4443:
2472:
3833:
1874:
4438:{\displaystyle {\begin{aligned}H\star W&=\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\star W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x+{\frac {i\hbar }{2}}{\vec {\partial }}_{p}\right)^{2}+{\frac {1}{2m}}\left(p-{\frac {i\hbar }{2}}{\vec {\partial }}_{x}\right)^{2}\right)W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\vec {\partial }}_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\vec {\partial }}_{x}^{2}\right)\right)W\\&\quad {}+{\frac {i\hbar }{2}}\left(m\omega ^{2}x{\vec {\partial }}_{p}-{\frac {p}{m}}{\vec {\partial }}_{x}\right)W\\&=E\cdot W.\end{aligned}}}
2467:{\displaystyle {\begin{aligned}(f\star g)(x,p)&=f\left(x+{\tfrac {i\hbar }{2}}{\vec {\partial }}_{p},p-{\tfrac {i\hbar }{2}}{\vec {\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{p},p+{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{x}\right)\\&=f\left(x+{\tfrac {i\hbar }{2}}{\vec {\partial }}_{p},p\right)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{p},p\right)\\&=f\left(x,p-{\tfrac {i\hbar }{2}}{\vec {\partial }}_{x}\right)\cdot g\left(x,p+{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{x}\right).\end{aligned}}}
5164:
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8134:
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3634:
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4454:
1672:
1851:
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1512:
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6244:
3629:{\displaystyle {\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({\frac {\hbar }{2}}({\overset {_{\gets }}{\partial }}_{x}{\vec {\partial }}_{p}-{\overset {_{\gets }}{\partial }}_{p}{\vec {\partial }}_{x})\right)H=-\{W,H\}+O(\hbar ^{2}),}
5957:
4903:
5648:{\displaystyle W(\mathbf {x} ,\mathbf {p} ;t)={\frac {1}{(\pi \hbar )^{3}}}\exp {\left(-\alpha ^{2}r^{2}-{\frac {p^{2}}{\alpha ^{2}\hbar ^{2}}}\left(1+\left({\frac {t}{\tau }}\right)^{2}\right)+{\frac {2t}{\hbar \tau }}\mathbf {x} \cdot \mathbf {p} \right)}~,}
2790:
1037:
of a quantum state is a quasiprobability distribution. In the phase-space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.
6170:
5158:
5829:
Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence:
3669:
vigorously denied the physical existence of such trajectories on the microscopic scale. By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method and the
3377:
4745:
3146:
2491:
6067:
1428:. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the SegalâBargmann transform is a
3662:; the probability fluid appears "diffusive" and compressible. The concept of quantum trajectory is therefore a delicate issue here. See the movie for the Morse potential, below, to appreciate the nonlocality of quantum phase flow.
1199:
1083:, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta and over the position interval:
106:
5685:
However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically,
1409:, and thus renders such "negative probabilities" less paradoxical. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of
5813:
1667:{\displaystyle {\begin{aligned}f{\overset {_{\gets }}{\partial }}_{x}g&=\left({\frac {\partial f}{\partial x}}\right)g,\\f{\vec {\partial }}_{x}g&=f\left({\frac {\partial g}{\partial x}}\right).\end{aligned}}}
5243:
can represent a macroscopic quantity and appear like a classical object rotating uniformly in phase space, a plain mechanical oscillator (see the animated figures). Integrating over all phases (starting positions at
3814:
1344:
6807:
Agarwal, G. S.; Wolf, E. (1970). "Calculus for
Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators".
4978:
4459:
Time evolution of combined ground and 1st excited state Wigner function for the simple harmonic oscillator. Note the rigid motion in phase space corresponding to the conventional oscillations in coordinate
6086:
1846:{\displaystyle f\star g=f\,\exp {\left({\frac {i\hbar }{2}}\left({\overset {_{\gets }}{\partial }}_{x}{\vec {\partial }}_{p}-{\overset {_{\gets }}{\partial }}_{p}{\vec {\partial }}_{x}\right)\right)}g,}
4497:
3838:
1879:
1517:
6243:
5836:
4768:
5682:
Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel.
5342:
3038:{\displaystyle \exp {\big (}{-a}(x^{2}+p^{2}){\big )}\star \exp {\big (}{-b}(x^{2}+p^{2}){\big )}={\frac {1}{1+\hbar ^{2}ab}}\exp \left(-{\frac {a+b}{1+\hbar ^{2}ab}}(x^{2}+p^{2})\right),}
5399:
6169:
6163:
is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics.
3244:
992:
The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as
Kontsevich's deformation-quantization (see
6686:
Agarwal, G. S.; Wolf, E. (1970). "Calculus for
Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space".
5020:
5227:
3290:
2770:{\displaystyle (f\star g)(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\tfrac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''.}
4623:
1424:
An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of the
3054:
5963:
5197:
1072:
for the
GlauberâSudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified.
1459:
5413:, with the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state propagating freely is
1089:
1068:, KirkwoodâRihaczek, Mehta, Rivier, and BornâJordan representations. These alternatives are most useful when the Hamiltonian takes a particular form, such as
7230:
Krivoruchenko, M. I.; Faessler, Amand (2007). "Weyl's symbols of
Heisenberg operators of canonical coordinates and momenta as quantum characteristics".
7378:
5270:
The eigenfunctions can also be characterized by being rotationally symmetric (thus time-invariant) pure states. That is, they are functions of form
7181:
36:
7931:
5692:
7579:
372:
3735:
1491:
characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the WignerâWeyl representation.
1255:
873:
1479:
The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is the
4918:
8102:
7428:
7128:
M. Oliva, D. Kakofengitis, O. Steuernagel (2018). "Anharmonic quantum mechanical systems do not feature phase space trajectories".
4755:
124:
8114:
4603:{\displaystyle {\frac {\hbar }{2}}\left(m\omega ^{2}x{\vec {\partial }}_{p}-{\frac {p}{m}}{\vec {\partial }}_{x}\right)\cdot W=0}
989:
and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.
580:
6953:
Baker, George A. (1958). "Formulation of
Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space".
1425:
7798:
7371:
1061:
353:
5248: = 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static
8173:
7732:
6635:
536:
2781:
7655:
6721:
Sudarshan, E. C. G. (1963). "Equivalence of
Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams".
6322:
Hamiltonian. The corresponding momentum and coordinate projections are plotted to the right and under the phase space plot.
6252:
6178:
6095:
3266:
1418:
1060:, discovered first. Other representations (in approximately descending order of prevalence in the literature) include the
1042:
1009:
459:
5952:{\displaystyle K_{\text{rad}}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}\left({\frac {3}{2}}-{\frac {1}{1+(t/\tau )^{2}}}\right)}
4898:{\displaystyle F_{n}(u)={\frac {(-1)^{n}}{\pi \hbar }}L_{n}\left(4{\frac {u}{\hbar \omega }}\right)e^{-2u/\hbar \omega },}
8168:
6918:
Lax, Melvin (1968). "Quantum Noise. XI. Multitime
Correspondence between Quantum and Classical Stochastic Processes".
7401:
7364:
5273:
3671:
119:
969:". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical
7511:
5410:
5229:
in order to show that the Wigner function oscillates within that radius, and decays rapidly outside of that radius.
1393:
Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few
208:
5347:
8069:
7778:
7773:
7496:
4476:. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating the
3729:
The
Hamiltonian for the simple harmonic oscillator in one spatial dimension in the WignerâWeyl representation is
993:
866:
315:
295:
163:
8158:
8081:
7753:
1365:
555:
285:
1041:
There are several different ways to represent the distribution, all interrelated. The most noteworthy is the
8059:
7836:
7758:
7541:
7411:
1013:
920:
595:
333:
233:
7283:
4472:
Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i.e., a
8163:
7793:
7727:
7722:
7693:
7406:
1678:
965:
as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the
961:
The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to
531:
526:
497:
348:
129:
5153:{\displaystyle W(x,p;t)=W(m\omega x\cos \omega t-p\sin \omega t,p\cos \omega t+\omega mx\sin \omega t;0).}
4983:
For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initial
7861:
7768:
3688:
1414:
916:
565:
310:
300:
6524:
Ali, S. Twareque; EngliĆĄ, Miroslav (2005). "Quantization
Methods: A Guide for Physicists and Analysts".
3208:
8137:
7899:
7707:
7678:
3274:
859:
511:
482:
1860:. Additional differential relations allow this to be written in terms of a change in the arguments of
7921:
7788:
7712:
7673:
7630:
7604:
7561:
7454:
3372:{\displaystyle {\frac {\partial f}{\partial t}}=-{\frac {1}{i\hbar }}\left(f\star H-H\star f\right),}
3253:
is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian.
1242:
1236:
1017:
919:). The two key features of the phase-space formulation are that the quantum state is described by a
516:
477:
430:
405:
328:
188:
4740:{\displaystyle W(x,p)=F\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\equiv F(u).}
7988:
7968:
7958:
7948:
7904:
7479:
5202:
4477:
3651:
1387:
1369:
600:
6844:
6314:
3141:{\displaystyle \delta (x)\star \delta (p)={\frac {2}{h}}\exp \left(2i{\frac {xp}{\hbar }}\right),}
7683:
6988:
Fairlie, D. B. (1964). "The formulation of quantum mechanics in terms of phase space functions".
5819:
1462:
1379:
1065:
997:
769:
487:
8097:
7645:
7034:
Curtright, T.; Fairlie, D.; Zachos, C. (1998). "Features of time-independent Wigner functions".
