2923:
301:. Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the
140:
5816:
2656:
43:
3975:
2918:{\displaystyle {\begin{aligned}\langle r|u\rangle &=\left({\frac {1}{\sqrt {2}}}\langle u|+{\frac {1}{\sqrt {2}}}\langle d|\right)\cdot |u\rangle \\&=\left({\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\end{pmatrix}}+{\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\1\end{pmatrix}}\right)\cdot {\begin{pmatrix}1\\0\end{pmatrix}}\\&={\frac {1}{\sqrt {2}}}\end{aligned}}}
3781:
4335:
3650:
3728:
that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each
1443:
3723:
is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the
3655:
The probability density function does not vary with time as the evolution of the wave function is dictated by the Schrödinger equation and is therefore entirely deterministic. This is key to understanding the importance of this interpretation: for a given particle constant
331:(so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of
3136:
4177:
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.
4197:
4053:
respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term
4143:
3970:{\displaystyle P=\left|\psi _{\text{first}}+\psi _{\text{second}}\right|^{2}=\left|\psi _{\text{first}}\right|^{2}+\left|\psi _{\text{second}}\right|^{2}+2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2}).}
4815:
A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain
2371:
3370:
3509:
4416:
712:
1090:
3452:
1318:
903:
3696:
Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic
2111:
3701:, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that
979:
347:
always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to
2591:
2526:
2661:
3024:
2213:
respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is
541:
4330:{\displaystyle \mathbf {j} ={\hbar \over m}{1 \over {2i}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)={\hbar \over m}\operatorname {Im} \left(\psi ^{*}\nabla \psi \right),}
1509:
4731:
The spanning set of a
Hilbert space does not suffice for defining coordinates as wave functions form rays in a projective Hilbert space (rather than an ordinary Hilbert space). See:
627:
4057:
1310:
409:
4462:
2960:
2046:
1826:
1627:
1140:
489:
2248:
4029:
4002:
2285:
2425:
2398:
2211:
2183:
2014:
1986:
1603:
222:
for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and
1573:
2648:
2621:
2459:
1762:
3016:
2988:
2135:
2155:
801:
767:
569:
2293:
4474:
corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local
4468:, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where
3285:
1534:
1249:
1225:
430:
3682:, the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at all subsequent times.
3645:{\displaystyle P_{\mathbf {x} \in V}(t)=\int _{V}|\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =\int _{V}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } .}
1275:
4823:
4349:
2434:
Another example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (
382:
no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of
643:
5060:
987:
5613:
1438:{\displaystyle \langle x|{\hat {\mathrm {x} }}|\Psi \rangle ={\hat {\mathrm {x} }}\langle x|\Psi \rangle =x_{0}\psi (x),\quad x\in \mathbb {R} ,}
3385:
218:
being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954
5261:
820:
2428:
4148:
However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then, due to
2054:
5784:
107:
5110:
1634:
214:
of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as
4913:
79:
5796:
5480:
5053:
60:
5414:
5029:
5010:
4983:
4677:
920:
86:
5337:
3131:{\displaystyle P(|u\rangle )=\langle r|u\rangle \langle u|r\rangle =\left({\frac {1}{\sqrt {2}}}\right)^{2}={\frac {1}{2}}}
235:
2532:
2467:
630:
3778:
of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
93:
4824:"Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates"
4155:
One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a
497:
450:
5083:
5046:
4957:
4938:
4902:
3241:, describing states of quantum particles, has solutions that describe a system and determine precisely how the state
126:
4643:. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of
271:, states on which the value of the observable is uniquely defined, for different possible values of the observable.
5193:
17:
454:
75:
5751:
5460:
5455:
5178:
4929:
4820:
realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf.
3725:
580:
4872:
4138:{\textstyle 2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2})}
5763:
5435:
1457:
327:) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of
64:
1530:
5741:
5518:
5440:
5223:
5093:
4975:
1449:
590:
1205:
5850:
5845:
5840:
5475:
5409:
5404:
5375:
5088:
4620:
3488:
3208:, that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained
1284:
1147:
1143:
215:
196:
160:
5543:
5450:
4032:
1937:
1903:
172:
5819:
5581:
5389:
5360:
4424:
3231:
780:
210:, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the
31:
3202:
the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one
309:
5603:
5470:
5394:
5355:
5312:
5286:
5243:
5136:
4160:
4156:
3458:
3376:
2931:
1705:
279:
211:
3461:, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence,
2023:
1785:
1608:
1099:
460:
5670:
5650:
5640:
5630:
5586:
5161:
4687:
4648:
4644:
2216:
1949:
1775:
100:
4007:
3980:
2253:
226:, were vigorously contested at the time by the original physicists working on the theory, such as
5365:
5291:
4831:
4817:
4697:
3242:
289:, returning the eigenvalue belonging to that eigenstate. The system may always be described by a
53:
5779:
5327:
4145:
is called the "interference term", and this would be missing if we had added the probabilities.
3238:
2403:
2376:
2188:
2160:
1991:
1963:
1582:
206:
vector of a system and the results of observations of that system, a link was first proposed by
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5445:
5370:
5238:
3698:
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1548:
446:
283:
219:
5281:
4895:
Lie
Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics
2366:{\textstyle |\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle }
1738:
442:
5715:
5228:
5208:
4702:
4149:
3771:
2626:
2599:
2437:
2120:
2017:
1921:
1576:
294:
262:
258:
3365:{\displaystyle \int |\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty .}
2993:
2965:
2140:
5746:
5675:
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5350:
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5131:
4993:
4840:
4608:
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the wave function still represents the same state and is therefore equal by definition to
227:
8:
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5345:
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4974:. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.:
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4465:
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4172:
1932:
transformation calculations. Discrete dynamical variables are used in such problems as a
1925:
1201:
910:
572:
247:
223:
4844:
378:
is observed the probability amplitudes are changed. A second, subsequent observation of
234:. It is the source of the mysterious consequences and philosophical difficulties in the
5791:
5660:
5558:
5266:
5213:
5105:
4864:
3215:
One can always divide any non-zero element of a
Hilbert space by its norm and obtain a
1953:
1234:
1210:
415:
290:
30:
This article is about probability amplitude in quantum mechanics. For other uses, see
5801:
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5680:
5608:
5571:
5566:
5548:
5513:
5503:
5218:
5183:
5166:
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3205:
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of measurements are made. The order of such results, is, however, completely random.
1933:
1279:
776:
180:
4411:{\displaystyle \nabla \cdot \mathbf {j} +{\partial \over \partial t}|\psi |^{2}=0.}
1675:
403:
5528:
5523:
5380:
5276:
4868:
4852:
4848:
3679:
1151:
707:{\displaystyle \mu =\mu _{\mathrm {ac} }+\mu _{\mathrm {sc} }+\mu _{\mathrm {pp} }}
576:
4924:
5758:
5685:
5665:
5635:
5598:
5593:
5498:
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4989:
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4604:
3775:
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298:
231:
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5307:
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4656:
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152:
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event. The complex amplitudes which represent the electron passing each slit (
5834:
5655:
5508:
5399:
5233:
5203:
5156:
4682:
4543:
3246:
1146:
for a measurement of the particle's position at a given time, defined as the
747:
433:
203:
156:
144:
4166:
1085:{\displaystyle \int _{X}|\psi (x)|^{2}\,dx\equiv \int _{X}\,d\mu _{ac}(x)=1}
5538:
5151:
5146:
4860:
2461:), the following must be true for the measurement of spin "up" and "down":
1259:
of an observable need not necessarily be measurable functions belonging to
1252:
3447:{\displaystyle \psi (\mathbf {x} ,t):={\frac {\psi (\mathbf {x} ,t)}{a}}.}
5576:
4967:
4712:
4692:
4652:
1728:
1161:
1256:
548:
391:
374:(all probability amplitudes zero except for one eigenstate), then when
348:
267:
251:
4659:
specifies transition probabilities between a finite number of states.
1960:. When the polarization is measured, it could be the horizontal state
5430:
5126:
4163:, the case A applies again and the interference pattern is restored.
1197:
302:
164:
5038:
4624:
2596:
If one assumes that system is prepared, so that +1 is registered in
898:{\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }
42:
4972:
Orthogonal polynomials on the unit circle. Part 1. Classical theory
4640:
3219:
state vector. Not every wave function belongs to the
Hilbert space
1948:
An example of the discrete case is a quantum system that can be in
1929:
731:
is singular with respect to the
Lebesgue measure and atomless, and
308:
Clearly, the sum of the probabilities, which equals the sum of the
207:
148:
4655:
specifies a finite probability distribution, a finite-dimensional
4340:
measured in units of (probability)/(area × time).
