5254:. The quantum state of the two slit experiment is a superposition of two single-slit quantum states, one corresponding to the left slit, and the other corresponding to the right slit. In the detector plane, the relative phase of those two single-slit states depends on the difference of the distances from the two slits. Depending on that relative phase, the interference is constructive at some locations and destructive in others, creating the interference pattern. We may say that superposed states are in
1203:
8515:
1832:. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
5903:
1816:
5241:
1077:
corresponding to that measurement and the value measured. Other aspects of the state may be unknown. Repeating the measurement will not alter the state. In some cases, compatible measurements can further refine the state, causing it to be an eigenstate corresponding to all these measurements. A full
1886:
of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of
1018:
The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the
5314:
with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited
975:
values at each instant of time. For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use
1033:
Measurements, macroscopic operations on quantum states, filter the state. Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state.
1109:. The position wave function is one representation often seen first in introductions to quantum mechanics. The equivalent momentum wave function is another wave function based representation. Representations are analogous to coordinate systems or similar mathematical devices like
1019:
equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.
5315:
knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.
5708:
1034:
Subsequent measurements may either further prepare the system â these are compatible measurements â or it may alter the state, redefining it â these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the
1544:
of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding
1054:
axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the
2834:
degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with
4982:
1732:
6865:
5103:
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i.e. separated by a zero delay. One can think of it as stopping the time, then making the two measurements one after the other, then resuming the time. Thus, the measurements occurred at the same time, but it is still possible to tell which was
1662:
The coefficient that corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the
4572:
4359:
6309:
Dirac (1958), p. 20: "The bra vectors, as they have been here introduced, are quite a different kind of vector from the kets, and so far there is no connexion between them except for the existence of a scalar product of a bra and a
6792:
3110:
2142:
Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called
5098:
3308:
particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).
913:
of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
6994:
4037:
3219:
117:
1193:
for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.
5620:
1660:
4816:
4221:
6916:
2284:
1145:. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed. The wave function is a complex-valued function of any complete set of commuting or compatible
2847:
SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number
3505:
1675:
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. A
935:
Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into
3647:
2791:
4878:
3783:
2068:
twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences, however, as follows.
3725:
6343:
2004:
1389:
4483:
5898:{\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}
4478:
4148:
2964:
4284:
6688:
6499:
6469:
6411:
4251:
3927:
1124:', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists.
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5356:
4600:
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The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system. The set will contain
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4987:
4870:
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1966:
1912:
1707:
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1007:. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the
1493:
8312:
6797:
5965:
5445:
5414:
5387:
4369:
2553:
1939:
1811:{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigl (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigr )},}
1179:
6226:
3986:
3124:
5306:
Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a
5236:{\displaystyle e^{i\theta _{\alpha }}\left(A_{\alpha }|\alpha \rangle +{\sqrt {1-A_{\alpha }^{2}}}e^{i\theta _{\beta }-i\theta _{\alpha }}|\beta \rangle \right)}
3304:-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all
7960:
6050:
5995:
5538:
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4640:
4620:
2805:
2445:
2387:
2367:
2310:
1499:
1052:
383:
6932:
is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion is
995:
would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.
5546:
2012:
is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.
1575:
979:
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are
6097:
4759:
884:
4153:
6120:
8483:
6721:
2147:, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see
5675:
to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.
1097:
can be formed into pure states. Experiments rarely produce pure states. Therefore statistical mixtures of solutions must be compared to experiments.
2493:. In order to make such calculations flow smoothly, and to make it unnecessary (in some contexts) to fully understand the underlying linear algebra,
8495:
7809:
7147:
591:
135:
6935:
3439:
1879:: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
991:
for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a
47:
7337:. Also translated as 'The physical content of quantum kinematics and mechanics' at pp. 62â84 by editors John Wheeler and Wojciech Zurek, in
5695:
The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (
8179:
7752:
1553:) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no
364:
8113:
547:
7698:
7667:
7526:
7494:
7469:
7157:
5696:
2293:
is usually formulated in terms of the Schrödinger picture, the
Heisenberg picture is often preferred in a relativistic context, that is, for
2064:; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure
8036:
7093:
3951:
of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single,
470:
7298:
6874:
3730:
5470:
pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a
Hilbert space
2202:
3004:
particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3
7782:
7745:
7414:
7185:
7127:
7102:
6130:
5243:
The overall phase factor in front has no physical effect. Only the relative phase affects the physical nature of the superposition.
3687:
130:
1325:
7892:
3348:
1864:
1554:
1028:
910:
219:
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8154:
326:
306:
8450:
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5251:
2901:
1664:
1247:
877:
174:
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2195:.) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the
1856:
296:
8462:
7441:
7211:
7048:
4375:
2840:
566:
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notation, where vectors are usually lower-case Latin letters, and it is clear from the context that they are indeed vectors.
1011:. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical
8440:
8217:
8139:
7922:
7792:
7272:
5491:
5300:
1883:
606:
344:
244:
8174:
3596:
2748:
2286:.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.
1283:
359:
8108:
8103:
8074:
7787:
2642:
1068:
542:
537:
508:
140:
8149:
321:
311:
8242:
6150:
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is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the
1875:
describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the
1218:
1113:. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult.
576:
8059:
2349:
is physically inconsequential (as long as the state is considered by itself). If a vector in a complex
Hilbert space
493:
8518:
8280:
8088:
6110:
1974:
870:
522:
8011:
7942:
441:
8302:
8169:
8093:
8054:
7985:
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7378:
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can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state.
1871:
for the outcome, and the form that this distribution takes is completely determined by the quantum state and the
527:
488:
416:
339:
199:
2555:" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual
1882:
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a
8369:
8349:
8339:
8329:
8285:
7860:
4448:
4105:
2973:
2394:
2072:
1868:
1852:
1840:
1238:
1181:
spatial coordinates of an electron. Preparing a system by measuring the complete set of compatible produces a
988:
917:
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
611:
7709:
For a discussion of purifications of mixed quantum states, see
Chapter 2 of John Preskill's lecture notes for
8539:
8478:
8064:
8026:
7990:
5672:
5270:
4697:
2979:
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1256:
1190:
1094:
780:
498:
456:
406:
354:
120:
6663:
6474:
6444:
6386:
4977:{\displaystyle c_{\alpha }=A_{\alpha }e^{i\theta _{\alpha }}\ \ c_{\beta }=A_{\beta }e^{i\theta _{\beta }}}
4226:
3902:
2045:) are known exactly; at least one of them will have a range of possible values. This is the content of the
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8144:
8069:
7980:
7937:
7659:
6504:
6165:
5908:
5679:
5247:
3787:
Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the
3364:
2196:
1714:
992:
785:
503:
411:
331:
301:
264:
7584:
Lev Landau (1927). "Das DĂ€mpfungsproblem in der
Wellenmechanik (The Damping Problem in Wave Mechanics)".
5678:
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the
5636:
8414:
7927:
7907:
7334:
6000:
5321:
4691:
2507:
The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form
2490:
2046:
1876:
1819:
1056:
984:
254:
239:
7830:
4577:
4419:
2499:
2476:
2369:
can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in
1718:
1565:
166:
6084:
States can be formulated in terms of observables, rather than as vectors in a vector space. These are
4703:
3790:
3555:
3374:
2986:
variable (for the spin) is added to the usual three continuous variables (for the position in space).
2155:) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.
1293:
8445:
8374:
8319:
8049:
7872:
7593:
7230:
6868:
5359:
5311:
5307:
5296:
4731:
4567:{\displaystyle \int _{K}|\phi (\mathbf {r} ,t)|^{2}\,\mathrm {d} ^{3}\mathbf {r} \geq 1-\varepsilon }
2399:
2390:
2346:
2294:
2148:
1836:
1726:
1677:
956:
952:
815:
571:
483:
209:
7354:
6693:
6550:
6416:
6358:
4387:
4256:
4048:
3958:
3890:. (The normalization condition above mandates that the total sum of probabilities is equal to one.)
3843:
3659:
3523:
3411:
2712:
2676:
2648:
2608:
2576:
2510:
2503:. Although the details of this are beyond the scope of this article, some consequences of this are:
2173:. (This approach was taken in the later part of the discussion above, with time-varying observables
1286:, there are two possible results: up or down. A pure state here is represented by a two-dimensional
8430:
8399:
8344:
8324:
8232:
8189:
8044:
7970:
7897:
7887:
7799:
7722:
5689:
5259:
4074:
3289:
3005:
2995:
2882:
2335:
2321:
1206:
1110:
1008:
968:
670:
478:
396:
224:
204:
156:
4083:
3932:
2199:. (This approach was taken in the earlier part of the discussion above, with a time-varying state
8359:
8257:
7965:
7912:
7804:
7609:
7254:
7220:
6155:
6135:
4821:
3368:
2170:
1860:
1569:
1527:
1146:
1012:
391:
316:
249:
161:
8490:
5497:
4848:
4354:{\displaystyle |\psi \rangle =\int d^{3}\mathbf {r} \,\psi (\mathbf {r} )|\mathbf {r} \rangle .}
3255:
2342:
in the
Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.
976:
equations containing the force of gravity to predict the trajectory of a cannon ball precisely.
7302:
5273:. The resulting superposition ends up oscillating back and forth between two different states.
1944:
1890:
1686:
1394:
8500:
8409:
8379:
8307:
8270:
8265:
8247:
8212:
8202:
7917:
7882:
7865:
7768:
7694:
7686:
7663:
7522:
7490:
7489:. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books.
7465:
7410:
7323:
7246:
7191:
7181:
7153:
7123:
7098:
7044:
6860:{\displaystyle \langle \mathbf {r} |\mathbf {r} '\rangle =\delta (\mathbf {r} '-\mathbf {r} )}
6140:
5977:" denotes trace. It is important to note that two types of averaging are occurring, one (over
3894:
3590:
3335:
3224:
2706:
2634:
2290:
2152:
1424:
906:
825:
800:
740:
735:
635:
601:
581:
179:
38:
7541:
3105:{\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}
1458:
8227:
8222:
8079:
7975:
7601:
7238:
7173:
6160:
5310:
of possible preparations; and second, when one wants to describe a physical system which is
5266:
2811:
2019:
observables. For example, we cannot prepare a state such that both the position measurement
1710:
1683:
A mixed state for electron spins, in the density-matrix formulation, has the structure of a
1478:
1282:. For another example, if the spin of an electron is measured in any direction, e.g. with a
948:
830:
820:
810:
710:
690:
675:
645:
513:
401:
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5423:
5392:
5365:
2538:
1917:
1152:
8457:
8384:
8364:
8334:
8297:
8292:
8197:
8021:
7510:
7437:
7143:
6926:
Note that this criterion works when the density matrix is normalized so that the trace of
2827:
2482:
2464:
2456:
1872:
1513:
1222:
980:
898:
855:
725:
705:
451:
291:
1105:
The same physical quantum state can be expressed mathematically in different ways called
971:. A classical dynamical state consists of a set of dynamical variables with well-defined
7597:
7234:
27:
Mathematical entity to describe the probability of each possible measurement on a system
8435:
8404:
8394:
8016:
8006:
7840:
7652:
7398:
7326:(1927). Ăber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,
6035:
5980:
5523:
5473:
5282:
4665:
4645:
4625:
4605:
4602:. The integral represents the probability that a particle is found in a bounded region
3952:
3517:
3009:
2983:
2430:
2372:
2352:
2330:, as follows. Any given system is identified with some finite- or infinite-dimensional
2327:
1848:
1844:
1537:
1516:
1452:
1287:
1265:
1202:
1037:
944:
921:
790:
750:
730:
700:
680:
630:
596:
446:
436:
229:
7276:
4374:
Though closely related, pure states are not the same as bound states belonging to the
8533:
8354:
8207:
8098:
7932:
7855:
7613:
7558:
6227:"If a system is small, we cannot observe it without producing a serious disturbance."
6069:
5543:
The density matrix describing a mixed state is defined to be an operator of the form
5417:
4041:
3360:
2331:
2298:
1722:
1557:. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
1519:
1234:
1138:
1133:
850:
845:
775:
745:
715:
586:
532:
259:
7645:
For a discussion of conceptual aspects and a comparison with classical states, see:
7449:. Course of Theoretical Physics. Vol. 3 (2nd ed.). London: Pergamon Press.
