4991:
3264:). In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos. For example, this is evident in the perturbation-induced quantum scarring: More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart.
4983:
6340:
3287:) dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the
2456:(energy levels), one can use standard quantum mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.
900:
1624:) of diamagnetic hydrogen showing peaks corresponding to periodic orbits of the classical system. Spectrum is at a scaled energy of −0.6. Peaks labeled R and V are repetitions of the closed orbit perpendicular and parallel to the field, respectively. Peaks labeled O correspond to the near circular periodic orbit that goes around the nucleus.
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and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration.
1641:
of action quantization, which applies only to integrable or near-integrable systems and computes individual eigenvalues from each trajectory, periodic-orbit theory is applicable to both integrable and non-integrable systems and asserts that each periodic orbit produces a sinusoidal fluctuation in the
2624:
Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified
2468:
Realize that for caustics the description diverges and use the insight by Maslov (approximately
Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however
2383:
Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits,
1594:
Many
Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a
1488:
In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic systems have been shown to be in good agreement with the
2455:
The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states
1417:
A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees
1327:
Other approaches have been developed in recent years. One is to express the
Hamiltonian in different coordinate systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions, and eigenvalues are obtained by
1323:
Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible) analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are regular classical solutions of the same
Hamiltonian, then there are
3255:
The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic
Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill
1383:
A given
Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a
950:
of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential
1331:
Another approach is numerical matrix diagonalization. If the
Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and still obtain accurate
1413:
theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown
Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly
2603:
Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a
1379:
is the dimension of the matrix, so it is important to choose the smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a basis in which the matrix is sparse and/or the matrix elements are given by simple algebraic expressions because
1400:
energy level spectra in an electric field as quantum defect is increased from 0.04 (a) to 0.32 (h). The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a
Poisson distribution (a) to that of
1422:
and Tabor have put forward strong arguments for a
Poisson distribution in the case of regular motion and Heusler et al. present a semiclassical explanation of the so-called Bohigas–Giannoni–Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The
1384:
Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos.
1599:, thus supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos, demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for the odd-parity energy level distribution.
1833:
1319:
is a parameter which cannot be considered small. Physicists have historically approached problems of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then treating the non-separable Hamiltonian as a perturbation.
1426:
Qualitative observations of level repulsions can be quantified and related to the classical dynamics using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a
2804:
2472:
Transform the Greens function to energy space to get the energy dependent Greens function (again approximate Fourier transform using the stationary phase approximation). New divergences might pop up that need to be cured using the same method as step
3260:). However, a quantum eigenstate of a classically chaotic system can be scarred: the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit (
1632:
Relative recurrence amplitudes of even and odd recurrences of the near circular orbit. Diamonds and plus signs are for odd and even quarter periods, respectively. Solid line is A/cosh(nX/8). Dashed line is A/sinh(nX/8) where A = 14.75 and X =
1489:
predictions of the Gaussian orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all chaotic systems with this symmetry. If the normalized spacing between two energy levels is
3180:
2628:
Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For
2897:). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as
2568:
1018:), but not well understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.
119:
2985:
Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the
1414:
predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.
1098:, dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in
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1651:
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1149:
energy level spectra of lithium in an electric field near n=15. Note that energy levels cannot cross due to the ionic core (and resulting quantum defect) breaking symmetries of dynamical motion.
2084:, represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity
3348:
is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting
1210:
1110:
of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular
1645:
The principal result of this development is an expression for the density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula:
1332:
results. These techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. The computational time required to diagonalize a matrix scales as
1091:. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics.
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in an electric field. The peaks labeled 1–5 are repetitions of the electron orbit parallel to the field going from the nucleus to the classical turning point in the uphill direction.
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2370:
1970:
of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices
1121:). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.
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2018:
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and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact
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1948:
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Courtney, Michael; Jiao, Hong; Spellmeyer, Neal; Kleppner, Daniel; Gao, J.; Delos, J. B. (February 1995). "Closed Orbit Bifurcations in Continuum Stark Spectra".
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can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and
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1988:
1968:
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and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.
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4696:
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For conservative systems, the goal of quantum mechanics in non-perturbative regimes is to find the eigenvalues and eigenvectors of a Hamiltonian of the form
3522:
Courtney, Michael; Spellmeyer, Neal; Jiao, Hong; Kleppner, Daniel (May 1995). "Classical, semiclassical, and quantum dynamics in the lithium Stark system".
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by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the
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1638:
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Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g.
3449:
3368:, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely
1137:
energy level spectra of hydrogen in an electric field near n=15. Note that energy levels can cross due to underlying symmetries of dynamical motion.
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5619:
593:
137:
49:
959:
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Heusler, Stefan; MĂĽller, Sebastian; Altland, Alexander; Braun, Petr; Haake, Fritz (January 2007). "Periodic-Orbit Theory of Level Correlations".
2452:) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies.
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is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to
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1423:
nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos.
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Experimental recurrence spectrum (circles) is compared with the results of the closed orbit theory of John Delos and Jing Gao for lithium
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5223:
5265:
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2570:(tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states
132:
1828:{\displaystyle g_{c}(E)=\sum _{k}T_{k}\sum _{n=1}^{\infty }{\frac {1}{2\sinh {(\chi _{nk}/2)}}}\,e^{i(nS_{k}-\alpha _{nk}\pi /2)}.}
951:
sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?
5702:
5140:
3056:
the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by
221:
5984:
5979:
328:
308:
6275:
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2314:
is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because
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ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.
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Experimental recurrence spectra of lithium in an electric field showing birth of quantum recurrences corresponding to
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Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the
495:
5285:
2376:. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo-
1030:
Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of
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872:
524:
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5752:
3573:
Yan, Bin; Sinitsyn, Nikolai A. (2020). "Recovery of Damaged Information and the Out-of-Time-Ordered Correlators".
2799:{\displaystyle f(w)=\sum _{k}\sum _{n=1}^{\infty }D_{\it {nk}}^{i}\sin(2\pi nw{\tilde {S_{k}}}-\phi _{\it {nk}}).}
1409:
Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems.
1324:(at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them.
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such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
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3791:"Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems"
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613:
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2027:
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Start with the semiclassical approximation of the time-dependent Green's function (the Van Vleck propagator).
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Questions related to the correspondence principle arise in many different branches of physics, ranging from
965:
Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same
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Luukko, Perttu J. J.; Drury, Byron; Klales, Anna; Kaplan, Lev; Heller, Eric J.; Räsänen, Esa (2016-11-28).
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for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such
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quantum systems far from semi-classical regimes as well as a large effort in quantum chaotic scattering.
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Development of methods for solving quantum problems where the perturbation cannot be considered small in
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During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the
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5150:
4723:
4591:. Springer series in synergetics (2nd rev. and enlarged ed.). Berlin Heidelberg Paris : Springer.
4384:
Barba, J.C.; et al. (2008). "The Berry–Tabor conjecture for spin chains of Haldane–Shastry type".
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168:
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Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a
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One open question remains understanding quantum chaos in systems that have finite-dimensional local
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Quantum chaos Y2K: proceedings of Nobel Symposium 116, Bäckaskog Castle, Sweden, June 13 - 17, 2000
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is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action
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908:
672:
480:
398:
226:
206:
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4548:(1951). "On the statistical distribution of the widths and spacings of nuclear resonance levels".
1993:
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In seeking to address the basic question of quantum chaos, several approaches have been employed:
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3958:"Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits"
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3859:
3828:
3802:
3738:
Courtney, Michael; Kleppner, Daniel (January 1996). "Core-induced chaos in diamagnetic lithium".
3686:
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3582:
3410:. Springer series in synergetics (2nd rev. and enl. ed.). Berlin ; New York: Springer.
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Important observations often associated with classically chaotic quantum systems are spectral
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Keski-Rahkonen, J.; Luukko, P. J. J.; Kaplan, L.; Heller, E. J.; Räsänen, E. (2017-09-20).
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27:
Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory
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3924:
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5290:
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5175:
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5022:
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4812:
4292:
Doron, Cohen (2004). "Driven chaotic mesoscopic systems, dissipation and decoherence".