6062:{\displaystyle K_{\text{ang}}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}{\frac {1}{1+(t/\tau )^{2}}}~.}
5163:
3284:
In any representation of the phase space distribution with its associated star product, this is
3195:
445:
343:
109:
8008:
7783:
7763:
7688:
7556:
1495:
943:
774:
492:
320:
290:
253:
7599:
5010:
evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simply
908:
400:
8033:
7546:
7526:
7175:
6276:
6206:
6122:
3270:
1461:. There is a quasiprobability density associated to the phase-space wave function; it is the
1402:
970:
962:
243:
228:
5167:
The Wigner function of a simple harmonic oscillator at different levels of excitations. The
947:
8064:
7993:
7938:
7668:
7491:
7449:
7319:
7297:
7249:
7204:
7147:
7100:
7053:
6997:
6962:
6927:
6891:
6859:
6817:
6765:
6730:
6695:
6657:
6488:
6442:
6399:
6355:
5170:
4759:
3278:
1429:
1374:
560:
472:
198:
155:
1435:
1368:, because regions under it do not represent mutually exclusive states, as required in the
804:
8:
8049:
8018:
7963:
7943:
7851:
7808:
7663:
7589:
7516:
7506:
7418:
3643:, the Wigner transform of the quantum commutator, while { , } is the classical
1194:{\displaystyle \operatorname {P} =\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx.}
982:
659:
467:
385:
213:
193:
145:
7323:
7253:
7208:
7151:
7104:
7057:
7001:
6966:
6931:
6895:
6863:
6821:
6769:
6734:
6699:
6661:
6492:
6446:
6403:
6359:
1249:
The expectation value of the observable with respect to the phase-space distribution is
1075:
The phase-space distribution possesses properties akin to the probability density in a 2
8109:
7978:
7876:
7584:
7531:
7423:
7343:
7309:
7265:
7239:
7163:
7137:
7069:
7043:
7013:
6599:
6581:
6551:
6533:
6458:
6415:
380:
305:
238:
150:
6879:
6367:
8119:
8028:
7998:
7926:
7889:
7884:
7866:
7831:
7821:
7536:
7501:
7484:
7387:
7335:
7216:
7017:
6631:
6603:
6555:
6462:
6419:
6182:
6160:
2482:
1417:. (For details on the properties and interpretation of the Wigner function, see its
894:
889:
814:
789:
729:
724:
624:
590:
570:
168:
27:
7347:
7269:
7167:
3654:: this equation manifestly reduces to the classical Liouville equation in the limit
1405:, which does not allow precise localization within phase-space regions smaller than
7846:
7841:
7698:
7594:
7327:
7257:
7212:
7155:
7108:
7061:
7005:
6970:
6935:
6899:
6867:
6825:
6773:
6738:
6703:
6665:
6627:
6591:
6543:
6504:
6496:
6450:
6407:
6390:
Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory".
6363:
819:
809:
799:
699:
679:
664:
634:
502:
390:
7073:
8076:
8003:
7983:
7953:
7916:
7911:
7816:
7640:
6756:
Glauber, Roy J. (1963). "Coherent and Incoherent States of the Radiation Field".
6099:
6078:
5264:
3665:
N.B. Given the restrictions placed by the uncertainty principle on localization,
3644:
1398:
1383:
974:
844:
714:
694:
440:
280:
7159:
8054:
8023:
8013:
7635:
7625:
7459:
7331:
7065:
6742:
6346:
Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics".
4473:
3693:
3262:
1410:
978:
932:
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
7009:
6595:
6547:
6509:
6479:
Wigner, E. (1932). "On the Quantum Correction for Thermodynamic Equilibrium".
6411:
1219:
is an operator representing an observable, it may be mapped to phase space as
8152:
7973:
7826:
7717:
7551:
7521:
7474:
7339:
6974:
6903:
6871:
6777:
6623:
6619:
3640:
1480:
1474:
966:
955:
936:
928:
924:
839:
834:
764:
734:
704:
575:
521:
248:
223:
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
6939:
6829:
6707:
6572:
Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space".
4750:
With this change of variables, it is possible to write the real part of the
7856:
7469:
7464:
6500:
6268:
6198:
6114:
5808:{\displaystyle W\longrightarrow {\frac {1}{(\pi \hbar )^{3}}}\exp \left\,.}
4466:
1857:
1069:
951:
829:
824:
759:
744:
709:
203:
7894:
7314:
7244:
7127:
6630:, "Quantum Mechanics in Phase Space" (World Scientific, Singapore, 2005)
6081:
is used to approximate the vibrational structure of a diatomic molecule.
5823:
986:
904:
794:
749:
684:
639:
6538:
6318:
Wigner function of two interfering coherent states evolving through the
7048:
6454:
3666:
3155:
1856:
where the argument of the exponential function can be interpreted as a
784:
754:
674:
649:
644:
629:
7261:
7113:
7088:
6669:
3809:{\displaystyle H={\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}.}
3265:
of the phase space distribution is given by a quantum modification of
7748:
7444:
6308:
6272:
6202:
6118:
1339:{\displaystyle \langle {\hat {A}}\rangle =\int A(x,p)W(x,p)\,dp\,dx.}
275:
7356:
6790:
KĂŽdi Husimi (1940). "Some Formal Properties of the Density Matrix",
6648:
Cohen, L. (1966). "Generalized Phase-Space Distribution Functions".
4754:-genvalue equation in the form of a modified Laguerre equation (not
7142:
1349:
A point of caution, however: despite the similarity in appearance,
900:
654:
7195:
Marinov, M. S. (1991). "A new type of phase-space path integral".
6586:
7300:(2002-01-15). "Concepts of radial and angular kinetic energies".
5662:
is a parameter describing the initial width of the Gaussian, and
4973:{\displaystyle E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right).}
2481:-product in a convolution integral form, essentially through the
7089:"Phase-Space Formulation of the Dynamics of Canonical Variables"
3658: â 0. In the quantum extension of the flow, however,
977:). Quantum mechanics in phase space is often favored in certain
6990:
Mathematical Proceedings of the Cambridge Philosophical Society
6392:
Mathematical Proceedings of the Cambridge Philosophical Society
3674:, although there are severe practical obstacles in both cases.
1378:
even for pure states, with the unique exception of (optionally
4453:
1487:. Each representation of the phase-space distribution has a
5409:
Suppose a particle is initially in a minimally uncertain
3194:. These are solved, analogously to the time-independent
6433:
Weyl, H. (1927). "Quantenmechanik und Gruppentheorie".
1494:
For notational convenience, we introduce the notion of
7033:
6880:"Density Operators and Quasiprobability Distributions"
2656:
2413:
2349:
2268:
2204:
2135:
2086:
1979:
1934:
7229:
5966:
5839:
5695:
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5350:
5276:
5205:
5173:
5023:
4921:
4771:
4626:
4500:
3836:
3738:
3660:
the density of points in phase space is not conserved
3391:
3293:
3211:
3057:
2793:
2494:
1877:
1690:
1515:
1438:
1258:
1240:. Conversely, this operator may be recovered by the
1092:
973:, enabling a natural comparison between the two (see
39:
6681:
6679:
6061:
5951:
5807:
5647:
5393:
5336:
5233:Typically, a "bump" (or coherent state) of energy
5221:
5191:
5152:
4972:
4897:
4739:
4602:
4437:
3808:
3628:
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3238:
3140:
3037:
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2466:
1845:
1666:
1453:
1338:
1193:
100:
6845:"Ordered Expansions in Boson Amplitude Operators"
5404:
8150:
6341:
6339:
6337:
6335:
1506:, the left and right derivatives are defined as
946:in 1946 in his PhD thesis, and independently by
6676:
6571:
6319:
5337:{\displaystyle W(x,p)=f({\sqrt {x^{2}+p^{2}}})}
935:) and operator multiplication is replaced by a
4617:-genstates as functions of a single argument:
1079:-dimensional phase space. For example, it is
7372:
6877:
6842:
6389:
6332:
3682:
2906:
2862:
2846:
2802:
867:
7295:
7180:: CS1 maint: multiple names: authors list (
6474:
6472:
5394:{\displaystyle W\star W=(2\pi \hbar )^{-1}W}
3709: = 19. Marginal distributions for
3598:
3586:
3436:
3433:
3421:
3418:
1274:
1259:
95:
69:
6806:
6685:
6385:
6383:
6381:
6379:
6377:
4487:Consider, first, the imaginary part of the
3382:or, for the Wigner function in particular,
3269:. This formula results from applying the
3182:, and the associated energies are known as
1003:
7379:
7365:
7029:
7027:
6345:
4908:introduced by Groenewold, with associated
3650:This yields a concise illustration of the
874:
860:
7313:
7243:
7141:
7112:
7047:
6720:
6615:
6613:
6585:
6537:
6523:
6508:
6469:
5801:
2719:
2642:
2601:
2557:
1706:
1326:
1319:
1181:
1174:
6567:
6565:
6374:
6313:
6306:
6240:
6165:
6083:
5162:
3692:
7194:
7024:
6987:
6755:
6426:
4758:!), the solution of which involves the
1372:. Moreover, it can, in general, take
8151:
6878:Cahill, K. E.; Glauber, R. J. (1969).
6843:Cahill, K. E.; Glauber, R. J. (1969).
6610:
6478:
6271:(a.u.). The solid lines represent the
6201:(a.u.). The solid lines represent the
7386:
7360:
7086:
6952:
6647:
6562:
4480:. From the general pedagogy web-site.