7258:
5318:
Mixed states inevitably arise from pure states when, for a composite quantum system
2633:, which corresponds to the same physical quantum state. Technically, the bra is the
8237:
7850:
7845:
7682:
7562:
7402:
6787:{\textstyle \int d^{3}\mathbf {r} '\,\langle \mathbf {r} |\mathbf {r} '\rangle =1,}
6125:
6115:
5093:{\displaystyle |c_{\alpha }|^{2}+|c_{\beta }|^{2}=A_{\alpha }^{2}+A_{\beta }^{2}=1}
2969:
2868:
1509:
1121:
972:
840:
835:
770:
755:
720:
214:
5454:
be described with a single ket vector. Instead, it is described by its associated
2451:
is used, properly speaking, a point the projective
Hilbert space corresponds to a
1141:
represent quantum states, particularly when they are functions of position or of
7710:
5997:) being the usual expected value of the observable when the quantum is in state
5490:
can be always represented as the partial trace of a pure quantum state (called a
4384:
if it remains localized in a bounded region of space for all times. A pure state
1560:
On the other hand, a system in a superposition of multiple different eigenstates
8275:
7732:
7353:(1927/1928). The quantum postulate and the recent development of atomic theory,
7088:
6145:
5291:
is a state which can be described by a single ket vector, as described above. A
5265:
Another example of the importance of relative phase in quantum superposition is
4443:
4380:
3825:
2339:
2123:
in the same run of the experiment, the system will transfer to an eigenstate of
1546:
805:
760:
695:
650:
3367:
is chosen for the
Hilbert space of a system, then any ket can be expanded as a
2169:
was fixed once at the beginning of the experiment. This approach is called the
1522:, while each measurable physical quantity (such as the energy or momentum of a
7433:
7350:
7242:
6089:
6073:
3829:
2968:
As a consequence, the quantum state of a particle with spin is described by a
2638:
2494:
2486:
2129:
after the first measurement, and we will generally notice that the results of
1968:. Unlike the linear combination case each system is in a definite eigenstate.
1541:
1532:
1004:
795:
765:
685:
660:
655:
640:
7250:
7195:
6989:{\displaystyle \operatorname {Tr} (\rho ^{2})=(\operatorname {Tr} \rho )^{2}}
4032:{\displaystyle \psi (\mathbf {r} )\equiv \langle \mathbf {r} |\psi \rangle .}
3214:{\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,S_{\nu }-1,\,S_{\nu }\}}
2111:, all runs of the experiment will yield the same result. If we measure first
112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
8129:
7825:
7518:
4378:
of an observable with no quantum uncertainty. A particle is said to be in a
3840:) of an observable, and that observable is measured on the normalized state
2872:
2844:
2702:
1941:
represents the probability of a randomly selected system being in the state
286:
17:
905:
is a mathematical entity that embodies the knowledge of a quantum system.
7225:
4370:
Decomposition of spectrum (functional analysis) § Quantum mechanics
2645:. In a finite-dimensional space with a chosen orthonormal basis, writing
1523:
1230:
1226:
1213:
Numerical or analytic solutions in quantum mechanics can be expressed as
1142:
665:
7638:
The concept of quantum states, in particular the content of the section
5671:
The density matrix can be thought of as a way of using the one-particle
5615:{\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}
3983:
is associated with a complex-valued function of three-dimensional space
7605:
6170:
3297:
2864:
2163:
One can take the observables to be dependent on time, while the state
1655:{\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}
6575:
can be found at different positions with corresponding probabilities.
4811:{\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }
3331:
3320:
2836:
2819:
2151:. These entangled states lead to experimentally testable properties (
2052:
Moreover, in contrast to classical mechanics, it is unavoidable that
7737:
7335:'The actual content of quantum theoretical kinematics and mechanics'
4216:{\displaystyle \int d^{3}\mathbf {r} \,|\psi (\mathbf {r} )|^{2}=1.}
1564:
in general have quantum uncertainty for the given observable. Using
7654:
Lectures on
Quantum Theory: Mathematical and Structural Foundations
7642:
above, is covered in most standard textbooks on quantum mechanics.
7209:
Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem".
5269:, where the relative phase of two states varies in time due to the
2744:) are written so as to look like a bra and ket next to each other:
2289:
Both viewpoints are used in quantum theory. While non-relativistic
1859:
classifies the multitude of ways to write a given mixed state as a
924:
describing quantum systems using position or momentum variables and
6911:{\displaystyle \langle \mathbf {r} |\mathbf {r} \rangle =\infty .}
3293:
2279:{\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }
2054:
performing a measurement on the system generally changes its state
947:
and incoherent states. Categories with special properties include
7515:
Mastering
Quantum Mechanics: Essentials, Theory, and Applications
6346:: "to define the scalar products as being between bras and kets."
5692:
is 0 for a pure state, and strictly positive for a mixed state.
2823:
2815:
1209:
for the electron of a hydrogen atom in different quantum states.
7741:
2455:
passing through the origin of the Hilbert space, rather than a
5967:
are eigenkets and eigenvalues, respectively, for the operator
7064:
Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (1977).
6092:, or sometimes other classes of algebras of observables. See
3868:, then the probability that the result of the measurement is
3500:{\displaystyle |\psi \rangle =\sum _{i}c_{i}|{k_{i}}\rangle }
7485:
Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014).
6471:
are elements of the same Hilbert space. A particle in state
967:
As a tool for physics, quantum states grew out of states in
7677:
For a more detailed coverage of mathematical aspects, see:
3341:
When symmetrization or anti-symmetrization is unnecessary,
2015:
There is no state that is simultaneously an eigenstate for
2806:
Mathematical formulation of quantum mechanics § Spin
2139:, and the order in which they are performed is important.
1867:
is performed on a quantum system, the theory gives only a
2795:(The phrase "bra-ket" is supposed to resemble "bracket".)
2497:
invented a notation to describe quantum states, known as
3371:
of those basis elements. Symbolically, given basis kets
2326:
Quantum physics is most commonly formulated in terms of
7122:. Vol. 2. Courier Dover Publications. p. 87.
5389:
is inaccessible to the observer. The state of the part
3351:
of one-particle spaces, to which we will return later.
2481:
Calculations in quantum mechanics make frequent use of
6724:
4756:
are two kets corresponding to quantum states, the ket
3520:. In physical terms, this is described by saying that
2338:
1. Thus the set of all pure states corresponds to the
2205:
7691:
Operator Algebras and Quantum Statistical Mechanics 1
6938:
6877:
6800:
6696:
6666:
6553:
6507:
6477:
6447:
6419:
6389:
6361:
6038:
6003:
5983:
5946:
5911:
5711:
5639:
5549:
5526:
5500:
5476:
5426:
5395:
5368:
5324:
5106:
4990:
4881:
4851:
4824:
4762:
4734:
4706:
4668:
4648:
4628:
4608:
4580:
4486:
4451:
4422:
4390:
4287:
4259:
4229:
4156:
4108:
4086:
4051:
3989:
3961:
3935:
3905:
3846:
3793:
3733:
3690:
3662:
3599:
3558:
3526:
3442:
3414:
3377:
3347:-particle spaces of states can be obtained simply by
3258:
3227:
3127:
3018:
2904:
2751:
2715:
2679:
2651:
2611:
2579:
2541:
2513:
2433:
2402:
2375:
2355:
2137:
Quantum mechanical measurements influence one another
1977:
1947:
1920:
1893:
1735:
1689:
1578:
1481:
1461:
1427:
1397:
1328:
1296:
1155:
1040:
50:
3338:
and can't be described with Schrödinger mechanics).
8471:
8423:
8256:
8188:
8122:
8035:
7999:
7953:
7818:
7775:
7393:
7391:
7120:
A History of the Theories of Aether and Electricity
5633:is the fraction of the ensemble in each pure state
3642:{\displaystyle c_{i}=\langle {k_{i}}|\psi \rangle }
2786:{\displaystyle \langle \psi _{1}|\psi _{2}\rangle }
1087:
7651:
7385:, 4th edition, Oxford University Press, Oxford UK.
7180:. Lecture Notes in Physics. Springer. p. 15.
6988:
6910:
6859:
6786:
6710:
6682:
6567:
6539:
6493:
6463:
6433:
6405:
6375:
6044:
6024:
5989:
5959:
5932:
5897:
5699:) of a measurement corresponding to an observable
5663:
5614:
5532:
5512:
5482:
5439:
5408:
5381:
5350:
5235:
5092:
4976:
4864:
4837:
4810:
4748:
4720:
4674:
4654:
4642:. If the probability remains arbitrarily close to
4634:
4614:
4594:
4566:
4472:
4434:
4404:
4353:
4273:
4245:
4215:
4142:
4094:
4065:
4031:
3975:
3943:
3921:
3860:
3816:
3778:{\displaystyle \sum _{i}\left|c_{i}\right|^{2}=1.}
3777:
3719:
3676:
3641:
3581:
3540:
3499:
3428:
3400:
3277:
3240:
3213:
3104:
2958:
2785:
2729:
2693:
2665:
2625:
2593:
2547:
2527:
2439:
2419:
2381:
2361:
2278:
2105:at the experiment's beginning. If we measure only
2099:. Suppose that the system is in an eigenstate of
2093:corresponds to a measurement earlier in time than
1998:
1960:
1933:
1906:
1810:
1701:
1654:
1487:
1467:
1443:
1413:
1383:
1314:
1173:
1046:
111:
7639:
7443:Quantum Mechanics — Non-Relativistic Theory
7341:(1983), Princeton University Press, Princeton NJ.
4818:is also a quantum state of the same system. Both
4223:In terms of the continuous set of position basis
1822:of joint spin states for two particles with spin
1237:, the relevant pure states are identified by the
1185:. More common, incomplete preparation produces a
1117:
928:
7428:
7426:
2056:. More precisely: After measuring an observable
7374:
7372:
7370:
3899:, which is the basis consisting of eigenstates
1221:, are labeled with quantized values, typically
1073:The quantum state after a measurement is in an
6333:now appears as a complete bracket expression."
6190:To avoid misunderstandings: Here we mean that
3720:{\displaystyle \langle \psi |\psi \rangle =1,}
3300:(particles with half-integer spin). The above
2316:Pure states as rays in a complex Hilbert space
7753:
6383:is a superposition of different basis states
3727:and for orthonormal basis this translates to
2311:Mathematical formulation of quantum mechanics
1999:{\displaystyle {\langle A\rangle }_{\sigma }}
1835:A pure quantum state can be represented by a
1800:
1764:
1384:{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}
878:
8:
7487:Quantum mechanics: the theoretical minimum;
7460:Blanchard, Philippe; BrĂŒning, Erwin (2015).
6896:
6878:
6824:
6801:
6772:
6749:
6705:
6677:
6562:
6488:
6458:
6428:
6400:
6370:
6319:Dirac (1958), p. 19: "A scalar product
6072:, the concept of mixture was put forward by
6019:
5927:
5859:
5831:
5780:
5744:
5718:
5712:
5655:
5594:
5591:
5225:
5150:
4805:
4781:
4743:
4715:
4399:
4345:
4296:
4268:
4240:
4073:. Similarly to the discrete case above, the
4060:
4023:
4007:
3970:
3916:
3855:
3811:
3705:
3691:
3671:
3636:
3613:
3576:
3535:
3494:
3451:
3423:
3395:
3208:
3128:
3096:
2953:
2905:
2839:. In non-relativistic quantum mechanics the
2780:
2752:
2724:
2701:is a row vector; to obtain it just take the
2680:
2660:
2612:
2588:
2522:
2273:
2223:
1986:
1980:
1680:of quantum states is again a quantum state.
1646:
1596:
1120:) the theory develops in terms of abstract '
106:
80:
7716:For a discussion of geometric aspects see:
5466:. Density matrices can describe both mixed
3285:for a particle that does not exhibit spin.
2334:. The pure states correspond to vectors of
1540:that acts on the states of the system. The
909:specifies the construction, evolution, and
7760:
7746:
7738:
7034:
7032:
4696:As mentioned above, quantum states may be
3066:
3059:
2322:Wigner's theorem § Rays and ray space
2159:Schrödinger picture vs. Heisenberg picture
1082:. Any state that is not pure is called a
1078:set of compatible measurements produces a
885:
871:
29:
7224:
7030:
7028:
7026:
7024:
7022:
7020:
7018:
7016:
7014:
7012:
6980:
6952:
6937:
6891:
6886:
6881:
6876:
6849:
6837:
6815:
6809:
6804:
6799:
6763:
6757:
6752:
6748:
6739:
6732:
6723:
6697:
6695:
6672:
6667:
6665:
6554:
6552:
6508:
6506:
6483:
6478:
6476:
6453:
6448:
6446:
6420:
6418:
6395:
6390:
6388:
6362:
6360:
6037:
6013:
6004:
6002:
5982:
5951:
5945:
5921:
5912:
5910:
5868:
5863:
5853:
5844:
5838:
5826:
5820:
5810:
5800:
5790:
5774:
5765:
5757:
5751:
5738:
5728:
5710:
5649:
5640:
5638:
5607:
5601:
5585:
5576:
5570:
5560:
5548:
5525:
5499:
5475:
5431:
5425:
5400:
5394:
5373:
5367:
5342:
5329:
5323:
5217:
5209:
5193:
5185:
5173:
5168:
5156:
5142:
5136:
5119:
5111:
5105:
5100:and extracting the common factors gives:
5078:
5073:
5060:
5055:
5042:
5037:
5030:
5021:
5012:
5007:
5000:
4991:
4989:
4966:
4958:
4948:
4935:
4917:
4909:
4899:
4886:
4880:
4856:
4850:
4829:
4823:
4797:
4791:
4773:
4767:
4761:
4735:
4733:
4707:
4705:
4667:
4647:
4627:
4607:
4588:
4587:
4579:
4547:
4541:
4536:
4534:
4528:
4523:
4508:
4497:
4491:
4485:
4473:{\displaystyle K\subset \mathbb {R} ^{3}}
4464:
4460:
4459:
4450:
4421:
4391:
4389:
4340:
4335:
4327:
4320:
4315:
4309:
4288:
4286:
4260:
4258:
4235:
4230:
4228:
4201:
4196:
4187:
4176:
4175:
4170:
4164:
4155:
4143:{\displaystyle |\psi (\mathbf {r} )|^{2}}
4134:
4129:
4120:
4109:
4107:
4087:
4085:
4052:
4050:
4015:
4010:
3996:
3988:
3962:
3960:
3936:
3934:
3911:
3906:
3904:
3847:
3845:
3804:
3799:
3794:
3792:
3763:
3753:
3738:
3732:
3697:
3689:
3663:
3661:
3628:
3621:
3616:
3604:
3598:
3569:
3564:
3559:
3557:
3527:
3525:
3487:
3482:
3477:
3471:
3461:
3443:
3441:
3415:
3413:
3388:
3383:
3378:
3376:
3263:
3257:
3232:
3226:
3202:
3197:
3182:
3177:
3170:
3155:
3147:
3138:
3126:
3087:
3082:
3073:
3068:
3050:
3045:
3036:
3031:
3019:
3017:
2959:{\displaystyle \{-S,-S+1,\ldots ,S-1,S\}}
2903:
2774:
2765:
2759:
2750:
2716:
2714:
2686:
2678:
2652:
2650:
2618:
2610:
2580:
2578:
2540:
2514:
2512:
2432:
2403:
2401:
2374:
2354:
2267:
2258:
2243:
2233:
2206:
2204:
2060:, the system will be in an eigenstate of
1990:
1979:
1976:
1952:
1946:
1925:
1919:
1898:
1892:
1799:
1798:
1763:
1762:
1750:
1734:
1688:
1640:
1631:
1616:
1606:
1579:
1577:
1480:
1460:
1436:
1428:
1426:
1406:
1398:
1396:
1366:
1361:
1352:
1343:
1338:
1329:
1327:
1295:
1154:
1039:
98:
87:
86:
72:
57:
49:
7729:. Cambridge: Cambridge University Press.