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3957:
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3261:
3175:{\displaystyle {\frac {d^{1/2}}{dx^{1/2}}}V^{-1}(x)=2{\sqrt {\pi }}{\frac {dN(x)}{dx}}}
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2880:
2297:
1973:
1953:
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448:
438:
231:
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4469:
Martin C. Gutzwiller (1971). "Periodic Orbits and Classical Quantization Conditions".
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4136:
3790:
3298:
Works are also focused in the study of driven chaotic systems, where the Hamiltonian
2636:
According to closed-orbit theory, the average oscillator strength density at constant
2604:
coordinate system parallel to the direction of movement, as it is done in many books.
2426:) semiclassically. He found agreement with quantum computations for low lying (up to
17:
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the available phase space evenly, up to random fluctuations and energy conservation (
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1410:
1402:
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777:
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2563:{\displaystyle d(E)=-{\frac {1}{\pi }}\Im (\operatorname {Tr} (G(x,x^{\prime },E))}
2021:
1397:
1146:
1134:
1118:
1103:
923:
842:
837:
772:
757:
722:
216:
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The transition to chaos: conservative classical systems and quantum manifestations
3708:
3634:
Berry, M. V.; Tabor, M. (1977-09-15). "Level clustering in the regular spectrum".
3228:
is the density of states and V(x) is the classical potential of the particle, the
1388:
Correlating statistical descriptions of quantum mechanics with classical behaviour
4632:. Institute for nonlinear science (2. ed.). New York Heidelberg: Springer.
4366:
3636:
Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
2877:
is the recurrence amplitude of a closed orbit for a given initial state (labeled
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5338:
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2388:
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in phase space, and neighboring trajectories wind around it. For stable orbits,
1839:
807:
762:
697:
652:
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1102:). In the semiclassical approach of quantum chaos, phenomena are identified in
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3847:
3232:
of the inverse of the potential is related to the density of states as in the
1842:-like term stemming from spin or other internal degrees of freedom. The index
1099:
797:
767:
687:
662:
657:
642:
4610:. Stockholm, Sweden: Physica Scripta, the Royal Swedish Academy of Sciences.
4348:
4235:
4103:
3942:
3767:
3716:
3663:
3551:
3543:
3495:
2840:
is a phase that depends on the Maslov index and other details of the orbits.
114:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
5954:
5635:
5433:
5393:
5135:
4797:
4782:
4266:
3759:
1111:
1084:
1080:
288:
4243:
4193:
4121:
4038:
3724:
3655:
3612:
3503:
903:
Quantum chaos is the field of physics attempting to bridge the theories of
899:
3995:
Keski-Rahkonen, J.; Ruhanen, A.; Heller, E. J.; Räsänen, E. (2019-11-21).
3775:
3559:
2625:
initial state whereas periodic-orbit theory yields the density of states.
5170:
4298:
667:
3447:
http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf
4840:
4792:
919:
4492:
4095:
3848:"Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory"
2114:
characterizes the instability of the orbit. A stable orbit moves on a
1628:
1612:
5413:
4192:
Keski-Rahkonen, J; Luukko, P J J; Åberg, S; Räsänen, E (2019-01-21).
3691:
2423:
2391:
1088:
5547:
4504:. Interdisciplinary applied mathematics. New York: Springer-Verlag.
2656:
is given by a smooth background plus an oscillatory sum of the form
1026:
998:
942:
of quantum mechanics, specifically in the limit as the ratio of the
4210:
4153:
4078:
4013:
3915:
3864:
3807:
3587:
3021:
For the case of one-dimensional system with the boundary condition
2612:
1866:: the shortest period orbits of a given set of initial conditions.
1141:
1129:
4398:
4194:"Effects of scarring on quantum chaos in disordered quantum wells"
2611:
2115:
1627:
1611:
1509:, the normalized distribution of spacings is well approximated by
1392:
1391:
1140:
1128:
1025:
997:
898:
972:
Study of probability distribution of individual eigenstates (see
5551:
4732:
4728:
4551:
Mathematical Proceedings of the Cambridge Philosophical Society
1380:
computing matrix elements can also be a computational burden.
2460:
Rough sketch on how to arrive at the Gutzwiller trace formula
2961:
2958:
2856:
2853:
2824:
2821:
2784:
2781:
2721:
2718:
2543:
911:. The figure shows the main ideas running in each direction.
4268:
Quantum Chaos in Disordered Two-Dimensional Nanostructures
4137:"Controllable quantum scars in semiconductor quantum dots"
3897:
Chan, Amos; De Luca, Andrea; Chalker, J. T. (2018-11-08).
3846:
Kos, Pavel; Ljubotina, Marko; Prosen, TomaĹľ (2018-06-08).
3899:"Solution of a Minimal Model for Many-Body Quantum Chaos"
1584:{\displaystyle P(s)={\frac {\pi }{2}}se^{-\pi s^{2}/4}.}
3789:
Vogl, M.; Pankratov, O.; Shallcross, S. (2017-07-27).
3304:
3273:
3188:
3062:
3027:
2992:
2951:
2931:
2911:
2905:
of an oscillator strength spectrum computed at fixed
2883:
2846:
2814:
2665:
2642:
2576:
2482:
2432:
2400:
2387:
Gutzwiller applied the trace formula to approach the
2320:
2300:
2258:
2228:
2176:
2124:
2090:
2030:
1996:
1976:
1956:
1926:
1899:
1872:
1848:
1654:
1518:
1495:
1440:
1365:
1338:
1305:
1278:
1248:
1221:
1162:
1036:
52:
6296:
6248:
6081:
6013:
5947:
5860:
5809:
5763:
5628:
5585:
5507:
5324:
5251:
5189:
5059:
5046:
4998:
4929:
4773:
4766:
4606:Berggren, Karl-Fredrik; °Aberg, Sven, eds. (2001).
1272:is non-separable in the coordinate system in which
989:
Direct application of the correspondence principle.
4708:Volume 91, Number 4, July–August, 2003 pp. 296–300
3340:
3279:
3220:
3174:
3048:
3005:
2974:
2937:
2917:
2889:
2869:
2832:
2798:
2648:
2591:
2562:
2444:
2414:
2364:
2306:
2286:
2244:
2214:
2162:
2106:
2076:
2012:
1982:
1962:
1942:
1912:
1893:is the period of the primitive periodic orbit and
1885:
1854:
1827:
1583:
1501:
1477:
1371:
1351:
1311:
1291:
1264:
1234:
1204:
1051:
113:
2252:is the winding number of the periodic orbit.
5563:
4744:
4062:"Strong quantum scarring by local impurities"
3517:
3515:
3513:
1205:{\displaystyle H=H_{s}+\varepsilon H_{ns},\,}
1125:Quantum mechanics in non-perturbative regimes
880:
8:
4697:Notices of the American Mathematical Society
108:
82:
4704:Brian Hayes, "The Spectrum of Riemannium";
1950:and a period which is an integral multiple
5570:
5556:
5548:
5056:
4770:
4751:
4737:
4729:
887:
873:
31:
4718:Eigenfunctions in chaotic quantum systems
4526:. Cambridge: Cambridge university press.
4397:
4338:
4297:
4209:
4152:
4111:
4077:
4012:
3932:
3914:
3881:
3863:
3806:
3690:
3586:
3303:
3295:are considered to be prototype problems.
3272:
3189:
3187:
3143:
3136:
3112:
3095:
3091:
3073:
3069:
3063:
3061:
3026:
2997:
2991:
2966:
2957:
2956:
2950:
2930:
2910:
2882:
2861:
2852:
2851:
2845:
2820:
2819:
2813:
2780:
2779:
2760:
2754:
2753:
2726:
2717:
2716:
2706:
2695:
2685:
2664:
2641:
2575:
2542:
2501:
2481:
2431:
2404:
2399:
2344:
2335:
2327:
2319:
2299:
2263:
2257:
2233:
2227:
2200:
2191:
2183:
2175:
2148:
2139:
2131:
2123:
2095:
2089:
2062:
2053:
2045:
2034:
2029:
2001:
1995:
1975:
1955:
1934:
1925:
1904:
1898:
1877:
1871:
1847:
1809:
1797:
1784:
1770:
1765:
1750:
1741:
1733:
1718:
1712:
1701:
1691:
1681:
1659:
1653:
1568:
1562:
1551:
1534:
1517:
1494:
1460:
1439:
1364:
1343:
1337:
1304:
1283:
1277:
1253:
1247:
1226:
1220:
1201:
1189:
1173:
1161:
1035:
100:
89:
88:
74:
59:
51:
4502:Chaos in classical and quantum mechanics
3267:Further studies concern the parametric (
2077:{\displaystyle 1/\sinh {(\chi _{nk}/2)}}
1242:is separable in some coordinate system,
3398:
2365:{\displaystyle \sin {(\chi _{nk}/2)}=0}
938:states that classical mechanics is the
56:
39:
2163:{\displaystyle \sinh {(\chi _{nk}/2)}}
3017:One-dimensional systems and potential
2215:{\displaystyle \sin {(\chi _{nk}/2)}}
7:
4198:Journal of Physics: Condensed Matter
962:and where quantum numbers are large.