3230:, though. Wave functions that fulfill this constraint are called
2106:{\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }
4761:
1196:
Whereas a
Hilbert space is separable if and only if it admits a
370:. Therefore, if the system is known to be in some eigenstate of
250:
is the behaviour of a quantum state, for which the value of the
191:
used for describing the behaviour of systems. The square of the
4623:. This strengthens the probabilistic interpretation explicated
3685:
1957:
1190:
1179:
must have the inverse dimension of the variable of integration
746:
A usual presentation of the probability amplitude is that of a
724:
is absolutely continuous with respect to the
Lebesgue measure,
408:
In a formal setup, the state of an isolated physical system in
323:(e.g. after an observation of the corresponding eigenvalue of
4897:. Studies in Mathematical Physics. Amsterdam: North Holland.
4773:
2016:. Until its polarization is measured the photon can be in a
386:
depend on whether it comes before or after a measurement of
27:
Complex number whose squared absolute value is a probability
3657:
2928:
The probability amplitude of measuring spin up is given by
4821:
4152:, the interference pattern is not observed on the screen.
202:
Probability amplitudes provide a relationship between the
167:): this is calculated from the probability amplitude. The
4635:
The concept of amplitudes is also used in the context of
4167:
Conservation of probabilities and the continuity equation
2400:
to come out horizontally polarized, and a probability of
335:
is the same as the set of eigenstates for measurement of
312:
of the probability amplitudes, must equal 1. This is the
168:
4749:
1920:
A discrete probability amplitude may be considered as a
366:
produces a jump to a state that is not an eigenstate of
4651:
just as in a random process. Like a finite-dimensional
3491:
and the probability that the particle is in the volume
3149:
In the example above, the measurement must give either
2250:, and the probability of being vertically polarized is
974:{\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}
246:
Neglecting some technical complexities, the problem of
159:. The solid body shows the places where the electron's
4060:
2996:
2968:
2934:
2868:
2834:
2793:
2629:
2602:
2440:
2406:
2379:
2296:
1231:(i.e. the probability that the system is "at position
4797:
4427:
4352:
4200:
4010:
3983:
3784:
3512:
3388:
3288:
3027:
2659:
2535:
2470:
2256:
2219:
2191:
2163:
2143:
2123:
2057:
2026:
1994:
1966:
1788:
1741:
1727:
it means that all elements of the standard basis are
1611:
1585:
1551:
1460:
1321:
1287:
1237:
1213:
1102:
990:
923:
823:
789:
755:
646:
593:
557:
500:
463:
418:
4922:
4662:
The "transitional" interpretation may be applied to
2586:{\displaystyle \sigma _{z}|d\rangle =(-1)|d\rangle }
2521:{\displaystyle \sigma _{z}|u\rangle =(+1)|u\rangle }
1828:
gives the probability of the corresponding value of
551:
can be elucidated by generalizing the quantum state
4923:Feynman, R. P.; Leighton, R. B.; Sands, M. (1989).
4911:
4767:
3204:. If to understand "all the possible states" as an
1529:, however, the rigorous notion of eigenstates from
319:If the system is known to be in some eigenstate of
67:. Unsourced material may be challenged and removed.
4915:Quantum mechanics in rigged Hilbert space language
4737:
4456:
4410:
4329:
4137:
4023:
3996:
3969:
3644:
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3364:
3130:
3010:
2982:
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2615:
2585:
2520:
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2419:
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2365:
2279:
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2205:
2177:
2149:
2129:
2105:
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2008:
1980:
1820:
1756:
1621:
1597:
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1503:
1437:
1304:
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1219:
1134:
1084:
973:
897:
795:
761:
706:
621:
563:
535:
483:
424:
4619:on the systems 1 and 2 behave on these states as
3747:) follow the law of precisely the form expected:
5832:
4893:BĂ€uerle, Gerard G. A.; de Kerf, Eddy A. (1990).
1764:is the probability amplitude for the eigenstate
536:{\displaystyle \psi (x)=\langle x|\Psi \rangle }
453:the relation between state vector and "position
238:âtopics that continue to be debated even today.
5005:. Providence (R.I): American Mathematical Soc.
1164:). As probability is a dimensionless quantity,
4892:
4779:
3263:gives a description of the particle (position
1674:for uniformity with the previous case. If the
1650:. The amplitudes are composed of state vector
5054:
4933:. Vol. 3. Redwood City: Addison-Wesley.
2623:and then the apparatus is rotated to measure
1719:produces some unique value of the observable
1525:. By replacing the state space by a suitable
3686:In the context of the double-slit experiment
3082:
3068:
3065:
3051:
3042:
3005:
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2949:
2935:
2758:
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2705:
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2554:
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2305:
2200:
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2100:
2083:
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2035:
2003:
1975:
1389:
1375:
1355:
1322:
831:
824:
530:
516:
478:
475:
464:
3189:must be 1. This leads to a constraint that
397:
5061:
5047:
4630:
4605:product of respective probability measures
2157:the probability amplitudes for the states
5003:Mathematical Methods in Quantum Mechanics
3628:
3590:
3329:
2962:, since the system had the initial state
2427:to come out vertically polarized when an
1504:{\displaystyle \psi (x)=\delta (x-x_{0})}
1428:
1050:
1030:
958:
882:
339:, then subsequent measurements of either
127:Learn how and when to remove this message
5019:
4918:(PhD thesis). Universidad de Valladolid.
4822:Schmidt, L. Ph. H.; et al. (2013).
4803:
4343:Then the current satisfies the equation
3169:, so the total probability of measuring
741:
491:of the Hilbert space can be written as
138:
1934:particle in an idealized reflective box
1185:. For example, the above amplitude has
622:{\displaystyle (X,{\mathcal {A}},\mu )}
14:
5833:
5000:
4947:
4755:
4607:. In other words, amplitudes of a non-
1540:
5068:
5042:
4966:
4743:
4678:Expectation value (quantum mechanics)
1924:in the probability frequency domain (
1305:{\displaystyle {\hat {\mathrm {x} }}}
4486:For two quantum systems with spaces
4481:
241:
236:interpretations of quantum mechanics
163:is above a certain value (here 0.02
65:adding citations to reliable sources
36:
4534:respectively, their combined state
2020:of both these states, so its state
637:into three mutually singular parts
362:are different, then measurement of
354:By contrast, if the eigenstates of
24:
4668:s on non-discrete spaces as well.
4373:
4369:
4353:
4313:
4261:
4249:
3630:
3592:
3356:
3331:
1928:) for the purposes of simplifying
1614:
1386:
1365:
1352:
1337:
1292:
1271:
892:
698:
695:
680:
677:
662:
659:
629:. This allows for a refinement of
605:
527:
313:
297:of these eigenstates with unequal
261:. Such a state is thought to be a
25:
5862:
4457:{\displaystyle \rho =|\psi |^{2}}
3209:
1711:A convenient configuration space
1633:); specifying the measure of any
171:on the colored surface shows the
5815:
5814:
4912:de la Madrid Modino, R. (2001).
4360:
4202:
3774:. The probability, which is the
3634:
3621:
3596:
3565:
3519:
3422:
3396:
3335:
3304:
3144:
2955:{\textstyle \langle r|u\rangle }
1663:; its components are denoted by
631:Lebesgue's decomposition theorem
412:is represented, at a fixed time
41:
4930:The Feynman Lectures on Physics
4616:
4464:, this equation is exactly the
2990:. The probability of measuring
1514:which clearly do not belong to
1420:
278:is made, the system (under the
52:needs additional citations for
5764:Relativistic quantum mechanics
5024:. Cambridge, Mass: MIT Press.
4853:10.1103/PhysRevLett.111.103201
4809:
4785:
4725:
4444:
4435:
4392:
4383:
4132:
4106:
3961:
3935:
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3561:
3554:
3537:
3531:
3432:
3418:
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3392:
3319:
3314:
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3293:
3141:Which agrees with experiment.
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2165:
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2041:{\displaystyle |\psi \rangle }
2028:
1996:
1968:
1821:{\displaystyle |\psi (x)|^{2}}
1808:
1803:
1797:
1790:
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1745:
1622:{\displaystyle {\mathcal {A}}}
1498:
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1296:
1135:{\displaystyle |\psi (x)|^{2}}
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484:{\displaystyle \{|x\rangle \}}
468:
404:Bound state § Definition
195:of this quantity represents a
13:
1:
5742:Quantum statistical mechanics
5519:Quantum differential calculus
5441:Delayed-choice quantum eraser
5224:Symmetry in quantum mechanics
4976:American Mathematical Society
4952:. San Diego: Academic Press.
