1036:: DiĂłsi and Penrose formulated the idea that gravity is responsible for the collapse of the wave function. Penrose argued that, in a quantum gravity scenario where a spatial superposition creates the superposition of two different spacetime curvatures, gravity does not tolerate such superpositions and spontaneously collapses them. He also provided a phenomenological formula for the collapse time. Independently and prior to Penrose, DiĂłsi presented a dynamical model that collapses the wave function with the same time scale suggested by Penrose.
1151:. Two distinct problems have been discussed in the literature. The first is the “bare” tails problem: it is not clear how to interpret these tails because they amount to the system never being really fully localized in space. A special case of this problem is known as the “counting anomaly”. Supporters of collapse theories mostly dismiss this criticism as a misunderstanding of the theory, as in the context of dynamical collapse theories, the absolute square of the wave function is interpreted as an actual matter density. In this case, the
1028:: The Schrödinger equation is supplemented with a nonlinear and stochastic diffusion process driven by a suitably chosen universal noise coupled to the mass-density of the system, which counteracts the quantum spread of the wave function. As for the GRW model, the larger the system, the stronger the collapse, thus explaining the quantum-to-classical transition as a progressive breakdown of quantum linearity, when the system's mass increases. The CSL model is formulated in terms of identical particles.
1020:: It is assumed that each constituent of a physical system independently undergoes spontaneous collapses. The collapses are random in time, distributed according to a Poisson distribution; they are random in space and are more likely to occur where the wave function is larger. In between collapses, the wave function evolves according to the Schrödinger equation. For composite systems, the collapse on each constituent causes the collapse of the center of mass wave functions.
3881:
1116:
This is often presented as an unavoidable consequence of
Heisenberg's uncertainty principle: the collapse in position causes a larger uncertainty in momentum. This explanation is wrong; in collapse theories the collapse in position also determines a localization in momentum, driving the wave function
1094:
They are based on the fact that the collapse noise, besides collapsing the wave function, also induces a diffusion on top of particles’ motion, which acts always, also when the wave function is already localized. Experiments of this kind involve cold atoms, opto-mechanical systems, gravitational wave
1063:
on each constituent of a physical system. Accordingly, the kinetic energy increases at a faint but constant rate. Such a feature can be modified, without altering the collapse properties, by including appropriate dissipative effects in the dynamics. This is achieved for the GRW, CSL and QMUPL models,
931:
In collapse theories, the Schrödinger equation is supplemented with additional nonlinear and stochastic terms (spontaneous collapses) which localize the wave function in space. The resulting dynamics is such that for microscopic isolated systems, the new terms have a negligible effect; therefore, the
1146:
In all collapse theories, the wave function is never fully contained within one (small) region of space, because the Schrödinger term of the dynamics will always spread it outside. Therefore, wave functions always contain tails stretching out to infinity, although their “weight” is smaller in larger
1133:
One of the biggest challenges in collapse theories is to make them compatible with relativistic requirements. The GRW, CSL and DP models are not. The biggest difficulty is how to combine the nonlocal character of the collapse, which is necessary in order to make it compatible with the experimentally
1052:
in the continuous models. The models can be generalized to include arbitrary (colored) noises, possibly with a frequency cutoff: the CSL model has been extended to its colored version (cCSL), as well as the QMUPL model (cQMUPL). In these new models the collapse properties remain basically unaltered,
1075:
Collapse models modify the Schrödinger equation; therefore, they make predictions that differ from standard quantum mechanical predictions. Although the deviations are difficult to detect, there is a growing number of experiments searching for spontaneous collapse effects. They can be classified in
1124:
Still, in the dissipative model the energy is not strictly conserved. A resolution to this situation might come by considering also the noise a dynamical variable with its own energy, which is exchanged with the quantum system in such a way that the energy of the total system and noise together is
982:
Philip Pearle's 1976 paper pioneered the quantum nonlinear stochastic equations to model the collapse of the wave function in a dynamical way; this formalism was later used for the CSL model. However, these models lacked the character of “universality” of the dynamics, i.e. its applicability to an
935:
An inbuilt amplification mechanism makes sure that for macroscopic systems consisting of many particles, the collapse becomes stronger than the quantum dynamics. Then their wave function is always well-localized in space, so well-localized that it behaves, for all practical purposes, like a point
1120:
This is the same situation as in classical
Brownian motion, and similarly this increase can be stopped by adding dissipative effects. Dissipative versions of the QMUPL, GRW and CSL model exist, where the collapse properties are left unaltered with respect to the original models, while the energy
998:
In 1990 the efforts for the GRW group on one side, and of P. Pearle on the other side, were brought together in formulating the
Continuous Spontaneous Localization (CSL) model, where the Schrödinger dynamics and a randomly fluctuating classical field produce collapse into spatially localized
986:
The next major advance came in 1986, when
Ghirardi, Rimini and Weber published the paper with the meaningful title “Unified dynamics for microscopic and macroscopic systems”, where they presented what is now known as the GRW model, after the initials of the authors. The model has two guiding
1083:
They are refined versions of the double-slit experiment, showing the wave nature of matter (and light). The modern versions are meant to increase the mass of the system, the time of flight, and/or the delocalization distance in order to create ever larger superpositions. The most prominent
1159:
matter. This leads into the second problem, however, the so-called “structured tails problem”: it is not clear how to interpret these tails because even though their “amount of matter” is small, that matter is structured like a perfectly legitimate world. Thus, after the box is opened and
1160:
Schroedinger’s cat has collapsed to the “alive” state, there still exists a tail of the wavefunction containing “low matter” entity structured like a dead cat. Collapse theorists have offered a range of possible solutions to the structured tails problem, but it remains an open problem.