7149:Quantum Computing: A Gentle Introduction
7043:. North Holland, John Wiley & Sons.
4080:of the particle being found at position
3893:A particularly important example is the
2990:Many-body states and particle statistics
2601:is uniquely associated with a so-called
1847:, while mixed states are represented by
1717:1. A more complicated case is given (in
1502:state that pure states, at a given time
1201:
7409:(2nd, illustrated ed.). Springer.
7178:Statistical Structure of Quantum Theory
7083:
7081:
7079:
7077:
7075:
7008:
6660:In the continuous case, the basis kets
6183:
5520:for a sufficiently large Hilbert space
4984:and defining the norm of the state as:
4662:then the particle is said to remain in
2978:. Equivalently, it is represented by a
54:
37:
7572:. London: Heinemann. pp. 284â302.
7544:Density matrix theory and applications
6086:positive normalized linear functionals
3652:One property worth noting is that the
1572:of eigenstates can be represented as:
1322:, with a length of one; that is, with
963:From the states of classical mechanics
6683:{\displaystyle |\mathbf {r} \rangle }
6494:{\displaystyle |\mathbf {r} \rangle }
6464:{\displaystyle |\mathbf {r} \rangle }
6406:{\displaystyle |\mathbf {r} \rangle }
6247:For concreteness' sake, suppose that
6065:that the quantum is in those states.
4246:{\displaystyle |\mathbf {r} \rangle }
3922:{\displaystyle |\mathbf {r} \rangle }
3589:. If the basis kets are chosen to be
3296:(particles with integer spin) versus
2972:-valued wave function with values in
1863:of pure states. Before a particular
1225:. For example, when dealing with the
1005:compatible and incompatible variables
7:
7118:Whittaker, Sir Edmund (1989-01-01).
7094:Quantum Theory: Concepts and Methods
6718:): They are normalized according to
6690:are not unit kets (unlike the state
6540:{\displaystyle \mathbf {r} =(x,y,z)}
6212:are measured in the same state, but
5933:{\displaystyle |\alpha _{i}\rangle }
5246:One example of superposition is the
4875:Writing the superposed state using
3355:Basis states of one-particle systems
2830:and, at quantum scale, behaves as a
2637:of the ket. It is an element of the
1713:and positive semi-definite, and has
1526:) is associated with a mathematical
1149:. For example, one set could be the
7563:"Remarks on the mind-body question"
7383:The Principles of Quantum Mechanics
7146:; Polak, Wolfgang H. (2011-03-04).
5664:{\displaystyle |\psi _{s}\rangle .}
2562:Dirac defined two kinds of vector,
2389:are said to correspond to the same
7618:English translation reprinted in:
6902:
6216:in the same run of the experiment.
6098:GelfandâNaimarkâSegal construction
6025:{\displaystyle |\psi _{s}\rangle }
5351:{\displaystyle H_{1}\otimes H_{2}}
5250:, in which superposition leads to
4537:
3323:(quanta of light) are bosons with
2264:
2211:
1949:
1895:
1637:
1584:
1015:corresponding to the measurement.
417:Sum-over-histories (path integral)
103:
77:
33:Part of a series of articles about
25:
6501:is located precisely at position
6131:Introduction to quantum mechanics
6121:GreenbergerâHorneâZeilinger state
6056:) average with the probabilities
4595:{\displaystyle t\in \mathbb {R} }
4435:{\displaystyle \varepsilon >0}
3000:The quantum state of a system of
1191:Schrödinger's equations of motion
1116:In formal quantum mechanics (see
8514:
8513:
6892:
6882:
6850:
6838:
6816:
6805:
6764:
6753:
6740:
6673:
6617:" (the left side corresponds to
6509:
6484:
6454:
6396:
4721:{\displaystyle |\alpha \rangle }
4548:
4509:
4341:
4328:
4316:
4236:
4188:
4171:
4150:and the normalized states have
4121:
4088:
4011:
3997:
3937:
3912:
3817:{\displaystyle |{k_{i}}\rangle }
3582:{\displaystyle |{k_{i}}\rangle }
3401:{\displaystyle |{k_{i}}\rangle }
3069:
3032:
2982:of four variables: one discrete
2641:, and related to the ket by the
2404:
1855:that act on Hilbert spaces. The
1315:{\displaystyle (\alpha ,\beta )}
1217:. These solution states, called
1093:The eigenstate solutions to the
1029:Measurement in quantum mechanics
940:
7622:Collected papers of L.D. Landau
7462:Mathematical Methods in Physics
7407:Quantum Mechanics: Fundamentals
7273:"Statistical Mixture of States"
5494:) on a larger bipartite system
4749:{\displaystyle |\beta \rangle }
2420:{\displaystyle \mathbf {P} (H)}
2047:Heisenberg uncertainty relation
1853:positive semidefinite operators
1500:postulates of quantum mechanics
1248:angular momentum quantum number
936:
8463:Relativistic quantum mechanics
7733:second, revised edition (2017)
7339:Quantum Theory and Measurement
7212:Foundations of Physics Letters
7097:. Kluwer Academic Publishers.
6977:
6964:
6958:
6945:
6887:
6854:
6833:
6810:
6758:
6711:{\displaystyle |\psi \rangle }
6698:
6668:
6568:{\displaystyle |\psi \rangle }
6555:
6534:
6516:
6479:
6449:
6434:{\displaystyle |\psi \rangle }
6421:
6391:
6376:{\displaystyle |\psi \rangle }
6363:
6005:
5913:
5892:
5883:
5864:
5845:
5827:
5766:
5758:
5641:
5608:
5577:
5218:
5143:
5038:
5022:
5008:
4992:
4798:
4774:
4736:
4708:
4524:
4519:
4505:
4498:
4405:{\displaystyle |\phi \rangle }
4392:
4336:
4332:
4324:
4289:
4274:{\displaystyle |\psi \rangle }
4261:
4231:
4197:
4192:
4184:
4177:
4130:
4125:
4117:
4110:
4066:{\displaystyle |\psi \rangle }
4053:
4016:
4001:
3993:
3976:{\displaystyle |\psi \rangle }
3963:
3907:
3861:{\displaystyle |\psi \rangle }
3848:
3795:
3698:
3677:{\displaystyle |\psi \rangle }
3664:
3629:
3560:
3541:{\displaystyle |\psi \rangle }
3528:
3478:
3444:
3429:{\displaystyle |\psi \rangle }
3416:
3379:
3093:
3027:
3020:
2766:
2730:{\displaystyle |\psi \rangle }
2717:
2694:{\displaystyle \langle \psi |}
2687:
2666:{\displaystyle |\psi \rangle }
2653:
2626:{\displaystyle \langle \psi |}
2619:
2594:{\displaystyle |\psi \rangle }
2581:
2528:{\displaystyle |\psi \rangle }
2515:
2447:. Note that although the word
2414:
2408:
2345:Multiplying a pure state by a
2259:
2255:
2249:
2220:
2214:
2207:
1790:
1774:
1671:Mixed states of wave functions
1632:
1628:
1622:
1593:
1587:
1580:
1437:
1429:
1407:
1399:
1362:
1353:
1339:
1330:
1309:
1297:
1198:Pure states of wave functions
567:Relativistic quantum mechanics
99:
92:
73:
1:
8441:Quantum statistical mechanics
8218:Quantum differential calculus
8140:Delayed-choice quantum eraser
7923:Symmetry in quantum mechanics
5301:quantum statistical mechanics
3593:(as is often the case), then
2740:Scalar products (also called
2030:and the momentum measurement
1189:. Wave function solutions of
1128:Wave function representations
607:Quantum statistical mechanics
7705:In particular, see Sec. 2.3.
7640:Formalism in quantum physics
6547:, while a particle in state
6080:Mathematical generalizations
6052:) being a statistical (said
4686:Superposition of pure states
4364:Pure states vs. bound states
4095:{\displaystyle \mathbf {r} }
4039:This function is called the
3944:{\displaystyle \mathbf {r} }
3312:Electrons are fermions with
3121:assume values from the set
2898:possible values in the set
2643:Riesz representation theorem
2305:Formalism in quantum physics
1069:Eigenvalues and eigenvectors
8243:Quantum stochastic calculus
8233:Quantum measurement problem
8155:MachâZehnder interferometer
7464:. BirkhÀuser. p. 431.
6283:in the above example, with
6151:Quantum harmonic oscillator
4838:{\displaystyle c_{\alpha }}
2871:(1/2, 3/2, 5/2 ...). For a
1536:. The operator serves as a
1086:as discussed in more depth
1063:Eigenstates and pure states
577:Quantum information science
8556:
7727:Geometry of Quantum States
7358:Supplement April 14 1928,
7333:: 172â198. Translation as
7068:. Wiley. pp. 231â235.
6111:Atomic electron transition
5513:{\displaystyle H\otimes K}
5280:
4865:{\displaystyle c_{\beta }}
4689:
4367:
3278:{\displaystyle S_{\nu }=0}
2993:
2890:always assumes one of the
2803:
2474:
2319:
2308:
1549:(which physicists call an
1131:
1066:
1026:
951:for time independence and
8509:
8303:Quantum complexity theory
8281:Quantum cellular automata
7986:Path integral formulation
7624:. Oxford: Pergamon Press.
7620:D. Ter Haar, ed. (1965).
7243:10.1007/s10702-006-1852-1
6225:Dirac (1958), p. 4:
5416:is expressed then as the
5262:in other wave phenomena.
3114:Here, the spin variables
1961:{\displaystyle \Phi _{n}}
1907:{\displaystyle \Phi _{n}}
1702:{\displaystyle 2\times 2}
1414:{\displaystyle |\alpha |}
999:Role in quantum mechanics
8370:Quantum machine learning
8350:Quantum key distribution
8340:Quantum image processing
8330:Quantum error correction
8180:Wheeler's delayed choice
7570:The Scientist Speculates
7039:Messiah, Albert (1966).
4412:is called a bound state
3955:particle), then any ket
3548:has been expressed as a
3241:{\displaystyle S_{\nu }}
2814:has the same dimension (
2395:projective Hilbert space
2073:incompatible observables
1869:probability distribution
1841:projective Hilbert space
1444:{\displaystyle |\beta |}
1284:SternâGerlach experiment
1239:principal quantum number
989:probability distribution
983:, quantized, limited by
612:Quantum machine learning
365:Wheeler's delayed-choice
8286:Quantum finite automata
7650:Isham, Chris J (1995).
6584:Landau (1965), p. 17: "
2980:complex-valued function
2856:reduced Planck constant
2135:are statistical. Thus:
1857:SchrödingerâHJW theorem
1665:time evolution operator
1468:{\displaystyle \alpha }
1257:magnetic quantum number
322:LeggettâGarg inequality
8390:Quantum neural network
7660:Imperial College Press
7586:Zeitschrift fĂŒr Physik
7568:. In I.J. Good (ed.).