4886:
2243:{\displaystyle |\alpha |^{2}}
1774:. If it corresponds to a non-
4708:Ward's probability amplitude
4621:independent random variables
4615:of original amplitudes, and
4024:{\displaystyle \varphi _{2}}
3997:{\displaystyle \varphi _{1}}
3489:probability density function
2373:would have a probability of
2280:{\displaystyle |\beta |^{2}}
1144:probability density function
7:
5544:Quantum stochastic calculus
5534:Quantum measurement problem
5456:MachâZehnder interferometer
5022:Mastering Quantum Mechanics
4948:Gudder, Stanley P. (1988).
4671:
4647:squared are interpreted as
4421:The probability density is
3770:. This is the principle of
2420:{\textstyle {\frac {2}{3}}}
2393:{\textstyle {\frac {1}{3}}}
2290:Hence, a photon in a state
1943:
1938:quantum harmonic oscillator
1906:. Otherwise the modulus of
781:square integrable functions
10:
5867:
4780:BĂ€uerle & de Kerf 1990
4170:
3689:
2206:{\displaystyle |V\rangle }
2178:{\displaystyle |H\rangle }
2009:{\displaystyle |V\rangle }
1981:{\displaystyle |H\rangle }
1598:{\displaystyle A\subset X}
1206:continuous random variable
401:
390:, and the two observables
32:Amplitude (disambiguation)
29:
5810:
5772:
5724:
5604:Quantum complexity theory
5582:Quantum cellular automata
5557:
5489:
5423:
5336:
5300:
5287:Path integral formulation
5254:
5119:
5076:
5020:Zwiebach, Barton (2022).
4639:, notably in the form of
4161:Copenhagen interpretation
4159:. Then, according to the
3459:Copenhagen interpretation
1706:probability mass function
1568:{\displaystyle \mu _{pp}}
1448:whose eigenfunctions are
738:is a pure point measure.
287:to one of the eigenstates
280:Copenhagen interpretation
212:Copenhagen interpretation
5671:Quantum machine learning
5651:Quantum key distribution
5641:Quantum image processing
5631:Quantum error correction
5481:Wheeler's delayed choice
4925:"Probability Amplitudes"
4768:de la Madrid Modino 2001
4718:
4688:Finite potential barrier
4649:transition probabilities
3706:(through either slit) =
2643:{\textstyle \sigma _{z}}
2616:{\textstyle \sigma _{x}}
2454:{\textstyle \sigma _{z}}
1757:{\displaystyle \psi (x)}
1715:is such that each point
1148:RadonâNikodym derivative
398:Mathematical formulation
5587:Quantum finite automata
4832:Physical Review Letters
4698:Phase space formulation
4631:Amplitudes in operators
4476:conservation of charges
3710:(through first slit) +
3011:{\textstyle |u\rangle }
2983:{\textstyle |r\rangle }
2650:, the following holds:
2130:{\displaystyle \alpha }
1272:normalization condition
1200:orthonormal basis, the
76:"Probability amplitude"
5691:Quantum neural network
4617:respective observables
4458:
4412:
4331:
4139:
4025:
3998:
3971:
3699:double-slit experiment
3692:Double-slit experiment
3646:
3448:
3366:
3275:). A wave function is
3132:
3012:
2984:
2956:
2919:
2644:
2617:
2587:
2522:
2455:
2421:
2394:
2367:
2281:
2244:
2207:
2179:
2151:
2150:{\displaystyle \beta }
2131:
2107:
2042:
2010:
1988:or the vertical state
1982:
1867:the same quantum state
1832:for the initial state
1822:
1758:
1623:
1599:
1569:
1535:spectral decomposition
1505:
1439:
1306:
1245:
1221:
1136:
1086:
975:
899:
797:
763:
708:
623:
565:
537:
485:
426:
274:When a measurement of
263:coherent superposition
220:Nobel Prize in Physics
176:
5716:Quantum teleportation
5244:Waveâparticle duality
4703:Uncertainty principle
4459:
4413:
4332:
4150:wavefunction collapse
4140:
4026:
3999:
3972:
3772:quantum superposition
3714:(through second slit)
3647:
3449:
3367:
3133:
3013:
2985:
2957:
2920:
2645:
2618:
2588:
2523:
2456:
2422:
2395:
2368:
2282:
2245:
2208:
2180:
2152:
2132:
2108:
2043:
2011:
1983:
1922:fundamental frequency
1823:
1759:
1624:
1600:
1570:
1506:
1450:Dirac delta functions
1440:
1307:
1246:
1222:
1189:, where L represents
1137:
1087:
976:
900:
798:
796:{\displaystyle \psi }
764:
762:{\displaystyle \psi }
742:Continuous amplitudes
709:
624:
586:-finite measure space
566:
564:{\displaystyle \psi }
547:Its relation with an
538:
486:
427:
185:probability amplitude
175:of the wave function.
142:
5747:Quantum field theory
5676:Quantum metamaterial
5621:Quantum cryptography
5351:Consistent histories
4611:composite state are
4554:can be expressed as
4425:
4350:
4198:
4058:
4008:
3981:
3782:
3726:interference pattern
3510:
3386:
3286:
3239:Schrödinger equation
3025:
2994:
2966:
2932:
2657:
2627:
2600:
2533:
2468:
2438:
2404:
2377:
2294:
2254:
2217:
2189:
2161:
2141:
2121:
2055:
2048:could be written as
2024:
1992:
1964:
1917:is between 0 and 1.
1899:|Ψ⟩
1872:|Ψ⟩
1835:|Ψ⟩
1786:
1739:
1689:is equal to 1, then
1686:|Ψ⟩
1653:|Ψ⟩
1609:
1583:
1549:
1527:rigged Hilbert space
1458:
1319:
1285:
1235:
1211:
1150:with respect to the
1100:
988:
921:
821:
787:
753:
644:
591:
577:domain of definition
555:
498:
461:
438:|Ψ⟩
416:
265:of the observable's
224:quantum measurements
216:emissions from atoms
61:improve this article
5851:Particle statistics
5846:Physical quantities
5841:Quantum measurement
5732:Quantum fluctuation
5701:Quantum programming
5661:Quantum logic gates
5646:Quantum information
5626:Quantum electronics
5101:Classical mechanics
5001:Teschl, G. (2014).
4950:Quantum Probability
4845:2013PhRvL.111j3201S
4466:continuity equation
4183:probability current
4173:Probability current
3457:Under the standard
1950:two possible states
1926:spherical harmonics
1541:Discrete amplitudes
777:equivalence classes
573:measurable function
248:quantum measurement
197:probability density
161:probability density
5785:in popular culture
5567:Quantum algorithms
5415:Von NeumannâWigner
5395:Objective collapse
5106:Old quantum theory
4758:, p. 114-119.
