6526:
6512:
6486:
6455:
6429:
6415:
6380:
6368:
6208:. Therefore, this derivation as well as the
5713:
5612:
5564:
4883:
4877:
4854:
4839:
4524:
4506:
4481:
4475:
4416:
4410:
4381:
4360:
4333:
4327:
4321:
4315:
4279:
4273:
3933:
3927:
3748:
3742:
3686:
3653:
3601:
3583:
3562:
3556:
3313:
3298:
3034:
3014:
2813:
2795:
2672:
2666:
2010:
2004:
1623:
1610:
1584:
1577:
1346:
1340:
1334:
1328:
1200:
1182:
1161:
1155:
1060:
1054:
1031:
1025:
1019:
1013:
95:
69:
3428:For the very general example of a massive
1038:
1037:
874:
860:
18:
6708:
6589:
6588:
6586:
6559:
6535:
6518:
6492:
6478:
6467:
6466:
6461:
6435:
6421:
6400:
6398:
6359:
6353:
6162:
6161:
6136:
6125:
6124:
6121:
6107:
6106:
6104:
6028:
5983:
5978:
5949:
5948:
5934:
5933:
5913:
5912:
5910:
5866:
5865:
5831:
5830:
5816:
5815:
5794:
5793:
5773:
5772:
5758:
5757:
5746:
5696:
5685:
5684:
5657:
5645:
5604:
5590:
5589:
5570:
5532:
5526:
5525:
5514:
5513:
5507:
5506:
5481:
5480:
5469:
5468:
5462:
5461:
5441:
5415:
5414:
5399:
5397:
5391:
5390:
5350:
5344:
5343:
5332:
5331:
5325:
5324:
5299:
5298:
5287:
5286:
5280:
5279:
5259:
5250:
5248:
5184:
5183:
5107:
5106:
5069:
5029:
5028:
4966:
4965:
4928:
4921:
4919:
4867:
4818:
4797:
4778:
4737:
4702:
4696:
4666:
4604:
4598:
4568:
4534:
4491:
4460:
4456:
4454:
4402:
4358:
4310:
4232:
4164:
4153:
4106:
4082:
4022:
4011:
3974:
3950:
3912:
3908:
3906:
3828:
3822:
3785:
3761:
3727:
3725:
3638:
3611:
3568:
3541:
3539:
3476:
3470:
3441:
3359:
3323:
3283:
3281:
3187:
3130:
3075:
3060:
3040:
3032:
3006:
3004:
2947:
2915:
2885:
2883:
2823:
2780:
2753:
2737:
2700:
2678:
2651:
2649:
2599:
2580:
2568:
2558:
2539:
2516:
2506:
2504:
2499:By taking the complex conjugate we find
2466:
2443:
2441:
2409:
2404:
2380:
2367:
2333:
2317:
2312:
2285:
2275:
2249:
2244:
2220:
2207:
2188:
2183:
2156:
2146:
2127:
2122:
2095:
2085:
2059:
2054:
2042:
2020:
1989:
1985:
1983:
1936:
1919:
1871:
1827:
1806:
1800:
1753:
1709:
1684:
1664:
1641:
1617:
1608:
1587:
1575:
1554:
1548:
1523:
1502:
1496:
1467:
1417:
1367:
1323:
1210:
1167:
1140:
1138:
1039:
998:
993:
961:
918:
87:
76:
75:
61:
46:
38:
1257:is some quantum mechanical operator and
6549:
6506:
6449:
6228:
6198:Koopmanâvon Neumann classical mechanics
5851:
5790:
5650:
4845:
4827:
4746:
4675:
4577:
4500:
4390:{\displaystyle \langle F(x,t)\rangle ,}
3647:
3577:
3368:
3196:
2956:
2789:
2687:
2589:
2548:
2475:
1176:
43:
26:
6676:is the ordinary position operator and
4431:{\displaystyle F(\langle X\rangle ,t)}
1967:Suppose some system is presently in a
913:to the expectation value of the force
3534:. Using Ehrenfest's theorem, we have
1963:Derivation in the Schrödinger picture
1629:{\displaystyle \langle x^{2}\rangle }
1596:{\displaystyle \langle x\rangle ^{2}}
7:
6333:
6235:
2870:Derivation in the Heisenberg picture
2862:Often (but not always) the operator
1412:which is typically not the same as
1309:, provided fluctuations are small.
6556:
6406:
6402:
6054:
6031:
6009:
5986:
5701:
5663:
5609:
5567:
5538:
5503:
5487:
5447:
5421:
5387:
5356:
5321:
5305:
5265:
5203:
5152:
5123:
5089:
5045:
5011:
4982:
4948:
4545:
4537:
4238:
4234:
4202:
4170:
4166:
4150:
4121:
4112:
4108:
4079:
4060:
4028:
4024:
4008:
3989:
3980:
3976:
3947:
3864:
3834:
3830:
3819:
3800:
3791:
3787:
3758:
3622:
3614:
3343:
3326:
3231:
3179:
3163:
3150:
3133:
3123:
3107:
3067:
3037:
3011:
2935:
2918:
2834:
2826:
2764:
2756:
2727:
2697:
2596:
2555:
2524:
2513:
2509:
2484:
2454:
2449:
2446:
2391:
2386:
2383:
2364:
2344:
2336:
2309:
2293:
2282:
2278:
2231:
2226:
2223:
2204:
2180:
2167:
2159:
2143:
2119:
2103:
2092:
2088:
2051:
2039:
1295:instead of a commutator. Dirac's
1221:
1213:
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
6801:Quantum Theory for Mathematicians
6610:is the Hamiltonian operator, and
6293:Introduction to Quantum Mechanics
3527:is the position of the particle.