3697:Wigner function for number states a)
3273:to the density matrix version of the
6432:
6236:
6155:
5263:, an intuitive visualization of the
4613:This implies that one may write the
1010:Wigner quasiprobability distribution
950:, each building on earlier ideas by
6917:
5822:reflects the spreading of the free
915:momentum representations (see also
13:
6117:(a.u.). The solid lines represent
6072:
4568:
4536:
4386:
4354:
4268:
4186:
4075:
4000:
3717:are recovered by integrating over
3554:
3530:
3509:
3485:
3403:
3395:
3305:
3297:
3239:{\displaystyle H\star W=E\cdot W,}
2477:It is also possible to define the
2432:
2369:
2287:
2224:
2154:
2105:
1999:
1954:
1811:
1787:
1766:
1742:
1644:
1636:
1601:
1571:
1563:
1526:
1151:
1146:
1093:
942:The theory was fully developed by
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
8185:
6255:time evolution for the potential
5714:
5466:
5372:
4935:
4884:
4821:
4320:
4062:
3987:
3672:method of quantum characteristics
3450:
3329:
3256:
3125:
2666:
2419:
2355:
2274:
2210:
2141:
2092:
1985:
1940:
1725:
1370:third axiom of probability theory
43:
8133:
8132:
5772:
5761:
5629:
5621:
5438:
5430:
4465:
4452:
16:Formulation of quantum mechanics
7289:
7284:Time-dependent Wigner Functions
7276:
7232:Journal of Mathematical Physics
7223:
7188:
7121:
7093:Journal of Mathematical Physics
7080:
6981:
6946:
6911:
6836:
6800:
6784:
6749:
6714:
6650:Journal of Mathematical Physics
6574:Asia Pacific Physics Newsletter
6526:Reviews in Mathematical Physics
5012:rigidly rotating in phase space
4308:
3639:where {{ , }} is the
2780:(Thus, e.g., Gaussians compose
1468:
1465:of the position wave function.
994:Kontsevich quantization formula
8082:Relativistic quantum mechanics
6641:
6517:
6185:through the potential barrier
6041:
6026:
5932:
5917:
5718:
5708:
5699:
5470:
5460:
5448:
5426:
5405:Free particle angular momentum
5376:
5363:
5331:
5301:
5292:
5280:
5186:
5174:
5144:
5054:
5045:
5027:
4807:
4797:
4788:
4782:
4731:
4725:
4642:
4630:
4571:
4539:
4389:
4357:
4271:
4189:
4078:
4003:
3620:
3607:
3569:
3557:
3537:
3512:
3492:
3479:
3082:
3076:
3067:
3061:
3024:
2998:
2901:
2875:
2841:
2815:
2710:
2672:
2639:
2605:
2598:
2564:
2522:
2510:
2507:
2495:
2439:
2372:
2294:
2227:
2161:
2112:
2065:
2053:
2037:
2025:
2002:
1957:
1909:
1897:
1894:
1882:
1814:
1794:
1769:
1749:
1604:
1533:
1413:this equation is known as the
1366:joint probability distribution
1316:
1304:
1298:
1286:
1268:
1171:
1159:
1117:
1099:
903:variables on equal footing in
556:Relativistic quantum mechanics
88:
81:
62:
1:
8060:Quantum statistical mechanics
7837:Quantum differential calculus
7759:Delayed-choice quantum eraser
7542:Symmetry in quantum mechanics
6368:10.1016/S0031-8914(46)80059-4
6326:
5222:{\displaystyle {\sqrt {n+1}}}
1397:, and hence disappear in the
1022:The phase-space distribution
1014:Quasiprobability distribution
921:quasiprobability distribution
596:Quantum statistical mechanics
8174:Foundational quantum physics
7217:10.1016/0375-9601(91)90352-9
1483:, represented by the symbol
7:
7862:Quantum stochastic calculus
7852:Quantum measurement problem
7774:MachâZehnder interferometer
7160:10.1016/j.physa.2017.10.047
6792:Proc. Phys. Math. Soc. Jpn.
3827:Wigner function then reads
3823:-genvalue equation for the
3689:quantum harmonic oscillator
3677:
3158:distributions are known as
1415:optical equivalence theorem
1401:. They are shielded by the
917:position and momentum space
888:phase-space formulation of
566:Quantum information science
10:
8190:
7332:10.1103/physreva.65.022109
7066:10.1103/PhysRevD.58.025002
6743:10.1103/PhysRevLett.10.277
5267:for large-action systems.
3686:
3683:Simple harmonic oscillator
3275:quantum Liouville equation
1498:. For a pair of functions
1496:left and right derivatives
1472:
1007:
8169:Mathematical quantization
8128:
8090:
8042:
7922:Quantum complexity theory
7900:Quantum cellular automata
7875:
7807:
7741:
7654:
7618:
7605:Path integral formulation
7572:
7437:
7394:
7010:10.1017/S0305004100038068
6596:10.1142/S2251158X12000069
6548:10.1142/S0129055X05002376
6412:10.1017/S0305004100000487
7989:Quantum machine learning
7969:Quantum key distribution
7959:Quantum image processing
7949:Quantum error correction
7799:Wheeler's delayed choice
6975:10.1103/PhysRev.109.2198
6904:10.1103/PhysRev.177.1882
6872:10.1103/PhysRev.177.1857
6778:10.1103/PhysRev.131.2766
4478:correspondence principle
3652:correspondence principle
1426:SegalâBargmann transform
1004:Phase-space distribution
601:Quantum machine learning
354:Wheeler's delayed-choice
7905:Quantum finite automata
6940:10.1103/PhysRev.172.350
6830:10.1103/PhysRevD.2.2161
6723:Physical Review Letters
6708:10.1103/PhysRevD.2.2187
3705: = 1, and c)
1681:of the star product is
1679:differential definition
1463:Husimi Q representation
998:noncommutative geometry
311:LeggettâGarg inequality
8009:Quantum neural network
6501:10.1103/PhysRev.40.749
6435:Zeitschrift fĂŒr Physik
6323:
6303:
6233:
6152:
6098:time-evolution of the
6063:
5953:
5826:in coordinate space.)
5809:
5649:
5395:
5338:
5230:
5223:
5193:
5154:
4974:
4899:
4741:
4604:
4439:
3810:
3726:
3630:
3373:
3240:
3142:
3039:
2771:
2468:
1847:
1668:
1455:
1386:, in violation of the
1340:
1195:
985:), or in the study of
102:
8159:Hamiltonian mechanics
8034:Quantum teleportation
7562:Waveâparticle duality
7298:Schleich, Wolfgang P.
7087:Mehta, C. L. (1964).
6317:
6250:
6176:
6093:
6064:
5954:
5810:
5650:
5396:
5339:
5224:
5194:
5192:{\displaystyle (q,p)}
5166:
5155:
4975:
4900:
4742:
4605:
4440:
3811:
3696:
3631:
3374:
3271:Wigner transformation
3241:
3143:
3040:
2772:
2469:
1848:
1669:
1456:
1403:uncertainty principle
1341:
1196:
1043:Wigner representation
1018:WignerâWeyl transform
971:statistical mechanics
963:Hamiltonian mechanics
296:Elitzur–Vaidman
286:Davisson–Germer
103:
8065:Quantum field theory
7994:Quantum metamaterial
7939:Quantum cryptography
7669:Consistent histories
5964:
5837:
5693:
5420:
5348:
5274:
5203:
5171:
5021:
4919:
4769:
4760:Laguerre polynomials
4624:
4498:
4491:-genvalue equation,
3834:
3736:
3389:
3291:
3279:von Neumann equation
3209:
3202:-genvalue equation,
3196:Schrödinger equation
3055:
2791:
2492:
1875:
1688:
1513:
1454:{\displaystyle x+ip}
1436:
1430:holomorphic function
1256:
1090:
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
8164:Symplectic geometry
8050:Quantum fluctuation
8019:Quantum programming
7979:Quantum logic gates
7964:Quantum information
7944:Quantum electronics
7419:Classical mechanics
7324:2002PhRvA..65b2109D
7254:2007JMP....48e2107K
7209:1991PhLA..153....5M
7152:2018PhyA..502..201O
7105:1964JMP.....5..677M
7058:1998PhRvD..58b5002C
7002:1964PCPS...60..581F
6967:1958PhRv..109.2198B
6932:1968PhRv..172..350L
6896:1969PhRv..177.1882C
6864:1969PhRv..177.1857C
6822:1970PhRvD...2.2161A
6770:1963PhRv..131.2766G
6735:1963PhRvL..10..277S
6700:1970PhRvD...2.2187A
6662:1966JMP.....7..781C
6493:1932PhRv...40..749W
6447:1927ZPhy...46....1W
6404:1949PCPS...45...99M
6360:1946Phy....12..405G
4287:
4205:
3701: = 0, b)
1155:
1137:
1062:GlauberâSudarshan P
983:optical phase space
944:Hilbrand Groenewold
909:Schrödinger picture
907:. In contrast, the
349:Stern–Gerlach
146:Classical mechanics
8103:in popular culture
7885:Quantum algorithms
7733:Von NeumannâWigner
7713:Objective collapse
7424:Old quantum theory
7296:Dahl, Jens Peder;
6510:10338.dmlcz/141466
6455:10.1007/BF02055756
6324:
6304:
6234:
6153:
6059:
5949:
5805:
5645:
5391:
5334:
5231:
5219:
5189:
5150:
4970:
4895:
4756:Hermite's equation
4737:
4600:
4435:
4433:
4264:
4182:
3806:
3727:
3626:
3369:
3236:
3138:
3035:
2767:
2670:
2464:
2462:
2427:
2363:
2282:
2218:
2149:
2100:
1993:
1948:
1843:
1664:
1662:
1451:
1336:
1191:
1138:
1123:
981:applications (see
911:uses the position
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
8146:
8145:
8120:Quantum mysticism
8098:Schrödinger's cat
8029:Quantum simulator
7999:Quantum metrology
7927:Quantum computing
7890:Quantum amplifier
7867:Quantum spacetime
7832:Quantum cosmology
7822:Quantum chemistry
7537:Scattering theory
7485:Zero-point energy
7480:Degenerate levels
7388:Quantum mechanics
7302:Physical Review A
7282:Curtright, T. L.
7262:10.1063/1.2735816
7197:Physics Letters A
7114:10.1063/1.1704163
7036:Physical Review D
6816:(10): 2161â2186.
6810:Physical Review D
6694:(10): 2187â2205.