4891:Measure-preserving dynamical system
4683:Category:Quantum Chaos Scholarpedia
3221:{\displaystyle {\frac {dN(x)}{dx}}}
2422:potential with an anisotropic mass
1396:Nearest neighbour distribution for
3435:Quantum: a guide for the perplexed
3433:, "Quantum Chaology", pp 104-5 of
2707:
2511:
1713:
419:Sum-over-histories (path integral)
105:
79:
35:Part of a series of articles about
25:
5459:Oleksandr Mykolayovych Sharkovsky
2287:{\displaystyle \chi _{nk}=2\pi m}
1616:Even parity recurrence spectrum (
6339:
6338:
4989:
4981:
2975:{\displaystyle D_{\it {nk}}^{i}}
2870:{\displaystyle D_{\it {nk}}^{i}}
2833:{\displaystyle \phi _{\it {nk}}}
2394:problem (a single particle in a
1639:Einstein–Brillouin–Keller method
4472:Journal of Mathematical Physics
4265:Keski-Rahkonen, Joonas (2020).
1133:Computed regular (non-chaotic)
6288:Relativistic quantum mechanics
5224:Rabinovich–Fabrikant equations
4524:Quantum chaos: An introduction
4031:10.1103/PhysRevLett.123.214101
3956:Heller, Eric J. (1984-10-15).
3605:10.1103/PhysRevLett.125.040605
3335:
3332:
3326:
3308:
3204:
3198:
3158:
3152:
3127:
3121:
3037:
3031:
2790:
2766:
2738:
2675:
2669:
2586:
2580:
2557:
2554:
2529:
2523:
2514:
2492:
2486:
2352:
2328:
2208:
2184:
2156:
2132:
2070:
2046:
1817:
1774:
1758:
1734:
1671:
1665:
1528:
1522:
1478:{\displaystyle P(s)=e^{-s}.\ }
1450:
1444:
1328:matching boundary conditions.
1052:{\displaystyle \epsilon =-3.0}
569:Relativistic quantum mechanics
101:
94:
75:
1:
6266:Quantum statistical mechanics
6043:Quantum differential calculus
5965:Delayed-choice quantum eraser
5733:Symmetry in quantum mechanics
4678:doi:10.4249/scholarpedia.3146
3709:10.1103/PhysRevLett.98.044103
609:Quantum statistical mechanics
4710:. Discusses relation to the
4315:"Quantum chaotic scattering"
2013:{\displaystyle \alpha _{nk}}
1862:distinguishes the primitive
6068:Quantum stochastic calculus
6058:Quantum measurement problem
5980:Mach–Zehnder interferometer
4959:Poincaré recurrence theorem
4589:Quantum signatures of chaos
3982:10.1103/PhysRevLett.53.1515
3488:10.1103/PhysRevLett.74.1538
3408:Quantum signatures of chaos
3341:{\displaystyle H(x,p;R(t))}
579:Quantum information science
6391:
4954:Poincaré–Bendixson theorem
4500:Gutzwiller, M. C. (1990).
4416:10.1209/0295-5075/83/27005
4368:The Berry–Tabor conjecture
4171:10.1103/PhysRevB.96.094204
3825:10.1103/PhysRevB.96.035442
2245:{\displaystyle \chi _{nk}}
2107:{\displaystyle \chi _{nk}}
6334:
6128:Quantum complexity theory
6106:Quantum cellular automata
5796:Path integral formulation
5306:Swinging Atwood's machine
4979:
4949:Krylov–Bogolyubov theorem
4826:
4572:10.1017/S0305004100027237
4340:10.4249/scholarpedia.9806
3997:"Quantum Lissajous Scars"
3934:10.1103/PhysRevX.8.041019
3883:10.1103/PhysRevX.8.021062
3250:quantum many-body systems
2918:{\displaystyle \epsilon }
2649:{\displaystyle \epsilon }
1597:Wigner-Dyson distribution
1312:{\displaystyle \epsilon }
6195:Quantum machine learning
6175:Quantum key distribution
6165:Quantum image processing
6155:Quantum error correction
6005:Wheeler's delayed choice
5214:Lotka–Volterra equations
5038:Synchronization of chaos
4841:axiom A dynamical system
4688:What is... Quantum Chaos
4439:"What Is Quantum Chaos?"
4437:Rudnick, Z. (Jan 2008).
4313:Gaspard, Pierre (2014).
4228:10.1088/1361-648x/aaf9fb
3544:10.1103/PhysRevA.51.3604
2024:. The amplitude factor,
936:correspondence principle
614:Quantum machine learning
367:Wheeler's delayed-choice
6111:Quantum finite automata
5199:Double scroll attractor
4964:Stable manifold theorem
4871:False nearest neighbors
4001:Physical Review Letters
3962:Physical Review Letters
3760:10.1103/PhysRevA.53.178
3679:Physical Review Letters
3575:Physical Review Letters
3468:Physical Review Letters
3443:Weidenfeld and Nicolson
324:Leggett–Garg inequality
6215:Quantum neural network
5239:Van der Pol oscillator
5219:Mackey–Glass equations
4851:Box-counting dimension
4520:Stöckmann, Hans-Jürgen
4271:. Tampere University.
3656:10.1098/rspa.1977.0140
3356:Berry–Tabor conjecture
3342:
3281:
3222:
3176:
3050:
3049:{\displaystyle y(0)=0}
3007:
2976:
2939:
2919:
2901:in strong fields, the
2891:
2871:
2834:
2800:
2711:
2650:
2621:
2593:
2564:
2469:gives a phase factor).
2446:
2416:
2366:
2308:
2288:
2246:
2216:
2164:
2108:
2078:
2014:
1984:
1964:
1944:
1943:{\displaystyle nS_{k}}
1914:
1887:
1856:
1829:
1717:
1634:
1625:
1585:
1503:
1479:
1406:
1373:
1353:
1313:
1293:
1266:
1265:{\displaystyle H_{ns}}
1236:
1206:
1150:
1138:
1060:
1053:
1007:
912:
115:
18:Berry–Tabor conjecture
6240:Quantum teleportation
5753:Wave–particle duality
5389:Svetlana Jitomirskaya
5296:Multiscroll attractor
5141:Interval exchange map
5094:Dyadic transformation
5079:Complex quadratic map
4921:Topological conjugacy
4856:Correlation dimension
4831:Anosov diffeomorphism
4712:Riemann zeta function
4587:Haake, Fritz (2001).
3406:Haake, Fritz (2001).