4454:
4408:
4327:
4135:
4021:
3994:
3967:
3642:
3444:
3362:
3128:
3008:
2980:
2952:
2915:
2913:
2883:
2849:
2808:
2640:
2613:
2583:
2518:
2451:
2417:
2390:
2363:
2277:
2240:
2203:
2175:
2147:
2127:
2103:
2038:
2006:
1978:
1818:
1754:
1619:
1595:
1565:
1501:
1435:
1302:
1241:
1217:
1132:
1082:
971:
895:
793:
759:
704:
619:
561:
533:
481:
422:
291:linear combination
257:to be measured is
177:
5828:
5827:
5802:Quantum mysticism
5780:Schrödinger's cat
5711:Quantum simulator
5681:Quantum metrology
5609:Quantum computing
5572:Quantum amplifier
5549:Quantum spacetime
5514:Quantum cosmology
5504:Quantum chemistry
5219:Scattering theory
5167:Zero-point energy
5162:Degenerate levels
5070:Quantum mechanics
5031:978-0-262-04613-8
5012:978-1-4704-1704-8
4985:978-0-8218-3446-6
4637:scattering theory
4603:, that gives the
4514:and given states
4482:Composite systems
4380:
4290:
4232:
4217:
4093:
4075:
3922:
3904:
3874:
3846:
3817:
3804:
3439:
3277:square integrable
3243:changes with time
3206:orthonormal basis
3200:; more generally
3126:
3103:
3102:
2909:
2908:
2827:
2826:
2786:
2785:
2729:
2728:
2703:
2702:
2415:
2388:
2350:
2349:
2321:
2320:
1635:discrete variable
1372:
1344:
1299:
1280:position operator
1244:{\displaystyle x}
1220:{\displaystyle x}
1154:(e.g. on the set
769:belonging to the
425:{\displaystyle t}
410:quantum mechanics
242:Physical overview
181:quantum mechanics
137:
136:
129:
111:
16:(Redirected from
5858:
5818:
5817:
5529:Quantum geometry
5524:Quantum dynamics
5381:Superdeterminism
5277:Matrix mechanics
5132:Braâket notation
5063:
5056:
5049:
5040:
5039:
5035:
5016:
4997:
4963:
4944:
4919:
4908:
4880:
4879:
4877:
4871:. Archived from
4828:
4813:
4807:
4801:
4795:
4793:Wigner's theorem
4789:
4783:
4777:
4771:
4765:
4759:
4753:
4747:
4741:
4735:
4733:Projective frame
4729:
4667:
4602:
4584:
4553:
4552:
4542:
4533:
4532:
4523:
4522:
4513:
4499:
4473:
4463:
4461:
4460:
4455:
4453:
4452:
4447:
4438:
4417:
4415:
4414:
4409:
4401:
4400:
4395:
4386:
4381:
4379:
4368:
4363:
4336:
4334:
4333:
4328:
4323:
4319:
4312:
4311:
4291:
4283:
4278:
4274:
4273:
4272:
4248:
4247:
4233:
4231:
4220:
4218:
4210:
4205:
4190:
4157:"quantum eraser"
4144:
4142:
4141:
4136:
4131:
4130:
4118:
4117:
4099:
4095:
4094:
4091:
4081:
4077:
4076:
4073:
4052:
4043:
4030:
4028:
4027:
4022:
4020:
4019:
4003:
4001:
4000:
3995:
3993:
3992:
3976:
3974:
3973:
3968:
3960:
3959:
3947:
3946:
3928:
3924:
3923:
3920:
3910:
3906:
3905:
3902:
3886:
3885:
3880:
3876:
3875:
3872:
3858:
3857:
3852:
3848:
3847:
3844:
3830:
3829:
3824:
3820:
3819:
3818:
3815:
3806:
3805:
3802:
3769:
3746:
3737:
3722:
3715:
3677:
3651:
3649:
3648:
3643:
3638:
3637:
3624:
3613:
3612:
3600:
3599:
3589:
3588:
3583:
3568:
3557:
3552:
3551:
3530:
3529:
3522:
3502:
3496:
3486:
3484:
3453:
3451:
3450:
3445:
3440:
3435:
3425:
3413:
3399:
3371:
3369:
3368:
3363:
3352:
3351:
3339:
3338:
3328:
3327:
3322:
3307:
3296:
3274:
3269:at a given time
3268:
3262:
3229:
3199:
3188:
3187:
3178:
3177:
3168:
3167:
3158:
3157:
3137:
3135:
3134:
3129:
3127:
3119:
3114:
3113:
3108:
3104:
3098:
3094:
3078:
3061:
3038:
3017:
3015:
3014:
3009:
3001:
2989:
2987:
2986:
2981:
2973:
2961:
2959:
2958:
2953:
2945:
2924:
2922:
2921:
2916:
2914:
2910:
2904:
2900:
2892:
2888:
2887:
2859:
2855:
2854:
2853:
2828:
2822:
2818:
2813:
2812:
2787:
2781:
2777:
2764:
2754:
2746:
2742:
2741:
2730:
2724:
2720:
2715:
2704:
2698:
2694:
2674:
2649:
2647:
2646:
2641:
2639:
2638:
2622:
2620:
2619:
2614:
2612:
2611:
2592:
2590:
2589:
2584:
2576:
2550:
2545:
2544:
2527:
2525:
2524:
2519:
2511:
2485:
2480:
2479:
2460:
2458:
2457:
2452:
2450:
2449:
2426:
2424:
2423:
2418:
2416:
2408:
2399:
2397:
2396:
2391:
2389:
2381:
2372:
2370:
2369:
2364:
2356:
2351:
2342:
2341:
2327:
2322:
2313:
2312:
2301:
2286:
2284:
2283:
2278:
2276:
2275:
2270:
2261:
2249:
2247:
2246:
2241:
2239:
2238:
2233:
2224:
2212:
2210:
2209:
2204:
2196:
2184:
2182:
2181:
2176:
2168:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2112:
2110:
2109:
2104:
2096:
2079:
2062:
2047:
2045:
2044:
2039:
2031:
2015:
2013:
2012:
2007:
1999:
1987:
1985:
1984:
1979:
1971:
1916:
1901:
1900:
1895:
1894:
1885:
1874:
1873:
1864:
1863:
1854:
1852:
1837:
1836:
1831:
1827:
1825:
1824:
1819:
1817:
1816:
1811:
1793:
1781:
1773:
1772:
1763:
1761:
1760:
1755:
1734:
1726:
1722:
1718:
1714:
1703:
1701:
1688:
1687:
1680:
1673:
1662:
1655:
1654:
1649:
1645:
1628:
1626:
1625:
1620:
1618:
1617:
1604:
1602:
1601:
1596:
1574:
1572:
1571:
1566:
1564:
1563:
1531:spectral theorem
1524:
1510:
1508:
1507:
1502:
1497:
1496:
1444:
1442:
1441:
1436:
1431:
1404:
1403:
1385:
1374:
1373:
1368:
1363:
1351:
1346:
1345:
1340:
1335:
1332:
1311:
1309:
1308:
1303:
1301:
1300:
1295:
1290:
1269:
1250:
1248:
1247:
1242:
1226:
1224:
1223:
1218:
1184:
1178:
1176:
1159:
1152:Lebesgue measure
1141:
1139:
1138:
1133:
1131:
1130:
1125:
1107:
1091:
1089:
1088:
1083:
1066:
1065:
1049:
1048:
1029:
1028:
1023:
1005:
1000:
999:
980:
978:
977:
972:
970:
969:
961:
952:
951:
946:
928:
916:
904:
902:
901:
896:
881:
880:
875:
857:
852:
851:
839:
838:
813:
802:
800:
799:
794:
774:
768:
766:
765:
760:
713:
711:
710:
705:
703:
702:
701:
685:
684:
683:
667:
666:
665:
628:
626:
625:
620:
609:
608:
585:
570:
568:
567:
562:
542:
540:
539:
534:
526:
490:
488:
487:
482:
471:
451:braâket notation
440:
439:
431:
429:
428:
423:
389:
385:
381:
377:
373:
369:
365:
361:
357:
346:
342:
338:
334:
330:
326:
322:
310:absolute squares
277:
256:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
18:Born probability
5866:
5865:
5861:
5860:
5859:
5857:
5856:
5855:
5831:
5830:
5829:
5824:
5806:
5792:Wigner's friend
5768:
5759:Quantum gravity
5720:
5706:Quantum sensing
5686:Quantum network
5666:Quantum machine
5636:Quantum imaging
5599:Quantum circuit
5594:Quantum channel
5553:
5499:Quantum biology
5485:
5461:ElitzurâVaidman
5436:DavissonâGermer
5419:
5371:Hidden-variable
5361:de BroglieâBohm
5338:Interpretations
5332:
5296:
5250:
5137:Complementarity
5115:
5072:
5067:
5032:
5013:
4986:
4960:
4941:
4905:
4889:
4884:
4883:
4875:
4826:
4814:
4810:
4802:
4798:
4790:
4786:
4778:
4774:
4766:
4762:
4754:
4750:
4742:
4738:
4730:
4726:
4721:
4674:
4663:
4645:matrix elements
4633:
4601:
4592:
4586:
4582:
4575:
4568:
4561:
4555:
4550:
4546:
4540:
4536:
4535:
4530:
4526:
4525:
4520:
4516:
4515:
4511:
4501:
4497:
4487:
4484:
4469:
4448:
4443:
4442:
4434:
4426:
4423:
4422:
4396:
4391:
4390:
4382:
4372:
4367:
4359:
4351:
4348:
4347:
4307:
4303:
4302:
4298:
4282:
4268:
4264:
4243:
4239:
4238:
4234:
4224:
4219:
4209:
4201:
4199:
4196:
4195:
4186:
4175:
4169:
4126:
4122:
4113:
4109:
4090:
4086:
4082:
4072:
4068:
4064:
4059:
4056:
4055:
4051:
4045:
4042:
4036:
4015:
4011:
4009:
4006:
4005:
3988:
3984:
3982:
3979:
3978:
3955:
3951:
3942:
3938:
3919:
3915:
3911:
3901:
3897:
3893:
3881:
3871:
3867:
3863:
3862:
3853:
3843:
3839:
3835:
3834:
3825:
3814:
3810:
3801:
3797:
3796:
3792:
3791:
3783:
3780:
3779:
3776:modulus squared
3768:
3761:
3754:
3748:
3745:
3739:
3736:
3730:
3717:
3702:
3694:
3688:
3675:
3661:
3633:
3629:
3620:
3608:
3604:
3595:
3591:
3584:
3579:
3578:
3564:
3553:
3547:
3543:
3518:
3517:
3513:
3511:
3508:
3507:
3498:
3492:
3471:
3462:
3421:
3414:
3412:
3395:
3387:
3384:
3383:
3347:
3343:
3334:
3330:
3323:
3318:
3317:
3303:
3292:
3287:
3284:
3283:
3270:
3264:
3249:
3220:
3190:
3181:
3180:
3171:
3170:
3161:
3160:
3151:
3150:
3147:
3118:
3109:
3093:
3089:
3088:
3074:
3057:
3034:
3026:
3023:
3022:
2997:
2995:
2992:
2991:
2969:
2967:
2964:
2963:
2941:
2933:
2930:
2929:
2912:
2911:
2899:
2890:
2889:
2882:
2881:
2875:
2874:
2864:
2863:
2848:
2847:
2841:
2840:
2830:
2829:
2817:
2807:
2806:
2800:
2799:
2789:
2788:
2776:
2775:
2771:
2762:
2761:
2750:
2737:
2719:
2711:
2693:
2692:
2688:
2681:
2670:
2660:
2658:
2655:
2654:
2634:
2630:
2628:
2625:
2624:
2607:
2603:
2601:
2598:
2597:
2572:
2546:
2540:
2536:
2534:
2531:
2530:
2507:
2481:
2475:
2471:
2469:
2466:
2465:
2445:
2441:
2439:
2436:
2435:
2407:
2405:
2402:
2401:
2380:
2378:
2375:
2374:
2352:
2340:
2323:
2311:
2297:
2295:
2292:
2291:
2271:
2266:
2265:
2257:
2255:
2252:
2251:
2234:
2229:
2228:
2220:
2218:
2215:
2214:
2192:
2190:
2187:
2186:
2164:
2162:
2159:
2158:
2142:
2139:
2138:
2122:
2119:
2118:
2092:
2075:
2058:
2056:
2053:
2052:
2027:
2025:
2022:
2021:
1995:
1993:
1990:
1989:
1967:
1965:
1962:
1961:
1946:
1907:
1898:
1897:
1888:
1887:
1886:if and only if
1876:
1871:
1870:
1857:
1856:
1855:if and only if
1843:
1841:
1834:
1833:
1829:
1812:
1807:
1806:
1789:
1787:
1784:
1783:
1779:
1766:
1765:
1740:
1737:
1736:
1732:
1724:
1723:. For discrete
1720:
1716:
1712:
1692:
1690:
1685:
1684:
1676:
1664:
1660:
1652:
1651:
1647:
1637:
1613:
1612:
1610:
1607:
1606:
1584:
1581:
1580:
1556:
1552:
1550:
1547:
1546:
1543:
1515:
1492:
1488:
1459:
1456:
1455:
1427:
1399:
1395:
1381:
1364:
1362:
1361:
1347:
1336:
1334:
1333:
1328:
1320:
1317:
1316:
1291:
1289:
1288:
1286:
1283:
1282:
1276:typical example
1260:
1236:
1233:
1232:
1229:uncountable set
1212:
1209:
1208:
1180:
1167:
1165:
1155:
1126:
1121:
1120:
1103:
1101:
1098:
1097:
1058:
1054:
1044:
1040:
1024:
1019:
1018:
1001:
995:
991:
989:
986:
985:
962:
957:
956:
947:
942:
941:
924:
922:
919:
918:
914:
876:
871:
870:
853:
847:
843:
834:
830:
822:
819:
818:
814:if and only if
804:
788:
785:
784:
770:
754:
751:
750:
744:
737:
730:
723:
694:
693:
689:
676:
675:
671:
658:
657:
653:
645:
642:
641:
604:
603:
592:
589:
588:
581:
556:
553:
552:
522:
499:
496:
495:
467:
462:
459:
458:
441:belonging to a
437:
436:
417:
414:
413:
406:
400:
387:
383:
379:
375:
371:
367:
363:
359:
355:
344:
340:
336:
332:
328:
324:
320:
275:
254:
244:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5864:
5854:
5853:
5848:
5843:
5826:
5825:
5823:
5822:
5811:
5808:
5807:
5805:
5804:
5799:
5794:
5789:
5788:
5787:
5776:
5774:
5770:
5769:
5767:
5766:
5761:
5756:
5755:
5754:
5744:
5739:
5737:Casimir effect
5734:
5728:
5726:
5722:
5721:
5719:
5718:
5713:
5708:
5703:
5698:
5696:Quantum optics
5693:
5688:
5683:
5678:
5673:
5668:
5663:
5658:
5653:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5617:
5616:
5606:
5601:
5596:
5591:
5590:
5589:
5579:
5574:
5569:
5563:
5561:
5555:
5554:
5552:
5551:
5546:
5541:
5536:
5531:
5526:
5521:
5516:
5511:
5506:
5501:
5495:
5493:
5487:
5486:
5484:
5483:
5478:
5473:
5471:Quantum eraser
5468:
5463:
5458:
5453:
5448:
5443:
5438:
5433:
5427:
5425:
5421:
5420:
5418:
5417:
5412:
5407:
5402:
5397:
5392:
5387:
5386:
5385:
5384:
5383:
5368:
5363:
5358:
5353:
5348:
5342:
5340:
5334:
5333:
5331:
5330:
5325:
5320:
5315:
5310:
5304:
5302:
5298:
5297:
5295:
5294:
5289:
5284:
5279:
5274:
5269:
5264:
5258:
5256:
5252:
5251:
5249:
5248:
5247:
5246:
5241:
5231:
5226:
5221:
5216:
5211:
5206:
5201:
5196:
5191:
5186:
5181:
5176:
5171:
5170:
5169:
5164:
5159:
5154:
5144:
5142:Density matrix
5139:
5134:
5129:
5123:
5121:
5117:
5116:
5114:
5113:
5108:
5103:
5098:
5097:
5096:
5086:
5080:
5078:
5074:
5073:
5066:
5065:
5058:
5051:
5043:
5037:
5036:
5030:
5017:
5011:
4998:
4984:
4964:
4958:
4945:
4939:
4920:
4909:
4903:
4888:
4885:
4882:
4881:
4878:on 2019-03-07.
4839:(10): 103201.
4818:ensemble-alike
4808:
4796:
4784:
4782:, p. 330.