6990:
6912:
6861:
6788:
6712:
6684:
6569:
6541:
6495:
6465:
6435:
6407:
6377:
6166:Wave function collapse
6046:
6032:, and the other (over
6026:
5991:
5961:
5934:
5899:
5665:
5616:
5534:
5514:
5484:
5441:
5410:
5383:
5362:state on it, the part
5352:
5256:coherent superposition
5248:double-slit experiment
5237:
5094:
4978:
4866:
4839:
4812:
4750:
4722:
4676:
4656:
4636:
4616:
4596:
4568:
4474:
4436:
4406:
4355:
4275:
4247:
4217:
4144:
4096:
4067:
4033:
3977:
3945:
3923:
3862:
3818:
3779:
3721:
3678:
3643:
3583:
3542:
3501:
3430:
3402:
3292:is very different for
3279:
3242:
3215:
3106:
2960:
2854:that, in units of the
2787:
2731:
2695:
2667:
2627:
2595:
2549:
2529:
2441:
2421:
2383:
2363:
2280:
2000:
1971:The expectation value
1962:
1935:
1908:
1812:
1703:
1656:
1489:
1488:{\displaystyle \beta }
1469:
1445:
1415:
1385:
1316:
1210:
1175:
1048:
993:double-slit experiment
113:
8415:Quantum teleportation
7943:Waveâparticle duality
7279:on September 23, 2019
6991:
6913:
6862:
6789:
6713:
6685:
6570:
6542:
6496:
6466:
6436:
6408:
6378:
6094:State on a C*-algebra
6047:
6027:
5992:
5962:
5960:{\displaystyle a_{i}}
5935:
5900:
5666:
5617:
5535:
5515:
5485:
5442:
5440:{\displaystyle H_{2}}
5411:
5409:{\displaystyle H_{1}}
5384:
5382:{\displaystyle H_{2}}
5353:
5238:
5095:
4979:
4867:
4840:
4813:
4751:
4723:
4692:Quantum superposition
4677:
4657:
4637:
4617:
4597:
4569:
4475:
4437:
4407:
4356:
4276:
4248:
4218:
4145:
4097:
4068:
4034:
3978:
3946:
3924:
3863:
3819:
3780:
3722:
3684:are characterized by
3679:
3644:
3584:
3550:quantum superposition
3543:
3502:
3431:
3403:
3280:
3243:
3216:
3107:
2994:Further information:
2961:
2841:group representations
2788:
2732:
2696:
2668:
2628:
2596:
2570:, dual to each other.
2550:
2548:{\displaystyle \psi }
2530:
2491:Hermitian conjugation
2442:
2422:
2384:
2364:
2281:
2001:
1963:
1936:
1934:{\displaystyle P_{n}}
1909:
1877:uncertainty principle
1813:
1704:
1657:
1490:
1470:
1446:
1416:
1386:
1317:
1207:Probability densities
1205:
1176:
1174:{\displaystyle x,y,z}
1067:Further information:
1057:uncertainty principle
1049:
987:, and only provide a
985:uncertainty relations
953:quantum vacuum states
929:vector quantum states
307:Elitzur–Vaidman
297:Davisson–Germer
114:
8446:Quantum field theory
8375:Quantum metamaterial
8320:Quantum cryptography
8050:Consistent histories
7574:Footnote 13 on p.180
7299:"The Density Matrix"
7174:Holevo, Alexander S.
6936:
6875:
6871:), which means that
6869:Dirac delta function
6798:
6722:
6694:
6664:
6551:
6505:
6475:
6445:
6417:
6387:
6359:
6036:
6001:
5981:
5944:
5909:
5709:
5637:
5547:
5524:
5498:
5474:
5424:
5393:
5366:
5322:
5308:statistical ensemble
5299:of pure states (see
5297:statistical ensemble
5271:Schrödinger equation
5252:quantum interference
5104:
4988:
4879:
4849:
4822:
4760:
4732:
4704:
4666:
4646:
4626:
4606:
4578:
4484:
4449:
4420:
4388:
4285:
4257:
4227:
4154:
4106:
4084:
4049:
3987:
3959:
3933:
3903:
3844:
3791:
3731:
3688:
3660:
3597:
3556:
3524:
3440:
3412:
3375:
3256:
3225:
3125:
3016:
2902:
2749:
2713:
2677:
2673:as a column vector,
2649:
2609:
2577:
2539:
2511:
2431:
2400:
2373:
2353:
2295:quantum field theory
2203:
1975:
1945:
1918:
1891:
1884:statistical ensemble
1733:
1727:quantum entanglement
1725:, which exemplifies
1687:
1576:
1479:
1459:
1425:
1395:
1326:
1294:
1153:
1111:parametric equations
1095:Schrödinger equation
1038:
957:quantum field theory
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
48:
8431:Quantum fluctuation
8400:Quantum programming
8360:Quantum logic gates
8345:Quantum information
8325:Quantum electronics
7800:Classical mechanics
7598:1927ZPhy...45..430L
7517:. Cambridge, Mass:
7305:on January 15, 2012
7235:2006FoPhL..19...95K
7144:Rieffel, Eleanor G.
5690:von Neumann entropy
5462:), usually denoted
5293:mixed quantum state
5178:
5083:
5065:
4376:pure point spectrum
3290:identical particles
3006:spatial coordinates
2996:Particle statistics
2883:spin quantum number
2875:particle with spin
2867:(0, 1, 2 ...) or a
2485:, scalar products,
2197:Schrödinger picture
1555:quantum uncertainty
1187:mixed quantum state
1009:equations of motion
969:classical mechanics
360:Stern–Gerlach
157:Classical mechanics
8484:in popular culture
8266:Quantum algorithms
8114:Von NeumannâWigner
8094:Objective collapse
7805:Old quantum theory
7606:10.1007/bf01343064
6986:
6908:
6857:
6784:
6708:
6680:
6565:
6537:
6491:
6461:
6431:
6403:
6373:
6355:Note that a state
6342:Gottfried (2013),
6156:Quantum logic gate
6136:No-cloning theorem
6100:for more details.
6042:
6022:
5987:
5957:
5930:
5895:
5805:
5795:
5733:
5661:
5612:
5565:
5530:
5510:
5480:
5437:
5406:
5379:
5348:
5289:pure quantum state
5258:, by analogy with
5233:
5164:
5090:
5069:
5051:
4974:
4862:
4835:
4808:
4746:
4718:
4672:
4652:
4632:
4612:
4592:
4564:
4470:
4432:
4402:
4351:
4271:
4243:
4213:
4140:
4092:
4063:
4029:
3973:
3941:
3919:
3858:
3814:
3775:
3743:
3717:
3674:
3639:
3579:
3538:
3497:
3466:
3426:
3398:
3369:linear combination
3275:
3238:
3211:
3102:
2956:
2783:
2727:
2691:
2663:
2623:
2591:
2545:
2525:
2437:
2417:
2379:
2359:
2276:
2238:
2171:Heisenberg picture
2041:(at the same time
1996:
1958:
1931:
1904:
1861:convex combination
1808:
1791:↓ ↑
1775:↑ ↓
1699:
1652:
1611:
1570:linear combination
1485:
1465:
1441:
1411:
1381:
1312:
1211:
1183:pure quantum state
1171:
1147:degrees of freedom
1044:
927:the more abstract
548:Von NeumannâWigner
528:Objective-collapse
327:Mach–Zehnder
317:Leggett inequality
312:Franck–Hertz
162:Old quantum theory
109:
8527:
8526:
8501:Quantum mysticism
8479:Schrödinger's cat
8410:Quantum simulator
8380:Quantum metrology
8308:Quantum computing
8271:Quantum amplifier
8248:Quantum spacetime
8213:Quantum cosmology
8203:Quantum chemistry
7918:Scattering theory
7866:Zero-point energy
7861:Degenerate levels
7769:Quantum mechanics
7700:978-3-540-17093-8
7687:Robinson, Derek W
7669:978-1-86094-001-9
7528:978-0-262-04613-8
7496:978-0-465-06290-4
7471:978-3-319-14044-5
7159:978-0-262-01506-6
7066:Quantum Mechanics
7041:Quantum Mechanics
6141:Orthonormal basis
6045:{\displaystyle s}
5990:{\displaystyle i}
5796:
5786:
5724:
5697:expectation value
5556:
5533:{\displaystyle K}
5483:{\displaystyle H}
5267:Rabi oscillations
5179:
4930:
4927:
4675:{\displaystyle K}
4655:{\displaystyle 1}
4635:{\displaystyle t}
4615:{\displaystyle K}
4045:corresponding to
3929:with eigenvalues
3734:
3457:
3330:(although in the
3288:The treatment of
2707:complex conjugate
2465:geometrical sense
2440:{\displaystyle H}
2382:{\displaystyle H}
2362:{\displaystyle H}
2291:quantum mechanics
2229:
2006:of an observable
1887:different states
1760:
1759:
1602:
1508:, correspond to
1047:{\displaystyle x}
949:stationary states
907:Quantum mechanics
895:
894:
602:Scattering theory
582:Quantum computing
355:Schrödinger's cat
287:Bell's inequality
95:
70:
39:Quantum mechanics
16:(Redirected from
8547:
8517:
8516:
8228:Quantum geometry
8223:Quantum dynamics
8080:Superdeterminism
7976:Matrix mechanics
7831:Braâket notation
7762:
7755:
7748:
7739:
7730:
7704:
7673:
7657:
7627:
7625:
7617:
7592:(5â6): 430â441.
7581:
7575:
7573:
7567:
7555:
7549:
7539:
7533:
7532:
7511:Zwiebach, Barton
7507:
7501:
7500:
7482:
7476:
7475:
7457:
7451:
7450:
7448:
7430:
7421:
7420:
7395:
7386:
7376:
7365:
7348:
7342:
7321:
7315:
7314:
7312:
7310:
7301:. Archived from
7295:
7289:
7288:
7286:
7284:
7275:. Archived from
7269:
7263:
7262:
7228:
7226:quant-ph/0305068
7206:
7200:
7199:
7170:
7164:
7163:
7140:
7134:
7133:
7115:
7109:
7108:
7085:
7070:
7069:
7061:
7055:
7054:
7036:
6996:
6995:
6993:
6992:
6987:
6985:
6984:
6957:
6956:
6931:
6924:
6918:
6917:
6915:
6914:
6909:
6895:
6890:
6885:
6866:
6864:
6863:
6858:
6853:
6845:
6841:
6823:
6819:
6813:
6808:
6793:
6791:
6790:
6785:
6771:
6767:
6761:
6756:
6747:
6743:
6737:
6736:
6717:
6715:
6714:
6709:
6701:
6689:
6687:
6686:
6681:
6676:
6671:
6658:
6652:
6650:
6630:
6629:
6616:
6582:
6576:
6574:
6572:
6571:
6566:
6558:
6546:
6544:
6543:
6538:
6512:
6500:
6498:
6497:
6492:
6487:
6482:
6470:
6468:
6467:
6462:
6457:
6452:
6440:
6438:
6437:
6432:
6424:
6412:
6410:
6409:
6404:
6399:
6394:
6382:
6380:
6379:
6374:
6366:
6353:
6347:
6340:
6334:
6332:
6331:
6317:
6311:
6307:
6301:
6299:
6282:
6264:
6245:
6239:
6235:
6229:
6223:
6217:
6211:
6200:
6188:
6161:Stationary state
6064:
6051:
6049:
6048:
6043:
6031:
6029:
6028:
6023:
6018:
6017:
6008:
5996:
5994:
5993:
5988:
5976:
5972:
5966:
5964:
5963:
5958:
5956:
5955:
5939:
5937:
5936:
5931:
5926:
5925:
5916:
5904:
5902:
5901:
5896:
5873:
5872:
5867:
5858:
5857:
5848:
5843:
5842:
5830:
5825:
5824:
5815:
5814:
5804:
5794:
5779:
5778:
5769:
5761:
5756:
5755:
5743:
5742:
5732:
5704:
5687:
5670:
5668:
5667:
5662:
5654:
5653:
5644:
5632:
5621:
5619:
5618:
5613:
5611:
5606:
5605:
5590:
5589:
5580:
5575:
5574:
5564:
5539:
5537:
5536:
5531:
5519:
5517:
5516:
5511:
5489:
5487:
5486:
5481:
5460:density operator
5446:
5444:
5443:
5438:
5436:
5435:
5415:
5413:
5412:
5407:
5405:
5404:
5388:
5386:
5385:
5380:
5378:
5377:
5357:
5355:
5354:
5349:
5347:
5346:
5334:
5333:
5242:
5240:
5239:
5234:
5232:
5228:
5221:
5216:
5215:
5214:
5213:
5198:
5197:
5180:
5177:
5172:
5157:
5146:
5141:
5140:
5126:
5125:
5124:
5123:
5099:
5097:
5096:
5091:
5082:
5077:
5064:
5059:
5047:
5046:
5041:
5035:
5034:
5025:
5017:
5016:
5011:
5005:
5004:
4995:
4983:
4981:
4980:
4975:
4973:
4972:
4971:
4970:
4953:
4952:
4940:
4939:
4928:
4925:
4924:
4923:
4922:
4921:
4904:
4903:
4891:
4890:
4871:
4869:
4868:
4863:
4861:
4860:
4844:
4842:
4841:
4836:
4834:
4833:
4817:
4815:
4814:
4809:
4801:
4796:
4795:
4777:
4772:
4771:
4755:
4753:
4752:
4747:
4739:
4727:
4725:
4724:
4719:
4711:
4681:
4679:
4678:
4673:
4661:
4659:
4658:
4653:
4641:
4639:
4638:
4633:
4621:
4619:
4618:
4613:
4601:
4599:
4598:
4593:
4591:
4573:
4571:
4570:
4565:
4551:
4546:
4545:
4540:
4533:
4532:
4527:
4512:
4501:
4496:
4495:
4479:
4477:
4476:
4471:
4469:
4468:
4463:
4441:
4439:
4438:
4433:
4411:
4409:
4408:
4403:
4395:
4360:
4358:
4357:
4352:
4344:
4339:
4331:
4319:
4314:
4313:
4292:
4280:
4278:
4277:
4272:
4264:
4252:
4250:
4249:
4244:
4239:
4234:
4222:
4220:
4219:
4214:
4206:
4205:
4200:
4191:
4180:
4174:
4169:
4168:
4149:
4147:
4146:
4141:
4139:
4138:
4133:
4124:
4113:
4101:
4099:
4098:
4093:
4091:
4072:
4070:
4069:
4064:
4056:
4038:
4036:
4035:
4030:
4019:
4014:
4000:
3982:
3980:
3979:
3974:
3966:
3950:
3948:
3947:
3942:
3940:
3928:
3926:
3925:
3920:
3915:
3910:
3889:
3887:
3876:
3867:
3865:
3864:
3859:
3851:
3839:
3823:
3821:
3820:
3815:
3810:
3809:
3808:
3798:
3784:
3782:
3781:
3776:
3768:
3767:
3762:
3758:
3757:
3742:
3726:
3724:
3723:
3718:
3701:
3683:
3681:
3680:
3675:
3667:
3648:
3646:
3645:
3640:
3632:
3627:
3626:
3625:
3609:
3608:
3588:
3586:
3585:
3580:
3575:
3574:
3573:
3563:
3547:
3545:
3544:
3539:
3531:
3515:
3506:
3504:
3503:
3498:
3493:
3492:
3491:
3481:
3476:
3475:
3465:
3447:
3435:
3433:
3432:
3427:
3419:
3407:
3405:
3404:
3399:
3394:
3393:
3392:
3382:
3346:
3329:
3318:
3284:
3282:
3281:
3276:
3268:
3267:
3247:
3245:
3244:
3239:
3237:
3236:
3220:
3218:
3217:
3212:
3207:
3206:
3187:
3186:
3160:
3159:
3143:
3142:
3111:
3109:
3108:
3103:
3092:
3091:
3078:
3077:
3072:
3055:
3054:
3041:
3040:
3035:
3023:
2965:
2963:
2962:
2957:
2897:
2889:
2880:
2862:
2853:
2812:angular momentum
2794:
2792:
2790:
2789:
2784:
2779:
2778:
2769:
2764:
2763:
2736:
2734:
2733:
2728:
2720:
2700:
2698:
2697:
2692:
2690:
2672:
2670:
2669:
2664:
2656:
2632:
2630:
2629:
2624:
2622:
2600:
2598:
2597:
2592:
2584:
2554:
2552:
2551:
2546:
2534:
2532:
2531:
2526:
2518:
2500:braâket notation
2483:linear operators
2477:Braâket notation
2471:Braâket notation
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2407:
2388:
2386:
2385:
2380:
2368:
2366:
2365:
2360:
2285:
2283:
2282:
2277:
2272:
2271:
2262:
2248:
2247:
2237:
2210:
2194:
2183:
2168:
2145:entangled states
2134:
2128:
2122:
2116:
2110:
2104:
2098:
2092:
2086:
2080:
2044:
2040:
2029:
2011:
2005:
2003:
2002:
1997:
1995:
1994:
1989:
1967:
1965:
1964:
1959:
1957:
1956:
1940:
1938:
1937:
1932:
1930:
1929:
1913:
1911:
1910:
1905:
1903:
1902:
1873:linear operators
1849:density matrices
1831:
1830:
1826:
1817:
1815:
1814:
1809:
1804:
1803:
1797:
1793:
1781:
1777:
1768:
1767:
1761:
1755:
1751:
1746:
1719:braâket notation
1708:
1706:
1705:
1700:
1661:
1659:
1658:
1653:
1645:
1644:
1635:
1621:
1620:
1610:
1583:
1566:braâket notation
1507:
1494:
1492:
1491:
1486:
1474:
1472:
1471:
1466:
1450:
1448:
1447:
1442:
1440:
1432:
1420:
1418:
1417:
1412:
1410:
1402:
1390:
1388:
1387:
1382:
1371:
1370:
1365:
1356:
1348:
1347:
1342:
1333:
1321:
1319:
1318:
1313:
1281:
1263:
1254:
1245:
1180:
1178:
1177:
1172:
1053:
1051:
1050:
1045:
887:
880:
873:
514:Superdeterminism
167:Braâket notation
118:
116:
115:
110:
102:
97:
96:
88:
76:
71:
69:
58:
30:
21:
8555:
8554:
8550:
8549:
8548:
8546:
8545:
8544:
8530:
8529:
8528:
8523:
8505:
8491:Wigner's friend
8467:
8458:Quantum gravity
8419:
8405:Quantum sensing
8385:Quantum network
8365:Quantum machine
8335:Quantum imaging
8298:Quantum circuit
8293:Quantum channel
8252:
8198:Quantum biology
8184:
8160:ElitzurâVaidman
8135:DavissonâGermer
8118:
8070:Hidden-variable
8060:de BroglieâBohm
8037:Interpretations
8031:
7995:
7949:
7836:Complementarity
7814:
7771:
7766:
7720:
7701:
7681:
7670:
7649:
7636:
7634:Further reading
7631:
7630:
7619:
7583:
7582:
7578:
7565:
7557:
7556:
7552:
7540:
7536:
7529:
7509:
7508:
7504:
7497:
7484:
7483:
7479:
7472:
7459:
7458:
7454:
7446:
7438:Evgeny Lifshitz
7432:
7431:
7424:
7417:
7399:Gottfried, Kurt
7397:
7396:
7389:
7377:
7368:
7349:
7345:
7322:
7318:
7308:
7306:
7297:
7296:
7292:
7282:
7280:
7271:
7270:
7266:
7208:
7207:
7203:
7188:
7172:
7171:
7167:
7160:
7142:
7141:
7137:
7130:
7117:
7116:
7112:
7105:
7087:
7086:
7073:
7063:
7062:
7058:
7051:
7038:
7037:
7010:
7005:
7000:
6999:
6976:
6948:
6934:
6933:
6927:
6925:
6921:
6873:
6872:
6836:
6814:
6796:
6795:
6762:
6738:
6728:
6720:
6719:
6692:
6691:
6662:
6661:
6659:
6655:
6644:
6640:
6632:
6628:′⟩
6627:
6623:
6619:
6618:
6614:
6610:
6598:
6597:
6592:
6590:
6585:
6583:
6579:
6549:
6548:
6503:
6502:
6473:
6472:
6443:
6442:
6415:
6414:
6385:
6384:
6357:
6356:
6354:
6350:
6341:
6337:
6321:
6320:
6318:
6314:
6308:
6304:
6297:
6290:
6284:
6280:
6266:
6262:
6248:
6246:
6242:
6236:
6232:
6224:
6220:
6202:
6191:
6189:
6185:
6180:
6175:
6106:
6082:
6062:
6057:
6034:
6033:
6009:
5999:
5998:
5979:
5978:
5974:
5968:
5947:
5942:
5941:
5917:
5907:
5906:
5862:
5849:
5834:
5816:
5806:
5770:
5747:
5734:
5707:
5706:
5700:
5683:
5645:
5635:
5634:
5631:
5623:
5597:
5581:
5566:
5545:
5544:
5522:
5521:
5496:
5495:
5472:
5471:
5427:
5422:
5421:
5396:
5391:
5390:
5369:
5364:
5363:
5338:
5325:
5320:
5319:
5285:
5279:
5205:
5189:
5181:
5132:
5131:
5127:
5115:
5107:
5102:
5101:
5036:
5026:
5006:
4996:
4986:
4985:
4962:
4954:
4944:
4931:
4913:
4905:
4895:
4882:
4877:
4876:
4852:
4847:
4846:
4825:
4820:
4819:
4787:
4763:
4758:
4757:
4730:
4729:
4702:
4701:
4694:
4688:
4664:
4663:
4644:
4643:
4624:
4623:
4604:
4603:
4576:
4575:
4535:
4522:
4487:
4482:
4481:
4458:
4447:
4446:
4418:
4417:
4386:
4385:
4372:
4366:
4305:
4283:
4282:
4255:
4254:
4225:
4224:
4195:
4160:
4152:
4151:
4128:
4104:
4103:
4082:
4081:
4047:
4046:
3985:
3984:
3957:
3956:
3931:
3930:
3901:
3900:
3885:
3880:
3878:
3874:
3869:
3842:
3841:
3837:
3832:
3800:
3789:
3788:
3749:
3745:
3744:
3729:
3728:
3686:
3685:
3658:
3657:
3617:
3600:
3595:
3594:
3565:
3554:
3553:
3522:
3521:
3518:complex numbers
3513:
3508:
3483:
3467:
3438:
3437:
3436:can be written
3410:
3409:
3384:
3373:
3372:
3357:
3349:tensor products
3342:
3324:
3313:
3259:
3254:
3253:
3248:is the spin of
3228:
3223:
3222:
3198:
3178:
3151:
3134:
3123:
3122:
3119:
3083:
3067:
3046:
3030:
3014:
3013:
2998:
2992:
2900:
2899:
2891:
2885:
2876:
2863:, is either an
2858:
2849:
2828:Planck constant
2808:
2802:
2770:
2755:
2747:
2746:
2745:
2711:
2710:
2705:and entry-wise
2675:
2674:
2647:
2646:
2607:
2606:
2575:
2574:
2537:
2536:
2509:
2508:
2479:
2473:
2429:
2428:
2398:
2397:
2371:
2370:
2351:
2350:
2324:
2318:
2313:
2307:
2297:. Compare with
2263:
2239:
2201:
2200:
2185:
2174:
2164:
2161:
2130:
2124:
2118:
2112:
2106:
2100:
2094:
2088:
2082:
2076:
2042:
2031:
2020:
2007:
1978:
1973:
1972:
1948:
1943:
1942:
1921:
1916:
1915:
1894:
1889:
1888:
1845:complex numbers
1828:
1824:
1823:
1818:which involves
1789:
1785:
1773:
1769:
1736:
1731:
1730:
1709:matrix that is
1685:
1684:
1673:
1636:
1612:
1574:
1573:
1538:linear function
1503:
1477:
1476:
1457:
1456:
1453:absolute values
1423:
1422:
1393:
1392:
1360:
1337:
1324:
1323:
1292:
1291:
1280:
1272:
1259:
1250:
1241:
1227:energy spectrum
1223:quantum numbers
1200:
1151:
1150:
1136:
1130:
1107:representations
1103:
1101:Representations
1071:
1065:
1036:
1035:
1031:
1025:
1001:
981:complex numbers
965:
945:coherent states
899:quantum physics
891:
862:
861:
860:
625:
617:
616:
562:
561:Advanced topics
554:
553:
552:
504:Hidden-variable
494:de BroglieâBohm
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:
8553:
8551:
8543:
8542:
8540:Quantum states
8532:
8531:
8525:
8524:
8522:
8521:
8510:
8507:
8506:
8504:
8503:
8498:
8493:
8488:
8487:
8486:
8475:
8473:
8469:
8468:
8466:
8465:
8460:
8455:
8454:
8453:
8443:
8438:
8436:Casimir effect
8433:
8427:
8425:
8421:
8420:
8418:
8417:
8412:
8407:
8402:
8397:
8395:Quantum optics
8392:
8387:
8382:
8377:
8372:
8367:
8362:
8357:
8352:
8347:
8342:
8337:
8332:
8327:
8322:
8317:
8316:
8315:
8305:
8300:
8295:
8290:
8289:
8288:
8278:
8273:
8268:
8262:
8260:
8254:
8253:
8251:
8250:
8245:
8240:
8235:
8230:
8225:
8220:
8215:
8210:
8205:
8200:
8194:
8192:
8186:
8185:
8183:
8182:
8177:
8172:
8170:Quantum eraser
8167:
8162:
8157:
8152:
8147:
8142:
8137:
8132:
8126:
8124:
8120:
8119:
8117:
8116:
8111:
8106:
8101:
8096:
8091:
8086:
8085:
8084:
8083:
8082:
8067:
8062:
8057:
8052:
8047:
8041:
8039:
8033:
8032:
8030:
8029:
8024:
8019:
8014:
8009:
8003:
8001:
7997:
7996:
7994:
7993:
7988:
7983:
7978:
7973:
7968:
7963:
7957:
7955:
7951:
7950:
7948:
7947:
7946:
7945:
7940:
7930:
7925:
7920:
7915:
7910:
7905:
7900:
7895:
7890:
7885:
7880:
7875:
7870:
7869:
7868:
7863:
7858:
7853:
7843:
7841:Density matrix
7838:
7833:
7828:
7822:
7820:
7816:
7815:
7813:
7812:
7807:
7802:
7797:
7796:
7795:
7785:
7779:
7777:
7773:
7772:
7767:
7765:
7764:
7757:
7750:
7742:
7736:
7735:
7707:
7706:
7703:. 2nd edition.