3387:Statistical mechanics
3343:
3282:
3223:
3177:
3051:
3008:
3006:{\displaystyle H^{-}}
2977:
2940:
2920:
2892:
2872:
2835:
2801:
2691:
2651:
2615:
2594:
2565:
2447:
2417:
2367:
2309:
2289:
2247:
2217:
2165:
2109:
2079:
2015:
1985:
1965:
1945:
1915:
1913:{\displaystyle S_{k}}
1888:
1886:{\displaystyle T_{k}}
1857:
1830:
1697:
1631:
1615:
1608:Periodic orbit theory
1603:Semiclassical methods
1586:
1504:
1480:
1395:
1374:
1354:
1352:{\displaystyle N^{3}}
1314:
1294:
1292:{\displaystyle H_{s}}
1267:
1237:
1235:{\displaystyle H_{s}}
1207:
1144:
1132:
1054:
1029:
1001:
984:Semiclassical methods
902:
309:Elitzur–Vaidman
299:Davisson–Germer
116:
6375:Quantum chaos theory
6271:Quantum field theory
6200:Quantum metamaterial
6145:Quantum cryptography
5875:Consistent histories
5399:Edward Norton Lorenz
3302:
3271:
3186:
3060:
3025:
2990:
2949:
2929:
2909:
2881:
2844:
2812:
2663:
2640:
2592:{\displaystyle d(E)}
2574:
2480:
2430:
2398:
2318:
2298:
2256:
2226:
2174:
2122:
2088:
2028:
1994:
1974:
1954:
1924:
1897:
1870:
1846:
1652:
1516:
1493:
1438:
1429:Poisson distribution
1363:
1336:
1303:
1276:
1246:
1219:
1160:
1034:
1006:of classical orbits.
574:Quantum field theory
486:Consistent histories
123:Schrödinger equation
50:
6256:Quantum fluctuation
6225:Quantum programming
6185:Quantum logic gates
6170:Quantum information
6150:Quantum electronics
5610:Classical mechanics
5359:Mitchell Feigenbaum
5301:Population dynamics
5286:Hénon–Heiles system
5146:Irrational rotation
5099:Dynamical billiards
5084:Coupled map lattice
4944:Liouville's theorem
4876:Hausdorff dimension
4861:Conservative system
4846:Bifurcation diagram
4662:Scientific American
4564:1951PCPS...47..790W
4485:1971JMP....12..343G
4408:2008EL.....8327005B
4331:2014SchpJ...9.9806G
4220:2019JPCM...31j5301K
4163:2017PhRvB..96i4204K
4088:2016NatSR...637656L
4023:2019PhRvL.123u4101K
3974:1984PhRvL..53.1515H
3925:2018PhRvX...8d1019C
3874:2018PhRvX...8b1062K
3817:2017PhRvB..96c5442V
3752:1996PhRvA..53..178C
3701:2007PhRvL..98d4103H
3648:1977RSPSA.356..375B
3597:2020PhRvL.125d0605Y
3536:1995PhRvA..51.3604C
3480:1995PhRvL..74.1538C
3234:Wu–Sprung potential
2971:
2866:
2731:
2608:Closed orbit theory
2445:{\displaystyle n=6}
2415:{\displaystyle 1/r}
2378:absorption spectrum
1642:density of states.
1077:solid-state physics
1016:celestial mechanics
960:perturbation theory
909:classical mechanics
362:Stern–Gerlach
159:Classical mechanics
6309:in popular culture
6091:Quantum algorithms
5939:Von Neumann–Wigner
5919:Objective collapse
5615:Old quantum theory
5537:Santa Fe Institute
5404:Aleksandr Lyapunov
5234:Three-body problem
5121:Gingerbreadman map
5008:Bifurcation theory
4886:Lyapunov stability
4706:American Scientist
4542:Eugene Paul Wigner
4447:Notices of the AMS
4066:Scientific Reports
3452:2013-03-08 at the
3338:
3277:
3258:Quantum ergodicity
3218:
3172:
3046:
3003:
2972:
2952:
2935:
2915:
2887:
2867:
2847:
2830:
2796:
2712:
2690:
2646:
2622:
2589:
2560:
2442:
2412:
2362:
2304:
2284:
2242:
2212:
2160:
2104:
2074:
2010:
1980:
1960:
1940:
1910:
1883:
1852:
1825:
1686:
1635:
1626:
1581:
1499:
1475:
1431:of energy levels:
1407:
1369:
1349:
1309:
1299:is separated, and
1289:
1262:
1232:
1202:
1151:
1139:
1061:
1049:
1012:three-body problem
1008:
978:quantum ergodicity
913:
550:Von Neumann–Wigner
530:Objective-collapse
329:Mach–Zehnder
319:Leggett inequality
314:Franck–Hertz
164:Old quantum theory
111:
6370:Quantum mechanics
6352:
6351:
6326:Quantum mysticism
6304:Schrödinger's cat
6235:Quantum simulator
6205:Quantum metrology
6133:Quantum computing
6096:Quantum amplifier
6073:Quantum spacetime
6038:Quantum cosmology
6028:Quantum chemistry
5728:Scattering theory
5676:Zero-point energy
5671:Degenerate levels
5579:Quantum mechanics
5545:
5544:
5409:Benoît Mandelbrot
5374:Martin Gutzwiller
5364:Peter Grassberger
5247:
5246:
5229:Rössler attractor
4977:
4976:
4881:Invariant measure
4803:Lyapunov exponent
4671:Martin Gutzwiller
4658:Martin Gutzwiller
4639:978-0-387-98788-0
4617:978-981-02-4711-9
4598:978-3-540-67723-9
4533:978-0-521-59284-0
4511:978-0-387-97173-5
4493:10.1063/1.1665596
4463:Further resources
4278:978-952-03-1699-0
4141:Physical Review B
4096:10.1038/srep37656
3968:(16): 1515–1518.
3903:Physical Review X
3852:Physical Review X
3795:Physical Review B
3740:Physical Review A
3642:(1686): 375–394.
3524:Physical Review A
3417:978-3-540-67723-9
3280:{\displaystyle R}
3240:Recent directions
3216:
3170:
3141:
3106:
2938:{\displaystyle w}
2925:as a function of
2903:Fourier transform
2890:{\displaystyle i}
2769:
2681:
2509:
2307:{\displaystyle m}
1983:{\displaystyle n}
1963:{\displaystyle n}
1855:{\displaystyle k}
1763:
1677:
1622:density of states
1618:Fourier transform
1542:
1502:{\displaystyle s}
1474:
1372:{\displaystyle N}
1145:Computed chaotic
928:dynamical systems
905:quantum mechanics
897:
896:
604:Scattering theory
584:Quantum computing
357:Schrödinger's cat
289:Bell's inequality
97:
72:
41:Quantum mechanics
16:(Redirected from
6382:
6342:
6341:
6053:Quantum geometry
6048:Quantum dynamics
5905:Superdeterminism
5837:Rarita–Schwinger
5786:Matrix mechanics
5641:Bra–ket notation
5572:
5565:
5558:
5549:
5517:Butterfly effect
5429:Itamar Procaccia
5379:Brosl Hasslacher
5276:Elastic pendulum
5204:Duffing equation
5151:Kaplan–Yorke map
5069:Arnold's cat map
5057:
5033:Stability theory
5018:Dynamical system
5013:Control of chaos
4993:
4985:
4969:Takens's theorem
4901:Poincaré section
4771:
4753:
4746:
4739:
4730:
4660:(1992 and 2008,
4643:
4626:Reichl, Linda E.
4621:
4602:
4583:
4537:
4515:
4496:
4456:
4455:
4443:
4434:
4428:
4427:
4401:
4381:
4375:
4374:
4373:
4359:
4353:
4352:
4342:
4310:
4304:
4303:
4301:
4299:quant-ph/0403061
4289:
4283:
4282:
4262:
4256:
4255:
4213:
4189:
4183:
4182:
4156:
4132:
4126:
4125:
4115:
4081:
4057:
4051:
4050:
4016:
3992:
3986:
3985:
3953:
3947:
3946:
3936:
3918:
3894:
3888:
3887:
3885:
3867:
3843:
3837:
3836:
3810:
3786:
3780:
3779:
3735:
3729:
3728:
3694:
3674:
3668:
3667:
3631:
3625:
3624:
3590:
3570:
3564:
3563:
3530:(5): 3604–3620.
3519:
3508:
3507:
3474:(9): 1538–1541.