1134:
verified violation of Bell inequalities, with the relativistic principle of locality. Models exist that attempt to generalize in a relativistic sense the GRW and CSL models, but their status as relativistic theories is still unclear. The formulation of a proper
1040:
The
Quantum Mechanics with Universal Position Localization (QMUPL) model should also be mentioned; an extension of the GRW model for identical particles formulated by Tumulka, which proves several important mathematical results regarding the collapse equations.
117:
1117:
to an almost minimum uncertainty state both in position and in momentum, compatibly with
Heisenberg's principle. The reason the energy increases is that the collapse noise diffuses the particle, thus accelerating it.
1002:
In the late 1980s and 1990s, Diosi and
Penrose and others independently formulated the idea that the wave function collapse is related to gravity. The dynamical equation is structurally similar to the CSL equation.
1112:
According to collapse theories, energy is not conserved, also for isolated particles. More precisely, in the GRW, CSL and DP models the kinetic energy increases at a constant rate, which is small but non-zero.
932:
usual quantum properties are recovered, apart from very tiny deviations. Such deviations can potentially be detected in dedicated experiments, and efforts are increasing worldwide towards testing them.
939:
In this sense, collapse models provide a unified description of microscopic and macroscopic systems, avoiding the conceptual problems associated to measurements in quantum theory.
1668:
Ghirardi, Gian Carlo; Pearle, Philip; Rimini, Alberto (1990). "Markov processes in
Hilbert space and continuous spontaneous localization of systems of identical particles".
1256:
Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013). "Models of wave-function collapse, underlying theories, and experimental tests".
3110:
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Bassi, Angelo; Ferialdi, Luca (2009). "Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties".
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In all collapse models, the noise effect cannot be described within quantum-mechanics. Instead it must prevent quantum mechanical linearity and unitarity.
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916:, they are possible explanations of why and how quantum measurements always give definite outcomes, not a superposition of them as predicted by the
591:
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47:
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is approximate. It works well for microscopic systems, but progressively loses its validity when the mass / complexity of the system increases.
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920:, and more generally how the classical world emerges from quantum theory. The fundamental idea is that the unitary evolution of the
130:
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arbitrary physical system (at least at the non-relativistic level), a necessary condition for any model to become a viable option.
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219:
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326:
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877:
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Lastly, the QMUPL model was further generalized to include both colored noise as well as dissipative effects (dcQMUPL model).
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obtaining their dissipative counterparts (dGRW, dCSL, dQMUPL). In these new models, the energy thermalizes to a finite value.
296:
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566:
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2714:
Ghirardi, G. C.; Grassi, R.; Pearle, P. (1990). "Relativistic dynamical reduction models: General framework and examples".
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Carlesso, Matteo; Donadi, Sandro; Ferialdi, Luca; Paternostro, Mauro; Ulbricht, Hendrik; Bassi, Angelo (February 2022).
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508:
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321:
311:
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2083:
Adler, Stephen L; Bassi, Angelo (2008). "Collapse models with non-white noises: II. Particle-density coupled noises".
576:
3425:
2316:
Smirne, Andrea; Vacchini, Bassano; Bassi, Angelo (2014). "Dissipative extension of the
Ghirardi-Rimini-Weber model".
493:
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The modification must reduce superpositions for macroscopic objects without altering the microscopic predictions.
488:
416:
339:
199:
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Bassi, Angelo; Ippoliti, Emiliano; Vacchini, Bassano (2005). "On the energy increase in space-collapse models".
3735:
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3705:
3695:
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3211:
611:
3064:, Stanford Encyclopedia of Philosophy (First published Thu Mar 7, 2002; substantive revision Fri May 15, 2020)
1622:
Pearle, Philip (1989). "Combining stochastic dynamical state-vector reduction with spontaneous localization".
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1371:
917:
780:
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456:
406:
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Ghirardi, G. C.; Rimini, A.; Weber, T. (1986). "Unified dynamics for microscopic and macroscopic systems".
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331:
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Ferialdi, Luca; Bassi, Angelo (2012). "Exact
Solution for a Non-Markovian Dissipative Quantum Dynamics".
3780:
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2998:
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
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In all models listed so far, the noise responsible for the collapse is Markovian (memoryless): either a
967:
254:
239:
3181:
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In collapse models the energy is not conserved, because the noise responsible for the collapse induces
166:
1711:
DiĂłsi, L. (1987). "A universal master equation for the gravitational violation of quantum mechanics".
991:
The position basis states are used in the dynamic state reduction (the "preferred basis" is position);
3811:
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3685:
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3223:
3015:
2906:"Do dynamical reduction models imply that arithmetic does not apply to ordinary macroscopic objects?"
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Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013-04-02).
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2004:
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thermalizes to a finite value (therefore it can even decrease, depending on its initial value).
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1996:
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Pearle, Philip (1976). "Reduction of the state vector by a nonlinear Schr\"odinger equation".
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38:
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Bassi, Angelo; Ferialdi, Luca (2009). "Non-Markovian Quantum Trajectories: An Exact Result".
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Tumulka, Roderich (2006). "On spontaneous wave function collapse and quantum field theory".
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Ferialdi, Luca; Bassi, Angelo (2012). "Dissipative collapse models with nonwhite noises".
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DiĂłsi, L. (1989). "Models for universal reduction of macroscopic quantum fluctuations".