6688:Physical Review D
6670:10.1063/1.1931206
6636:978-981-238-384-6
6248:
6237:Quartic potential
6174:
6156:Quantum tunneling
6091:
6055:
6051:
6012:
5974:
5942:
5900:
5885:
5847:
5783:
5728:
5641:
5618:
5580:
5554:
5480:
5329:
5251:
5217:
4960:
4911:
4858:
4825:
4753:
4712:
4664:
4616:
4574:
4562:
4542:
4509:
4490:
4392:
4380:
4360:
4327:
4274:
4262:
4227:
4192:
4180:
4132:
4081:
4069:
4042:
4006:
3994:
3954:
3918:
3870:
3822:
3801:
3753:
3560:
3542:
3515:
3497:
3477:
3453:
3410:
3333:
3312:
3201:
3189:
3177:
3165:
3128:
3096:
2996:
2943:
2669:
2555:
2483:Fourier transform
2480:
2444:
2426:
2375:
2362:
2299:
2281:
2230:
2217:
2166:
2148:
2117:
2099:
2005:
1992:
1960:
1947:
1817:
1799:
1772:
1754:
1732:
1651:
1607:
1578:
1538:
1486:
1364:is not a genuine
1271:
890:quantum mechanics
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
8181:
8136:
8135:
7847:Quantum geometry
7842:Quantum dynamics
7699:Superdeterminism
7595:Matrix mechanics
7450:Braâket notation
7381:
7374:
7367:
7358:
7357:
7352:
7351:
7317:
7315:quant-ph/0110134
7293:
7287:
7280:
7274:
7273:
7247:
7245:quant-ph/0604075
7227:
7221:
7220:
7192:
7186:
7185:
7179:
7171:
7145:
7125:
7119:
7118:
7116:
7084:
7078:
7077:
7051:
7031:
7022:
7021:
6985:
6979:
6978:
6961:(6): 2198â2206.
6950:
6944:
6943:
6915:
6909:
6907:
6890:(5): 1882â1902.
6875:
6858:(5): 1857â1881.
6849:
6840:
6834:
6833:
6804:
6798:
6788:
6782:
6781:
6764:(6): 2766â2788.
6753:
6747:
6746:
6718:
6712:
6711:
6683:
6674:
6673:
6645:
6639:
6617:
6608:
6607:
6589:
6569:
6560:
6559:
6541:
6521:
6515:
6514:
6512:
6476:
6467:
6466:
6430:
6424:
6423:
6387:
6372:
6371:
6343:
6249:
6175:
6092:
6068:
6066:
6065:
6060:
6053:
6052:
6050:
6049:
6048:
6036:
6015:
6013:
6011:
6003:
6002:
6001:
5992:
5991:
5981:
5976:
5975:
5972:
5958:
5956:
5955:
5950:
5948:
5944:
5943:
5941:
5940:
5939:
5927:
5906:
5901:
5893:
5886:
5884:
5876:
5875:
5874:
5865:
5864:
5854:
5849:
5848:
5845:
5814:
5812:
5811:
5806:
5800:
5796:
5795:
5794:
5789:
5785:
5784:
5779:
5775:
5769:
5764:
5753:
5752:
5729:
5727:
5726:
5725:
5703:
5678:
5654:
5652:
5651:
5646:
5639:
5638:
5637:
5633:
5632:
5624:
5619:
5617:
5609:
5601:
5596:
5592:
5591:
5590:
5585:
5581:
5573:
5555:
5553:
5552:
5551:
5542:
5541:
5531:
5530:
5521:
5516:
5515:
5506:
5505:
5481:
5479:
5478:
5477:
5455:
5441:
5433:
5400:
5398:
5397:
5392:
5387:
5386:
5343:
5341:
5340:
5335:
5330:
5328:
5327:
5315:
5314:
5305:
5262:
5249:
5242:
5228:
5226:
5225:
5220:
5218:
5207:
5199:are rescaled by
5198:
5196:
5195:
5190:
5159:
5157:
5156:
5151:
5009:
4979:
4977:
4976:
4971:
4966:
4962:
4961:
4953:
4931:
4930:
4909:
4904:
4902:
4901:
4896:
4891:
4890:
4883:
4864:
4860:
4859:
4857:
4846:
4836:
4835:
4826:
4824:
4816:
4815:
4814:
4795:
4781:
4780:
4751:
4746:
4744:
4743:
4738:
4718:
4714:
4713:
4711:
4703:
4702:
4693:
4688:
4687:
4678:
4677:
4665:
4657:
4614:
4609:
4607:
4606:
4601:
4587:
4583:
4582:
4581:
4576:
4575:
4567:
4563:
4555:
4550:
4549:
4544:
4543:
4535:
4528:
4527:
4510:
4502:
4488:
4469:
4456:
4444:
4442:
4441:
4436:
4434:
4412:
4405:
4401:
4400:
4399:
4394:
4393:
4385:
4381:
4373:
4368:
4367:
4362:
4361:
4353:
4346:
4345:
4328:
4323:
4315:
4310:
4304:
4297:
4293:
4292:
4288:
4286:
4281:
4276:
4275:
4267:
4263:
4258:
4257:
4248:
4243:
4242:
4228:
4226:
4215:
4210:
4206:
4204:
4199:
4194:
4193:
4185:
4181:
4176:
4175:
4166:
4161:
4160:
4146:
4145:
4133:
4125:
4112:
4105:
4101:
4100:
4099:
4094:
4090:
4089:
4088:
4083:
4082:
4074:
4070:
4065:
4057:
4043:
4041:
4030:
4025:
4024:
4019:
4015:
4014:
4013:
4008:
4007:
3999:
3995:
3990:
3982:
3968:
3967:
3955:
3947:
3934:
3924:
3920:
3919:
3917:
3909:
3908:
3899:
3894:
3893:
3884:
3883:
3871:
3863:
3820:
3815:
3813:
3812:
3807:
3802:
3800:
3792:
3791:
3782:
3777:
3776:
3767:
3766:
3754:
3746:
3635:
3633:
3632:
3627:
3619:
3618:
3576:
3572:
3568:
3567:
3562:
3561:
3553:
3549:
3548:
3543:
3541:
3540:
3529:
3523:
3522:
3517:
3516:
3508:
3504:
3503:
3498:
3496:
3495:
3484:
3478:
3470:
3454:
3446:
3411:
3409:
3401:
3393:
3378:
3376:
3375:
3370:
3365:
3361:
3334:
3332:
3321:
3313:
3311:
3303:
3295:
3252:
3245:
3243:
3242:
3237:
3199:
3187:
3175:
3172:stargenfunctions
3163:
3147:
3145:
3144:
3139:
3134:
3130:
3129:
3124:
3116:
3097:
3089:
3044:
3042:
3041:
3036:
3031:
3027:
3023:
3022:
3010:
3009:
2997:
2995:
2988:
2987:
2971:
2960:
2944:
2942:
2935:
2934:
2915:
2910:
2909:
2900:
2899:
2887:
2886:
2874:
2866:
2865:
2850:
2849:
2840:
2839:
2827:
2826:
2814:
2806:
2805:
2776:
2774:
2773:
2768:
2763:
2752:
2741:
2730:
2718:
2717:
2713:
2709:
2701:
2690:
2682:
2671:
2665:
2657:
2638:
2621:
2597:
2580:
2556:
2554:
2553:
2552:
2543:
2542:
2529:
2478:
2473:
2471:
2470:
2465:
2463:
2456:
2452:
2451:
2450:
2445:
2443:
2442:
2431:
2428:
2422:
2414:
2388:
2384:
2383:
2382:
2377:
2376:
2368:
2364:
2358:
2350:
2321:
2317:
2313:
2306:
2305:
2300:
2298:
2297:
2286:
2283:
2277:
2269:
2249:
2245:
2238:
2237:
2232:
2231:
2223:
2219:
2213:
2205:
2182:
2178:
2174:
2173:
2172:
2167:
2165:
2164:
2153:
2150:
2144:
2136:
2124:
2123:
2118:
2116:
2115:
2104:
2101:
2095:
2087:
2043:
2018:
2014:
2013:
2012:
2007:
2006:
1998:
1994:
1988:
1980:
1968:
1967:
1962:
1961:
1953:
1949:
1943:
1935:
1852:
1850:
1849:
1844:
1836:
1835:
1831:
1830:
1826:
1825:
1824:
1819:
1818:
1810:
1806:
1805:
1800:
1798:
1797:
1786:
1780:
1779:
1774:
1773:
1765:
1761:
1760:
1755:
1753:
1752:
1741:
1733:
1728:
1720:
1673:
1671:
1670:
1665:
1663:
1656:
1652:
1650:
1642:
1634:
1615:
1614:
1609:
1608:
1600:
1583:
1579:
1577:
1569:
1561:
1545:
1544:
1539:
1537:
1536:
1525:
1484:
1460:
1458:
1457:
1452:
1408:
1396:
1363:
1345:
1343:
1342:
1337:
1273:
1272:
1264:
1237:Wigner transform
1233:
1218:
1200:
1198:
1197:
1192:
1154:
1149:
1136:
1131:
1059:
1036:
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
8189:
8188:
8184:
8183:
8182:
8180:
8179:
8178:
8149:
8148:
8147:
8142:
8124:
8110:Wigner's friend
8086:
8077:Quantum gravity
8038:
8024:Quantum sensing
8004:Quantum network
7984:Quantum machine
7954:Quantum imaging
7917:Quantum circuit
7912:Quantum channel
7871:
7817:Quantum biology
7803:
7779:ElitzurâVaidman