1704:is linear. In that special case,
1491:is cubic, (i.e. proportional to
6762:Journal of Mathematical Physics
6096:whose solution is the familiar
6027:
5808:
6727:10.1103/PhysRevLett.109.190403
6594:
6519:
6479:
6472:
6462:
6422:
6173:
6167:
6158:
6130:
6112:
6080:
6074:
6049:
6037:
6004:
5992:
5960:
5954:
5939:
5930:
5918:
5897:canonical commutation relation
5877:
5871:
5862:
5842:
5836:
5821:
5812:
5799:
5784:
5778:
5763:
5754:
5710:
5704:
5697:
5690:
5605:
5601:
5595:
5586:
5571:
5519:
5474:
5404:
5337:
5292:
5212:
5206:
5195:
5189:
5180:
5161:
5155:
5132:
5126:
5112:
5098:
5092:
5054:
5048:
5034:
5020:
5014:
4991:
4985:
4971:
4957:
4951:
4775:
4763:
4638:
4626:
4521:
4509:
4425:
4407:
4378:
4366:
4336:
4312:
4262:
4250:
4194:
4182:
4103:
4091:
4052:
4040:
3971:
3959:
3867:
3861:
3849:
3843:
3782:
3770:
3683:
3680:
3668:
3656:
3598:
3586:
3510:
3498:
3464:
3446:
3400:
3391:
3385:
3379:
3338:
3332:
3310:
3304:
3223:
3214:
3208:
3202:
3145:
3139:
3099:
3093:
3048:{\displaystyle \langle \Psi |}
3041:
3020:{\displaystyle |\Psi \rangle }
3007:
2983:
2974:
2968:
2962:
2930:
2924:
2909:
2903:
2810:
2798:
2724:
2706:
1942:
1929:
1892:
1886:
1774:
1768:
1543:is quadratic (proportional to
1478:
1472:
1462:If for example, the potential
1441:
1435:
1349:
1325:
1197:
1185:
1088:
1082:
972:
966:
943:
937:
556:Relativistic quantum mechanics
88:
81:
62:
1:
6827:Theorems in quantum mechanics
1313:Relation to classical physics
902:of the position and momentum
596:Quantum statistical mechanics
3901:on the second term, we have
3436:, the Hamiltonian is simply
1122:and the expectation of the
16:Theorem in quantum mechanics
1272:It is most apparent in the
566:Quantum information science
6848:
6603:{\displaystyle {\hat {H}}}
1656:and is therefore nonzero.
1126:of that operator with the
1978:, that is, by definition
1948:{\displaystyle V'(x_{0})}
6217:Ehrenfest time and chaos
4444:correspondence principle
1278:correspondence principle
949:{\displaystyle F=-V'(x)}
601:Quantum machine learning
354:Wheeler's delayed-choice
6799:Hall, Brian C. (2013),
6697:Physical Review Letters
311:LeggettâGarg inequality
6604:
6575:
6387:
6291:Smith, Henrik (1991).
6250:Zeitschrift fĂŒr Physik
6183:
6090:
5967:
5887:
5726:
5626:
5231:
4894:
4432:
4391:
4343:
4293:
3889:
3696:
3517:
3415:
3246:
3049:
3021:
2993:
2854:
2615:
2491:
2426:
1949:
1904:
1860:
1816:
1786:
1742:
1698:
1673:
1650:
1630:
1597:
1564:
1537:
1512:
1485:
1456:
1406:
1356:
1244:
1106:
979:
950:
102:
6605:
6576:
6388:
6184:
6091:
5968:
5888:
5727:
5627:
5232:
4895:
4433:
4392:
4344:
4294:
3890:
3697:
3518:
3416:
3247:
3050:
3022:
2994:
2855:
2616:
2492:
2427:
1950:
1905:
1861:
1817:
1815:{\displaystyle x_{0}}
1787:
1743:
1699:
1674:
1651:
1631:
1598:
1565:
1563:{\displaystyle x^{2}}
1538:
1513:
1511:{\displaystyle x^{3}}
1486:
1457:
1407:
1357:
1291:, which involves the
1289:Hamiltonian mechanics
1245:
1107:
980:
951:
296:Elitzur–Vaidman
286:Davisson–Germer
103:
6832:Mathematical physics
6585:
6397:
6352:
6193:Schrödinger equation
6103:
5977:
5909:
5745:
5644:
5247:
4918:
4912:Schrödinger equation
4453:
4401:
4357:
4309:
3905:
3724:
3538:
3440:
3280:
3059:
3031:
3003:
2882:
2648:
2503:
2440:
2434:Schrödinger equation
1982:
1918:
1870:
1826:
1799:
1752:
1708:
1683:
1663:
1640:
1607:
1574:
1547:
1522:
1495:
1484:{\displaystyle V(x)}
1466:
1416:
1366:
1322:
1137:
992:
978:{\displaystyle V(x)}
960:
917:
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
6774:2005JMP....46f3510T
6719:2012PhRvL.109s0403B
6316:Wheeler, Nicholas.
6262:1927ZPhy...45..455E
6206:classical mechanics
6098:quantum Hamiltonian
5239:Application of the
4351:Newton's second law
3897:After applying the
3704:since the operator
3027:from the right and
1285:Liouville's theorem
894:, relates the time
349:Stern–Gerlach
146:Classical mechanics
6600:
6571:
6383:
6270:10.1007/BF01329203
6179:
6086:
5963:
5883:
5722:
5622:
5620:
5227:
5225:
4890:
4888:
4428:
4387:
4339:
4305:say that the pair
4289:
4287:
3885:
3692:
3513:
3411:
3242:
3045:
3017:
2989:
2876:Heisenberg picture
2850:
2611:
2487:
2422:
2420:
1945:
1900:
1856:
1812:
1782:
1738:
1697:{\displaystyle V'}
1694:
1669:
1646:
1626:
1593:
1560:
1536:{\displaystyle V'}
1533:
1508:
1481:
1452:
1402:
1352:
1274:Heisenberg picture
1240:
1117:quantum mechanical
1102:
975:
946:
900:expectation values
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
6782:10.1063/1.1924703
6597:
6553:
6510:
6475:
6453:
6413:
6170:
6150:
6133:
6115:
6061:
6016:
5957:
5942:
5921:
5874:
5839:
5824:
5802:
5781:
5766:
5718:
5693:
5675:
5598:
5550:
5522:
5477:
5459:
5412:
5407:
5368:
5340:
5295:
5277:
5192:
5115:
5082:
5037:
4974:
4941:
4875:
4837:
4791:
4756:
4716:
4679:
4618:
4581:
4552:
4504:
4473:
4245:
4207:
4177:
4126:
4119:
4065:
4035:
3994:
3987:
3925:
3881:
3872:
3841:
3805:
3798:
3740:
3651:
3629:
3581:
3554:
3490:
3372:
3350:
3296:
3254:One may pull the
3200:
3157:
3088:
2960:
2942:
2898:
2841:
2793:
2771:
2732:
2691:
2664:
2593:
2552:
2531:
2479:
2461:
2398:
2351:
2300:
2238:
2174:
2110:
2033:
2002:
1679:is quadratic and
1672:{\displaystyle V}
1649:{\displaystyle x}
1267:expectation value
1236:
1228:
1180:
1153:
1098:
1052:
1011:
888:Ehrenfest theorem
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
6839:
6813:
6786:
6785:
6753:
6747:
6746:
6712:
6692:
6681:
6679:
6675:
6671:
6647:
6639:
6630:
6624:
6622:
6618:
6613:
6609:
6607:
6606:
6601:
6599:
6598:
6590:
6580:
6578:
6577:
6572:
6564:
6563:
6554:
6552:
6544:
6536:
6522:
6511:
6509:
6501:
6493:
6482:
6477:
6476:
6468:
6465:
6454:
6452:
6444:
6436:
6425:
6414:
6412:
6401:
6392:
6390:
6389:
6384:
6364:
6363:
6347:braâket notation
6343:
6337:
6331:
6325:
6324:
6322:
6313:
6307:
6306:
6288:
6282:
6281:
6256:(7â8): 455â457.