7699:
7675:
7674:
7668:
7635:
7632:
7629:
7628:
7576:
7550:
7534:
7527:
7502:
7495:
7477:
7470:
7452:
7422:
7415:
7387:
7366:
7343:
7324:Heisenberg, W.
7316:
7290:
7264:
7201:
7186:
7165:
7158:
7135:
7128:
7110:
7103:
7071:
7056:
7049:
7007:
7006:
7004:
7001:
6998:
6997:
6983:
6979:
6975:
6972:
6969:
6966:
6963:
6960:
6955:
6951:
6947:
6944:
6941:
6919:
6907:
6904:
6901:
6898:
6894:
6889:
6884:
6880:
6856:
6852:
6848:
6844:
6840:
6835:
6832:
6829:
6826:
6822:
6818:
6812:
6807:
6803:
6783:
6780:
6777:
6774:
6770:
6766:
6760:
6755:
6751:
6746:
6742:
6735:
6731:
6727:
6707:
6704:
6700:
6679:
6675:
6670:
6653:
6642:
6638:
6625:
6621:
6612:
6608:
6595:
6594:
6588:
6587:
6577:
6564:
6561:
6557:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6515:
6511:
6490:
6486:
6481:
6460:
6456:
6451:
6430:
6427:
6423:
6402:
6398:
6393:
6372:
6369:
6365:
6348:
6335:
6312:
6302:
6295:
6288:
6278:
6260:
6240:
6230:
6218:
6182:
6181:
6179:
6176:
6174:
6173:
6168:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6128:
6123:
6118:
6113:
6107:
6105:
6102:
6081:
6078:
6060:
6041:
6021:
6016:
6012:
6007:
5986:
5954:
5950:
5929:
5924:
5920:
5915:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5871:
5866:
5861:
5856:
5852:
5847:
5841:
5837:
5833:
5829:
5823:
5819:
5813:
5809:
5803:
5799:
5793:
5789:
5785:
5782:
5777:
5773:
5768:
5764:
5760:
5754:
5750:
5746:
5741:
5737:
5731:
5727:
5723:
5720:
5717:
5714:
5660:
5657:
5652:
5648:
5643:
5627:
5610:
5604:
5600:
5596:
5593:
5588:
5584:
5579:
5573:
5569:
5563:
5559:
5555:
5552:
5529:
5509:
5506:
5503:
5479:
5456:density matrix
5450:A mixed state
5434:
5430:
5403:
5399:
5376:
5372:
5345:
5341:
5337:
5332:
5328:
5283:Density matrix
5281:Main article:
5278:
5275:
5231:
5227:
5224:
5220:
5212:
5208:
5204:
5201:
5196:
5192:
5188:
5184:
5176:
5171:
5167:
5163:
5160:
5155:
5152:
5149:
5145:
5139:
5135:
5130:
5122:
5118:
5114:
5110:
5089:
5086:
5081:
5076:
5072:
5068:
5063:
5058:
5054:
5050:
5045:
5040:
5033:
5029:
5024:
5020:
5015:
5010:
5003:
4999:
4994:
4969:
4965:
4961:
4957:
4951:
4947:
4943:
4938:
4934:
4920:
4916:
4912:
4908:
4902:
4898:
4894:
4889:
4885:
4859:
4855:
4832:
4828:
4807:
4804:
4800:
4794:
4790:
4786:
4783:
4780:
4776:
4770:
4766:
4745:
4742:
4738:
4717:
4714:
4710:
4690:Main article:
4687:
4684:
4671:
4651:
4631:
4611:
4590:
4586:
4583:
4563:
4560:
4557:
4554:
4550:
4544:
4539:
4531:
4526:
4521:
4518:
4515:
4511:
4507:
4504:
4500:
4494:
4490:
4467:
4462:
4457:
4454:
4431:
4428:
4425:
4414:if and only if
4401:
4398:
4394:
4365:
4362:
4350:
4347:
4343:
4338:
4334:
4330:
4326:
4323:
4318:
4312:
4308:
4304:
4301:
4298:
4295:
4291:
4270:
4267:
4263:
4242:
4238:
4233:
4212:
4209:
4204:
4199:
4194:
4190:
4186:
4183:
4179:
4173:
4167:
4163:
4159:
4137:
4132:
4127:
4123:
4119:
4116:
4112:
4090:
4062:
4059:
4055:
4028:
4025:
4022:
4018:
4013:
4009:
4006:
4003:
3999:
3995:
3992:
3972:
3969:
3965:
3939:
3918:
3914:
3909:
3896:position basis
3883:
3872:
3857:
3854:
3850:
3835:
3813:
3807:
3803:
3797:
3774:
3771:
3766:
3761:
3756:
3752:
3748:
3741:
3737:
3716:
3713:
3710:
3707:
3704:
3700:
3696:
3693:
3673:
3670:
3666:
3638:
3635:
3631:
3624:
3620:
3615:
3612:
3607:
3603:
3578:
3572:
3568:
3562:
3552:of the states
3537:
3534:
3530:
3511:
3496:
3490:
3486:
3480:
3474:
3470:
3464:
3460:
3456:
3453:
3450:
3446:
3425:
3422:
3418:
3397:
3391:
3387:
3381:
3356:
3353:
3274:
3271:
3266:
3262:
3252:-th particle.
3235:
3231:
3210:
3205:
3201:
3196:
3193:
3190:
3185:
3181:
3176:
3173:
3169:
3166:
3163:
3158:
3154:
3150:
3146:
3141:
3137:
3133:
3130:
3117:
3101:
3098:
3095:
3090:
3086:
3081:
3076:
3071:
3065:
3062:
3058:
3053:
3049:
3044:
3039:
3034:
3029:
3026:
3022:
2991:
2988:
2984:quantum number
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2804:Main article:
2801:
2798:
2797:
2796:
2782:
2777:
2773:
2768:
2762:
2758:
2754:
2738:
2726:
2723:
2719:
2689:
2685:
2682:
2662:
2659:
2655:
2621:
2617:
2614:
2590:
2587:
2583:
2571:
2560:
2544:
2524:
2521:
2517:
2475:Main article:
2472:
2469:
2436:
2416:
2413:
2410:
2406:
2378:
2358:
2328:linear algebra
2317:
2314:
2306:
2303:
2275:
2270:
2266:
2261:
2257:
2254:
2251:
2246:
2242:
2236:
2232:
2228:
2225:
2222:
2219:
2216:
2213:
2209:
2160:
2157:
2153:Bell's theorem
1993:
1988:
1985:
1982:
1955:
1951:
1928:
1924:
1901:
1897:
1807:
1802:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1766:
1758:
1754:
1749:
1745:
1742:
1739:
1698:
1695:
1692:
1672:
1669:
1651:
1648:
1643:
1639:
1634:
1630:
1627:
1624:
1619:
1615:
1609:
1605:
1601:
1598:
1595:
1592:
1589:
1586:
1582:
1484:
1464:
1439:
1435:
1431:
1409:
1405:
1401:
1380:
1377:
1374:
1369:
1364:
1359:
1355:
1351:
1346:
1341:
1336:
1332:
1311:
1308:
1305:
1302:
1299:
1276:
1199:
1196:
1170:
1167:
1164:
1161:
1158:
1139:Wave functions
1132:Main article:
1129:
1126:
1102:
1099:
1064:
1061:
1043:
1027:Main article:
1024:
1021:
1000:
997:
964:
961:
933:
932:
925:
922:wave functions
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:
8552:
8541:
8538:
8537:
8535:
8520:
8512:
8511:
8508:
8502:
8499:
8497:
8494:
8492:
8489:
8485:
8482:
8481:
8480:
8477:
8476:
8474:
8470:
8464:
8461:
8459:
8456:
8452:
8449:
8448:
8447:
8444:
8442:
8439:
8437:
8434:
8432:
8429:
8428:
8426:
8422:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8398:
8396:
8393:
8391:
8388:
8386:
8383:
8381:
8378:
8376:
8373:
8371:
8368:
8366:
8363:
8361:
8358:
8356:
8355:Quantum logic
8353:
8351:
8348:
8346:
8343:
8341:
8338:
8336:
8333:
8331:
8328:
8326:
8323:
8321:
8318:
8314:
8311:
8310:
8309:
8306:
8304:
8301:
8299:
8296:
8294:
8291:
8287:
8284:
8283:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8259:
8255:
8249:
8246:
8244:
8241:
8239:
8236:
8234:
8231:
8229:
8226:
8224:
8221:
8219:
8216:
8214:
8211:
8209:
8208:Quantum chaos
8206:
8204:
8201:
8199:
8196:
8195:
8193:
8191:
8187:
8181:
8178:
8176:
8175:SternâGerlach
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8127:
8125:
8121:
8115:
8112:
8110:
8109:Transactional
8107:
8105:
8102:
8100:
8099:Quantum logic
8097:
8095:
8092:
8090:
8087:
8081:
8078:
8077:
8076:
8073:
8072:
8071:
8068:
8066:
8063:
8061:
8058:
8056:
8053:
8051:
8048:
8046:
8043:
8042:
8040:
8038:
8034:
8028:
8025:
8023:
8020:
8018:
8015:
8013:
8010:
8008:
8005:
8004:
8002:
7998:
7992:
7989:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7958:
7956:
7952:
7944:
7941:
7939:
7936:
7935:
7934:
7933:Wave function
7931:
7929:
7926:
7924:
7921:
7919:
7916:
7914:
7911:
7909:
7908:Superposition
7906:
7904:
7903:Quantum state
7901:
7899:
7896:
7894:
7891:
7889:
7886:
7884:
7881:
7879:
7876:
7874:
7871:
7867:
7864:
7862:
7859:
7857:
7856:Excited state
7854:
7852:
7849:
7848:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7827:
7824:
7823:
7821:
7817:
7811:
7808:
7806:
7803:
7801:
7798:
7794:
7791:
7790:
7789:
7786:
7784:
7781:
7780:
7778:
7774:
7770:
7763:
7758:
7756:
7751:
7749:
7744:
7743:
7740:
7734:
7728:
7724:
7721:Bengtsson I;
7719:
7718:
7717:
7714:
7712:
7702:
7696:
7692:
7688:
7684:
7683:Bratteli, Ola
7680:
7679:
7678:
7671:
7665:
7661:
7656:
7655:
7648:
7647:
7646:
7643:
7641:
7633:
7623:
7615:
7611:
7607:
7603:
7599:
7595:
7591:
7587:
7580:
7577:
7571:
7564:
7560:
7559:Eugene Wigner
7554:
7551:
7547:
7545:
7538:
7535:
7530:
7524:
7520:
7516:
7512:
7506:
7503:
7498:
7492:
7488:
7481:
7478:
7473:
7467:
7463:
7456:
7453:
7445:
7444:
7439:
7435:
7429:
7427:
7423:
7418:
7416:9780387955766
7412:
7408:
7404:
7403:Yan, Tung-Mow
7400:
7394:
7392:
7388:
7384:
7380:
7379:Dirac, P.A.M.
7375:
7373:
7371:
7367:
7363:
7361:
7357:
7352:
7347:
7344:
7340:
7336:
7332:
7329:
7325:
7320:
7317:
7304:
7300:
7294:
7291:
7278:
7274:
7268:
7265:
7260:
7256:
7252:
7248:
7244:
7240:
7236:
7232:
7227:
7222:
7219:(1): 95â102.
7218:
7214:
7213:
7205:
7202:
7197:
7193:
7189:
7187:3-540-42082-7
7183:
7179:
7175:
7169:
7166:
7161:
7155:
7152:. MIT Press.