1724:
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systems. Critics of collapse theories argue that it is not clear how to interpret these
3801:
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2765:
Tumulka, Roderich (2006). "A Relativistic Version of the Ghirardi–Rimini–Weber Model".
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436:
229:
2651:"Present status and future challenges of non-interferometric tests of collapse models"
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1992:
1373:
Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy
1348:
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259:
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2505:
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2008:
1952:
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3201:
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2236:
1907:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
840:
835:
770:
755:
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214:
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2022:
Adler, Stephen L; Bassi, Angelo (2007). "Collapse models with non-white noises".
1969:
Bassi, Angelo (2005). "Collapse models: analysis of the free particle dynamics".
1541:
Pearle, Philip (1984). "Experimental tests of dynamical state-vector reduction".
17:
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3027:
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805:
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695:
650:
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2550:
2347:
2276:"Testing the limits of quantum mechanics: motivation, state of play, prospects"
2175:
1416:"Models of wave-function collapse, underlying theories, and experimental tests"
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1287:
795:
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685:
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2000:
1944:
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1834:
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1295:
112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
3495:
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2970:
2882:
2843:
1600:
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286:
2931:
2627:
2441:
2252:
1936:
1689:
2955:"Discussion. More about dynamical reduction and the enumeration principle"
1788:
1697:
1651:
1608:
2922:
2779:
2472:
1983:
1919:
1331:
1317:
Bassi, Angelo; Ghirardi, GianCarlo (2003). "Dynamical reduction models".
665:
2735:
1826:
1805:
Penrose, Roger (1996). "On Gravity's role in Quantum State Reduction".
1511:
1138:
theory of continuous objective collapse is still a matter of research.
1085:
2415:
1858:"On the Gravitization of Quantum Mechanics 1: Quantum State Reduction"
3088:
3010:
2867:"Discussion. Losing your marbles in wavefunction collapse theories"
2667:
2594:
2533:
2398:
2330:
2219:
2158:
2097:
2036:
1432:
1270:
3092:
3071:"Physics Experiments Spell Doom for Quantum 'Collapse' Theory"
2382:"Dissipative Continuous Spontaneous Localization (CSL) model"
1490:
Pearle, Philip (1979). "Toward explaining why events occur".
1053:
but specific physical predictions can change significantly.
1011:
Three models are most widely discussed in the literature:
1084:
experiments of this kind are with atoms, molecules and
2994:"Four Tails Problems for Dynamical Collapse Theories"
50:
3837:
3789:
3622:
3554:
3488:
3401:
3350:
3304:
3169:
3126:
942:The most well-known examples of such theories are:
2085:Journal of Physics A: Mathematical and Theoretical
2024:Journal of Physics A: Mathematical and Theoretical
111:
2959:The British Journal for the Philosophy of Science
2910:The British Journal for the Philosophy of Science
2871:The British Journal for the Philosophy of Science
2832:The British Journal for the Philosophy of Science
1409:
1407:
1405:
1403:
1401:
1155:merely represent an immeasurably small amount of
2828:"Quantum Mechanics, Orthogonality, and Counting"
1025:Continuous spontaneous localization (CSL) model
952:Continuous spontaneous localization (CSL) model
2460:Journal of Physics A: Mathematical and General
1971:Journal of Physics A: Mathematical and General
3104:
878:
8:
1492:International Journal of Theoretical Physics
1100:Problems and criticisms to collapse theories
936:moving in space according to Newton's laws.
106:
80:
3111:
3097:
3089:
1376:(2 ed.). Cambridge University Press.
885:
871:
29:
3009:
2921:
2778:
2666:
2593:
2532:
2471:
2431:
2397:
2329:
2218:
2157:
2096:
2035:
1982:
1918:
1881:
1431:
1330:
1269:
962:Collapse theories stand in opposition to
98:
87:
86:
72:
57:
49:
910:measurement problem in quantum mechanics
1248:
54:
37:
2380:Smirne, Andrea; Bassi, Angelo (2015).
966:, in that they hold that a process of
2453:
2451:
2375:
2373:
2311:
2309:
1964:
1962:
1800:
1798:
1746:
1744:
1742:
1663:
1661:
7:
3906:Interpretations of quantum mechanics
2280:Journal of Physics: Condensed Matter
1574:
1572:
914:interpretations of quantum mechanics
2953:Bassi, A.; Ghirardi, G.-C. (1999).
2904:Ghirardi, G. C.; Bassi, A. (1999).
1171:Interpretation of quantum mechanics
1095:detectors, underground experiments.
964:many-worlds interpretation theories
27:Interpretation of quantum mechanics
1807:General Relativity and Gravitation
1105:Violation of the principle of the
974:and removes unobserved behaviour.
417:Sum-over-histories (path integral)
103:
77:
33:Part of a series of articles about
25:
1017:Ghirardi–Rimini–Weber (GRW) model
947:Ghirardi–Rimini–Weber (GRW) model
3880:
3879:
2865:Clifton, R.; Monton, B. (1999).
1226:Measurement in quantum mechanics
1092:Non-interferometric experiments.
1048:in the discrete GRW model, or a
908:, are proposed solutions to the
3829:Relativistic quantum mechanics
2767:Journal of Statistical Physics
2612:10.1103/PhysRevLett.108.170404
2237:10.1103/PhysRevLett.103.050403
2115:10.1088/1751-8113/41/39/395308
970:curtails the branching of the
567:Relativistic quantum mechanics
99:
92:
73:
1:
3807:Quantum statistical mechanics
3584:Quantum differential calculus
3506:Delayed-choice quantum eraser
3274:Symmetry in quantum mechanics
1349:10.1016/S0370-1573(03)00103-0
607:Quantum statistical mechanics
1733:10.1016/0375-9601(87)90681-5
1129:Relativistic collapse models
1081:Interferometric experiments.