4772:
4760:
4748:
4736:
4723:
4722:
4720:
4717:
4716:
4715:
4710:
4705:
4700:
4695:
4690:
4685:
4680:
4673:
4670:
4657:unitary matrix
4632:
4629:
4599:
4590:
4585:a function on
4580:
4573:
4566:
4559:
4548:
4538:
4528:
4518:
4509:
4495:
4483:
4480:
4451:
4446:
4441:
4437:
4433:
4430:
4419:
4418:
4407:
4404:
4399:
4394:
4389:
4385:
4378:
4375:
4371:
4366:
4362:
4358:
4355:
4338:
4337:
4326:
4322:
4318:
4315:
4310:
4306:
4301:
4297:
4294:
4289:
4286:
4281:
4277:
4271:
4267:
4263:
4260:
4257:
4254:
4251:
4246:
4242:
4237:
4230:
4227:
4223:
4216:
4213:
4208:
4204:
4171:Main article:
4168:
4165:
4134:
4129:
4125:
4121:
4116:
4112:
4108:
4105:
4102:
4098:
4089:
4085:
4080:
4071:
4067:
4063:
4049:
4040:
4018:
4014:
3991:
3987:
3966:
3963:
3958:
3954:
3950:
3945:
3941:
3937:
3934:
3931:
3927:
3918:
3914:
3909:
3900:
3896:
3892:
3889:
3884:
3879:
3870:
3866:
3861:
3856:
3851:
3842:
3838:
3833:
3828:
3823:
3813:
3809:
3800:
3795:
3790:
3787:
3766:
3759:
3752:
3743:
3734:
3690:Main article:
3687:
3684:
3673:
3653:
3652:
3641:
3636:
3632:
3627:
3623:
3619:
3616:
3611:
3607:
3603:
3598:
3594:
3587:
3582:
3577:
3574:
3571:
3567:
3563:
3560:
3556:
3550:
3546:
3542:
3539:
3536:
3533:
3528:
3525:
3521:
3516:
3497:at fixed time
3455:
3454:
3443:
3438:
3434:
3431:
3428:
3424:
3420:
3417:
3411:
3408:
3405:
3402:
3398:
3394:
3391:
3373:
3372:
3361:
3358:
3355:
3350:
3346:
3342:
3337:
3333:
3326:
3321:
3316:
3313:
3310:
3306:
3302:
3299:
3295:
3291:
3146:
3143:
3139:
3138:
3125:
3122:
3117:
3112:
3107:
3101:
3097:
3092:
3087:
3084:
3081:
3077:
3073:
3070:
3067:
3064:
3060:
3056:
3053:
3050:
3047:
3044:
3041:
3037:
3033:
3030:
3007:
3004:
3000:
2979:
2976:
2972:
2951:
2948:
2944:
2940:
2937:
2926:
2925:
2907:
2903:
2898:
2895:
2893:
2891:
2886:
2880:
2877:
2876:
2873:
2870:
2869:
2867:
2862:
2858:
2852:
2846:
2843:
2842:
2839:
2836:
2835:
2833:
2825:
2821:
2816:
2811:
2805:
2802:
2801:
2798:
2795:
2794:
2792:
2784:
2780:
2774:
2770:
2767:
2765:
2763:
2760:
2757:
2753:
2749:
2745:
2740:
2736:
2733:
2727:
2723:
2718:
2714:
2710:
2707:
2701:
2697:
2691:
2687:
2684:
2682:
2680:
2677:
2673:
2669:
2666:
2663:
2662:
2637:
2633:
2610:
2606:
2594:
2593:
2582:
2579:
2575:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2549:
2543:
2539:
2528:
2517:
2514:
2510:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2484:
2478:
2474:
2448:
2444:
2414:
2411:
2387:
2384:
2362:
2359:
2355:
2348:
2345:
2339:
2336:
2333:
2330:
2326:
2319:
2316:
2310:
2307:
2304:
2300:
2274:
2269:
2264:
2260:
2237:
2232:
2227:
2223:
2202:
2199:
2195:
2174:
2171:
2167:
2146:
2126:
2115:
2114:
2102:
2099:
2095:
2091:
2088:
2085:
2082:
2078:
2074:
2071:
2068:
2065:
2061:
2037:
2034:
2030:
2005:
2002:
1998:
1977:
1974:
1970:
1945:
1942:
1815:
1810:
1805:
1802:
1799:
1796:
1792:
1778:eigenvalue of
1753:
1750:
1747:
1744:
1616:
1594:
1591:
1588:
1579:(i.e. the set
1562:
1559:
1555:
1542:
1539:
1537:is preserved.
1512:
1511:
1500:
1495:
1491:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1446:
1445:
1434:
1430:
1426:
1423:
1419:
1416:
1413:
1410:
1407:
1402:
1398:
1394:
1391:
1388:
1384:
1380:
1377:
1371:
1367:
1360:
1357:
1354:
1350:
1343:
1339:
1331:
1327:
1324:
1298:
1294:
1251:" will always
1240:
1216:
1129:
1124:
1119:
1116:
1113:
1110:
1106:
1094:
1093:
1081:
1078:
1075:
1072:
1069:
1064:
1061:
1057:
1053:
1047:
1043:
1039:
1036:
1033:
1027:
1022:
1017:
1014:
1011:
1008:
1004:
998:
994:
968:
965:
960:
955:
950:
945:
940:
937:
934:
931:
927:
907:
906:
894:
891:
888:
885:
879:
874:
869:
866:
863:
860:
856:
850:
846:
842:
837:
833:
829:
826:
792:
758:
743:
740:
735:
728:
721:
715:
714:
700:
697:
692:
688:
682:
679:
674:
670:
664:
661:
656:
652:
649:
633:, decomposing
618:
615:
612:
607:
602:
599:
596:
560:
545:
544:
532:
529:
525:
521:
518:
515:
512:
509:
506:
503:
480:
477:
474:
470:
466:
421:
399:
396:
392:do not commute
243:
240:
189:complex number
153:atomic orbital
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
5863:
5852:
5849:
5847:
5844:
5842:
5839:
5838:
5836:
5821:
5813:
5812:
5809:
5803:
5800:
5798:
5795:
5793:
5790:
5786:
5783:
5782:
5781:
5778:
5777:
5775:
5771:
5765:
5762:
5760:
5757:
5753:
5750:
5749:
5748:
5745:
5743:
5740:
5738:
5735:
5733:
5730:
5729:
5727:
5723:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5672:
5669:
5667:
5664:
5662:
5659:
5657:
5656:Quantum logic
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5615:
5612:
5611:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5588:
5585:
5584:
5583:
5580:
5578:
5575:
5573:
5570:
5568:
5565:
5564:
5562:
5560:
5556:
5550:
5547:
5545:
5542:
5540:
5537:
5535:
5532:
5530:
5527:
5525:
5522:
5520:
5517:
5515:
5512:
5510:
5509:Quantum chaos
5507:
5505:
5502:
5500:
5497:
5496:
5494:
5492:
5488:
5482:
5479:
5477:
5476:SternâGerlach
5474:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5428:
5426:
5422:
5416:
5413:
5411:
5410:Transactional
5408:
5406:
5403:
5401:
5400:Quantum logic
5398:
5396:
5393:
5391:
5388:
5382:
5379:
5378:
5377:
5374:
5373:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5344:
5343:
5341:
5339:
5335:
5329:
5326:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5305:
5303:
5299:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5259:
5257:
5253:
5245:
5242:
5240:
5237:
5236:
5235:
5234:Wave function
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5209:Superposition
5207:
5205:
5204:Quantum state
5202:
5200:
5197:
5195:
5192:
5190:
5187:
5185:
5182:
5180:
5177:
5175:
5172:
5168:
5165:
5163:
5160:
5158:
5157:Excited state
5155:
5153:
5150:
5149:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5124:
5122:
5118:
5112:
5109:
5107:
5104:
5102:
5099:
5095:
5092:
5091:
5090:
5087:
5085:
5082:
5081:
5079:
5075:
5071:
5064:
5059:
5057:
5052:
5050:
5045:
5044:
5041:
5033:
5027:
5023:
5018:
5014:
5008:
5004:
4999:
4995:
4991:
4987:
4981:
4977:
4973:
4969:
4965:
4961:
4959:0-12-305340-4
4955:
4951:
4946:
4942:
4940:0-201-51005-7
4936:
4932:
4931:
4926:
4921:
4917:
4916:
4910:
4906:
4904:0-444-88776-8
4900:
4896:
4891:
4890:
4874:
4870:
4866:
4862:
4858:
4854:
4850:
4846:
4842:
4838:
4834:
4833:
4825:
4819:
4812:
4806:, p. 78.
4805:
4804:Zwiebach 2022
4800:
4794:
4788:
4781:
4776:
4770:, p. 97.
4769:
4764:
4757:
4752:
4746:, p. 43.