7151:
7150:
7145:
7139:
7136:
7131:
7129:0-486-26126-3
7125:
7121:
7114:
7111:
7106:
7104:0-7923-2549-4
7100:
7096:
7095:
7090:
7084:
7082:
7080:
7078:
7076:
7072:
7067:
7060:
7057:
7052:
7046:
7042:
7035:
7033:
7031:
7029:
7027:
7025:
7023:
7021:
7019:
7017:
7015:
7013:
7009:
7002:
6981:
6973:
6970:
6967:
6961:
6953:
6949:
6942:
6939:
6930:
6923:
6920:
6905:
6899:
6870:
6846:
6842:
6830:
6827:
6820:
6781:
6778:
6775:
6768:
6744:
6733:
6729:
6725:
6702:
6657:
6654:
6648:
6636:
6606:
6602:
6581:
6578:
6559:
6531:
6528:
6525:
6522:
6519:
6513:
6425:
6367:
6352:
6349:
6345:
6339:
6336:
6329:
6325:
6316:
6313:
6306:
6303:
6294:
6287:
6277:
6273:
6269:
6259:
6255:
6251:
6244:
6241:
6234:
6231:
6228:
6222:
6219:
6215:
6209:
6205:
6198:
6194:
6187:
6184:
6177:
6172:
6169:
6167:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6109:
6108:
6103:
6101:
6099:
6095:
6091:
6087:
6079:
6077:
6075:
6071:
6070:Eugene Wigner
6068:According to
6066:
6063:
6055:
6039:
6014:
6010:
5984:
5971:
5952:
5948:
5922:
5918:
5889:
5886:
5880:
5877:
5874:
5869:
5854:
5850:
5839:
5835:
5821:
5817:
5811:
5807:
5801:
5797:
5791:
5787:
5783:
5775:
5771:
5762:
5752:
5748:
5739:
5735:
5729:
5725:
5721:
5715:
5703:
5698:
5693:
5691:
5686:
5681:
5676:
5674:
5658:
5650:
5646:
5630:
5626:
5602:
5598:
5586:
5582:
5571:
5567:
5561:
5557:
5553:
5550:
5541:
5527:
5507:
5504:
5501:
5493:
5477:
5469:
5465:
5461:
5457:
5453:
5448:
5432:
5428:
5419:
5418:partial trace
5401:
5397:
5374:
5370:
5361:
5343:
5339:
5335:
5330:
5326:
5316:
5313:
5309:
5304:
5302:
5298:
5294:
5290:
5284:
5276:
5274:
5272:
5268:
5263:
5261:
5257:
5253:
5249:
5244:
5229:
5222:
5210:
5206:
5202:
5199:
5194:
5190:
5186:
5182:
5174:
5169:
5165:
5161:
5158:
5153:
5147:
5137:
5133:
5128:
5120:
5116:
5112:
5108:
5087:
5084:
5079:
5074:
5070:
5066:
5061:
5056:
5052:
5048:
5043:
5031:
5027:
5018:
5013:
5001:
4997:
4967:
4963:
4959:
4955:
4949:
4945:
4941:
4936:
4932:
4918:
4914:
4910:
4906:
4900:
4896:
4892:
4887:
4883:
4873:
4857:
4853:
4830:
4826:
4802:
4792:
4788:
4784:
4778:
4768:
4764:
4740:
4712:
4699:
4693:
4685:
4683:
4669:
4649:
4629:
4609:
4584:
4581:
4561:
4558:
4555:
4552:
4542:
4529:
4516:
4513:
4502:
4492:
4488:
4465:
4455:
4452:
4445:
4429:
4426:
4423:
4415:
4396:
4383:
4382:
4377:
4371:
4363:
4361:
4348:
4321:
4310:
4306:
4302:
4299:
4293:
4265:
4210:
4207:
4202:
4181:
4165:
4161:
4157:
4135:
4114:
4079:
4078:
4057:
4044:
4043:
4042:wave function
4026:
4020:
4004:
3990:
3967:
3954:
3898:
3897:
3891:
3886:
3875:
3852:
3838:
3831:
3827:
3805:
3801:
3785:
3772:
3769:
3764:
3759:
3754:
3750:
3746:
3739:
3735:
3714:
3711:
3708:
3702:
3694:
3668:
3655:
3650:
3633:
3622:
3618:
3610:
3605:
3601:
3592:
3570:
3566:
3551:
3532:
3519:
3514:
3488:
3484:
3472:
3468:
3462:
3458:
3454:
3448:
3420:
3389:
3385:
3370:
3366:
3362:
3361:Hilbert space
3354:
3352:
3350:
3345:
3339:
3337:
3333:
3327:
3322:
3316:
3310:
3307:
3303:
3299:
3295:
3291:
3286:
3272:
3269:
3264:
3260:
3251:
3233:
3229:
3203:
3199:
3194:
3191:
3188:
3183:
3179:
3174:
3171:
3167:
3164:
3161:
3156:
3152:
3148:
3144:
3139:
3135:
3131:
3120:
3112:
3099:
3088:
3084:
3079:
3074:
3063:
3060:
3056:
3051:
3047:
3042:
3037:
3024:
3011:
3007:
3003:
2997:
2989:
2987:
2985:
2981:
2977:
2976:
2971:
2966:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2895:
2888:
2884:
2879:
2874:
2870:
2866:
2861:
2857:
2852:
2846:
2842:
2838:
2833:
2829:
2825:
2821:
2817:
2813:
2807:
2799:
2775:
2771:
2760:
2756:
2743:
2739:
2721:
2708:
2704:
2683:
2657:
2644:
2640:
2636:
2615:
2604:
2585:
2572:
2569:
2565:
2561:
2558:
2542:
2519:
2506:
2505:
2504:
2502:
2501:
2496:
2492:
2488:
2484:
2478:
2470:
2468:
2466:
2462:
2458:
2454:
2450:
2434:
2411:
2396:
2392:
2376:
2356:
2348:
2343:
2341:
2337:
2333:
2332:Hilbert space
2329:
2323:
2315:
2312:
2304:
2302:
2300:
2299:Dirac picture
2296:
2292:
2287:
2268:
2252:
2244:
2240:
2234:
2230:
2226:
2217:
2198:
2192:
2188:
2181:
2177:
2172:
2167:
2158:
2156:
2154:
2150:
2146:
2140:
2138:
2133:
2127:
2121:
2115:
2109:
2103:
2097:
2091:
2085:
2079:
2074:
2071:Consider two
2069:
2067:
2063:
2059:
2055:
2050:
2048:
2038:
2034:
2027:
2023:
2018:
2013:
2010:
1991:
1983:
1969:
1953:
1926:
1922:
1899:
1885:
1880:
1878:
1874:
1870:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1833:
1821:
1820:superposition
1805:
1794:
1786:
1782:
1778:
1770:
1756:
1752:
1747:
1743:
1740:
1737:
1728:
1724:
1723:singlet state
1720:
1716:
1712:
1696:
1693:
1690:
1681:
1679:
1670:
1668:
1666:
1649:
1641:
1625:
1617:
1613:
1607:
1603:
1599:
1590:
1571:
1567:
1563:
1558:
1556:
1552:
1548:
1543:
1539:
1535:
1534:
1529:
1525:
1521:
1520:Hilbert space
1518:
1515:
1511:
1506:
1501:
1496:
1482:
1462:
1454:
1433:
1403:
1378:
1375:
1372:
1367:
1357:
1349:
1344:
1334:
1306:
1303:
1300:
1289:
1285:
1279:
1275:
1270:
1267:
1262:
1258:
1253:
1249:
1244:
1240:
1236:
1235:hydrogen atom
1232:
1228:
1224:
1220:
1216:
1208:
1204:
1197:
1195:
1192:
1188:
1184:
1168:
1165:
1162:
1159:
1156:
1148:
1144:
1140:
1135:
1134:Wave function
1127:
1125:
1123:
1119:
1114:
1112:
1108:
1100:
1098:
1096:
1091:
1089:
1085:
1081:
1076:
1070:
1062:
1060:
1058:
1041:
1030:
1022:
1020:
1016:
1014:
1010:
1006:
998:
996:
994:
990:
986:
982:
977:
974:
970:
962:
960:
958:
954:
950:
946:
942:
938:
930:
926:
923:
920:
919:
918:
915:
912:
908:
904:
903:quantum state
900:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
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:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
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:
506:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
476:
472:
467:
466:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
434:
427:
426:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
385:
382:
381:
375:
374:
366:
363:
361:
358:
356:
353:
352:
346:
343:
342:
341:
338:
337:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
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:
211:
208:
206:
203:
201:
198:
197:
190:
189:
181:
178:
176:
173:
172:
168:
165:
163:
160:
158:
155:
154:
148:
147:
142:
139:
137:
134:
132:
129:
128:
127:
126:
122:
89:
83:
66:
63:
59:
51:
44:
43:
40:
36:
32:
31:
19:
8238:Quantum mind
8150:FranckâHertz
8012:KleinâGordon
7961:Formulations
7954:Formulations
7902:
7883:Interference
7873:Entanglement
7851:Ground state
7846:Energy level
7819:Fundamentals
7783:Introduction
7726:
7723:ƻyczkowski K
7715:
7713:at Caltech.
7711:Physics 219
7708:
7693:. Springer.
7690:
7676:
7653:
7644:
7637:
7626:p.8–18
7621:
7589:
7585:
7579:
7569:
7553:
7543:
7537:
7514:
7505:
7486:
7480:
7461:
7455:
7442:
7406:
7382:
7359:
7355:
7346:
7338:
7330:
7327:
7319:
7307:. Retrieved
7303:the original
7293:
7281:. Retrieved
7277:the original
7267:
7216:
7210:
7204:
7177:
7168:
7148:
7138:
7119:
7113:
7092:
7089:Peres, Asher
7065:
7059:
7040:
6928:
6922:
6656:
6646:
6634:
6604:
6600:
6580:
6351:
6338:
6327:
6323:
6315:
6305:
6292:
6285:
6275:
6271:
6267:
6257:
6253:
6249:
6243:
6233:
6221:
6213:
6207:
6203:
6196:
6192:
6186:
6126:Ground state
6116:Bloch sphere
6083:
6067:
6058:
6053:
5969:
5705:is given by
5701:
5694:
5684:
5677:
5628:
5624:
5542:
5492:purification
5467:
5463:
5459:
5455:
5451:
5449:
5317:
5305:
5292:
5288:
5286:
5277:Mixed states
5264:
5255:
5245:
4874:
4695:
4622:at any time
4413:
4379:
4373:
4253:, the state
4076:
4075:probability
4040:
3895:
3892:
3881:
3870:
3833:
3786:
3653:
3651:
3549:
3509:
3359:As with any
3358:
3343:
3340:
3325:
3314:
3311:
3305:
3301:
3287:
3249:
3115:
3113:
3001:
2999:
2974:
2967:
2893:
2886:
2877:
2869:half-integer
2859:
2850:
2831:
2809:
2741:
2602:
2567:
2563:
2557:mathematical
2556:
2535:(where the "
2498:
2480:
2460:
2452:
2448:
2344:
2325:
2288:
2190:
2186:
2179:
2175:
2165:
2162:
2149:entanglement
2144:
2141:
2136:
2131:
2125:
2119:
2113:
2107:
2101:
2095:
2089:
2083:
2077:
2070:
2065:
2061:
2057:
2053:
2051:
2036:
2032:
2025:
2021:
2016:
2014:
2008:
1970:
1881:
1851:, which are
1834:
1682:
1674:
1561:
1559:
1550:
1531:
1504:
1497:
1277:
1273:
1268:
1260:
1251:
1242:
1214:
1212:
1186:
1182:
1137:
1122:vector space
1115:
1106:
1104:
1092:
1083:
1079:
1074:
1072:
1032:
1023:Measurements
1017:
1002:
978:
966:
941:mixed states
934:
916:
902:
896:
442:KleinâGordon
378:Formulations
234:
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
8496:EPR paradox
8276:Quantum bus
8145:Double-slit
8123:Experiments
8089:Many-worlds
8027:Schrödinger
7991:Phase space
7981:Schrödinger
7971:Interaction
7928:Uncertainty
7898:Nonlocality
7893:Measurement
7888:Decoherence
7878:Hamiltonian
7309:January 24,
7283:November 9,
6146:PBR theorem
4444:compact set
4442:there is a
4381:bound state
3830:eigenvalues
3826:eigenstates
3591:orthonormal
2487:dual spaces
2340:unit sphere
1914:. A number
1865:measurement
1547:eigenvector
1542:eigenvalues
1530:called the
1271:-component
1219:eigenstates
1215:pure states
1084:mixed state
911:measurement
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
18:Eigenstates
8424:Extensions
8258:Technology
8104:Relational
8055:Copenhagen
7966:Heisenberg
7913:Tunnelling
7776:Background
7434:Lev Landau
7050:0486409244
7003:References
6641:′ â
6611:′ â
6344:p. 31
6090:C*-algebra
6074:Lev Landau
6054:incoherent
4698:superposed
4480:such that
4416:for every
4368:See also:
3654:normalized
3408:, any ket
2639:dual space
2605:, denoted
2495:Paul Dirac
2320:See also:
2309:See also:
1551:eigenstate
1533:observable
1264:, and the
1080:pure state
1075:eigenstate
943:, or into
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
8130:Bell test
8000:Equations
7826:Born rule
7614:125732617
7546:, page 39
7519:MIT Press
7362:: 580â590
7251:0894-9875
7196:318268606
6974:ρ
6971:
6950:ρ
6943:
6903:∞
6897:⟩
6879:⟨
6847:−
6831:δ
6825:⟩
6802:⟨
6773:⟩
6750:⟨
6726:∫
6706:⟩
6703:ψ
6678:⟩
6647:df′
6563:⟩
6560:ψ
6489:⟩
6459:⟩
6429:⟩
6426:ψ
6401:⟩
6371:⟩
6368:ψ
6020:⟩
6011:ψ
5928:⟩
5919:α
5887:ρ
5881:
5860:⟩
5851:ψ
5836:α
5832:⟨
5798:∑
5788:∑
5781:⟩
5772:ψ
5749:ψ
5745:⟨
5726:∑
5719:⟩
5713:⟨
5673:formalism
5656:⟩
5647:ψ
5599:ψ
5595:⟨
5592:⟩
5583:ψ
5558:∑
5551:ρ
5505:⊗
5360:entangled
5336:⊗
5312:entangled
5260:coherence
5226:⟩
5223:β
5211:α
5207:θ
5200:−
5195:β
5191:θ
5170:α
5162:−
5151:⟩
5148:α
5138:α
5121:α
5117:θ
5075:β
5057:α
5032:β
5002:α
4968:β
4964:θ
4950:β
4937:β
4919:α
4915:θ
4901:α
4888:α
4858:β
4831:α
4806:⟩
4803:β
4793:β
4782:⟩
4779:α
4769:α
4744:⟩
4741:β
4716:⟩
4713:α
4585:∈
4562:ε
4559:−
4553:≥
4503:ϕ
4489:∫
4456:⊂
4424:ε
4400:⟩
4397:ϕ
4346:⟩
4322:ψ
4303:∫
4297:⟩
4294:ψ
4269:⟩
4266:ψ
4241:⟩
4182:ψ
4158:∫
4115:ψ
4061:⟩
4058:ψ
4024:⟩
4021:ψ
4008:⟨
4005:≡
3991:ψ
3971:⟩
3968:ψ
3917:⟩
3856:⟩
3853:ψ
3812:⟩
3736:∑
3706:⟩
3703:ψ
3695:ψ
3692:⟨
3672:⟩
3669:ψ
3637:⟩
3634:ψ
3614:⟨
3577:⟩
3536:⟩
3533:ψ
3495:⟩
3459:∑
3452:⟩
3449:ψ
3424:⟩
3421:ψ
3396:⟩
3334:they are
3265:ν
3234:ν
3204:ν
3189:−
3184:ν
3172:…
3157:ν
3149:−
3140:ν
3132:−
3097:⟩
3061:…
3025:ψ
2942:−
2933:…
2918:−
2909:−
2845:Lie group
2826:) as the
2781:⟩
2772:ψ
2757:ψ
2753:⟨
2725:⟩
2722:ψ
2703:transpose
2684:ψ
2681:⟨
2661:⟩
2658:ψ
2616:ψ
2613:⟨
2589:⟩
2586:ψ
2573:Each ket
2543:ψ
2523:⟩
2520:ψ
2457:half-line
2274:⟩
2265:Φ
2231:∑
2224:⟩
2212:Ψ
2117:and then
1992:σ
1987:⟩
1981:⟨
1950:Φ
1896:Φ
1843:over the
1783:−
1741:ψ
1721:) by the
1711:Hermitian
1694:×
1647:⟩
1638:Φ
1604:∑
1597:⟩
1585:Ψ
1514:separable
1483:β
1463:α
1434:β
1404:α
1358:β
1335:α
1307:β
1301:α
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
8534:Category
8519:Category
8313:Timeline
8065:Ensemble
8045:Bayesian
7938:Collapse
7810:Glossary
7793:Timeline
7725:(2006).