978:History of collapse theories
3609:Quantum stochastic calculus
3599:Quantum measurement problem
3521:Mach–Zehnder interferometer
3028:10.1016/j.shpsb.2014.12.001
2992:McQueen, Kelvin J. (2015).
2490:10.1088/0305-4470/38/37/007
2292:10.1088/0953-8984/14/15/201
2274:Leggett, A J (2002-04-22).
2054:10.1088/1751-8113/40/50/012
1993:10.1088/0305-4470/38/14/008
902:spontaneous collapse models
898:Objective-collapse theories
577:Quantum information science
3927:
2685:10.1038/s41567-021-01489-5
2551:10.1103/PhysRevA.86.022108
2348:10.1103/PhysRevA.90.062135
2176:10.1103/PhysRevA.80.012116
1176:Many-worlds interpretation
924:describing the state of a
906:dynamical reduction models
3875:
3669:Quantum complexity theory
3647:Quantum cellular automata
3337:Path integral formulation
2797:10.1007/s10955-006-9227-3
1883:10.1007/s10701-013-9770-0
1442:10.1103/RevModPhys.85.471
1420:Reviews of Modern Physics
1288:10.1103/RevModPhys.85.471
1258:Reviews of Modern Physics
1181:Philosophy of information
3736:Quantum machine learning
3716:Quantum key distribution
3706:Quantum image processing
3696:Quantum error correction
3546:Wheeler's delayed choice
2826:Lewis, Peter J. (1997).
1773:10.1103/PhysRevA.40.1165
1644:10.1103/PhysRevA.39.2277
1382:10.1017/cbo9780511815676
1071:Tests of collapse models
1033:Diósi–Penrose (DP) model
957:Diósi–Penrose (DP) model
612:Quantum machine learning
365:Wheeler's delayed-choice
3652:Quantum finite automata
2582:Physical Review Letters
2207:Physical Review Letters
1856:Penrose, Roger (2014).
1601:10.1103/PhysRevD.34.470
1563:10.1103/PhysRevD.29.235
1477:10.1103/PhysRevD.13.857
322:Leggett–Garg inequality
3756:Quantum neural network
2716:Foundations of Physics
1937:10.1098/rspa.2005.1636
1862:Foundations of Physics
1690:10.1103/PhysRevA.42.78
1231:Wave function collapse
1107:conservation of energy
968:wave function collapse
113:
3781:Quantum teleportation
3294:Wave–particle duality
2971:10.1093/bjps/50.4.719
2883:10.1093/bjps/50.4.697
2844:10.1093/bjps/48.3.313
1186:Philosophy of physics
307:Elitzur–Vaidman
297:Davisson–Germer
114:
3812:Quantum field theory
3741:Quantum metamaterial
3686:Quantum cryptography
3416:Consistent histories
3060:Giancarlo Ghirardi,
2932:10.1093/bjps/50.1.49
1370:Bell, J. S. (2004).
1196:Quantum entanglement
918:Schrödinger equation
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
48:
3911:Quantum measurement
3797:Quantum fluctuation
3766:Quantum programming
3726:Quantum logic gates
3711:Quantum information
3691:Quantum electronics
3151:Classical mechanics
3020:2015SHPMP..49...10M
2789:2006JSP...125..821T
2728:1990FoPh...20.1271G
2677:2022NatPh..18..243C
2604:2012PhRvL.108q0404F
2543:2012PhRvA..86b2108F
2482:2005JPhA...38.8017B
2408:2015NatSR...512518S
2340:2014PhRvA..90f2135S
2229:2009PhRvL.103e0403B
2168:2009PhRvA..80a2116B
2107:2008JPhA...41M5308A
2046:2007JPhA...4015083A
2030:(50): 15083–15098.
1929:2006RSPSA.462.1897T
1913:(2070): 1897–1908.
1874:2014FoPh...44..557P
1819:1996GReGr..28..581P
1765:1989PhRvA..40.1165D
1725:1987PhLA..120..377D
1682:1990PhRvA..42...78G
1636:1989PhRvA..39.2277P
1593:1986PhRvD..34..470G
1555:1984PhRvD..29..235P
1504:1979IJTP...18..489P
1469:1976PhRvD..13..857P
1341:2003PhR...379..257B
1280:2013RvMP...85..471B
1221:Measurement problem
1216:Quantum Zeno effect
1206:Quantum decoherence
1201:Coherence (physics)
1191:Quantum information
1007:Most popular models
360:Stern–Gerlach
157:Classical mechanics
3850:in popular culture
3632:Quantum algorithms
3480:Von Neumann–Wigner
3460:Objective collapse
3156:Old quantum theory
2736:10.1007/BF01883487
2386:Scientific Reports
1827:10.1007/BF02105068
1512:10.1007/BF00670504
548:Von Neumann–Wigner
528:Objective-collapse
327:Mach–Zehnder
317:Leggett inequality
312:Franck–Hertz
162:Old quantum theory
109:
3893:
3892:
3867:Quantum mysticism
3845:Schrödinger's cat
3776:Quantum simulator
3746:Quantum metrology
3674:Quantum computing
3637:Quantum amplifier
3614:Quantum spacetime
3579:Quantum cosmology
3569:Quantum chemistry
3269:Scattering theory
3217:Zero-point energy
3212:Degenerate levels
3120:Quantum mechanics
3062:Collapse Theories
2722:(11): 1271–1316.