4745:
4740:
4734:
4728:
4724:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4684:
4683:Free particle
4681:
4679:
4676:
4675:
4669:
4666:
4660:
4658:
4654:
4650:
4646:
4642:
4638:
4628:
4626:
4622:
4618:
4614:
4610:
4606:
4598:
4595:
4589:
4579:
4572:
4565:
4558:
4545:
4508:
4504:
4494:
4490:
4479:
4477:
4472:
4467:
4449:
4439:
4431:
4428:
4405:
4402:
4397:
4387:
4376:
4364:
4356:
4346:
4345:
4344:
4341:
4324:
4320:
4316:
4308:
4304:
4299:
4295:
4292:
4287:
4284:
4279:
4275:
4269:
4265:
4258:
4255:
4252:
4244:
4240:
4235:
4228:
4225:
4221:
4214:
4211:
4206:
4194:
4193:
4192:
4189:
4184:
4179:
4174:
4164:
4162:
4158:
4153:
4151:
4146:
4127:
4123:
4119:
4114:
4110:
4103:
4100:
4096:
4087:
4083:
4078:
4069:
4065:
4061:
4048:
4039:
4034:
4016:
4012:
3989:
3985:
3964:
3956:
3952:
3948:
3943:
3939:
3932:
3929:
3925:
3916:
3912:
3907:
3898:
3894:
3890:
3887:
3882:
3877:
3868:
3864:
3859:
3854:
3849:
3840:
3836:
3831:
3826:
3821:
3811:
3807:
3798:
3793:
3788:
3785:
3777:
3773:
3765:
3758:
3751:
3742:
3733:
3727:
3720:
3713:
3709:
3705:
3700:
3693:
3683:
3681:
3672:
3668:
3664:
3659:
3639:
3614:
3609:
3605:
3601:
3585:
3572:
3569:
3558:
3548:
3544:
3540:
3534:
3526:
3523:
3514:
3506:
3505:
3504:
3501:
3495:
3490:
3482:
3478:
3474:
3469:
3465:
3460:
3441:
3436:
3429:
3426:
3415:
3409:
3403:
3400:
3389:
3382:
3381:
3380:
3378:
3377:normalization
3359:
3353:
3348:
3344:
3340:
3324:
3311:
3308:
3297:
3289:
3282:
3281:
3280:
3278:
3273:
3267:
3260:
3256:
3252:
3248:
3247:wave function
3244:
3240:
3235:
3233:
3227:
3223:
3218:
3213:
3211:
3207:
3203:
3197:
3193:
3185:
3175:
3165:
3155:
3145:Normalization
3142:
3123:
3120:
3115:
3110:
3105:
3099:
3095:
3090:
3085:
3079:
3071:
3062:
3054:
3048:
3039:
3028:
3021:
3020:
3019:
3002:
2974:
2946:
2938:
2905:
2901:
2896:
2894:
2884:
2878:
2871:
2865:
2860:
2856:
2850:
2844:
2837:
2831:
2823:
2819:
2814:
2809:
2803:
2796:
2790:
2782:
2778:
2772:
2768:
2766:
2755:
2747:
2743:
2734:
2725:
2721:
2716:
2708:
2699:
2695:
2689:
2685:
2683:
2675:
2667:
2653:
2652:
2651:
2635:
2631:
2608:
2604:
2577:
2566:
2563:
2557:
2551:
2541:
2537:
2529:
2512:
2501:
2498:
2492:
2486:
2476:
2472:
2464:
2463:
2462:
2446:
2442:
2432:
2430:
2412:
2409:
2385:
2382:
2357:
2346:
2343:
2337:
2334:
2328:
2317:
2314:
2308:
2302:
2288:
2272:
2262:
2235:
2225:
2197:
2169:
2144:
2124:
2097:
2089:
2086:
2080:
2072:
2069:
2063:
2051:
2050:
2049:
2032:
2019:
2018:superposition
2000:
1972:
1959:
1955:
1951:
1941:
1939:
1935:
1931:
1927:
1923:
1918:
1914:
1910:
1905:
1892:
1883:
1879:
1868:
1861:
1850:
1846:
1839:
1813:
1800:
1794:
1777:
1770:
1748:
1742:
1730:
1709:
1707:
1699:
1695:
1682:
1679:
1671:
1667:
1658:
1644:
1640:
1636:
1632:
1592:
1589:
1586:
1578:
1560:
1557:
1553:
1538:
1536:
1532:
1528:
1522:
1518:
1493:
1489:
1485:
1482:
1476:
1473:
1467:
1461:
1454:
1453:
1452:
1451:
1432:
1424:
1421:
1417:
1411:
1405:
1400:
1396:
1392:
1378:
1358:
1325:
1315:
1314:
1313:
1281:
1277:
1273:
1267:
1263:
1258:
1254:
1238:
1230:
1214:
1207:
1203:
1199:
1194:
1192:
1188:
1183:
1174:
1170:
1163:
1158:
1153:
1149:
1145:
1127:
1114:
1108:
1079:
1076:
1070:
1062:
1059:
1055:
1051:
1045:
1041:
1037:
1034:
1031:
1025:
1012:
1006:
996:
992:
984:
983:
982:
966:
963:
953:
948:
935:
929:
912:
889:
886:
883:
877:
864:
858:
848:
844:
840:
835:
827:
817:
816:
815:
811:
807:
790:
782:
778:
773:
756:
749:
748:wave function
739:
734:
727:
720:
690:
686:
672:
668:
654:
650:
647:
640:
639:
638:
636:
632:
613:
610:
600:
597:
587:
584:
578:
574:
558:
550:
519:
513:
507:
501:
494:
493:
492:
472:
456:
452:
448:
447:Hilbert space
444:
435:
419:
411:
405:
395:
393:
352:
350:
317:
316:requirement.
315:
314:normalization
311:
306:
304:
300:
296:
295:superposition
292:
288:
286:
281:
272:
270:
269:
264:
260:
253:
249:
239:
237:
233:
229:
225:
221:
217:
213:
209:
205:
204:quantum state
200:
198:
194:
190:
186:
182:
174:
173:complex phase
170:
166:
162:
158:
157:hydrogen atom
154:
150:
147:for a single
146:
145:wave function
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: â
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
5539:Quantum mind
5451:FranckâHertz
5313:KleinâGordon
5262:Formulations
5255:Formulations
5184:Interference
5174:Entanglement
5152:Ground state
5147:Energy level
5120:Fundamentals
5084:Introduction
5021:
5002:
4971:
4968:Simon, Barry
4949:
4928:
4914:
4894:
4873:the original
4836:
4830:
4811:
4799:
4787:
4775:
4763:
4751:
4739:
4727:
4664:
4661:
4634:
4596:
4587:
4577:
4570:
4563:
4556:
4547:|Ψ
4537:|Ψ
4527:|Ψ
4517:|Ψ
4506:
4502:
4492:
4488:
4485:
4470:
4420:
4342:
4339:
4187:
4180:
4176:
4154:
4147:
4046:
4037:
3763:
3756:
3749:
3740:
3731:
3718:
3711:
3707:
3703:
3695:
3670:
3666:
3662:
3654:
3503:is given by
3499:
3493:
3480:
3476:
3472:
3467:
3463:
3456:
3374:
3271:
3265:
3258:
3254:
3250:
3245:. Suppose a
3236:
3232:normalizable
3225:
3221:
3216:
3214:
3201:
3195:
3191:
3183:
3173:
3163:
3153:
3148:
3140:
3018:is given by
2927:
2595:
2433:
2289:
2116:
1954:polarization
1947:
1919:
1912:
1908:
1890:
1881:
1877:
1859:
1848:
1844:
1840:
1768:
1729:eigenvectors
1710:
1697:
1693:
1677:
1669:
1665:
1642:
1638:
1630:
1544:
1520:
1516:
1513:
1447:
1265:
1261:
1255:). As such,
1195:
1181:
1172:
1168:
1162:real numbers
1156:
1095:
913:is equal to
908:
809:
805:
771:
745:
732:
725:
718:
716:
634:
582:
546:
434:state vector
407:
353:
318:
307:
284:
273:
266:
245:
201:
184:
178:
123:
117:January 2014
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
5797:EPR paradox
5577:Quantum bus
5446:Double-slit
5424:Experiments
5390:Many-worlds
5328:Schrödinger
5292:Phase space
5282:Schrödinger
5272:Interaction
5229:Uncertainty
5199:Nonlocality
5194:Measurement
5189:Decoherence
5179:Hamiltonian
4756:Teschl 2014
4713:Wave packet
4693:Matter wave
4653:unit vector
4181:Define the
1952:, e.g. the
1533:as well as
1312:defined as
1257:eigenstates
803:belongs to
579:to a given
268:eigenstates
228:Schrödinger
5835:Categories
5725:Extensions
5559:Technology
5405:Relational
5356:Copenhagen
5267:Heisenberg
5214:Tunnelling
5077:Background
4887:References
4744:Simon 2005
4641:S-matrices
4185:(or flux)
3660:, initial
3470:) = |
3217:normalized
1904:orthogonal
1853:| = 1
1776:degenerate
1274:below). A
981:such that
775:space of (
549:observable
402:See also:
252:observable
87:newspapers
5431:Bell test
5301:Equations
5127:Born rule
4791:See also
4609:entangled
4440:ψ
4429:ρ
4388:ψ
4374:∂
4370:∂
4357:⋅
4354:∇
4317:ψ
4314:∇
4309:∗
4305:ψ
4296:
4285:ℏ
4270:∗
4266:ψ
4262:∇
4259:ψ
4256:−
4253:ψ
4250:∇
4245:∗
4241:ψ
4212:ℏ
4124:φ
4120:−
4111:φ
4104:
4088:ψ
4070:ψ
4033:arguments
4013:φ
3986:φ
3953:φ
3949:−
3940:φ
3933:
3917:ψ
3899:ψ
3869:ψ
3841:ψ
3812:ψ
3799:ψ
3680:potential
3615:ρ
3606:∫
3559:ψ
3545:∫
3524:∈
3416:ψ
3390:ψ
3357:∞
3298:ψ
3290:∫
3083:⟩
3069:⟨
3066:⟩
3052:⟨
3043:⟩
3006:⟩
2978:⟩
2950:⟩
2936:⟨
2861:⋅
2759:⟩
2748:⋅
2732:⟨
2706:⟨
2679:⟩
2665:⟨
2632:σ
2605:σ
2581:⟩
2564:−
2555:⟩
2538:σ
2516:⟩
2490:⟩
2473:σ
2443:σ
2361:⟩
2335:−
2332:⟩
2306:⟩
2303:ψ
2263:β
2226:α
2201:⟩
2173:⟩
2145:β
2125:α
2101:⟩
2090:β
2084:⟩
2073:α
2067:⟩
2064:ψ
2036:⟩
2033:ψ
2004:⟩
1976:⟩
1795:ψ
1743:ψ
1646:equal to
1590:⊂
1554:μ
1486:−
1477:δ
1462:ψ
1425:∈
1406:ψ
1390:⟩
1387:Ψ
1376:⟨
1370:^
1356:⟩
1353:Ψ
1342:^
1323:⟨
1297:^
1198:countable
1187:dimension
1109:ψ
1056:μ
1042:∫
1038:≡
1007:ψ
993:∫
964:≥
954:∈
930:ψ
893:∞
859:ψ
845:∫
832:‖
828:ψ
825:‖
791:ψ
757:ψ
691:μ
673:μ
655:μ
648:μ
614:μ
559:ψ
531:⟩
528:Ψ
517:⟨
502:ψ
476:⟩
443:separable
303:Born rule
299:"weights"
259:uncertain
5820:Category
5614:Timeline
5366:Ensemble
5346:Bayesian
5239:Collapse
5111:Glossary
5094:Timeline
4970:(2005).