7754:DavissonâGermer
7737:
7689:Hidden-variable
7679:de BroglieâBohm
7656:Interpretations
7650:
7614:
7568:
7455:Complementarity
7433:
7390:
7385:
7355:
7294:
7290:
7281:
7277:
7228:
7224:
7193:
7189:
7173:
7172:
7126:
7122:
7085:
7081:
7032:
7025:
6986:
6982:
6955:Physical Review
6951:
6947:
6920:Physical Review
6916:
6912:
6884:Physical Review
6852:Physical Review
6847:
6841:
6837:
6805:
6801:
6789:
6785:
6758:Physical Review
6754:
6750:
6719:
6715:
6684:
6677:
6646:
6642:
6618:
6611:
6570:
6563:
6539:math-ph/0405065
6522:
6518:
6481:Physical Review
6477:
6470:
6431:
6427:
6388:
6375:
6344:
6333:
6329:
6312:
6253:Wigner function
6241:
6239:
6179:Wigner function
6166:
6158:
6100:Morse potential
6096:Wigner function
6084:
6079:Morse potential
6075:
6073:Morse potential
6044:
6040:
6032:
6019:
6014:
6004:
5997:
5993:
5987:
5983:
5982:
5980:
5971:
5967:
5965:
5962:
5961:
5935:
5931:
5923:
5910:
5905:
5892:
5891:
5887:
5877:
5870:
5866:
5860:
5856:
5855:
5853:
5844:
5840:
5838:
5835:
5834:
5818:(This relative
5790:
5771:
5770:
5768:
5760:
5759:
5755:
5754:
5748:
5744:
5740:
5736:
5721:
5717:
5707:
5702:
5694:
5691:
5690:
5663:
5628:
5620:
5610:
5602:
5600:
5586:
5572:
5568:
5567:
5560:
5556:
5547:
5543:
5537:
5533:
5532:
5526:
5522:
5520:
5511:
5507:
5501:
5497:
5493:
5489:
5488:
5473:
5469:
5459:
5454:
5437:
5429:
5421:
5418:
5417:
5407:
5379:
5375:
5349:
5346:
5345:
5323:
5319:
5310:
5306:
5304:
5275:
5272:
5271:
5265:classical limit
5253:
5234:
5206:
5204:
5201:
5200:
5172:
5169:
5168:
5022:
5019:
5018:
4984:
4952:
4945:
4941:
4926:
4922:
4920:
4917:
4916:
4879:
4869:
4865:
4850:
4845:
4841:
4837:
4831:
4827:
4817:
4810:
4806:
4796:
4794:
4776:
4772:
4770:
4767:
4766:
4704:
4698:
4694:
4692:
4683:
4679:
4673:
4669:
4656:
4655:
4651:
4625:
4622:
4621:
4577:
4566:
4565:
4564:
4554:
4545:
4534:
4533:
4532:
4523:
4519:
4515:
4511:
4501:
4499:
4496:
4495:
4485:
4484:
4483:
4482:
4481:
4470:
4462:
4461:
4457:
4432:
4431:
4410:
4409:
4395:
4384:
4383:
4382:
4372:
4363:
4352:
4351:
4350:
4341:
4337:
4333:
4329:
4316:
4314:
4309:
4302:
4301:
4282:
4277:
4266:
4265:
4253:
4249:
4247:
4238:
4234:
4233:
4229:
4219:
4214:
4200:
4195:
4184:
4183:
4171:
4167:
4165:
4156:
4152:
4151:
4147:
4141:
4137:
4124:
4123:
4119:
4110:
4109:
4095:
4084:
4073:
4072:
4071:
4058:
4056:
4049:
4045:
4044:
4034:
4029:
4020:
4009:
3998:
3997:
3996:
3983:
3981:
3974:
3970:
3969:
3963:
3959:
3946:
3945:
3941:
3932:
3931:
3910:
3904:
3900:
3898:
3889:
3885:
3879:
3875:
3862:
3861:
3857:
3850:
3837:
3835:
3832:
3831:
3793:
3787:
3783:
3781:
3772:
3768:
3762:
3758:
3745:
3737:
3734:
3733:
3691:
3685:
3680:
3645:Poisson bracket
3614:
3610:
3563:
3552:
3551:
3550:
3544:
3536:
3533:
3528:
3527:
3518:
3507:
3506:
3505:
3499:
3491:
3488:
3483:
3482:
3469:
3468:
3464:
3445:
3402:
3394:
3392:
3390:
3387:
3386:
3339:
3335:
3325:
3320:
3304:
3296:
3294:
3292:
3289:
3288:
3259:
3250:
3210:
3207:
3206:
3117:
3115:
3108:
3104:
3088:
3056:
3053:
3052:
3018:
3014:
3005:
3001:
2983:
2979:
2972:
2961:
2959:
2955:
2951:
2930:
2926:
2919:
2914:
2905:
2904:
2895:
2891:
2882:
2878:
2867:
2861:
2860:
2845:
2844:
2835:
2831:
2822:
2818:
2807:
2801:
2800:
2792:
2789:
2788:
2756:
2745:
2734:
2723:
2702:
2694:
2683:
2675:
2658:
2655:
2654:
2650:
2649:
2631:
2614:
2590:
2573:
2548:
2544:
2538:
2534:
2533:
2528:
2493:
2490:
2489:
2461:
2460:
2446:
2438:
2435:
2430:
2429:
2415:
2412:
2399:
2395:
2378:
2367:
2366:
2365:
2351:
2348:
2335:
2331:
2319:
2318:
2301:
2293:
2290:
2285:
2284:
2270:
2267:
2260:
2256:
2233:
2222:
2221:
2220:
2206:
2203:
2196:
2192:
2180:
2179:
2168:
2160:
2157:
2152:
2151:
2137:
2134:
2119:
2111:
2108:
2103:
2102:
2088:
2085:
2078:
2074:
2041:
2040:
2008:
1997:
1996:
1995:
1981:
1978:
1963:
1952:
1951:
1950:
1936:
1933:
1926:
1922:
1912:
1878:
1876:
1873:
1872:
1820:
1809:
1808:
1807:
1801:
1793:
1790:
1785:
1784:
1775:
1764:
1763:
1762:
1756:
1748:
1745:
1740:
1739:
1738:
1734:
1721:
1719:
1718:
1714:
1713:
1689:
1686:
1685:
1661:
1660:
1643:
1635:
1633:
1629:
1619:
1610:
1599:
1598:
1597:
1591:
1590:
1570:
1562:
1560:
1556:
1549:
1540:
1532:
1529:
1524:
1523:
1516:
1514:
1511:
1510:
1477:
1471:
1437:
1434:
1433:
1406:
1399:classical limit
1394:
1384:coherent states
1375:negative values
1350:
1263:
1262:
1257:
1254:
1253:
1220:
1205:
1150:
1142:
1132:
1127:
1091:
1088:
1087:
1046:
1023:
1020:
1008:Main articles:
1006:
975:classical limit
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:
87:
76:
75:
61:
51:
46:
38:
35:
34:
17:
12:
11:
5:
8187:
8177:
8176:
8171:
8166:
8161:
8144:
8143:
8141:
8140:
8129:
8126:
8125:
8123:
8122:
8117:
8112:
8107:
8106:
8105:
8094:
8092:
8088:
8087:
8085:
8084:
8079:
8074:
8073:
8072:
8062:
8057:
8055:Casimir effect
8052:
8046:
8044:
8040:
8039:
8037:
8036:
8031:
8026:
8021:
8016:
8014:Quantum optics
8011:
8006:
8001:
7996:
7991:
7986:
7981:
7976:
7971:
7966:
7961:
7956:
7951:
7946:
7941:
7936:
7935:
7934:
7924:
7919:
7914:
7909:
7908:
7907:
7897:
7892:
7887:
7881:
7879:
7873:
7872:
7870:
7869:
7864:
7859:
7854:
7849:
7844:
7839:
7834:
7829:
7824:
7819:
7813:
7811:
7805:
7804:
7802:
7801:
7796:
7791:
7789:Quantum eraser
7786:
7781:
7776:
7771:
7766:
7761:
7756:
7751:
7745:
7743:
7739:
7738:
7736:
7735:
7730:
7725:
7720:
7715:
7710:
7705:
7704:
7703:
7702:
7701:
7686:
7681:
7676:
7671:
7666:
7660:
7658:
7652:
7651:
7649:
7648:
7643:
7638:
7633:
7628:
7622:
7620:
7616:
7615:
7613:
7612:
7607:
7602:
7597:
7592:
7587:
7582:
7576:
7574:
7570:
7569:
7567:
7566:
7565:
7564:
7559:
7549:
7544:
7539:
7534:
7529:
7524:
7519:
7514:
7509:
7504:
7499:
7494:
7489:
7488:
7487:
7482:
7477:
7472:
7462:
7460:Density matrix
7457:
7452:
7447:
7441:
7439:
7435:
7434:
7432:
7431:
7426:
7421:
7416:
7415:
7414:
7404:
7398:
7396:
7392:
7391:
7384:
7383:
7376:
7369:
7361:
7354:
7353:
7288:
7275:
7222:
7187:
7120:
7099:(5): 677â686.
7079:
7049:hep-th/9711183
7023:
6996:(3): 581â586.
6980:
6945:
6926:(2): 350â361.
6910:
6835:
6799:
6783:
6748:
6729:(7): 277â279.
6713:
6675:
6656:(5): 781â786.
6640:
6609:
6561:
6532:(4): 391â490.
6516:
6487:(5): 749â759.
6468:
6425:
6373:
6354:(7): 405â460.