2521:Physical Review A
2466:(37): 8017–8038.
2416:10.1038/srep12518
2318:Physical Review A
2286:(15): R415–R451.
2146:Physical Review A
1977:(14): 3173–3192.
1753:Physical Review A
1713:Physics Letters A
1670:Physical Review A
1624:Physical Review A
1581:Physical Review D
1543:Physical Review D
1457:Physical Review D
1391:978-0-521-52338-7
1136:Lorentz-covariant
912:. As with other
895:
894:
602:Scattering theory
582:Quantum computing
355:Schrödinger's cat
287:Bell's inequality
95:
70:
39:Quantum mechanics
18:Collapse theories
16:(Redirected from
3918:
3883:
3882:
3594:Quantum geometry
3589:Quantum dynamics
3446:Superdeterminism
3378:Rarita–Schwinger
3327:Matrix mechanics
3182:Bra–ket notation
3113:
3106:
3099:
3090:
3085:
3083:
3082:
3048:
3047:
3013:
2989:
2983:
2982:
2950:
2944:
2943:
2925:
2923:quant-ph/9810041
2901:
2895:
2894:
2862:
2856:
2855:
2823:
2817:
2816:
2782:
2780:quant-ph/0406094
2762:
2756:
2755:
2711:
2705:
2704:
2670:
2646:
2640:
2639:
2597:
2577:
2571:
2570:
2536:
2516:
2510:
2509:
2475:
2473:quant-ph/0506083
2455:
2446:
2445:
2435:
2401:
2377:
2368:
2367:
2333:
2313:
2304:
2303:
2271:
2265:
2264:
2222:
2202:
2196:
2195:
2161:
2141:
2135:
2134:
2100:
2080:
2074:
2073:
2039:
2019:
2013:
2012:
1986:
1984:quant-ph/0410222
1966:
1957:
1956:
1922:
1920:quant-ph/0508230
1902:
1896:
1895:
1885:
1853:
1847:
1846:
1802:
1793:
1792:
1759:(3): 1165–1174.
1748:
1737:
1736:
1708:
1702:
1701:
1665:
1656:
1655:
1630:(5): 2277–2289.
1619:
1613:
1612:
1576:
1567:
1566:
1538:
1532:
1531:
1487:
1481:
1480:
1452:
1446:
1445:
1435:
1411:
1396:
1395:
1367:
1361:
1360:
1334:
1332:quant-ph/0302164
1325:(5–6): 257–426.
1314:
1308:
1307:
1273:
1253:
887:
880:
873:
514:Superdeterminism
167:Bra–ket notation
118:
116:
115:
110:
102:
97:
96:
88:
76:
71:
69:
58:
30:
21:
3926:
3925:
3921:
3920:
3919:
3917:
3916:
3915:
3896:
3895:
3894:
3889:
3871:
3857:Wigner's friend
3833:
3824:Quantum gravity
3785:
3771:Quantum sensing
3751:Quantum network
3731:Quantum machine
3701:Quantum imaging
3664:Quantum circuit
3659:Quantum channel
3618:
3564:Quantum biology
3550:
3526:Elitzur–Vaidman
3501:Davisson–Germer
3484:
3436:Hidden-variable
3426:de Broglie–Bohm
3403:Interpretations
3397:
3346:
3300:
3187:Complementarity
3165:
3122:
3117:
3080:
3078:
3075:Quanta Magazine
3069:
3057:
3052:
3051:
2991:
2990:
2986:
2952:
2951:
2947:
2903:
2902:
2898:
2864:
2863:
2859:
2825:
2824:
2820:
2764:
2763:
2759:
2713:
2712:
2708:
2648:
2647:
2643:
2579:
2578:
2574:
2518:
2517:
2513:
2457:
2456:
2449:
2379:
2378:
2371:
2315:
2314:
2307:
2273:
2272:
2268:
2204:
2203:
2199:
2143:
2142:
2138:
2082:
2081:
2077:
2021:
2020:
2016:
1968:
1967:
1960:
1904:
1903:
1899:
1855:
1854:
1850:
1804:
1803:
1796:
1750:
1749:
1740:
1710:
1709:
1705:
1667:
1666:
1659:
1621:
1620:
1616:
1578:
1577:
1570:
1540:
1539:
1535:
1489:
1488:
1484:
1454:
1453:
1449:
1413:
1412:
1399:
1392:
1369:
1368:
1364:
1319:Physics Reports
1316:
1315:
1311:
1255:
1254:
1250:
1245:
1240:
1236:Quantum gravity
1166:
1144:
1131:
1110:
1102:
1073:
1061:Brownian motion
1046:Poisson process
1009:
980:
891:
862:
861:
860:
625:
617:
616:
562:
561:Advanced topics
554:
553:
552:
504:Hidden-variable
494:de Broglie–Bohm
473:
471:Interpretations
463:
462:
461:
431:
423:
422:
421:
379:
371:
370:
369:
336:
292:CHSH inequality
281:
273:
272:
271:
200:Complementarity
194:
186:
185:
184:
152:
123:
62:
46:
45:
28:
23:
22:
15:
12:
11:
5:
3924:
3922:
3914:
3913:
3908:
3898:
3897:
3891:
3890:
3888:
3887:
3876:
3873:
3872:
3870:
3869:
3864:
3859:
3854:
3853:
3852:
3841:
3839:
3835:
3834:
3832:
3831:
3826:
3821:
3820:
3819:
3809:
3804:
3802:Casimir effect
3799:
3793:
3791:
3787:
3786:
3784:
3783:
3778:
3773:
3768:
3763:
3761:Quantum optics
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3682:
3681:
3671:
3666:
3661:
3656:
3655:
3654:
3644:
3639:
3634:
3628:
3626:
3620:
3619:
3617:
3616:
3611:
3606:
3601:
3596:
3591:
3586:
3581:
3576:
3571:
3566:
3560:
3558:
3552:
3551:
3549:
3548:
3543:
3538:
3536:Quantum eraser
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3498:
3492:
3490:
3486:
3485:
3483:
3482:
3477:
3472:
3467:
3462:
3457:
3452:
3451:
3450:
3449:
3448:
3433:
3428:
3423:
3418:
3413:
3407:
3405:
3399:
3398:
3396:
3395:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3354:
3352:
3348:
3347:
3345:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3308:
3306:
3302:
3301:
3299:
3298:
3297:
3296:
3291:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3221:
3220:
3219:
3214:
3209:
3204:
3194:
3192:Density matrix
3189:
3184:
3179:
3173:
3171:
3167:
3166:
3164:
3163:
3158:
3153:
3148:
3147:
3146:
3136:
3130:
3128:
3124:
3123:
3118:
3116:
3115:
3108:
3101:
3093:
3087:
3086:
3066:
3065:
3056:
3055:External links
3053:
3050:
3049:
2984:
2965:(4): 719–734.