6245:
6239:
6233:
6188:
6186:
6185:
6180:
6172:
6171:
6163:
6151:
6149:
6141:
6140:
6135:
6134:
6126:
6122:
6117:
6116:
6108:
6095:
6093:
6092:
6087:
6073:
6062:
6060:
6052:
6029:
6017:
6015:
6007:
5984:
5972:
5970:
5969:
5964:
5959:
5958:
5950:
5944:
5943:
5935:
5923:
5922:
5914:
5904:
5892:
5890:
5889:
5884:
5876:
5875:
5867:
5861:
5841:
5840:
5832:
5826:
5825:
5817:
5804:
5803:
5795:
5783:
5782:
5774:
5768:
5767:
5759:
5740:
5735:
5731:
5729:
5728:
5723:
5716:
5700:
5695:
5694:
5686:
5680:
5676:
5674:
5666:
5658:
5639:
5631:
5629:
5628:
5623:
5621:
5608:
5600:
5599:
5591:
5585:
5574:
5556:
5552:
5551:
5549:
5541:
5533:
5531:
5530:
5524:
5523:
5515:
5512:
5511:
5494:
5490:
5486:
5485:
5479:
5478:
5470:
5467:
5466:
5460:
5458:
5450:
5442:
5428:
5424:
5420:
5419:
5413:
5408:
5400:
5398:
5396:
5395:
5374:
5370:
5369:
5367:
5359:
5351:
5349:
5348:
5342:
5341:
5333:
5330:
5329:
5312:
5308:
5304:
5303:
5297:
5296:
5288:
5285:
5284:
5278:
5276:
5268:
5260:
5236:
5234:
5233:
5228:
5226:
5219:
5215:
5194:
5193:
5185:
5179:
5168:
5164:
5139:
5135:
5117:
5116:
5108:
5105:
5101:
5083:
5081:
5070:
5061:
5057:
5039:
5038:
5030:
5027:
5023:
4998:
4994:
4976:
4975:
4967:
4964:
4960:
4942:
4940:
4929:
4899:
4897:
4896:
4891:
4889:
4876:
4868:
4860:
4838:
4836:
4819:
4811:
4807:
4803:
4802:
4801:
4792:
4790:
4779:
4757:
4755:
4738:
4730:
4726:
4722:
4718:
4717:
4715:
4707:
4706:
4697:
4680:
4678:
4667:
4659:
4649:
4645:
4641:
4619:
4617:
4609:
4608:
4599:
4582:
4580:
4569:
4561:
4557:
4553:
4551:
4543:
4535:
4505:
4503:
4492:
4474:
4472:
4461:
4437:
4435:
4434:
4429:
4396:
4394:
4393:
4388:
4348:
4346:
4345:
4340:
4298:
4296:
4295:
4290:
4288:
4269:
4265:
4246:
4244:
4233:
4217:
4205:
4201:
4197:
4178:
4176:
4165:
4158:
4157:
4136:
4124:
4120:
4118:
4107:
4087:
4086:
4063:
4059:
4055:
4036:
4034:
4023:
4016:
4015:
3992:
3988:
3986:
3975:
3955:
3954:
3926:
3924:
3913:
3894:
3892:
3891:
3886:
3879:
3870:
3842:
3840:
3829:
3827:
3826:
3803:
3799:
3797:
3786:
3766:
3765:
3741:
3739:
3728:
3719:
3711:
3707:
3701:
3699:
3698:
3693:
3652:
3650:
3639:
3634:
3630:
3628:
3620:
3612:
3582:
3580:
3569:
3555:
3553:
3542:
3533:
3526:
3522:
3520:
3519:
3514:
3491:
3489:
3481:
3480:
3471:
3420:
3418:
3417:
3412:
3407:
3403:
3373:
3371:
3360:
3355:
3351:
3349:
3341:
3324:
3297:
3295:
3284:
3275:
3274:
3272:
3271:
3266:
3263:
3251:
3249:
3248:
3243:
3238:
3234:
3230:
3226:
3201:
3199:
3188:
3170:
3166:
3162:
3158:
3156:
3148:
3131:
3114:
3110:
3106:
3102:
3089:
3087:
3076:
3054:
3052:
3051:
3046:
3044:
3026:
3024:
3023:
3018:
3010:
2998:
2996:
2995:
2990:
2961:
2959:
2948:
2943:
2941:
2933:
2916:
2899:
2897:
2886:
2865:
2859:
2857:
2856:
2851:
2846:
2842:
2840:
2832:
2824:
2794:
2792:
2781:
2776:
2772:
2770:
2762:
2754:
2742:
2741:
2730:
2705:
2704:
2692:
2690:
2679:
2665:
2663:
2652:
2633:
2620:
2618:
2617:
2612:
2604:
2603:
2594:
2592:
2581:
2573:
2572:
2563:
2562:
2553:
2551:
2540:
2532:
2530:
2522:
2521:
2520:
2507:
2496:
2494:
2493:
2488:
2480:
2478:
2467:
2462:
2460:
2452:
2444:
2431:
2429:
2428:
2423:
2421:
2414:
2413:
2403:
2399:
2397:
2389:
2381:
2372:
2371:
2356:
2352:
2350:
2342:
2334:
2322:
2321:
2305:
2301:
2299:
2291:
2290:
2289:
2276:
2261:
2254:
2253:
2243:
2239:
2237:
2229:
2221:
2212:
2211:
2193:
2192:
2179:
2175:
2173:
2165:
2157:
2151:
2150:
2132:
2131:
2115:
2111:
2109:
2101:
2100:
2099:
2086:
2071:
2064:
2063:
2047:
2046:
2034:
2032:
2021:
2003:
2001:
1990:
1977:
1973:
1954:
1952:
1951:
1946:
1941:
1940:
1928:
1909:
1907:
1906:
1901:
1899:
1895:
1885:
1865:
1863:
1862:
1857:
1855:
1851:
1836:
1821:
1819:
1818:
1813:
1811:
1810:
1791:
1789:
1788:
1783:
1781:
1777:
1767:
1747:
1745:
1744:
1739:
1737:
1733:
1718:
1703:
1701:
1700:
1695:
1693:
1678:
1676:
1675:
1670:
1655:
1653:
1652:
1647:
1635:
1633:
1632:
1627:
1622:
1621:
1602:
1600:
1599:
1594:
1592:
1591:
1569:
1567:
1566:
1561:
1559:
1558:
1542:
1540:
1539:
1534:
1532:
1517:
1515:
1514:
1509:
1507:
1506:
1490:
1488:
1487:
1482:
1461:
1459:
1458:
1453:
1448:
1444:
1434:
1411:
1409:
1408:
1403:
1398:
1394:
1379:
1361:
1359:
1358:
1353:
1304:
1264:
1256:
1249:
1247:
1246:
1241:
1234:
1233:
1229:
1227:
1219:
1211:
1181:
1179:
1168:
1154:
1152:
1141:
1111:
1109:
1108:
1103:
1096:
1095:
1091:
1081:
1053:
1051:
1040:
1012:
1010:
999:
984:
982:
981:
976:
955:
953:
952:
947:
936:
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
6847:
6846:
6842:
6841:
6840:
6838:
6837:
6836:
6817:
6816:
6811:
6798:
6795:
6790:
6789:
6755:
6754:
6750:
6694:
6693:
6684:
6677:
6673:
6665:
6650:linear operator
6645:
6633:
6631:
6627:
6620:
6616:
6611:
6583:
6582:
6555:
6545:
6537:
6502:
6494:
6445:
6437:
6405:
6395:
6394:
6355:
6350:
6349:
6344:
6340:
6332:
6328:
6320:
6315:
6314:
6310:
6303:
6290:
6289:
6285:
6247:
6246:
6242:
6234:
6230:
6225:
6204:formulation of
6200:, which is the
6142:
6123:
6101:
6100:
6066:
6053:
6030:
6008:
5985:
5975:
5974:
5907:
5906:
5899:
5854:
5743:
5742:
5738:
5733:
5667:
5659:
5653:
5642:
5641:
5637:
5634:Stone's theorem
5619:
5618:
5578:
5557:
5542:
5534:
5502:
5498:
5451:
5443:
5440:
5436:
5433:
5432:
5386:
5382:
5375:
5360:
5352:
5320:
5316:
5269:
5261:
5258:
5254:
5245:
5244:
5224:
5223:
5202:
5198:
5172:
5151:
5147:
5140:
5122:
5118:
5088:
5084:
5074:
5066:
5065:
5044:
5040:
5010:
5006:
4999:
4981:
4977:
4947:
4943:
4933:
4916:
4915:
4908:
4887:
4886:
4858:
4857:
4823:
4809:
4808:
4793:
4783:
4762:
4758:
4742:
4728:
4727:
4708:
4698:
4689:
4685:
4681:
4671:
4657:
4656:
4610:
4600:
4591:
4587:
4583:
4573:
4559:
4558:
4544:
4536:
4530:
4496:
4484:
4465:
4451:
4450:
4399:
4398:
4355:
4354:
4307:
4306:
4286:
4285:
4237:
4228:
4224:
4215:
4214:
4169:
4163:
4159:
4149:
4134:
4133:
4111:
4078:
4027:
4021:
4017:
4007:
3979:
3946:
3936:
3917:
3903:
3902:
3833:
3818:
3790:
3757:
3732:
3722:
3721:
3713:
3709:
3705:
3643:
3621:
3613:
3607:
3573:
3546:
3536:
3535:
3531:
3524:
3482:
3472:
3438:
3437:
3426:
3424:General example
3378:
3374:
3364:
3342:
3325:
3319:
3288:
3278:
3277:
3267:
3264:
3259:
3258:
3256:
3255:
3192:
3186:
3182:
3178:
3174:
3149:
3132:
3126:
3122:
3118:
3080:
3074:
3070:
3066:
3062:
3057:
3056:
3029:
3028:
3001:
3000:
2952:
2934:
2917:
2890:
2880:
2879:
2872:
2863:
2833:
2825:
2819:
2785:
2763:
2755:
2749:
2733:
2696:
2683:
2656:
2646:
2645:
2624:
2595:
2585:
2564:
2554:
2544:
2523:
2512:
2508:
2501:
2500:
2471:
2453:
2445:
2438:
2437:
2436:, we find that
2419:
2418:
2405:
2390:
2382:
2376:
2363:
2343:
2335:
2329:
2313:
2292:
2281:
2277:
2271:
2259:
2258:
2245:
2230:
2222:
2216:
2203:
2184:
2166:
2158:
2152:
2142:
2123:
2102:
2091:
2087:
2081:
2069:
2068:
2055:
2038:
2025:
2013:
1994:
1980:
1979:
1975:
1971:
1965:
1932:
1921:
1916:
1915:
1878:
1877:
1873:
1868:
1867:
1841:
1837:
1829:
1824:
1823:
1802:
1797:
1796:
1760:
1759:
1755:
1750:
1749:
1723:
1719:
1711:
1706:
1705:
1686:
1681:
1680:
1661:
1660:
1638:
1637:
1613:
1605:
1604:
1583:
1572:
1571:
1550:
1545:
1544:
1525:
1520:
1519:
1498:
1493:
1492:
1464:
1463:
1427:
1426:
1422:
1414:
1413:
1384:
1380:
1372:
1364:
1363:
1320:
1319:
1315:
1300:
1293:Poisson bracket
1258:
1254:
1251:
1220:
1212:
1206:
1172:
1145:
1135:
1134:
1113:
1074:
1073:
1069:
1044:
1003:
990:
989:
958:
957:
929:
915:
914:
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de BroglieâBohm
462:
460:Interpretations
452:
451:
450:
420:
412:
411:
410:
368:
360:
359:
358:
325:
281:CHSH inequality
270:
262:
261:
260:
189:Complementarity
183:
175:
174:
173:
141:
112:
51:
35:
34:
17:
12:
11:
5:
6845:
6843:
6835:
6834:
6829:
6819:
6818:
6815:
6814:
6810:978-1461471158
6809:
6794:
6791:
6788:
6787:
6748:
6703:(19): 190403.