6330:
6328:
6325:
6311:
6305:
6238:
6235:
6157:
6154:
6074:
6071:
6070:
6069:
6058:
6047:
6043:
6039:
6035:
6031:
6028:
6025:
6022:
6018:
6010:
6007:
6000:
5996:
5990:
5986:
5979:
5970:
5959:
5947:
5938:
5934:
5930:
5926:
5922:
5919:
5916:
5913:
5909:
5904:
5899:
5896:
5890:
5883:
5880:
5873:
5869:
5863:
5859:
5852:
5843:
5816:
5815:
5804:
5799:
5793:
5788:
5782:
5778:
5774:
5767:
5763:
5758:
5751:
5747:
5743:
5739:
5735:
5732:
5724:
5720:
5716:
5713:
5710:
5706:
5701:
5698:
5656:
5655:
5644:
5636:
5631:
5627:
5623:
5616:
5613:
5608:
5605:
5599:
5595:
5589:
5584:
5579:
5576:
5571:
5566:
5563:
5559:
5550:
5546:
5540:
5536:
5529:
5525:
5519:
5514:
5510:
5504:
5500:
5496:
5492:
5487:
5484:
5476:
5472:
5468:
5465:
5462:
5458:
5453:
5450:
5447:
5444:
5440:
5436:
5432:
5428:
5425:
5411:Gaussian state
5406:
5403:
5390:
5385:
5382:
5378:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5333:
5326:
5322:
5318:
5313:
5309:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5216:
5213:
5210:
5188:
5185:
5182:
5179:
5176:
5161:
5160:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
4981:
4980:
4969:
4965:
4959:
4956:
4951:
4948:
4944:
4940:
4937:
4934:
4929:
4925:
4906:
4905:
4894:
4889:
4886:
4882:
4878:
4875:
4872:
4868:
4863:
4856:
4853:
4849:
4844:
4840:
4834:
4830:
4823:
4820:
4813:
4809:
4805:
4802:
4799:
4793:
4790:
4787:
4784:
4779:
4775:
4748:
4747:
4736:
4733:
4730:
4727:
4724:
4721:
4717:
4710:
4707:
4701:
4697:
4691:
4686:
4682:
4676:
4672:
4668:
4663:
4660:
4654:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4611:
4610:
4599:
4596:
4593:
4590:
4586:
4580:
4573:
4570:
4561:
4558:
4553:
4548:
4541:
4538:
4531:
4526:
4522:
4518:
4514:
4508:
4505:
4474:coherent state
4471:
4464:
4463:
4458:
4451:
4450:
4449:
4448:
4447:
4446:
4445:
4430:
4427:
4424:
4421:
4418:
4415:
4413:
4411:
4408:
4404:
4398:
4391:
4388:
4379:
4376:
4371:
4366:
4359:
4356:
4349:
4344:
4340:
4336:
4332:
4326:
4322:
4319:
4313:
4307:
4305:
4303:
4300:
4296:
4291:
4285:
4280:
4273:
4270:
4261:
4256:
4252:
4246:
4241:
4237:
4232:
4225:
4222:
4218:
4213:
4209:
4203:
4198:
4191:
4188:
4179:
4174:
4170:
4164:
4159:
4155:
4150:
4144:
4140:
4136:
4131:
4128:
4122:
4118:
4115:
4113:
4111:
4108:
4104:
4098:
4093:
4087:
4080:
4077:
4068:
4064:
4061:
4055:
4052:
4048:
4040:
4037:
4033:
4028:
4023:
4018:
4012:
4005:
4002:
3993:
3989:
3986:
3980:
3977:
3973:
3966:
3962:
3958:
3953:
3950:
3944:
3940:
3937:
3935:
3933:
3930:
3927:
3923:
3916:
3913:
3907:
3903:
3897:
3892:
3888:
3882:
3878:
3874:
3869:
3866:
3860:
3856:
3853:
3851:
3849:
3846:
3843:
3840:
3839:
3817:
3816:
3805:
3799:
3796:
3790:
3786:
3780:
3775:
3771:
3765:
3761:
3757:
3752:
3749:
3744:
3741:
3687:Main article:
3684:
3681:
3679:
3676:
3637:
3636:
3625:
3622:
3617:
3613:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3575:
3571:
3566:
3559:
3556:
3547:
3539:
3535:
3532:
3526:
3521:
3514:
3511:
3502:
3494:
3490:
3487:
3481:
3476:
3473:
3467:
3463:
3460:
3457:
3452:
3449:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3408:
3405:
3400:
3397:
3380:
3379:
3368:
3364:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3338:
3331:
3328:
3324:
3319:
3316:
3310:
3307:
3302:
3299:
3267:Liouville flow
3263:time evolution
3258:
3257:Time evolution
3255:
3247:
3246:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3149:
3148:
3137:
3133:
3127:
3123:
3120:
3114:
3111:
3107:
3103:
3100:
3095:
3092:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3046:
3045:
3034:
3030:
3026:
3021:
3017:
3013:
3008:
3004:
3000:
2994:
2991:
2986:
2982:
2978:
2975:
2970:
2967:
2964:
2958:
2954:
2950:
2947:
2941:
2938:
2933:
2929:
2925:
2922:
2918:
2913:
2908:
2903:
2898:
2894:
2890:
2885:
2881:
2877:
2873:
2870:
2864:
2859:
2856:
2853:
2848:
2843:
2838:
2834:
2830:
2825:
2821:
2817:
2813:
2810:
2804:
2799:
2796:
2782:hyperbolically
2778:
2777:
2766:
2762:
2759:
2755:
2751:
2748:
2744:
2740:
2737:
2733:
2729:
2726:
2722:
2716:
2712:
2708:
2705:
2700:
2697:
2693:
2689:
2686:
2681:
2678:
2674:
2668:
2664:
2661:
2653:
2648:
2645:
2641:
2637:
2634:
2630:
2627:
2624:
2620:
2617:
2613:
2610:
2607:
2604:
2600:
2596:
2593:
2589:
2586:
2583:
2579:
2576:
2572:
2569:
2566:
2563:
2560:
2551:
2547:
2541:
2537:
2532:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2475:
2474:
2459:
2455:
2449:
2441:
2437:
2434:
2425:
2421:
2418:
2411:
2408:
2405:
2402:
2398:
2394:
2391:
2387:
2381:
2374:
2371:
2361:
2357:
2354:
2347:
2344:
2341:
2338:
2334:
2330:
2327:
2324:
2322:
2320:
2316:
2312:
2309:
2304:
2296:
2292:
2289:
2280:
2276:
2273:
2266:
2263:
2259:
2255:
2252:
2248:
2244:
2241:
2236:
2229:
2226:
2216:
2212:
2209:
2202:
2199:
2195:
2191:
2188:
2185:
2183:
2181:
2177:
2171:
2163:
2159:
2156:
2147:
2143:
2140:
2133:
2130:
2127:
2122:
2114:
2110:
2107:
2098:
2094:
2091:
2084:
2081:
2077:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2044:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2017:
2011:
2004:
2001:
1991:
1987:
1984:
1977:
1974:
1971:
1966:
1959:
1956:
1946:
1942:
1939:
1932:
1929:
1925:
1921:
1918:
1915:
1913:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1880:
1854:
1853:
1842:
1839:
1834:
1829:
1823:
1816:
1813:
1804:
1796:
1792:
1789:
1783:
1778:
1771:
1768:
1759:
1751:
1747:
1744:
1737:
1731:
1727:
1724:
1717:
1712:
1709:
1705:
1702:
1699:
1696:
1693:
1675:
1674:
1659:
1655:
1649:
1646:
1641:
1638:
1632:
1628:
1625:
1622:
1620:
1618:
1613:
1606:
1603:
1596:
1593:
1592:
1589:
1586:
1582:
1576:
1573:
1568:
1565:
1559:
1555:
1552:
1550:
1548:
1543:
1535:
1531:
1528:
1522:
1519:
1518:
1473:Main article:
1470:
1467:
1450:
1447:
1444:
1441:
1411:quantum optics
1347:
1346:
1335:
1332:
1329:
1325:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1270:
1267:
1261:
1243:Weyl transform
1202:
1201:
1190:
1187:
1184:
1180:
1177:
1173:
1170:
1167:
1164:
1161:
1158:
1153:
1148:
1145:
1141:
1135:
1130:
1126:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1005:
1002:
979:quantum optics
933:density matrix
923:(instead of a
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:
9:
6:
4:
3:
2:
8186:
8175:
8172:
8170:
8167:
8165:
8162:
8160:
8157:
8156:
8154:
8139:
8131:
8130:
8127:
8121:
8118:
8116:
8113:
8111:
8108:
8104:
8101:
8100:
8099:
8096:
8095:
8093:
8089:
8083:
8080:
8078:
8075:
8071:
8068:
8067:
8066:
8063:
8061:
8058:
8056:
8053:
8051:
8048:
8047:
8045:
8041:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8017:
8015:
8012:
8010:
8007:
8005:
8002:
8000:
7997:
7995:
7992:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7974:Quantum logic
7972:
7970:
7967:
7965:
7962:
7960:
7957:
7955:
7952:
7950:
7947:
7945:
7942:
7940:
7937:
7933:
7930:
7929:
7928:
7925:
7923:
7920:
7918:
7915:
7913:
7910:
7906:
7903:
7902:
7901:
7898:
7896:
7893:
7891:
7888:
7886:
7883:
7882:
7880:
7878:
7874:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7843:
7840:
7838:
7835:
7833:
7830:
7828:
7827:Quantum chaos
7825:
7823:
7820:
7818:
7815:
7814:
7812:
7810:
7806:
7800:
7797:
7795:
7794:SternâGerlach
7792:
7790:
7787:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7746:
7744:
7740:
7734:
7731:
7729:
7728:Transactional
7726:
7724:
7721:
7719:
7718:Quantum logic
7716:
7714:
7711:
7709:
7706:
7700:
7697:
7696:
7695:
7692:
7691:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7665:
7662:
7661:
7659:
7657:
7653:
7647:
7644:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7624:
7623:
7621:
7617:
7611:
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7577:
7575:
7571:
7563:
7560:
7558:
7555:
7554:
7553:
7552:Wave function
7550:
7548:
7545:
7543:
7540:
7538:
7535:
7533:
7530:
7528:
7527:Superposition
7525:
7523:
7522:Quantum state
7520:
7518:
7515:
7513:
7510:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7486:
7483:
7481:
7478:
7476:
7475:Excited state
7473:
7471:
7468:
7467:
7466:
7463:
7461:
7458:
7456:
7453:
7451:
7448:
7446:
7443:
7442:
7440:
7436:
7430:
7427:
7425:
7422:
7420:
7417:
7413:
7410:
7409:
7408:
7405:
7403:
7400:
7399:
7397:
7393:
7389:
7382:
7377:
7375:
7370:
7368:
7363:
7362:
7359:
7349:
7345:
7341:
7337:
7333:
7329:
7325:
7321:
7316:
7311:
7308:(2): 022109.