4861:25166663
4672:See also
4613:products
4551:⟩
4541:⟩
4531:⟩
4521:⟩
4031:are the
3716:, where
3186:⟩
3176:⟩
3166:⟩
3156:⟩
2429:ensemble
1944:Examples
1930:M-theory
1893:⟩
1862:⟩
1771:⟩
783:, i.e.,
575:and its
449:. Using
445:complex
232:Einstein
208:Max Born
149:electron
5773:Related
5752:History
5491:Science
5323:Rydberg
5089:History
4994:2105088
4869:2725093
4841:Bibcode
3721:(event)
3182:|
3172:|
3162:|
3152:|
1782:, then
1735:. Then
1657:indexed
1278:is the
1253:be zero
1160:of all
1142:is the
909:If the
432:, by a
349:commute
193:modulus
101:scholar
5466:Popper
5028:
5009:
4992:
4982:
4956:
4937:
4901:
4867:
4859:
4571:ψ
4557:ψ
4092:second
4050:second
4047:ψ
4038:ψ
3977:Here,
3921:second
3873:second
3816:second
3767:second
3764:ψ
3757:ψ
3750:ψ
3744:second
3741:ψ
3732:ψ
3663:ψ
3485:|
3473:ψ
3375:After
3251:ψ
1958:photon
1909:ψ
1889:|
1878:ψ
1858:|
1845:ψ
1842:|
1767:|
1702:|
1694:ψ
1691:|
1666:ψ
1629:is an
1577:atomic
1227:is an
1191:length
1177:|
1169:ψ
1166:|
717:where
583:σ
151:on 5d
103:
96:
89:
82:
74:
5376:Local
5318:Pauli
5308:Dirac
4876:(PDF)
4865:S2CID
4827:(PDF)
4719:Notes
4625:above
4074:first
4041:first
3903:first
3845:first
3803:first
3760:first
3753:total
3735:first
3487:is a
3210:above
2117:with
1956:of a
1884:) = 0
1704:is a
1681:-norm
1270:(see
1204:of a
1202:range
1096:then
571:to a
455:basis
285:jumps
187:is a
155:of a
108:JSTOR
94:books
5026:ISBN
5007:ISBN
4980:ISBN
4954:ISBN
4935:ISBN
4899:ISBN
4857:PMID
4524:and
4500:and
4044:and
4004:and
3738:and
3678:and
3658:mass
3354:<
3279:if
3237:The
2185:and
2137:and
1936:and
1902:are
1896:and
1631:atom
1545:Let
917:and
911:norm
890:<
779:of)
358:and
230:and
183:, a
80:news
4849:doi
4837:111
4191:as
4101:cos
4035:of
3930:cos
3198:= 1
3179:or
3159:or
1869:as
1865:is
1731:of
1683:of
1659:by
1605:in
1575:be
1193:.
343:or
293:or
179:In
169:hue
63:by
5837::
4990:MR
4988:.
4978:.
4927:.
4863:.
4855:.
4847:.
4835:.
4829:.
4627:.
4569:)
4478:.
4406:0.
4293:Im
3762:+
3755:=
3669:,
3479:,
3410::=
3257:,
3234:.
3212:.
3194:+
2287:.
1940:.
1875:.
1838:.
1708:.
1641:â
736:pp
729:sc
722:ac
457:"
394:.
351:.
305:.
282:)
199:.
165:nm
143:A
5062:e
5055:t
5048:v
5034:.
5015:.
4996:.
4962:.
4943:.
4907:.
4851::
4843::
4665:L
4600:2
4597:X
4594:Ă
4591:1
4588:X
4583:)
4581:2
4578:x
4576:(
4574:2
4567:1
4564:x
4562:(
4560:1
4549:2
4544:â
4539:1
4529:2
4519:1
4512:)
4510:2
4507:X
4505:(
4503:L
4498:)
4496:1
4493:X
4491:(
4489:L
4471:j
4450:2
4445:|
4436:|
4432:=
4403:=
4398:2
4393:|
4384:|
4377:t
4365:+
4361:j
4325:,
4321:)
4300:(
4288:m
4280:=
4276:)
4236:(
4229:i
4226:2
4222:1
4215:m
4207:=
4203:j
4188:j
4133:)
4128:2
4115:1
4107:(
4097:|
4084:|
4079:|
4066:|
4062:2
4017:2
3990:1
3965:.
3962:)
3957:2
3944:1
3936:(
3926:|
3913:|
3908:|
3895:|
3891:2
3888:+
3883:2
3878:|
3865:|
3860:+
3855:2
3850:|
3837:|
3832:=
3827:2
3822:|
3808:+
3794:|
3789:=
3786:P
3719:P
3712:P
3708:P
3704:P
3676:)
3674:0
3671:t
3667:x
3665:(
3640:.
3635:x
3631:d
3626:)
3622:x
3618:(
3610:V
3602:=
3597:x
3593:d
3586:2
3581:|
3576:)
3573:t
3570:,
3566:x
3562:(
3555:|
3549:V
3541:=
3538:)
3535:t
3532:(
3527:V
3520:x
3515:P
3500:t
3494:V
3483:)
3481:t
3477:x
3475:(
3468:x
3466:(
3464:Ï
3442:.
3437:a
3433:)
3430:t
3427:,
3423:x
3419:(
3407:)
3404:t
3401:,
3397:x
3393:(
3360:.
3349:2
3345:a
3341:=
3336:x
3332:d
3325:2
3320:|
3315:)
3312:t
3309:,
3305:x
3301:(
3294:|
3272:t
3266:x
3261:)
3259:t
3255:x
3253:(
3228:)
3226:X
3224:(
3222:L
3196:ÎČ
3192:α
3184:V
3174:H
3164:V
3154:H
3124:2
3121:1
3116:=
3111:2
3106:)
3100:2
3096:1
3091:(
3086:=
3080:r
3076:|
3072:u
3063:u
3059:|
3055:r
3049:=
3046:)
3040:u
3036:|
3032:(
3029:P
3003:u
2999:|
2975:r
2971:|
2947:u
2943:|
2939:r
2906:2
2902:1
2897:=
2885:)
2879:0
2872:1
2866:(
2857:)
2851:)
2845:1
2838:0
2832:(
2824:2
2820:1
2815:+
2810:)
2804:0
2797:1
2791:(
2783:2
2779:1
2773:(
2769:=
2756:u
2752:|
2744:)
2739:|
2735:d
2726:2
2722:1
2717:+
2713:|
2709:u
2700:2
2696:1
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