6682:
6658:time evolution
6625:
6596:
6593:
6570:
6567:
6562:
6558:
6551:
6548:
6543:
6540:
6534:
6531:
6528:
6525:
6521:
6517:
6514:
6508:
6505:
6500:
6497:
6491:
6488:
6485:
6481:
6474:
6471:
6464:
6460:
6457:
6451:
6448:
6443:
6440:
6434:
6431:
6428:
6424:
6420:
6417:
6411:
6408:
6404:
6382:
6379:
6376:
6373:
6370:
6367:
6362:
6358:
6338:
6326:
6308:
6302:978-9810204754
6301:
6283:
6240:
6227:
6226:
6224:
6221:
6178:
6175:
6169:
6166:
6160:
6157:
6154:
6148:
6145:
6139:
6132:
6129:
6120:
6114:
6111:
6085:
6082:
6079:
6076:
6072:
6069:
6065:
6059:
6056:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6026:
6023:
6020:
6014:
6011:
6006:
6003:
6000:
5997:
5994:
5991:
5988:
5982:
5962:
5956:
5953:
5947:
5941:
5938:
5932:
5929:
5926:
5920:
5917:
5882:
5879:
5873:
5870:
5864:
5860:
5857:
5853:
5850:
5847:
5844:
5838:
5835:
5829:
5823:
5820:
5814:
5811:
5807:
5801:
5798:
5792:
5789:
5786:
5780:
5777:
5771:
5765:
5762:
5756:
5753:
5750:
5721:
5715:
5712:
5709:
5706:
5703:
5699:
5692:
5689:
5683:
5679:
5673:
5670:
5665:
5662:
5656:
5652:
5649:
5617:
5614:
5611:
5607:
5603:
5597:
5594:
5588:
5584:
5581:
5577:
5573:
5569:
5566:
5563:
5560:
5558:
5555:
5548:
5545:
5540:
5537:
5529:
5521:
5518:
5510:
5505:
5501:
5497:
5493:
5489:
5484:
5476:
5473:
5465:
5457:
5454:
5449:
5446:
5439:
5435:
5434:
5431:
5427:
5423:
5418:
5411:
5406:
5403:
5394:
5389:
5385:
5381:
5378:
5376:
5373:
5366:
5363:
5358:
5355:
5347:
5339:
5336:
5328:
5323:
5319:
5315:
5311:
5307:
5302:
5294:
5291:
5283:
5275:
5272:
5267:
5264:
5257:
5253:
5252:
5222:
5218:
5214:
5211:
5208:
5205:
5201:
5197:
5191:
5188:
5182:
5178:
5175:
5171:
5167:
5163:
5160:
5157:
5154:
5150:
5146:
5143:
5141:
5138:
5134:
5131:
5128:
5125:
5121:
5114:
5111:
5104:
5100:
5097:
5094:
5091:
5087:
5080:
5077:
5073:
5068:
5067:
5064:
5060:
5056:
5053:
5050:
5047:
5043:
5036:
5033:
5026:
5022:
5019:
5016:
5013:
5009:
5005:
5002:
5000:
4997:
4993:
4990:
4987:
4984:
4980:
4973:
4970:
4963:
4959:
4956:
4953:
4950:
4946:
4939:
4936:
4932:
4927:
4924:
4923:
4907:
4904:
4885:
4882:
4879:
4874:
4871:
4866:
4863:
4861:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4835:
4832:
4829:
4826:
4822:
4817:
4814:
4812:
4810:
4806:
4800:
4796:
4789:
4786:
4782:
4777:
4774:
4771:
4768:
4765:
4761:
4754:
4751:
4748:
4745:
4741:
4736:
4733:
4731:
4729:
4725:
4721:
4714:
4711:
4705:
4701:
4695:
4692:
4688:
4684:
4677:
4674:
4670:
4665:
4662:
4660:
4658:
4655:
4652:
4648:
4644:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4616:
4613:
4607:
4603:
4597:
4594:
4590:
4586:
4579:
4576:
4572:
4567:
4564:
4562:
4560:
4556:
4550:
4547:
4542:
4539:
4533:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4502:
4499:
4495:
4490:
4487:
4485:
4483:
4480:
4477:
4471:
4468:
4464:
4459:
4458:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4284:
4281:
4278:
4275:
4272:
4268:
4264:
4261:
4258:
4255:
4252:
4249:
4243:
4240:
4236:
4231:
4227:
4223:
4220:
4218:
4216:
4213:
4210:
4204:
4200:
4196:
4193:
4190:
4187:
4184:
4181:
4175:
4172:
4168:
4162:
4156:
4152:
4148:
4145:
4142:
4139:
4137:
4135:
4132:
4129:
4123:
4117:
4114:
4110:
4105:
4102:
4099:
4096:
4093:
4090:
4085:
4081:
4077:
4074:
4071:
4068:
4062:
4058:
4054:
4051:
4048:
4045:
4042:
4039:
4033:
4030:
4026:
4020:
4014:
4010:
4006:
4003:
4000:
3997:
3991:
3985:
3982:
3978:
3973:
3970:
3967:
3964:
3961:
3958:
3953:
3949:
3945:
3942:
3939:
3937:
3935:
3932:
3929:
3923:
3920:
3916:
3911:
3910:
3884:
3878:
3875:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3839:
3836:
3832:
3825:
3821:
3817:
3814:
3811:
3808:
3802:
3796:
3793:
3789:
3784:
3781:
3778:
3775:
3772:
3769:
3764:
3760:
3756:
3753:
3750:
3747:
3744:
3738:
3735:
3731:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3649:
3646:
3642:
3637:
3633:
3627:
3624:
3619:
3616:
3610:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3579:
3576:
3572:
3567:
3564:
3561:
3558:
3552:
3549:
3545:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3488:
3485:
3479:
3475:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3425:
3422:
3410:
3406:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3377:
3370:
3367:
3363:
3358:
3354:
3348:
3345:
3340:
3337:
3334:
3331:
3328:
3322:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3294:
3291:
3287:
3241:
3237:
3233:
3229:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3198:
3195:
3191:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3155:
3152:
3147:
3144:
3141:
3138:
3135:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3101:
3098:
3095:
3092:
3086:
3083:
3079:
3073:
3069:
3065:
3043:
3039:
3036:
3016:
3013:
3009:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2958:
2955:
2951:
2946:
2940:
2937:
2932:
2929:
2926:
2923:
2920:
2914:
2911:
2908:
2905:
2902:
2896:
2893:
2889:
2871:
2868:
2849:
2845:
2839:
2836:
2831:
2828:
2822:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2791:
2788:
2784:
2779:
2775:
2769:
2766:
2761:
2758:
2752:
2748:
2745:
2740:
2736:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2703:
2699:
2695:
2689:
2686:
2682:
2677:
2674:
2671:
2668:
2662:
2659:
2655:
2634:, because the
2610:
2607:
2602:
2598:
2591:
2588:
2584:
2579:
2576:
2571:
2567:
2561:
2557:
2550:
2547:
2543:
2538:
2535:
2529:
2526:
2519:
2515:
2511:
2486:
2483:
2477:
2474:
2470:
2465:
2459:
2456:
2451:
2448:
2417:
2412:
2408:
2402:
2396:
2393:
2388:
2385:
2379:
2375:
2370:
2366:
2362:
2359:
2355:
2349:
2346:
2341:
2338:
2332:
2328:
2325:
2320:
2316:
2311:
2308:
2304:
2298:
2295:
2288:
2284:
2280:
2274:
2270:
2267:
2264:
2262:
2260:
2257:
2252:
2248:
2242:
2236:
2233:
2228:
2225:
2219:
2215:
2210:
2206:
2202:
2199:
2196:
2191:
2187:
2182:
2178:
2172:
2169:
2164:
2161:
2155:
2149:
2145:
2141:
2138:
2135:
2130:
2126:
2121:
2118:
2114:
2108:
2105:
2098:
2094:
2090:
2084:
2080:
2077:
2074:
2072:
2070:
2067:
2062:
2058:
2053:
2050:
2045:
2041:
2037:
2031:
2028:
2024:
2019:
2016:
2014:
2012:
2009:
2006:
2000:
1997:
1993:
1988:
1987:
1964:
1961:
1944:
1939:
1935:
1931:
1927:
1924:
1898:
1894:
1891:
1888:
1884:
1881:
1876:
1854:
1850:
1847:
1844:
1840:
1835:
1832:
1809:
1805:
1780:
1776:
1773:
1770:
1766:
1763:
1758:
1736:
1732:
1729:
1726:
1722:
1717:
1714:
1692:
1689:
1668:
1645:
1625:
1620:
1616:
1612:
1590:
1586:
1582:
1579:
1557:
1553:
1531:
1528:
1505:
1501:
1480:
1477:
1474:
1471:
1451:
1447:
1443:
1440:
1437:
1433:
1430:
1425:
1421:
1401:
1397:
1393:
1390:
1387:
1383:
1378:
1375:
1371:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1314:
1311:
1239:
1232:
1226:
1223:
1218:
1215:
1209:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1178:
1175:
1171:
1166:
1163:
1160:
1157:
1151:
1148:
1144:
1132:
1130:of the system
1101:
1094:
1090:
1087:
1084:
1080:
1077:
1072:
1068:
1065:
1062:
1059:
1056:
1050:
1047:
1043:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1009:
1006:
1002:
997:
987:
974:
971:
968:
965:
945:
942:
939:
935:
932:
928:
925:
922:
892:Paul Ehrenfest
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
612:
611:
608:
607:
604:
603:
598:
593:
588:
586:Density matrix
583:
578:
573:
568:
563:
558:
552:
549:
548:
545:
544:
540:
539:
534:
529:
524:
519:
514:
509:
508:
507:
506:
505:
490:
485:
480:
475:
470:
464:
463:
458:
457:
454:
453:
449:
448:
443:
438:
433:
428:
422:
421:
418:
417:
414:
413:
409:
408:
403:
398:
393:
388:
383:
377:
376:
375:
369:
366:
365:
362:
361:
357:
356:
351:
346:
340:
339:
338:
337:
336:
334:Delayed-choice
329:Quantum eraser
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
272:
271:
268:
267:
264:
263:
259:
258:
257:
256:
246:
241:
236:
231:
226:
221:
219:Quantum number
216:
211:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
160:
159:
158:
153:
148:
142:
139:
138:
135:
134:
133:
132:
127:
122:
114:
113:
108:
97:
94:
90:
83:
80:
74:
71:
68:
64:
57:
54:
50:
45:
42:
31:
30:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
6844:
6833:
6830:
6828:
6825:
6824:
6822:
6812:
6806:
6802:
6797:
6796:
6792:
6783:
6779:
6775:
6771:
6768:(6): 063510.
6767:
6763:
6759:
6752:
6749:
6744:
6740:
6736:
6732:
6728:
6724:
6720:
6716:
6711:
6706:
6702:
6698:
6691:
6689:
6687:
6683:
6669:
6663:
6659:
6655:
6654:Hilbert space
6651:
6643:
6640:, which is a
6637:
6629:
6626:
6591:
6568:
6565:
6560:
6546:
6541:
6538:
6532:
6529:
6523:
6515:
6503:
6498:
6495:
6489:
6483:
6469:
6458:
6446:
6441:
6438:
6432:
6426:
6418:
6409:
6377:
6374:
6371:
6365:
6360:
6356:
6348:
6342:
6339:
6335:
6330:
6327:
6319:
6312:
6309:
6304:
6298:
6294:
6287:
6284:
6279:
6275:
6271:
6267:
6263:
6259:
6255:
6251:
6244:
6241:
6238:Section 3.7.5
6237:
6232:
6229:
6222:
6220:
6218:
6213:
6211:
6207:
6203:
6202:Hilbert space
6199:
6194:
6189:
6176:
6164:
6155:
6152:
6146:
6143:
6137:
6127:
6118:
6109:
6099:
6083:
6077:
6070:
6067:
6063:
6057:
6046:
6043:
6040:
6034:
6024:
6021:
6018:
6012:
6001:
5998:
5995:
5989:
5980:
5951:
5945:
5936:
5927:
5924:
5915:
5903:
5898:
5893:
5880:
5868:
5858:
5855:
5848:
5845:
5833:
5827:
5818:
5809:
5805:
5796:
5787:
5775:
5769:
5760:
5751:
5748:
5741:are derived:
5719:
5707:
5687:
5681:
5677:
5671:
5668:
5660:
5654:
5647:
5635:
5615:
5592:
5582:
5579:
5575:
5561:
5559:
5553:
5546:
5543:
5535:
5516:
5499:
5495:
5491:
5471:
5455:
5452:
5444:
5437:
5429:
5425:
5409:
5401:
5383:
5379:
5377:
5371:
5364:
5361:
5353:
5334:
5317:
5313:
5309:
5289:
5273:
5270:
5262:
5255:
5242:
5237:
5220:
5216:
5209:
5199:
5186:
5176:
5173:
5169:
5165:
5158:
5148:
5144:
5142:
5136:
5129:
5119:
5109:
5102:
5095:
5085:
5078:
5075:
5071:
5062:
5058:
5051:
5041:
5031:
5024:
5017:
5007:
5003:
5001:
4995:
4988:
4978:
4968:
4961:
4954:
4944:
4937:
4934:
4930:
4925:
4913:
4905:
4903:
4900:
4880:
4872:
4869:
4864:
4862:
4851:
4848:
4842:
4833:
4830:
4824:
4820:
4815:
4813:
4804:
4798:
4794:
4787:
4784:
4780:
4772:
4769:
4766:
4759:
4752:
4749:
4743:
4739:
4734:
4732:
4723:
4719:
4712:
4709:
4703:
4699:
4693:
4690:
4686:
4682:
4672:
4668:
4663:
4661:
4653:
4650:
4646:
4642:
4635:
4632:
4629:
4623:
4620:
4614:
4611:
4605:
4601:
4595:
4592:
4588:
4584:
4574:
4570:
4565:
4563:
4554:
4548:
4540:
4531:
4527:
4518:
4515:
4512:
4497:
4493:
4488:
4486:
4478:
4469:
4466:
4462:
4447:
4445:
4441:
4440:approximately
4422:
4419:
4413:
4404:
4384:
4375:
4372:
4369:
4363:
4352:
4330:
4324:
4318:
4304:
4299:
4282:
4276:
4270:
4266:
4259:
4256:
4253:
4247:
4241:
4229:
4225:
4221:
4219:
4211:
4208:
4198:
4191:
4188:
4185:
4179:
4173:
4160:
4154:
4146:
4143:
4140:
4138:
4130:
4127:
4115:
4100:
4097:
4094:
4088:
4083:
4075:
4072:
4069:
4066:
4056:
4049:
4046:
4043:
4037:
4031:
4018:
4012:
4004:
4001:
3998:
3995:
3983:
3968:
3965:
3962:
3956:
3951:
3943:
3940:
3938:
3930:
3921:
3918:
3914:
3900:
3895:
3882:
3876:
3873:
3858:
3855:
3852:
3846:
3837:
3823:
3815:
3812:
3809:
3806:
3794:
3779:
3776:
3773:
3767:
3762:
3754:
3751:
3745:
3736:
3733:
3729:
3717:
3702:
3689:
3677:
3674:
3671:
3665:
3662:
3659:
3644:
3640:
3635:
3631:
3625:
3617:
3608:
3604:
3595:
3592:
3589:
3574:
3570:
3565:
3559:
3550:
3547:
3543:
3528:
3507:
3504:
3501:
3495:
3492:
3486:
3483:
3477:
3473:
3467:
3461:
3458:
3455:
3452:
3449:
3443:
3435:
3431:
3423:
3421:
3408:
3404:
3397:
3394:
3388:
3382:
3375:
3365:
3361:
3356:
3352:
3346:
3335:
3329:
3320:
3316:
3307:
3301:
3292:
3289:
3285:
3270:
3262:
3252:
3239:
3235:
3227:
3220:
3217:
3211:
3205:
3193:
3189:
3183:
3175:
3171:
3167:
3159:
3153:
3142:
3136:
3127:
3119:
3115:
3111:
3103:
3096:
3090:
3084:
3081:
3077:
3071:
3063:
2986:
2980:
2977:
2971:
2965:
2953:
2949:
2944:
2938:
2927:
2921:
2912:
2906:
2900:
2894:
2891:
2887:
2877:
2869:
2867:
2860:
2847:
2843:
2837:
2829:
2820:
2816:
2807:
2804:
2801:
2786:
2782:
2777:
2773:
2767:
2759:
2750:
2746:
2743:
2738:
2734:
2721:
2718:
2715:
2712:
2709:
2701:
2693:
2684:
2680:
2675:
2669:
2660:
2657:
2653:
2643:
2641:
2637:
2631:
2627:
2621:
2608:
2605:
2600:
2586:
2582:
2577:
2574:
2569:
2565:
2559:
2545:
2541:
2536:
2533:
2527:
2517:
2497:
2481:
2472:
2468:
2463:
2457:
2435:
2415:
2410:
2406:
2400:
2394:
2377:
2373:
2368:
2360:
2357:
2353:
2347:
2339:
2330:
2326:
2323:
2318:
2314:
2306:
2302:
2296:
2286:
2272:
2268:
2265:
2263:
2255:
2250:
2246:
2240:
2234:
2217:
2213:
2208:
2200:
2197:
2194:
2189:
2185:
2176:
2170:
2162:
2153:
2147:
2139:
2136:
2133:
2128:
2124:
2116:
2112:
2106:
2096:
2082:
2078:
2075:
2073:
2065:
2060:
2056:
2048:
2043:
2035:
2029:
2026:
2022:
2017:
2015:
2007:
1998:
1995:
1991:
1970:
1969:quantum state
1962:
1960:
1958:
1957:approximately
1937:
1933:
1925:
1922:
1913:
1896:
1889:
1882:
1879:
1874:
1852:
1848:
1845:
1842:
1838:
1833:
1830:
1807:
1803:
1793:
1778:
1771:
1764:
1761:
1756:
1734:
1730:
1727:
1724:
1720:
1715:
1712:
1690:
1687:
1666:
1657:
1643:
1618:
1614:
1588:
1580:
1555:
1551:
1529:
1526:
1503:
1499:
1475:
1469:
1449:
1445:
1438:
1431:
1428:
1423:
1419:
1399:
1395:
1391:
1388:
1385:
1381:
1376:
1373:
1369:
1343:
1337:
1331:
1312:
1310:
1308:
1307:approximately
1303:
1298:
1297:rule of thumb
1294:
1290:
1286:
1281:
1279:
1275:
1270:
1268:
1262:
1250:
1237:
1230:
1224:
1216:
1207:
1203:
1194:
1191:
1188:
1173:
1169:
1164:
1158:
1149:
1146:
1142:
1131:
1129:
1125:
1121:
1118:
1112:
1099:
1092:
1085:
1078:
1075:
1070:
1066:
1063:
1057:
1048:
1045:
1041:
1034:
1028:
1022:
1016:
1007:
1004:
1000:
995:
986:
969:
963:
940:
933:
930:
926:
923:
920:
912:
908:
905:
901:
897:
893:
889:
877:
872:
870:
865:
863:
858:
857:
855:
854:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
610:
609:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
576:Quantum chaos
574:
572:
569:
567:
564:
562:
559:
557:
554:
553:
547:
546:
538:
535:
533:
532:Transactional
530:
528:
525:
523:
522:Quantum logic
520:
518:
515:
513:
510:
504:
501:
500:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
461:
456:
455:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
423:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
378:
374:
371:
370:
364:
363:
355:
352:
350:
347:
345:
342:
341:
335:
332:
331:
330:
327:
326:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
266:
265:
255:
252:
251:
250:
249:Wave function
247:
245:
242:
240:
237:
235:
232:
230:
229:Superposition
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
161:
157:
154:
152:
149:
147:
144:
143:
137:
136:
131:
128:
126:
123:
121:
118:
117:
116:
115:
111:
78:
72:
55:
52:
48:
40:
33:
32:
29:
25:
21:
20:
6800:
6765:
6761:
6751:
6700:
6696:
6667:
6635:
6628:
6341:
6329:
6311:
6292:
6286:
6253:
6249:
6243:
6231:
6214:
6191:Whence, the
6190:
5901:
5894:
5632:Here, apply
5241:product rule
5238:
4909:
4901:
4448:
4439:
4397:rather than
4302:
4300:
3899:product rule
3896:
3715:
3703:
3529:
3432:moving in a
3427:
3268:
3260:
3253:
2873:
2861:
2644:
2629:
2625:
2622:
2498:
1966:
1956:
1911:
1794:
1658:
1316:
1301:
1282:
1271:
1260:
1252:
1133:
1114:
988:
910:
906:
887:
885:
431:KleinâGordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
6642:real-number
2636:Hamiltonian
1128:Hamiltonian
820:von Neumann
805:Schrödinger
581:EPR paradox
512:Many-worlds
446:Schrödinger
401:Schrödinger
396:Phase-space
386:Interaction
291:Double-slit
269:Experiments
244:Uncertainty
214:Nonlocality
209:Measurement
194:Decoherence
164:Hamiltonian
6821:Categories
6793:References
5905:. Setting
4349:satisfies
1124:commutator
896:derivative
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
6710:1105.4014
6595:^
6561:∗
6557:Φ
6550:ℏ
6539:−
6527:⟩
6516:ϕ
6513:⟨
6507:ℏ
6496:−
6487:⟩
6473:^
6459:ϕ
6456:⟨
6450:ℏ
6439:−
6430:⟩
6419:ϕ
6416:⟨
6407:∂
6403:∂
6381:⟩
6372:ϕ
6369:⟨
6361:∗
6357:ϕ
6334:Hall 2013
6278:123011242
6236:Hall 2013
6168:^
6131:^
6113:^
6055:∂
6032:∂
6010:∂
5987:∂
5955:^
5940:^
5919:^
5872:^
5852:ℏ
5849:−
5837:^
5822:^
5800:^
5791:ℏ
5779:^
5764:^
5714:⟩
5702:Ψ
5691:^
5664:Ψ
5651:ℏ
5613:⟩
5610:Ψ
5596:^
5576:−
5568:Ψ
5565:⟨
5539:Ψ
5520:^
5504:Ψ
5488:Ψ
5475:^
5448:Ψ
5422:Ψ
5405:^
5388:Ψ
5357:Ψ
5338:^
5322:Ψ
5306:Ψ
5293:^
5266:Ψ
5243:leads to
5204:Ψ
5190:^
5170:−
5153:Ψ
5124:Ψ
5113:^
5090:Ψ
5046:Ψ
5035:^
5012:Ψ
4983:Ψ
4972:^
4949:Ψ
4884:⟩
4878:⟨
4855:⟩
4846:ℏ
4840:⟨
4828:ℏ
4747:ℏ
4676:ℏ
4578:ℏ
4546:∂
4538:∂
4525:⟩
4507:⟨
4501:ℏ
4482:⟩
4476:⟨
4417:⟩
4411:⟨
4382:⟩
4361:⟨
4334:⟩
4328:⟨
4322:⟩
4316:⟨
4280:⟩
4274:⟨
4239:∂
4235:∂
4230:−
4203:Φ
4171:∂
4167:∂
4155:∗
4151:Φ
4147:∫
4144:−
4122:Φ
4113:∂
4109:∂
4084:∗
4080:Φ
4076:∫
4073:−
4061:Φ
4029:∂
4025:∂
4013:∗
4009:Φ
4005:∫
4002:−
3990:Φ
3981:∂
3977:∂
3952:∗
3948:Φ
3944:∫
3934:⟩
3928:⟨
3865:Φ
3835:∂
3831:∂
3824:∗
3820:Φ
3816:∫
3813:−
3801:Φ
3792:∂
3788:∂
3763:∗
3759:Φ
3755:∫
3749:⟩
3743:⟨
3720:, we get
3687:⟩
3654:⟨
3648:ℏ
3623:∂
3615:∂
3602:⟩
3584:⟨
3578:ℏ
3563:⟩
3557:⟨
3434:potential
3369:ℏ
3344:∂
3327:∂
3314:⟩
3299:⟨
3232:Ψ
3197:ℏ
3180:Ψ
3164:Ψ
3151:∂
3134:∂
3124:Ψ
3108:Ψ
3068:Ψ
3038:Ψ
3035:⟨
3015:⟩
3012:Ψ
2957:ℏ
2936:∂
2919:∂
2835:∂
2827:∂
2814:⟩
2796:⟨
2790:ℏ
2765:∂
2757:∂
2728:Φ
2716:−
2702:∗
2698:Φ
2694:∫
2688:ℏ
2673:⟩
2667:⟨
2640:Hermitian
2601:∗
2597:Φ
2590:ℏ
2578:−
2570:∗
2560:∗
2556:Φ
2549:ℏ
2537:−
2525:∂
2518:∗
2514:Φ
2510:∂
2485:Φ
2476:ℏ
2455:∂
2450:Φ
2447:∂
2392:∂
2387:Φ
2384:∂
2369:∗
2365:Φ
2361:∫
2345:∂
2337:∂
2310:Φ
2294:∂
2287:∗
2283:Φ
2279:∂
2269:∫
2232:∂
2227:Φ
2224:∂
2209:∗
2205:Φ
2201:∫
2181:Φ
2168:∂
2160:∂
2148:∗
2144:Φ
2140:∫
2120:Φ
2104:∂
2097:∗
2093:Φ
2089:∂
2079:∫
2052:Φ
2044:∗
2040:Φ
2036:∫
2011:⟩
2005:⟨
1624:⟩
1611:⟨
1585:⟩
1578:⟨
1420:−
1370:−
1347:⟩
1341:⟨
1335:⟩
1329:⟨
1222:∂
1214:∂
1201:⟩
1183:⟨
1177:ℏ
1162:⟩
1156:⟨
1067:−
1061:⟩
1055:⟨
1032:⟩
1026:⟨
1020:⟩
1014:⟨
927:−
904:operators
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
6743:19605000
6735:23215365
6672:, where
6071:′
5859:′
5678:⟩
5636:, using
5583:′
5554:⟩
5500:⟨
5492:⟩
5438:⟨
5426:⟩
5384:⟨
5372:⟩
5318:⟨
5310:⟩
5256:⟨
5217:⟩
5177:′
5149:⟨
5137:⟩
5086:⟨
5059:⟩
5008:⟨
4996:⟩
4945:⟨
4805:⟩
4760:⟨
4724:⟩
4683:⟨
4647:⟩
4585:⟨
4555:⟩
4532:⟨
4267:⟩
4226:⟨
3632:⟩
3609:⟨
3430:particle
3405:⟩
3376:⟨
3353:⟩
3321:⟨
3236:⟩
3176:⟨
3168:⟩
3120:⟨
3112:⟩
3064:⟨
2844:⟩
2821:⟨
2774:⟩
2751:⟨
2354:⟩
2331:⟨
1926:′
1910:will be
1897:⟩
1883:′
1875:⟨
1849:⟩
1843:⟨
1834:′
1779:⟩
1765:′
1757:⟨
1731:⟩
1725:⟨
1716:′
1691:′
1530:′
1518:), then
1446:⟩
1432:′
1424:⟨
1392:⟩
1386:⟨
1377:′
1231:⟩
1208:⟨
1120:operator
1093:⟩
1079:′
1071:⟨
934:′
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
6770:Bibcode
6715:Bibcode
6652:on the
6258:Bibcode
3273:
3257:
2874:In the
2632:
1822:, then
1265:is its
898:of the
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
6807:
6741:
6733:
6662:Ad hoc
6581:where
6299:
6276:
5732:where
5717:
4206:
4125:
4064:
3993:
3880:
3871:
3804:
3523:where
2731:
1912:almost
1253:where
1235:
1097:
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
6739:S2CID
6705:arXiv
6336:p. 78
6321:(PDF)
6274:S2CID
6223:Notes
2623:Note
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
6805:ISBN
6731:PMID
6619:and
6393:, so
6297:ISBN
3712:by
1866:and
1748:and
1269:.
909:and
886:The
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
6778:doi
6723:doi
6701:109
6345:In
6266:doi
4303:not
2638:is
1287:of
6823::
6776:.
6766:46
6764:.
6760:.
6737:.
6729:.
6721:.
6713:.
6699:.
6685:^
6668:xt
6272:.
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5900:=
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2628:=
1302:iħ
1280:.
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6784:.
6780::
6772::
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6717::
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6678:t
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3409:.
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2018:=
2008:A
1999:t
1996:d
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1972:Ί
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1938:0
1934:x
1930:(
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