2945:
2896:
2877:(4): 697–717.
2857:
2838:(3): 313–328.
2818:
2773:(4): 821–840.
2757:
2706:
2661:(3): 243–250.
2655:Nature Physics
2641:
2588:(17): 170404.
2572:
2511:
2447:
2369:
2305:
2266:
2197:
2136:
2091:(39): 395308.
2075:
2014:
1958:
1897:
1868:(5): 557–575.
1848:
1813:(5): 581–600.
1794:
1738:
1719:(8): 377–381.
1703:
1657:
1614:
1587:(2): 470–491.
1568:
1549:(2): 235–240.
1533:
1498:(7): 489–518.
1482:
1463:(4): 857–868.
1447:
1426:(2): 471–527.
1397:
1390:
1362:
1309:
1264:(2): 471–527.
1247:
1246:
1244:
1241:
1239:
1238:
1233:
1228:
1223:
1218:
1213:
1208:
1203:
1198:
1193:
1188:
1183:
1178:
1173:
1167:
1165:
1162:
1143:
1140:
1130:
1127:
1109:
1103:
1101:
1098:
1097:
1096:
1089:
1072:
1069:
1038:
1037:
1029:
1021:
1008:
1005:
999:eigentstates.
996:
995:
992:
979:
976:
960:
959:
954:
949:
926:quantum system
893:
892:
890:
889:
882:
875:
867:
864:
863:
859:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
627:
626:
623:
622:
619:
618:
615:
614:
609:
604:
599:
597:Density matrix
594:
589:
584:
579:
574:
569:
563:
560:
559:
556:
555:
551:
550:
545:
540:
535:
530:
525:
520:
519:
518:
517:
516:
501:
496:
491:
486:
481:
475:
474:
469:
468:
465:
464:
460:
459:
454:
449:
444:
439:
433:
432:
429:
428:
425:
424:
420:
419:
414:
409:
404:
399:
394:
388:
387:
386:
380:
377:
376:
373:
372:
368:
367:
362:
357:
351:
350:
349:
348:
347:
345:Delayed-choice
340:Quantum eraser
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
282:
279:
278:
275:
274:
270:
269:
268:
267:
257:
252:
247:
242:
237:
232:
230:Quantum number
227:
222:
217:
212:
207:
202:
196:
195:
192:
191:
188:
187:
183:
182:
177:
171:
170:
169:
164:
159:
153:
150:
149:
146:
145:
144:
143:
138:
133:
125:
124:
119:
108:
105:
101:
94:
91:
85:
82:
79:
75:
68:
65:
61:
56:
53:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3923:
3912:
3909:
3907:
3904:
3903:
3901:
3886:
3878:
3877:
3874:
3868:
3865:
3863:
3860:
3858:
3855:
3851:
3848:
3847:
3846:
3843:
3842:
3840:
3836:
3830:
3827:
3825:
3822:
3818:
3815:
3814:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3794:
3792:
3788:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3721:Quantum logic
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3680:
3677:
3676:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3653:
3650:
3649:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3629:
3627:
3625:
3621:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3574:Quantum chaos
3572:
3570:
3567:
3565:
3562:
3561:
3559:
3557:
3553:
3547:
3544:
3542:
3541:Stern–Gerlach
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3493:
3491:
3487:
3481:
3478:
3476:
3475:Transactional
3473:
3471:
3468:
3466:
3465:Quantum logic
3463:
3461:
3458:
3456:
3453:
3447:
3444:
3443:
3442:
3439:
3438:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3408:
3406:
3404:
3400:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3355:
3353:
3349:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3309:
3307:
3303:
3295:
3292:
3290:
3287:
3286:
3285:
3284:Wave function
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3259:Superposition
3257:
3255:
3254:Quantum state
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3218:
3215:
3213:
3210:
3208:
3207:Excited state
3205:
3203:
3200:
3199:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3174:
3172:
3168:
3162:
3159:
3157:
3154:
3152:
3149:
3145:
3142:
3141:
3140:
3137:
3135:
3132:
3131:
3129:
3125:
3121:
3114:
3109:
3107:
3102:
3100:
3095:
3094:
3091:
3076:
3072:
3068:
3067:
3063:
3059:
3058:
3054:
3045:
3041:
3037:
3033:
3029:
3025:
3021:
3017:
3012:
3007:
3003:
2999:
2995:
2988:
2985:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2949:
2946:
2941:
2937:
2933:
2929:
2924:
2919:
2915:
2911:
2907:
2900:
2897:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2861:
2858:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2822:
2819:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2781:
2776:
2772:
2768:
2761:
2758:
2753:
2749:
2745:
2741:
2737:
2733:
2729:
2725:
2721:
2717:
2710:
2707:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2669:
2664:
2660:
2656:
2652:
2645:
2642:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2601:
2596:
2591:
2587:
2583:
2576:
2573:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2535:
2530:
2527:(2): 022108.