3463:
3457:
3428:
3422:
3421:
3403:
3347:
3345:
3344:
3339:
3286:
3284:
3283:
3278:
3227:
3225:
3224:
3219:
3217:
3215:
3207:
3190:
3181:
3179:
3178:
3173:
3171:
3169:
3161:
3144:
3142:
3137:
3120:
3119:
3107:
3105:
3104:
3103:
3099:
3082:
3081:
3077:
3064:
3055:
3053:
3052:
3047:
3012:
3010:
3009:
3004:
3002:
3001:
2981:
2979:
2978:
2973:
2970:
2965:
2964:
2944:
2942:
2941:
2936:
2924:
2922:
2921:
2916:
2896:
2894:
2893:
2888:
2876:
2874:
2873:
2868:
2865:
2860:
2859:
2839:
2837:
2836:
2831:
2829:
2828:
2827:
2805:
2803:
2802:
2797:
2789:
2788:
2787:
2771:
2770:
2765:
2764:
2755:
2730:
2725:
2724:
2710:
2705:
2689:
2655:
2653:
2652:
2647:
2598:
2596:
2595:
2590:
2569:
2567:
2566:
2561:
2547:
2546:
2510:
2502:
2451:
2449:
2448:
2443:
2421:
2419:
2418:
2413:
2408:
2371:
2369:
2368:
2363:
2355:
2348:
2343:
2342:
2313:
2311:
2310:
2305:
2293:
2291:
2290:
2285:
2271:
2270:
2251:
2249:
2248:
2243:
2241:
2240:
2221:
2219:
2218:
2213:
2211:
2204:
2199:
2198:
2169:
2167:
2166:
2161:
2159:
2152:
2147:
2146:
2113:
2111:
2110:
2105:
2103:
2102:
2083:
2081:
2080:
2075:
2073:
2066:
2061:
2060:
2038:
2019:
2017:
2016:
2011:
2009:
2008:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1949:
1947:
1946:
1941:
1939:
1938:
1919:
1917:
1916:
1911:
1909:
1908:
1892:
1890:
1889:
1884:
1882:
1881:
1861:
1859:
1858:
1853:
1834:
1832:
1831:
1826:
1821:
1820:
1813:
1805:
1804:
1789:
1788:
1764:
1762:
1761:
1754:
1749:
1748:
1719:
1716:
1711:
1696:
1695:
1685:
1664:
1663:
1590:
1588:
1587:
1582:
1577:
1576:
1572:
1567:
1566:
1543:
1535:
1508:
1506:
1505:
1500:
1484:
1482:
1481:
1476:
1472:
1468:
1467:
1403:Wigner's surmise
1378:
1376:
1375:
1370:
1358:
1356:
1355:
1350:
1348:
1347:
1318:
1316:
1315:
1310:
1298:
1296:
1295:
1290:
1288:
1287:
1271:
1269:
1268:
1263:
1261:
1260:
1241:
1239:
1238:
1233:
1231:
1230:
1211:
1209:
1208:
1203:
1197:
1196:
1178:
1177:
1058:
1056:
1055:
1050:
889:
882:
875:
516:Superdeterminism
169:Bra–ket notation
120:
118:
117:
112:
104:
99:
98:
90:
78:
73:
71:
60:
32:
21:
6390:
6389:
6385:
6384:
6383:
6381:
6380:
6379:
6355:
6354:
6353:
6348:
6330:
6316:Wigner's friend
6292:
6283:Quantum gravity
6244:
6230:Quantum sensing
6210:Quantum network
6190:Quantum machine
6160:Quantum imaging
6123:Quantum circuit
6118:Quantum channel
6077:
6023:Quantum biology
6009:
5985:Elitzur–Vaidman
5960:Davisson–Germer
5943:
5895:Hidden-variable
5885:de Broglie–Bohm
5862:Interpretations
5856:
5805:
5759:
5646:Complementarity
5624:
5581:
5576:
5546:
5541:
5509:
5503:
5449:Caroline Series
5344:Mary Cartwright
5326:
5320:
5271:Double pendulum
5253:
5243:
5192:
5185:
5111:Exponential map
5062:
5048:
5042:
5000:
4994:
4987:
4973:
4939:Ergodic theorem
4932:
4925:
4916:Stable manifold
4906:Recurrence plot
4822:
4776:
4762:
4757:
4720:by Arnd Bäcker.
4694:(January 2008,
4650:
4640:
4624:
4618:
4605:
4599:
4586:
4546:Dirac, P. A. M.
4540:
4534:
4518:
4512:
4499:
4468:
4465:
4460:
4459:
4441:
4436:
4435:
4431:
4383:
4382:
4378:
4371:
4361:
4360:
4356:
4312:
4311:
4307:
4291:
4290:
4286:
4279:
4264:
4263:
4259:
4191:
4190:
4186:
4134:
4133:
4129:
4059:
4058:
4054:
3994:
3993:
3989:
3955:
3954:
3950:
3896:
3895:
3891:
3845:
3844:
3840:
3788:
3787:
3783:
3737:
3736:
3732:
3676:
3675:
3671:
3633:
3632:
3628:
3572:
3571:
3567:
3521:
3520:
3511:
3465:
3464:
3460:
3454:Wayback Machine
3429:
3425:
3418:
3405:
3404:
3400:
3395:
3378:
3366:Riemann surface
3358:
3300:
3299:
3269:
3268:
3242:
3230:half derivative
3208:
3191:
3184:
3183:
3162:
3145:
3108:
3087:
3083:
3065:
3058:
3057:
3023:
3022:
3019:
2993:
2988:
2987:
2947:
2946:
2927:
2926:
2907:
2906:
2879:
2878:
2842:
2841:
2815:
2810:
2809:
2775:
2756:
2661:
2660:
2638:
2637:
2610:
2572:
2571:
2538:
2478:
2477:
2462:
2428:
2427:
2396:
2395:
2372:at a classical
2331:
2316:
2315:
2296:
2295:
2259:
2254:
2253:
2229:
2224:
2223:
2187:
2172:
2171:
2135:
2120:
2119:
2091:
2086:
2085:
2049:
2026:
2025:
2020:is the orbit's
1997:
1992:
1991:
1972:
1971:
1952:
1951:
1930:
1922:
1921:
1900:
1895:
1894:
1873:
1868:
1867:
1864:periodic orbits
1844:
1843:
1793:
1780:
1766:
1737:
1723:
1687:
1655:
1650:
1649:
1610:
1605:
1558:
1547:
1514:
1513:
1491:
1490:
1456:
1436:
1435:
1390:
1361:
1360:
1339:
1334:
1333:
1301:
1300:
1279:
1274:
1273:
1249:
1244:
1243:
1222:
1217:
1216:
1185:
1169:
1158:
1157:
1127:
1114:distributions.