7689:(1987).
7561:(1962).
7513:(2022).
7440:(1965).
7405:(2003).
7381:(1958).
7351:Bohr, N.
7328:Z. Phys.
7259:15995449
7176:(2001).
7091:(1995).
6843:′
6821:′
6769:′
6745:′
6620:⟨
6330:⟩
6322:⟨
6104:See also
5358:with an
4574:for all
3953:spinless
3336:massless
3298:fermions
2832:discrete
2742:brackets
2087:, where
1795:⟩
1779:⟩
1744:⟩
1568:, this
1528:operator
1524:particle
1451:are the
1231:electron
1143:momentum
1013:operator
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
8472:Related
8451:History
8190:Science
8022:Rydberg
7788:History
7594:Bibcode
7231:Bibcode
6591:′
6171:W state
5973:, and "
4077:density
3656:states
3363:, if a
3321:photons
3012:, e.g.
2873:massive
2865:integer
2843:of the
2837:spinors
2635:adjoint
2463:in the
2393:in the
1827:⁄
1678:mixture
1517:complex
1510:vectors
1290:vector
1288:complex
1229:of the
939:versus
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
8165:Popper
7697:
7666:
7612:
7542:Blum,
7525:
7493:
7468:
7413:
7356:Nature
7257:
7249:
7194:
7184:
7156:
7126:
7101:
7047:
6794:i.e.,
6624:|
6326:|
6298:> 0
6238:first.
5905:where
5622:where
5452:cannot
4929:
4926:
3888:|
3879:|
3828:(with
3507:where
3332:vacuum
3294:bosons
3250:ν
3221:where
2970:vector
2881:, its
2347:scalar
1391:where
1255:, the
1246:, the
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
8075:Local
8017:Pauli
8007:Dirac
7610:S2CID
7566:(PDF)
7447:(PDF)
7255:S2CID
7221:arXiv
6413:, so
6310:ket."
6291:>
6178:Notes
6088:on a
5680:trace
5420:over
5295:is a
4700:. If
3365:basis
3317:= 1/2
2459:, or
1839:in a
1715:trace
1512:in a
1233:in a
1118:below
1088:below
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
7695:ISBN
7664:ISBN
7523:ISBN
7491:ISBN
7466:ISBN
7411:ISBN
7311:2012
7285:2021
7247:ISSN
7192:OCLC
7182:ISBN
7154:ISBN
7124:ISBN
7099:ISBN
7045:ISBN
6631:), "
6441:and
6265:and
6201:and
6096:and
5940:and
5458:(or
4845:and
4728:and
4427:>
4281:is:
3824:are
3516:are
3010:spin
3008:and
2810:The
2800:Spin
2566:and
2489:and
2453:line
2336:norm
2081:and
1562:does
1498:The
1475:and
1421:and
1266:spin
973:real
937:pure
901:, a
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
7602:doi
7360:121
7239:doi
6867:(a
6649:= 1
6586:⫠Κ
6214:not
5682:of
5468:and
5303:).
4102:is
3877:is
3328:= 1
2896:+ 1
2709:of
2603:bra
2568:ket
2564:bra
2461:ray
2449:ray
2427:of
2391:ray
2017:all
1837:ray
1455:of
955:in
897:In
8536::
7731:,
7685:;
7662:.
7658:.
7608:.
7600:.
7590:45
7588:.
7521:.
7436:;
7425:^
7401:;
7390:^
7369:^
7331:43
7253:.
7245:.
7237:.
7229:.
7217:19
7215:.
7190:.
7074:^
7011:^
6968:Tr
6940:Tr
6651:".
6645:)
6633:â«
6603:=
6601:dq
6599:*
6270:=
6252:=
6076:.
5975:tr
5878:tr
5540:.
5447:.
5287:A
4682:.
4211:1.
3773:1.
3649:.
3319:,
2467:.
2301:.
2184:,
2075:,
2049:.
1729::
1667:.
1495:.
1090:.
1059:.
959:.
7761:e
7754:t
7747:v
7672:.
7616:.
7604::
7596::
7548:.
7531:.
7499:.
7474:.
7419:.
7364:.
7313:.
7287:.
7261:.
7241::
7233::
7223::
7198:.
7162:.
7132:.
7107:.
7053:.
6982:2
6978:)
6965:(
6962:=
6959:)
6954:2
6946:(
6929:Ï
6906:.
6900:=
6893:r
6888:|
6883:r
6855:)
6851:r
6839:r
6834:(
6828:=
6817:r
6811:|
6806:r
6782:,
6779:1
6776:=
6765:r
6759:|
6754:r
6741:r
6734:3
6730:d
6699:|
6674:r
6669:|
6643:f
6639:f
6637:(
6635:ÎŽ
6626:f
6622:f
6615:)
6613:f
6609:f
6607:(
6605:ÎŽ
6596:f
6593:Κ
6589:f
6556:|
6535:)
6532:z
6529:,
6526:y
6523:,
6520:x
6517:(
6514:=
6510:r
6485:r
6480:|
6455:r
6450:|
6422:|
6397:r
6392:|
6364:|
6328:A
6324:B
6300:.
6296:1
6293:t
6289:2
6286:t
6281:)
6279:2
6276:t
6274:(
6272:P
6268:B
6263:)
6261:1
6258:t
6256:(
6254:Q
6250:A
6210:)
6208:t
6206:(
6204:P
6199:)
6197:t
6195:(
6193:Q
6061:s
6059:p
6040:s
6015:s
6006:|
5985:i
5970:A
5953:i
5949:a
5923:i
5914:|
5893:)
5890:A
5884:(
5875:=
5870:2
5865:|
5855:s
5846:|
5840:i
5828:|
5822:i
5818:a
5812:s
5808:p
5802:i
5792:s
5784:=
5776:s
5767:|
5763:A
5759:|
5753:s
5740:s
5736:p
5730:s
5722:=
5716:A
5702:A
5685:Ï
5659:.
5651:s
5642:|
5629:s
5625:p
5609:|
5603:s
5587:s
5578:|
5572:s
5568:p
5562:s
5554:=
5528:K
5508:K
5502:H
5478:H
5464:Ï
5433:2
5429:H
5402:1
5398:H
5375:2
5371:H
5344:2
5340:H
5331:1
5327:H
5230:)
5219:|
5203:i
5187:i
5183:e
5175:2
5166:A
5159:1
5154:+
5144:|
5134:A
5129:(
5113:i
5109:e
5088:1
5085:=
5080:2
5071:A
5067:+
5062:2
5053:A
5049:=
5044:2
5039:|
5028:c
5023:|
5019:+
5014:2
5009:|
4998:c
4993:|
4960:i
4956:e
4946:A
4942:=
4933:c
4911:i
4907:e
4897:A
4893:=
4884:c
4854:c
4827:c
4799:|
4789:c
4785:+
4775:|
4765:c
4737:|
4709:|
4670:K
4650:1
4630:t
4610:K
4589:R
4582:t
4556:1
4549:r
4543:3
4538:d
4530:2
4525:|
4520:)
4517:t
4514:,
4510:r
4506:(
4499:|
4493:K
4466:3
4461:R
4453:K
4430:0
4393:|
4349:.
4342:r
4337:|
4333:)
4329:r
4325:(
4317:r
4311:3
4307:d
4300:=
4290:|
4262:|
4237:r
4232:|
4208:=
4203:2
4198:|
4193:)
4189:r
4185:(
4178:|
4172:r
4166:3
4162:d
4136:2
4131:|
4126:)
4122:r
4118:(
4111:|
4089:r
4054:|
4027:.
4017:|
4012:r
4002:)
3998:r
3994:(
3964:|
3938:r
3913:r
3908:|
3884:i
3882:c
3873:i
3871:k
3849:|
3836:i
3834:k
3806:i
3802:k
3796:|
3770:=
3765:2
3760:|
3755:i
3751:c
3747:|
3740:i
3715:,
3712:1
3709:=
3699:|
3665:|
3630:|
3623:i
3619:k
3611:=
3606:i
3602:c
3571:i
3567:k
3561:|
3529:|
3512:i
3510:c
3489:i
3485:k
3479:|
3473:i
3469:c
3463:i
3455:=
3445:|
3417:|
3390:i
3386:k
3380:|
3344:N
3326:S
3315:S
3306:N
3302:N
3273:0
3270:=
3261:S
3230:S
3209:}
3200:S
3195:,
3192:1
3180:S
3175:,
3168:,
3165:1
3162:+
3153:S
3145:,
3136:S
3129:{
3118:Μ
3116:m
3100:.
3094:)
3089:N
3085:m
3080:,
3075:N
3070:r
3064:;
3057:;
3052:1
3048:m
3043:,
3038:1
3033:r
3028:(
3021:|
3002:N
2975:C
2954:}
2951:S
2948:,
2945:1
2939:S
2936:,
2930:,
2927:1
2924:+
2921:S
2915:,
2912:S
2906:{
2894:S
2892:2
2887:m
2878:S
2860:ħ
2851:S
2824:T
2822:·
2820:L
2818:·
2816:M
2793:.
2776:2
2767:|
2761:1
2737:.
2718:|
2688:|
2654:|
2620:|
2582:|
2516:|
2435:H
2415:)
2412:H
2409:(
2405:P
2377:H
2357:H
2269:n
2260:|
2256:)
2253:t
2250:(
2245:n
2241:C
2235:n
2227:=
2221:)
2218:t
2215:(
2208:|
2193:)
2191:t
2189:(
2187:Q
2182:)
2180:t
2178:(
2176:P
2166:Ï
2132:B
2126:A
2120:B
2114:A
2108:B
2102:B
2096:B
2090:A
2084:B
2078:A
2066:A
2062:A
2058:A
2043:t
2039:)
2037:t
2035:(
2033:P
2028:)
2026:t
2024:(
2022:Q
2009:A
1984:A
1954:n
1927:n
1923:P
1900:n
1829:2
1825:1
1806:,
1801:)
1787:|
1771:|
1765:(
1757:2
1753:1
1748:=
1738:|
1697:2
1691:2
1650:.
1642:n
1633:|
1629:)
1626:t
1623:(
1618:n
1614:C
1608:n
1600:=
1594:)
1591:t
1588:(
1581:|
1505:t
1438:|
1430:|
1408:|
1400:|
1379:,
1376:1
1373:=
1368:2
1363:|
1354:|
1350:+
1345:2
1340:|
1331:|
1310:)
1304:,
1298:(
1278:z
1274:s
1269:z
1261:m
1252:â
1243:n
1169:z
1166:,
1163:y
1160:,
1157:x
1042:x
931:.
886:e
879:t
872:v
100:|
90:H
84:=
74:|
67:t
64:d
60:d
52:i
20:)
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