2526:
2522:
2515:
2512:
2507:
2503:
2499:
2495:
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2353:
2349:
2345:
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2327:
2324:(6): 062135.
2323:
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2312:
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2306:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
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2258:
2254:
2250:
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2234:
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2216:
2213:(5): 050403.
2212:
2208:
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2173:
2169:
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2160:
2155:
2152:(1): 012116.
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2132:
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2063:
2059:
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2033:
2029:
2025:
2018:
2015:
2010:
2006:
2002:
1998:
1994:
1990:
1985:
1980:
1976:
1972:
1965:
1963:
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1954:
1950:
1946:
1942:
1938:
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1930:
1926:
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1916:
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1234:
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1197:
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1179:
1177:
1174:
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1158:
1154:
1150:
1142:Tails problem
1141:
1139:
1137:
1128:
1126:
1122:
1118:
1114:
1108:
1104:
1099:
1093:
1090:
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1013:
1012:
1006:
1004:
1000:
993:
990:
989:
988:
984:
977:
975:
973:
972:wave function
969:
965:
958:
955:
953:
950:
948:
945:
944:
943:
940:
937:
933:
929:
927:
923:
922:wave function
919:
915:
911:
907:
903:
900:, also known
899:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
849:
847:
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794:
792:
789:
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782:
779:
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774:
772:
769:
767:
764:
762:
759:
757:
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752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
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682:
679:
677:
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672:
669:
667:
664:
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654:
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647:
644:
642:
639:
637:
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629:
628:
621:
620:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
587:Quantum chaos
585:
583:
580:
578:
575:
573:
570:
568:
565:
564:
558:
557:
549:
546:
544:
543:Transactional
541:
539:
536:
534:
533:Quantum logic
531:
529:
526:
524:
521:
515:
512:
511:
510:
507:
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502:
500:
497:
495:
492:
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293:
290:
288:
285:
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277:
276:
266:
263:
262:
261:
260:Wave function
258:
256:
253:
251:
248:
246:
243:
241:
240:Superposition
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
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208:
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129:
128:
127:
126:
122:
89:
83:
66:
63:
59:
51:
44:
43:
40:
36:
32:
31:
19:
3604:Quantum mind
3516:Franck–Hertz
3459:
3358:Klein–Gordon
3312:Formulations
3305:Formulations
3234:Interference
3224:Entanglement
3202:Ground state
3197:Energy level
3170:Fundamentals
3134:Introduction
3079:. Retrieved
3077:. 2022-10-20
3074:
3001:
2997:
2987:
2962:
2958:
2948:
2916:(1): 49–64.
2913:
2909:
2899:
2874:
2870:
2860:
2835:
2831:
2821:
2770:
2766:
2760:
2719:
2715:
2709:
2658:
2654:
2644:
2585:
2581:
2575:
2524:
2520:
2514:
2463:
2459:
2392:(1): 12518.
2389:
2385:
2321:
2317:
2283:
2279:
2269:
2210:
2206:
2200:
2149:
2145:
2139:
2088:
2084:
2078:
2027:
2023:
2017:
1974:
1970:
1910:
1906:
1900:
1865:
1861:
1851:
1810:
1806:
1756:
1752:
1716:
1712:
1706:
1676:(1): 78–89.
1673:
1669:
1627:
1623:
1617:
1584:
1580:
1546:
1542:
1536:
1495:
1491:
1485:
1460:
1456:
1450:
1423:
1419:
1372:
1365:
1322:
1318:
1312:
1261:
1257:
1251:
1156:
1152:
1148:
1145:
1132:
1123:
1119:
1115:
1111:
1091:
1080:
1076:two groups:
1074:
1066:
1058:
1055:
1043:
1039:
1031:
1023:
1015:
1010:
1001:
997:
987:principles:
985:
981:
961:
941:
938:
934:
930:
905:
901:
897:
896:
527:
442:Klein–Gordon
378:Formulations
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
3862:EPR paradox
3642:Quantum bus
3511:Double-slit
3489:Experiments
3455:Many-worlds
3393:Schrödinger
3342:Phase space
3332:Schrödinger
3322:Interaction
3279:Uncertainty
3249:Nonlocality
3244:Measurement
3239:Decoherence
3229:Hamiltonian
1211:EPR paradox
1157:smeared-out
1125:conserved.