7307:
7303:
7299:
7292:
7285:
7279:
7271:
7267:
7263:
7259:
7255:
7251:
7246:
7241:
7238:(5): 052107.
7237:
7233:
7226:
7218:
7214:
7210:
7206:
7202:
7198:
7191:
7183:
7177:
7169:
7165:
7161:
7157:
7153:
7149:
7144:
7139:
7135:
7131:
7124:
7115:
7110:
7106:
7102:
7098:
7094:
7090:
7083:
7075:
7071:
7067:
7063:
7059:
7055:
7050:
7045:
7042:(2): 025002.
7041:
7037:
7030:
7028:
7019:
7015:
7011:
7007:
7003:
6999:
6995:
6991:
6984:
6976:
6972:
6968:
6964:
6960:
6956:
6949:
6941:
6937:
6933:
6929:
6925:
6921:
6914:
6905:
6901:
6897:
6893:
6889:
6885:
6881:
6873:
6869:
6865:
6861:
6857:
6853:
6846:
6839:
6831:
6827:
6823:
6819:
6815:
6811:
6803:
6796:
6793:
6787:
6779:
6775:
6771:
6767:
6763:
6759:
6752:
6744:
6740:
6736:
6732:
6728:
6724:
6717:
6709:
6705:
6701:
6697:
6693:
6689:
6682:
6680:
6671:
6667:
6663:
6659:
6655:
6651:
6644:
6637:
6633:
6629:
6625:
6621:
6616:
6614:
6605:
6601:
6597:
6593:
6588:
6583:
6579:
6575:
6568:
6566:
6557:
6553:
6549:
6545:
6540:
6535:
6531:
6527:
6520:
6511:
6506:
6502:
6498:
6494:
6490:
6486:
6482:
6475:
6473:
6464:
6460:
6456:
6452:
6448:
6444:
6441:(1â2): 1â46.
6440:
6437:(in German).
6436:
6429:
6421:
6417:
6413:
6409:
6405:
6401:
6398:(1): 99â124.
6397:
6393:
6386:
6384:
6382:
6380:
6378:
6369:
6365:
6361:
6357:
6353:
6349:
6342:
6340:
6338:
6336:
6331:
6321:
6316:
6310:
6301:
6297:
6293:
6289:
6285:
6281:
6278:
6274:
6270:
6266:
6262:
6258:
6254:
6231:
6227:
6223:
6219:
6215:
6211:
6208:
6204:
6200:
6196:
6192:
6188:
6184:
6180:
6164:
6162:
6151:
6147:
6143:
6139:
6135:
6131:
6127:
6124:
6120:
6116:
6112:
6108:
6104:
6101:
6097:
6082:
6080:
6056:
6045:
6037:
6033:
6029:
6023:
6020:
6016:
6008:
6005:
5998:
5994:
5988:
5984:
5977:
5968:
5960:
5945:
5936:
5928:
5924:
5920:
5914:
5911:
5907:
5902:
5897:
5894:
5888:
5881:
5878:
5871:
5867:
5861:
5857:
5850:
5841:
5833:
5832:
5831:
5827:
5825:
5821:
5802:
5797:
5791:
5786:
5780:
5776:
5765:
5756:
5749:
5745:
5741:
5737:
5733:
5730:
5722:
5711:
5704:
5696:
5689:
5688:
5687:
5683:
5680:
5677:
5674:
5670:
5666:
5661:
5642:
5634:
5625:
5614:
5611:
5606:
5603:
5597:
5593:
5587:
5582:
5577:
5574:
5569:
5564:
5561:
5557:
5548:
5544:
5538:
5534:
5527:
5523:
5517:
5512:
5508:
5502:
5498:
5494:
5490:
5485:
5482:
5474:
5463:
5456:
5451:
5445:
5442:
5434:
5423:
5416:
5415:
5414:
5412:
5402:
5388:
5383:
5380:
5369:
5366:
5360:
5357:
5354:
5351:
5344:that satisfy
5324:
5320:
5316:
5311:
5307:
5298:
5295:
5289:
5286:
5283:
5277:
5268:
5266:
5260:
5256:
5247:
5241:
5237:
5214:
5211:
5208:
5183:
5180:
5177:
5165:
5147:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5051:
5048:
5042:
5039:
5036:
5033:
5030:
5024:
5017:
5016:
5015:
5013:
5007:
5003:
4999:
4995:
4991:
4987:
4967:
4963:
4957:
4954:
4949:
4946:
4942:
4938:
4932:
4927:
4923:
4915:
4914:
4913:
4892:
4887:
4880:
4876:
4873:
4870:
4866:
4861:
4854:
4851:
4847:
4842:
4838:
4832:
4828:
4818:
4811:
4803:
4800:
4791:
4785:
4777:
4773:
4765:
4764:
4763:
4761:
4757:
4734:
4728:
4722:
4719:
4715:
4708:
4705:
4699:
4695:
4689:
4684:
4680:
4674:
4670:
4666:
4661:
4658:
4652:
4648:
4645:
4639:
4636:
4633:
4627:
4620:
4619:
4618:
4597:
4594:
4591:
4588:
4584:
4578:
4559:
4556:
4551:
4546:
4529:
4524:
4520:
4516:
4512:
4506:
4503:
4494:
4493:
4492:
4479:
4475:
4468:
4455:
4428:
4425:
4422:
4419:
4416:
4414:
4406:
4402:
4396:
4377:
4374:
4369:
4364:
4347:
4342:
4338:
4334:
4330:
4324:
4317:
4311:
4306:
4298:
4294:
4289:
4283:
4278:
4259:
4254:
4250:
4244:
4239:
4235:
4230:
4223:
4220:
4216:
4211:
4207:
4201:
4196:
4177:
4172:
4168:
4162:
4157:
4153:
4148:
4142:
4138:
4134:
4129:
4126:
4120:
4116:
4114:
4106:
4102:
4096:
4091:
4085:
4066:
4059:
4053:
4050:
4046:
4038:
4035:
4031:
4026:
4021:
4016:
4010:
3991:
3984:
3978:
3975:
3971:
3964:
3960:
3956:
3951:
3948:
3942:
3938:
3936:
3928:
3925:
3921:
3914:
3911:
3905:
3901:
3895:
3890:
3886:
3880:
3876:
3872:
3867:
3864:
3858:
3854:
3852:
3847:
3844:
3841:
3830:
3829:
3828:
3826:
3803:
3797:
3794:
3788:
3784:
3778:
3773:
3769:
3763:
3759:
3755:
3750:
3747:
3742:
3739:
3732:
3731:
3730:
3725:respectively.
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3695:
3690:
3675:
3673:
3668:
3663:
3661:
3657:
3653:
3648:
3646:
3642:
3641:Moyal bracket
3623:
3615:
3611:
3604:
3601:
3595:
3592:
3589:
3583:
3580:
3577:
3573:
3564:
3545:
3534:
3524:
3519:
3500:
3489:
3474:
3471:
3465:
3461:
3458:
3455:
3447:
3442:
3439:
3430:
3427:
3424:
3415:
3412:
3406:
3398:
3385:
3384:
3383:
3366:
3362:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3336:
3326:
3322:
3317:
3314:
3308:
3300:
3287:
3286:
3285:
3282:
3280:
3276:
3272:
3268:
3264:
3254:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3205:
3204:
3203:
3197:
3193:
3185:
3184:stargenvalues
3181:
3173:
3169:
3161:
3160:stargenstates
3157:
3152:
3135:
3131:
3121:
3118:
3112:
3109:
3105:
3101:
3098:
3093:
3090:
3085:
3079:
3073:
3070:
3064:
3058:
3051:
3050:
3049:
3032:
3028:
3019:
3015:
3011:
3006:
3002:
2992:
2989:
2984:
2980:
2976:
2973:
2968:
2965:
2962:
2956:
2952:
2948:
2945:
2939:
2936:
2931:
2927:
2923:
2920:
2916:
2911:
2896:
2892:
2888:
2883:
2879:
2871:
2868:
2857:
2854:
2851:
2836:
2832:
2828:
2823:
2819:
2811:
2808:
2797:
2794:
2787:
2786:
2785:
2783:
2764:
2760:
2757:
2753:
2749:
2746:
2742:
2738:
2735:
2731:
2727:
2724:
2720:
2714:
2706:
2703:
2698:
2695:
2691:
2687:
2684:
2679:
2676:
2662:
2659:
2651:
2646:
2643:
2635:
2632:
2628:
2625:
2622:
2618:
2615:
2611:
2608:
2602:
2594:
2591:
2587:
2584:
2581:
2577:
2574:
2570:
2567:
2561:
2558:
2549:
2545:
2539:
2535:
2530:
2525:
2519:
2516:
2513:
2504:
2501:
2498:
2488:
2487:
2486:
2484:
2457:
2453:
2447:
2436:
2423:
2416:
2409:
2406:
2403:
2400:
2396:
2392:
2389:
2385:
2379:
2359:
2352:
2345:
2342:
2339:
2336:
2332:
2328:
2325:
2323:
2314:
2310:
2307:
2302:
2291:
2278:
2271:
2264:
2261:
2257:
2253:
2250:
2246:
2242:
2239:
2234:
2214:
2207:
2200:
2197:
2193:
2189:
2186:
2184:
2175:
2169:
2158:
2145:
2138:
2131:
2128:
2125:
2120:
2109:
2096:
2089:
2082:
2079:
2075:
2071:
2068:
2062:
2059:
2056:
2050:
2047:
2045:
2034:
2031:
2028:
2022:
2019:
2015:
2009:
1989:
1982:
1975:
1972:
1969:
1964:
1944:
1937:
1930:
1927:
1923:
1919:
1916:
1914:
1906:
1903:
1900:
1891:
1888:
1885:
1871:
1870:
1869:
1867:
1863:
1859:
1840:
1837:
1832:
1827:
1821:
1802:
1791:
1781:
1776:
1757:
1746:
1735:
1729:
1722:
1715:
1710:
1707:
1703:
1700:
1697:
1694:
1691:
1684:
1683:
1682:
1680:
1657:
1653:
1647:
1639:
1630:
1626:
1623:
1621:
1616:
1611:
1594:
1587:
1584:
1580:
1574:
1566:
1557:
1553:
1551:
1546:
1541:
1530:
1520:
1509:
1508:
1507:
1505:
1501:
1497:
1492:
1490:
1482:
1476:
1475:Moyal product
1466:
1464:
1448:
1445:
1442:
1439:
1431:
1427:
1422:
1420:
1416:
1412:
1404:
1400:
1391:
1389:
1385:
1381:
1377:
1376:
1371:
1367:
1361:
1357:
1353:
1333:
1330:
1327:
1323:
1320:
1313:
1310:
1307:
1301:
1295:
1292:
1289:
1283:
1280:
1277:
1265:
1252:
1251:
1250:
1247:
1245:
1244:
1239:
1238:
1231:
1227:
1223:
1216:
1212:
1208:
1188:
1185:
1182:
1178:
1175:
1168:
1165:
1162:
1156:
1143:
1139:
1133:
1128:
1124:
1120:
1114:
1111:
1108:
1105:
1102:
1096:
1086:
1085:
1084:
1082:
1078:
1073:
1071:
1067:
1063:
1057:
1053:
1049:
1044:
1039:
1034:
1030:
1026:
1019:
1015:
1011:
1001:
999:
995:
990:
988:
984:
980:
976:
972:
968:
967:Hilbert space
964:
959:
957:
956:Eugene Wigner
953:
949:
945:
940:
938:
934:
930:
926:
925:wave function
922:
918:
914:
910:
906:
902:
899:
896:
892:
891:
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:
26:
25:
21:
20:
7857:Quantum mind
7769:FranckâHertz
7631:KleinâGordon
7609:
7580:Formulations
7573:Formulations
7502:Interference
7492:Entanglement
7470:Ground state
7465:Energy level
7438:Fundamentals
7402:Introduction
7305:
7301:
7291:
7278:
7235:
7231:
7225:
7200:
7196:
7190:
7176:cite journal
7133:
7129:
7123:
7096:
7092:
7082:
7039:
7035:
6993:
6989:
6983:
6958:
6954:
6948:
6923:
6919:
6913:
6887:
6883:
6855:
6851:
6838:
6813:
6809:
6802:
6794:
6791:
6786:
6761:
6757:
6751:
6726:
6722:
6716:
6691:
6687:
6653:
6649:
6643:
6628:T. Curtright
6577:
6573:
6529:
6525:
6519:
6484:
6480:
6438:
6434:
6428:
6395:
6391:
6351:
6347:
6307:Schrödinger
6299:
6295:
6291:
6287:
6283:
6279:
6269:atomic units
6264:
6260:
6256:
6229:
6225:
6221:
6217:
6213:
6209:
6199:atomic units
6194:
6190:
6186:
6159:
6149:
6145:
6141:
6137:
6133:
6129:
6125:
6115:atomic units
6110:
6106:
6102:
6076:
5828:
5817:
5684:
5681:
5675:
5672:
5668:
5664:
5659:
5657:
5408:
5269:
5258:
5254:
5245:
5239:
5235:
5232:
5011:
5005:
5001:
4997:
4993:
4989:
4985:
4982:
4907:
4749:
4612:
4486:
3824:
3818:
3728:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3664:
3659:
3655:
3649:
3638:
3381:
3283:
3260:
3248:
3191:
3183:
3180:genfunctions
3179:
3171:
3167:
3159:
3153:
3150:
3047:
2779:
2476:
1865:
1861:
1858:power series
1855:
1676:
1503:
1499:
1493:
1488:
1481:star product
1478:
1469:Star product
1423:
1419:main article
1392:
1373:
1359:
1355:
1351:
1348:
1248:
1241:
1235:
1234:through the
1229:
1225:
1221:
1214:
1210:
1206:
1203:
1080:
1076:
1074:
1070:normal order
1055:
1051:
1047:
1040:
1032:
1028:
1024:
1021:
991:
960:
952:Hermann Weyl
941:
937:star product
929:state vector
912:
897:
887:
885:
431:KleinâGordon
395:
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
8115:EPR paradox
7895:Quantum bus
7764:Double-slit
7742:Experiments
7708:Many-worlds
7646:Schrödinger
7610:Phase space
7600:Schrödinger
7590:Interaction
7547:Uncertainty
7517:Nonlocality
7512:Measurement
7507:Decoherence
7497:Hamiltonian
7203:(1): 5â11.
7136:: 201â210.
6277:Hamiltonian
6207:Hamiltonian
6123:Hamiltonian
6109:) = 20(1 â
5824:wave packet
5820:"squeezing"
5252:-genstates
4912:-genvalues
3154:The energy
1388:first axiom
1081:real-valued
987:decoherence
905:phase space
893:places the
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
8153:Categories
8043:Extensions
7877:Technology
7723:Relational
7674:Copenhagen
7585:Heisenberg
7532:Tunnelling
7395:Background
7143:1611.03303
6797:: 264â314.
6624:D. Fairlie
6327:References
3667:Niels Bohr
3156:eigenstate
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
7749:Bell test
7619:Equations
7445:Born rule
7340:1050-2947
7130:Physica A
7018:122039228
6620:C. Zachos
6604:119230734
6587:1104.5269
6580:: 37â46.
6556:119152724
6463:121036548
6420:124183640
6309:cat state
6273:level set
6203:level set
6183:tunneling
6161:Tunneling
6119:level set
6038:τ
5995:ℏ
5985:α
5929:τ
5903:−
5868:ℏ
5858:α
5766:−
5746:α
5742:−
5734:
5715:ℏ
5712:π
5700:⟶
5626:⋅
5615:τ
5612:ℏ
5578:τ
5545:ℏ
5535:α
5518:−
5499:α
5495:−
5486:
5467:ℏ
5464:π
5381:−
5373:ℏ
5370:π
5355:⋆
5133:ω
5130:
5118:ω
5109:ω
5106:
5091:ω
5088:
5079:−
5073:ω
5070:
5061:ω
4939:ω
4936:ℏ
4888:ω
4885:ℏ
4871:−
4855:ω
4852:ℏ
4822:ℏ
4819:π
4801:−
4720:≡
4671:ω
4589:⋅
4572:→
4569:∂
4552:−
4540:→
4537:∂
4521:ω
4504:ℏ
4423:⋅
4390:→
4387:∂
4370:−
4358:→
4355:∂
4339:ω
4321:ℏ
4272:→
4269:∂
4251:ℏ
4245:−
4190:→
4187:∂
4169:ℏ
4163:−
4139:ω
4079:→
4076:∂
4063:ℏ
4054:−
4004:→
4001:∂
3988:ℏ
3961:ω
3926:⋆
3877:ω
3845:⋆
3760:ω
3612:ℏ
3584:−
3558:→
3555:∂
3538:←
3531:∂
3525:−
3513:→
3510:∂
3493:←
3486:∂
3472:ℏ
3462:
3451:ℏ
3443:−
3416:−
3404:∂
3396:∂
3356:⋆
3350:−
3344:⋆
3330:ℏ
3318:−
3306:∂
3298:∂
3228:⋅
3216:⋆
3198:, by the
3192:genvalues
3168:genstates
3126:ℏ
3102:
3074:δ
3071:⋆
3059:δ
2981:ℏ
2957:−
2949:
2928:ℏ
2869:−
2858:
2852:⋆
2809:−
2798:
2692:−
2667:ℏ
2647:
2559:∫
2546:ℏ
2536:π
2502:⋆
2440:←
2433:∂
2420:ℏ
2390:⋅
2373:→
2370:∂
2356:ℏ
2346:−
2295:←
2288:∂
2275:ℏ
2265:−
2251:⋅
2228:→
2225:∂
2211:ℏ
2162:←
2155:∂
2142:ℏ
2113:←
2106:∂
2093:ℏ
2083:−
2069:⋅
2020:⋅
2003:→
2000:∂
1986:ℏ
1976:−
1958:→
1955:∂
1941:ℏ
1889:⋆
1815:→
1812:∂
1795:←
1788:∂
1782:−
1770:→
1767:∂
1750:←
1743:∂
1726:ℏ
1711:
1695:⋆
1645:∂
1637:∂
1605:→
1602:∂
1572:∂
1564:∂
1534:←
1527:∂
1489:different
1281:∫
1275:⟩
1269:^
1260:⟨
1152:∞
1147:∞
1144:−
1140:∫
1125:∫
1112:≤
1106:≤
1097:
948:Joe Moyal
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
8138:Category
7932:Timeline
7684:Ensemble
7664:Bayesian
7557:Collapse
7429:Glossary
7412:Timeline
7348:39409789
7270:42068076
7168:53691877
3678:Examples
2761:″
2750:″
2739:′
2728:′
2707:′
2699:″
2688:″
2680:′
2636:″
2619:″
2595:′
2578:′
1380:squeezed
1066:Husimi Q
901:momentum
895:position
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
8091:Related
8070:History
7809:Science
7641:Rydberg
7407:History
7320:Bibcode
7250:Bibcode
7205:Bibcode
7148:Bibcode
7101:Bibcode
7054:Bibcode
6998:Bibcode
6963:Bibcode
6928:Bibcode
6892:Bibcode
6860:Bibcode
6818:Bibcode
6766:Bibcode
6731:Bibcode
6696:Bibcode
6658:Bibcode
6489:Bibcode
6443:Bibcode
6400:Bibcode
6356:Bibcode
6348:Physica
6275:of the
6263:) = 0.1
6205:of the
6121:of the
5000:= 0) =
4996:;
4992:,
1358:,
1213:,
1054:,
1031:,
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
7784:Popper
7346:
7338:
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