1050:white noise
831:von Neumann
816:Schrödinger
592:EPR paradox
523:Many-worlds
457:Schrödinger
412:Schrödinger
407:Phase-space
397:Interaction
302:Double-slit
280:Experiments
255:Uncertainty
225:Nonlocality
220:Measurement
205:Decoherence
175:Hamiltonian
3900:Categories
3790:Extensions
3624:Technology
3470:Relational
3421:Copenhagen
3317:Heisenberg
3264:Tunnelling
3127:Background
3081:2022-10-21
3011:1501.05778
2668:2203.04231
1243:References
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
3496:Bell test
3351:Equations
3177:Born rule
3036:1355-2198
3004:: 10–18.
2979:0007-0882
2940:0007-0882
2891:0007-0882
2852:0007-0882
2805:0022-4715
2752:123661865
2744:0015-9018
2701:246949254
2693:1745-2481
2620:0031-9007
2595:1204.4348
2567:119216571
2559:1050-2947
2534:1112.5065
2498:0305-4470
2424:2045-2322
2399:1408.6446
2356:1050-2947
2331:1408.6115
2300:0953-8984
2245:0031-9007
2220:0907.1615
2192:119297164
2184:1050-2947
2159:0901.1254
2131:118551622
2123:1751-8113
2098:0807.2846
2070:118366772
2062:1751-8113
2037:0708.3624
2001:0305-4470
1945:1364-5021
1892:0015-9018
1835:0001-7701
1781:0556-2791
1528:119407617
1520:0020-7748
1433:1204.4325
1357:119076099
1304:119261020
1296:0034-6861
1271:1204.4325
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
3885:Category
3679:Timeline
3431:Ensemble
3411:Bayesian
3373:Majorana
3289:Collapse
3161:Glossary
3144:Timeline
3044:55718585
2813:13923422
2636:16746767
2628:22680843
2506:43241594
2442:26243034
2364:52232273
2261:25021141
2253:19792469
2009:37142667
1953:16123332
1843:44038399
1164:See also
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
3838:Related
3817:History
3556:Science
3388:Rydberg
3139:History
3016:Bibcode
2785:Bibcode
2724:Bibcode
2673:Bibcode
2600:Bibcode
2539:Bibcode
2478:Bibcode
2433:4525142
2404:Bibcode
2336:Bibcode
2225:Bibcode
2164:Bibcode
2103:Bibcode
2042:Bibcode
1925:Bibcode
1870:Bibcode
1815:Bibcode
1789:9902248
1761:Bibcode
1721:Bibcode
1698:9903779
1678:Bibcode
1652:9901493
1632:Bibcode
1609:9957165
1589:Bibcode
1551:Bibcode
1500:Bibcode
1465:Bibcode
1337:Bibcode
1276:Bibcode
1086:phonons
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
3531:Popper
3042:
3034:
2977:
2938:
2889:
2850:
2811:
2803:
2750:
2742:
2699:
2691:
2634:
2626:
2618:
2565:
2557:
2504:
2496:
2440:
2430:
2422:
2362:
2354:
2298:
2259:
2251:
2243:
2190:
2182:
2129:
2121:
2068:
2060:
2007:
1999:
1951:
1943:
1890:
1841:
1833:
1787:
1779:
1696:
1650:
1607:
1526:
1518:
1388:
1355:
1302:
1294:
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
3441:Local
3383:Pauli
3363:Dirac
3040:S2CID
3006:arXiv
2918:arXiv
2809:S2CID
2775:arXiv
2748:S2CID
2697:S2CID
2663:arXiv
2632:S2CID
2590:arXiv
2563:S2CID
2529:arXiv
2502:S2CID
2468:arXiv
2394:arXiv
2360:S2CID
2326:arXiv
2257:S2CID
2215:arXiv
2188:S2CID
2154:arXiv
2127:S2CID
2093:arXiv
2066:S2CID
2032:arXiv
2005:S2CID
1979:arXiv
1949:S2CID
1915:arXiv
1839:S2CID
1524:S2CID
1428:arXiv
1353:S2CID
1327:arXiv
1300:S2CID
1266:arXiv
1153:tails
1149:tails
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
3368:Weyl
3032:ISSN
2975:ISSN
2936:ISSN
2887:ISSN
2848:ISSN
2801:ISSN
2740:ISSN
2689:ISSN
2624:PMID
2616:ISSN
2555:ISSN
2494:ISSN
2438:PMID
2420:ISSN
2352:ISSN
2296:ISSN
2249:PMID
2241:ISSN
2180:ISSN
2119:ISSN
2058:ISSN
1997:ISSN
1941:ISSN
1888:ISSN
1831:ISSN
1785:PMID
1777:ISSN
1694:PMID
1648:PMID
1605:PMID
1516:ISSN
1386:ISBN
1292:ISSN
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
3024:doi
2967:doi
2928:doi
2879:doi
2840:doi
2793:doi
2771:125
2732:doi
2681:doi
2608:doi
2586:108
2547:doi
2486:doi
2428:PMC
2412:doi
2344:doi
2288:doi
2233:doi
2211:103
2172:doi
2111:doi
2050:doi
1989:doi
1933:doi
1911:462
1878:doi
1823:doi
1769:doi
1729:doi
1717:120
1686:doi
1640:doi
1597:doi
1559:doi
1508:doi
1473:doi
1438:doi
1378:doi
1345:doi
1323:379
1284:doi
904:or
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2436:.
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2334:.
2322:90
2320:.
2308:^
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2003:.
1995:.
1987:.
1975:38
1973:.
1961:^
1947:.
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2011:.
1991::
1981::
1955:.
1935::
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1088:.
886:e
879:t
872:v
100:|
90:H
84:=
74:|
67:t
64:d
60:d
52:i
20:)
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