1096:level repulsion
1032:
1031:
1024:
996:
944:Planck constant
940:classical limit
932:classical chaos
922:focused on how
918:is a branch of
893:
864:
863:
862:
627:
619:
618:
564:
563:Advanced topics
556:
555:
554:
506:Hidden-variable
496:de Broglie–Bohm
475:
473:Interpretations
465:
464:
463:
433:
425:
424:
423:
381:
373:
372:
371:
338:
294:CHSH inequality
283:
275:
274:
273:
202:Complementarity
196:
188:
187:
186:
154:
125:
64:
48:
47:
28:
23:
22:
15:
12:
11:
5:
6388:
6386:
6378:
6377:
6372:
6367:
6357:
6356:
6350:
6349:
6347:
6346:
6335:
6332:
6331:
6329:
6328:
6323:
6318:
6313:
6312:
6311:
6300:
6298:
6294:
6293:
6291:
6290:
6285:
6280:
6279:
6278:
6268:
6263:
6261:Casimir effect
6258:
6252:
6250:
6246:
6245:
6243:
6242:
6237:
6232:
6227:
6222:
6220:Quantum optics
6217:
6212:
6207:
6202:
6197:
6192:
6187:
6182:
6177:
6172:
6167:
6162:
6157:
6152:
6147:
6142:
6141:
6140:
6130:
6125:
6120:
6115:
6114:
6113:
6103:
6098:
6093:
6087:
6085:
6079:
6078:
6076:
6075:
6070:
6065:
6060:
6055:
6050:
6045:
6040:
6035:
6030:
6025:
6019:
6017:
6011:
6010:
6008:
6007:
6002:
5997:
5995:Quantum eraser
5992:
5987:
5982:
5977:
5972:
5967:
5962:
5957:
5951:
5949:
5945:
5944:
5942:
5941:
5936:
5931:
5926:
5921:
5916:
5911:
5910:
5909:
5908:
5907:
5892:
5887:
5882:
5877:
5872:
5866:
5864:
5858:
5857:
5855:
5854:
5849:
5844:
5839:
5834:
5829:
5824:
5819:
5813:
5811:
5807:
5806:
5804:
5803:
5798:
5793:
5788:
5783:
5778:
5773:
5767:
5765:
5761:
5760:
5758:
5757:
5756:
5755:
5750:
5740:
5735:
5730:
5725:
5720:
5715:
5710:
5705:
5700:
5695:
5690:
5685:
5680:
5679:
5678:
5673:
5668:
5663:
5653:
5651:Density matrix
5648:
5643:
5638:
5632:
5630:
5626:
5625:
5623:
5622:
5617:
5612:
5607:
5606:
5605:
5595:
5589:
5587:
5583:
5582:
5577:
5575:
5574:
5567:
5560:
5552:
5543:
5542:
5540:
5539:
5534:
5532:Predictability
5529:
5524:
5519:
5513:
5511:
5505:
5504:
5502:
5501:
5499:Lai-Sang Young
5496:
5494:James A. Yorke
5491:
5489:Amie Wilkinson
5486:
5481:
5476:
5471:
5466:
5461:
5456:
5451:
5446:
5441:
5436:
5431:
5426:
5424:Henri Poincaré
5421:
5416:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5371:
5366:
5361:
5356:
5351:
5346:
5341:
5336:
5330:
5328:
5322:
5321:
5319:
5318:
5313:
5308:
5303:
5298:
5293:
5291:Kicked rotator
5288:
5283:
5278:
5273:
5268:
5263:
5261:Chua's circuit
5257:
5255:
5249:
5248:
5245:
5244:
5242:
5241:
5236:
5231:
5226:
5221:
5216:
5211:
5206:
5201:
5195:
5193:
5190:
5187:
5186:
5184:
5183:
5181:Zaslavskii map
5178:
5176:Tinkerbell map
5173:
5168:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5123:
5118:
5113:
5108:
5107:
5106:
5096:
5091:
5086:
5081:
5076:
5071:
5065:
5063:
5060:
5054:
5044:
5043:
5041:
5040:
5035:
5030:
5025:
5023:Ergodic theory
5020:
5015:
5010:
5004:
5002:
4996:
4995:
4980:
4978:
4975:
4974:
4972:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4935:
4933:
4930:
4927:
4926:
4924:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4888:
4883:
4878:
4873:
4868:
4863:
4858:
4853:
4848:
4843:
4838:
4833:
4827:
4824:
4823:
4821:
4820:
4815:
4813:Periodic point
4810:
4805:
4800:
4795:
4790:
4785:
4779:
4777:
4774:
4768:
4764:
4763:
4758:
4756:
4755:
4748:
4741:
4733:
4727:
4726:
4721:
4715:
4701:
4685:
4680:
4665:
4649:
4648:External links
4646:
4645:
4644:
4638:
4622:
4616:
4603:
4597:
4584:
4538:
4532:
4516:
4510:
4497:
4479:(3): 343–358.
4464:
4461:
4458:
4457:
4429:
4376:
4354:
4305:
4284:
4277:
4257:
4204:(10): 105301.
4184:
4127:
4052:
4007:(21): 214101.
3987:
3948:
3889:
3838:
3781:
3746:(1): 178–191.
3730:
3669:
3626:
3565:
3509:
3458:
3439:Jim Al-Khalili
3423:
3416:
3397:
3396:
3394:
3391:
3390:
3389:
3384:
3382:Scar (physics)
3377:
3374:
3357:
3354:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3293:kicked rotator
3276:
3246:Hilbert spaces
3241:
3238:
3214:
3211:
3206:
3203:
3200:
3197:
3194:
3168:
3165:
3160:
3157:
3154:
3151:
3148:
3140:
3135:
3132:
3129:
3126:
3123:
3118:
3115:
3111:
3102:
3098:
3094:
3090:
3086:
3080:
3076:
3072:
3068:
3045:
3042:
3039:
3036:
3033:
3030:
3018:
3015:
3000:
2996:
2969:
2963:
2960:
2955:
2934:
2914:
2886:
2864:
2858:
2855:
2850:
2826:
2823:
2818:
2807:
2806:
2795:
2792:
2786:
2783:
2778:
2774:
2768:
2763:
2759:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2729:
2723:
2720:
2715:
2709:
2704:
2701:
2698:
2694:
2688:
2684:
2680:
2677:
2674:
2671:
2668:
2645:
2609:
2606:
2601:
2600:
2588:
2585:
2582:
2579:
2559:
2556:
2553:
2550:
2545:
2541:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2508:
2505:
2500:
2497:
2494:
2491:
2488:
2485:
2474:
2470:
2466:
2461:
2458:
2441:
2438:
2435:
2411:
2407:
2403:
2361:
2358:
2354:
2351:
2347:
2341:
2338:
2334:
2330:
2326:
2323:
2303:
2283:
2280:
2277:
2274:
2269:
2266:
2262:
2239:
2236:
2232:
2210:
2207:
2203:
2197:
2194:
2190:
2186:
2182:
2179:
2158:
2155:
2151:
2145:
2142:
2138:
2134:
2130:
2127:
2101:
2098:
2094:
2072:
2069:
2065:
2059:
2056:
2052:
2048:
2044:
2041:
2037:
2033:
2007:
2004:
2000:
1979:
1959:
1937:
1933:
1929:
1907:
1903:
1880:
1876:
1851:
1836:
1835:
1824:
1819:
1816:
1812:
1808:
1803:
1800:
1796:
1792:
1787:
1783:
1779:
1776:
1773:
1769:
1760:
1757:
1753:
1747:
1744:
1740:
1736:
1732:
1729:
1726:
1722:
1715:
1710:
1707:
1704:
1700:
1694:
1690:
1684:
1680:
1676:
1673:
1670:
1667:
1662:
1658:
1609:
1606:
1604:
1601:
1592:
1591:
1580:
1575:
1571:
1565:
1561:
1557:
1554:
1550:
1546:
1541:
1538:
1533:
1530:
1527:
1524:
1521:
1498:
1486:
1485:
1471:
1466:
1463:
1459:
1455:
1452:
1449:
1446:
1443:
1389:
1386:
1368:
1346:
1342:
1308:
1286:
1282:
1259:
1256:
1252:
1229:
1225:
1213:
1212:
1200:
1195:
1192:
1188:
1184:
1181:
1176:
1172:
1168:
1165:
1126:
1123:
1108:time evolution
1079:, and even to
1048:
1045:
1042:
1039:
1023:
1020:
995:
992:
991:
990:
987:
981:
970:
963:
895:
894:
892:
891:
884:
877:
869:
866:
865:
861:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
629:
628:
625:
624:
621:
620:
617:
616:
611:
606:
601:
599:Density matrix
596:
591:
586:
581:
576:
571:
565:
562:
561:
558:
557:
553:
552:
547:
542:
537:
532:
527:
522:
521:
520:
519:
518:
503:
498:
493:
488:
483:
477:
476:
471:
470:
467:
466:
462:
461:
456:
451:
446:
441:
435:
434:
431:
430:
427:
426:
422:
421:
416:
411:
406:
401:
396:
390:
389:
388:
382:
379:
378:
375:
374:
370:
369:
364:
359:
353:
352:
351:
350:
349:
347:Delayed-choice
342:Quantum eraser
337:
336:
331:
326:
321:
316:
311:
306:
301:
296:
291:
285:
284:
281:
280:
277:
276:
272:
271:
270:
269:
259:
254:
249:
244:
239:
234:
232:Quantum number
229:
224:
219:
214:
209:
204:
198:
197:
194:
193:
190:
189:
185:
184:
179:
173:
172:
171:
166:
161:
155:
152:
151:
148:
147:
146:
145:
140:
135:
127:
126:
121:
110:
107:
103:
96:
93:
87:
84:
81:
77:
70:
67:
63:
58:
55:
44:
43:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6387:
6376:
6373:
6371:
6368:
6366:
6363:
6362:
6360:
6345:
6337:
6336:
6333:
6327:
6324:
6322:
6319:
6317:
6314:
6310:
6307:
6306:
6305:
6302:
6301:
6299:
6295:
6289:
6286:
6284:
6281:
6277:
6274:
6273:
6272:
6269:
6267:
6264:
6262:
6259:
6257:
6254:
6253:
6251:
6247:
6241:
6238:
6236:
6233:
6231:
6228:
6226:
6223:
6221:
6218:
6216:
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6180:Quantum logic
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6139:
6136:
6135:
6134:
6131:
6129:
6126:
6124:
6121:
6119:
6116:
6112:
6109:
6108:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6088:
6086:
6084:
6080:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6033:Quantum chaos
6031:
6029:
6026:
6024:
6021:
6020:
6018:
6016:
6012:
6006:
6003:
6001:
6000:Stern–Gerlach
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5971:
5968:
5966:
5963:
5961:
5958:
5956:
5953:
5952:
5950:
5946:
5940:
5937:
5935:
5934:Transactional
5932:
5930:
5927:
5925:
5924:Quantum logic
5922:
5920:
5917:
5915:
5912:
5906:
5903:
5902:
5901:
5898:
5897:
5896:
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5867:
5865:
5863:
5859:
5853:
5850:
5848:
5845:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5814:
5812:
5808:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5772:
5769:
5768:
5766:
5762:
5754:
5751:
5749:
5746:
5745:
5744:
5743:Wave function
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5718:Superposition
5716:
5714:
5713:Quantum state
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5677:
5674:
5672:
5669:
5667:
5666:Excited state
5664:
5662:
5659:
5658:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5633:
5631:
5627:
5621:
5618:
5616:
5613:
5611:
5608:
5604:
5601:
5600:
5599:
5596:
5594:
5591:
5590:
5588:
5584:
5580:
5573:
5568:
5566:
5561:
5559:
5554:
5553:
5550:
5538:
5535:
5533:
5530:
5528:
5527:Edge of chaos
5525:
5523:
5520:
5518:
5515:
5514:
5512:
5506:
5500:
5497:
5495:
5492:
5490:
5487:
5485:
5484:Marcelo Viana
5482:
5480:
5477:
5475:
5474:Audrey Terras
5472:
5470:
5469:Floris Takens
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5369:Celso Grebogi
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5350:
5349:Chen Guanrong
5347:
5345:
5342:
5340:
5337:
5335:
5334:Michael Berry
5332:
5331:
5329:
5323:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5258:
5256:
5250:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5209:Lorenz system
5207:
5205:
5202:
5200:
5197:
5196:
5194:
5188:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5156:Langton's ant
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5131:Horseshoe map
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5105:
5102:
5101:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5066:
5064:
5058:
5055:
5052:
5045:
5039:
5036:
5034:
5031:
5029:
5028:Quantum chaos
5026:
5024:
5021:
5019:
5016:
5014:
5011:
5009:
5006:
5005:
5003:
4997:
4992:
4988:
4984:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4936:
4934:
4928:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4836:Arnold tongue
4834:
4832:
4829:
4828:
4825:
4819:
4816:
4814:
4811:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4791:
4789:
4786:
4784:
4781:
4780:
4778:
4772:
4769:
4765:
4761:
4754:
4749:
4747:
4742:
4740:
4735:
4734:
4731:
4725:
4724:ChaosBook.org
4722:
4719:
4716:
4713:
4709:
4707:
4702:
4699:
4698:
4693:
4692:Ze'ev Rudnick
4689:
4686:
4684:
4681:
4679:
4675:
4672:
4669:
4668:Quantum Chaos
4666:
4663:
4659:
4655:
4654:Quantum Chaos
4652:
4651:
4647:
4641:
4635:
4631:
4627:
4623:
4619:
4613:
4609:
4604:
4600:
4594:
4590:
4585:
4581:
4577:
4573:
4569:
4565:
4561:
4557:
4553:
4552:
4547:
4543:
4539:
4535:
4529:
4525:
4521:
4517:
4513:
4507:
4503:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4473:
4467:
4466:
4462:
4453:
4449:
4448:
4440:
4433:
4430:
4425:
4421:
4417:
4413:
4409:
4405:
4400:
4395:
4391:
4387:
4380:
4377:
4370:
4369:
4364:
4363:Marklof, Jens
4358:
4355:
4350:
4346:
4341:
4336:
4332:
4328:
4324:
4320:
4316:
4309:
4306:
4300:
4295:
4288:
4285:
4280:
4274:
4270:
4269:
4261:
4258:
4253:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4221:
4217:
4212:
4207:
4203:
4199:
4195:
4188:
4185:
4180:
4176:
4172:
4168:
4164:
4160:
4155:
4150:
4147:(9): 094204.
4146:
4142:
4138:
4131:
4128:
4123:
4119:
4114:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4080:
4075:
4071:
4067:
4063:
4056:
4053:
4048:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4015:
4010:
4006:
4002:
3998:
3991:
3988:
3983:
3979:
3975:
3971:
3967:
3963:
3959:
3952:
3949:
3944:
3940:
3935:
3930:
3926:
3922:
3917:
3912:
3909:(4): 041019.
3908:
3904:
3900:
3893:
3890:
3884:
3879:
3875:
3871:
3866:
3861:
3858:(2): 021062.
3857:
3853:
3849:
3842:
3839:
3834:
3830:
3826:
3822:
3818:
3814:
3809:
3804:
3801:(3): 035442.
3800:
3796:
3792:
3785:
3782:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3734:
3731:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3693:
3688:
3685:(4): 044103.
3684:
3680:
3673:
3670:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3630:
3627:
3622:
3618:
3614:
3610:
3606:
3602:
3598:
3594:
3589:
3584:
3581:(4): 040605.
3580:
3576:
3569:
3566:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3529:
3525:
3518:
3516:
3514:
3510:
3505:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3469:
3462:
3459:
3455:
3451:
3448:
3444:
3440:
3436:
3432:
3431:Michael Berry
3427:
3424:
3419:
3413:
3409:
3402:
3399:
3392:
3388:
3385:
3383:
3380:
3379:
3375:
3373:
3371:
3367:
3363:
3355:
3353:
3351:
3329:
3323:
3320:
3317:
3314:
3311:
3305:
3296:
3294:
3290:
3274:
3265:
3263:
3259:
3253:
3251:
3247:
3239:
3237:
3235:
3231:
3212:
3209:
3201:
3195:
3192:
3166:
3163:
3155:
3149:
3146:
3138:
3133:
3130:
3124:
3116:
3113:
3109:
3100:
3096:
3092:
3088:
3084:
3078:
3074:
3070:
3066:
3043:
3040:
3034:
3028:
3016:
3014:
2998:
2994:
2983:
2967:
2953:
2932:
2912:
2904:
2900:
2899:Rydberg atoms
2884:
2862:
2848:
2816:
2793:
2776:
2772:
2761:
2757:
2750:
2747:
2744:
2741:
2735:
2732:
2727:
2713:
2702:
2699:
2696:
2692:
2686:
2682:
2678:
2672:
2666:
2659:
2658:
2657:
2643:
2634:
2632:
2631:Rydberg atoms
2626:
2619:
2618:Rydberg atoms
2614:
2607:
2605:
2583:
2577:
2551:
2548:
2539:
2535:
2532:
2526:
2520:
2517:
2506:
2503:
2498:
2495:
2489:
2483:
2475:
2471:
2467:
2464:
2463:
2459:
2457:
2453:
2439:
2436:
2433:
2425:
2409:
2405:
2401:
2393:
2390:
2385:
2381:
2379:
2375:
2359:
2356:
2349:
2345:
2339:
2336:
2332:
2324:
2321:
2301:
2281:
2278:
2275:
2272:
2267:
2264:
2260:
2237:
2234:
2230:
2205:
2201:
2195:
2192:
2188:
2180:
2177:
2153:
2149:
2143:
2140:
2136:
2128:
2125:
2117:
2099:
2096:
2092:
2067:
2063:
2057:
2054:
2050:
2042:
2039:
2035:
2031:
2023:
2005:
2002:
1998:
1977:
1957:
1935:
1931:
1927:
1905:
1901:
1878:
1874:
1865:
1849:
1841:
1822:
1814:
1810:
1806:
1801:
1798:
1794:
1790:
1785:
1781:
1777:
1771:
1767:
1755:
1751:
1745:
1742:
1738:
1730:
1727:
1724:
1720:
1708:
1705:
1702:
1698:
1692:
1688:
1682:
1678:
1674:
1668:
1660:
1656:
1648:
1647:
1646:
1643:
1640:
1630:
1623:
1619:
1614:
1607:
1602:
1600:
1598:
1578:
1573:
1569:
1563:
1559:
1555:
1552:
1548:
1544:
1539:
1536:
1531:
1525:
1519:
1512:
1511:
1510:
1496:
1469:
1464:
1461:
1457:
1453:
1447:
1441:
1434:
1433:
1432:
1430:
1424:
1421:
1415:
1412:
1411:Random matrix
1404:
1399:
1394:
1387:
1385:
1381:
1366:
1344:
1340:
1329:
1325:
1321:
1306:
1284:
1280:
1257:
1254:
1250:
1227:
1223:
1198:
1193:
1190:
1186:
1182:
1179:
1174:
1170:
1166:
1163:
1156:
1155:
1154:
1148:
1143:
1136:
1131:
1124:
1122:
1120:
1115:
1113:
1109:
1105:
1101:
1097:
1092:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1046:
1043:
1040:
1037:
1028:
1021:
1019:
1017:
1013:
1005:
1000:
993:
988:
985:
982:
979:
975:
971:
968:
964:
961:
957:
956:
955:
952:
949:
945:
941:
937:
933:
929:
925:
921:
917:
916:Quantum chaos
910:
906:
901:
890:
885:
883:
878:
876:
871:
870:
868:
867:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
630:
623:
622:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
589:Quantum chaos
587:
585:
582:
580:
577:
575:
572:
570:
567:
566:
560:
559:
551:
548:
546:
545:Transactional
543:
541:
538:
536:
535:Quantum logic
533:
531:
528:
526:
523:
517:
514:
513:
512:
509:
508:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
478:
474:
469:
468:
460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
436:
429:
428:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
391:
387:
384:
383:
377:
376:
368:
365:
363:
360:
358:
355:
354:
348:
345:
344:
343:
340:
339:
335:
332:
330:
327:
325:
322:
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262:Wave function
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242:Superposition
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30:
19:
6365:Chaos theory
6063:Quantum mind
6032:
5975:Franck–Hertz
5817:Klein–Gordon
5771:Formulations
5764:Formulations
5693:Interference
5683:Entanglement
5661:Ground state
5656:Energy level
5629:Fundamentals
5593:Introduction
5479:Mary Tsingou
5444:David Ruelle
5439:Otto Rössler
5384:Michel HĂ©non
5354:Leon O. Chua
5311:Tilt-A-Whirl
5281:FPUT problem
5166:Standard map
5161:Logistic map
5027:
4986:
4760:Chaos theory
4705:
4695:
4676:2(12):3146.
4674:Scholarpedia
4661:
4629:
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4319:Scholarpedia
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3407:
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3289:standard map
3266:
3254:
3243:
3020:
2984:
2808:
2635:
2627:
2623:
2602:
2454:
2386:
2382:
2022:Maslov index
1837:
1644:
1636:
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1418:of freedom (
1416:
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1398:Rydberg atom
1382:
1330:
1326:
1322:
1214:
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1147:Rydberg atom
1135:Rydberg atom
1119:Dyson series
1116:
1104:spectroscopy
1093:
1062:
1009:
1004:bifurcations
953:
915:
914:
588:
444:Klein–Gordon
380:Formulations
217:Energy level
212:Entanglement
195:Fundamentals
182:Interference
133:Introduction
29:
6321:EPR paradox
6101:Quantum bus
5970:Double-slit
5948:Experiments
5914:Many-worlds
5852:Schrödinger
5801:Phase space
5791:Schrödinger
5781:Interaction
5738:Uncertainty
5708:Nonlocality
5703:Measurement
5698:Decoherence
5688:Hamiltonian
5464:Nina Snaith
5454:Yakov Sinai
5339:Rufus Bowen
5089:Duffing map
5074:Baker's map
4999:Theoretical
4911:SRB measure
4818:Phase space
4788:Bifurcation
4454:(1): 32–34.
4325:(6): 9806.
2389:anisotropic
2374:bifurcation
1840:Berry phase
967:Hamiltonian
833:von Neumann
818:Schrödinger
594:EPR paradox
525:Many-worlds
459:Schrödinger
414:Schrödinger
409:Phase-space
399:Interaction
304:Double-slit
282:Experiments
257:Uncertainty
227:Nonlocality
222:Measurement
207:Decoherence
177:Hamiltonian
6359:Categories
6249:Extensions
6083:Technology
5929:Relational
5880:Copenhagen
5776:Heisenberg
5723:Tunnelling
5586:Background
5522:Complexity
5419:Edward Ott
5266:Convection
5191:Continuous
4866:Ergodicity
4558:(4): 790.
4211:1806.02598
4154:1710.00585
4079:1511.04198
4014:1911.09729
3916:1712.06836
3865:1712.02665
3808:1611.08879
3588:2003.07267
3393:References
3370:integrable
1100:scattering
1085:microwaves
1022:Approaches
926:classical
828:Sommerfeld
743:Heisenberg
738:Gutzwiller
678:de Broglie
626:Scientists
540:Relational
491:Copenhagen
394:Heisenberg
252:Tunnelling
153:Background
5955:Bell test
5810:Equations
5636:Born rule
5434:Mary Rees
5394:Bryna Kra
5327:theorists
5136:Ikeda map
5126:HĂ©non map
5116:Gauss map
4798:Limit set
4783:Attractor
4580:120852535
4399:0804.3685
4349:1941-6016
4236:0953-8984
4179:119083672
4104:2045-2322
4047:208248295
3943:2160-3308
3833:119028983
3768:1050-2947
3717:0031-9007
3664:0080-4630
3621:212725801
3552:1050-2947
3496:0031-9007
3360:In 1977,
3350:many-body
3139:π
3114:−
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2817:ϕ
2777:ϕ
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1183:ε
1112:amplitude
1081:acoustics
1073:molecular
1044:−
1038:ϵ
969:(system).
858:Zeilinger
703:Ehrenfest
432:Equations
109:⟩
106:Ψ
95:^
83:⟩
80:Ψ
57:ℏ
6344:Category
6138:Timeline
5890:Ensemble
5870:Bayesian
5832:Majorana
5748:Collapse
5620:Glossary
5603:Timeline
5510:articles
5252:Physical
5171:Tent map
5061:Discrete
5001:branches
4931:Theorems
4767:Concepts
4628:(2004).
4522:(1999).
4424:53550992
4252:51693305
4244:30566927
4122:27892510
4039:31809168
3725:17358777
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3504:10059054
3450:Archived
3376:See also
3291:and the
2294:, where
2222:, where
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1359:, where
783:Millikan
708:Einstein
693:Davisson
648:Blackett
633:Aharonov
501:Ensemble
481:Bayesian
386:Overview
267:Collapse
247:Symmetry
138:Glossary
6297:Related
6276:History
6015:Science
5847:Rydberg
5598:History
5508:Related
5316:Weather
5254:systems
5047:Chaotic
4793:Fractal
4560:Bibcode
4481:Bibcode
4404:Bibcode
4327:Bibcode
4216:Bibcode
4159:Bibcode
4113:5124902
4084:Bibcode
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3970:Bibcode
3921:Bibcode
3870:Bibcode
3813:Bibcode
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994:History
946:to the
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924:chaotic
920:physics
823:Simmons
813:Rydberg
778:Moseley
758:Kramers
748:Hilbert
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713:Everett
683:Compton
454:Rydberg
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2392:Kepler
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1089:optics
1069:atomic
948:action
853:Zeeman
848:Wigner
798:Planck
768:Landau
753:Jordan
404:Matrix
334:Popper
5900:Local
5842:Pauli
5822:Dirac
5325:Chaos
5104:outer
4808:Orbit
4576:S2CID
4442:(PDF)
4420:S2CID
4394:arXiv
4372:(PDF)
4294:arXiv
4248:S2CID
4206:arXiv
4175:S2CID
4149:arXiv
4074:arXiv
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4009:arXiv
3911:arXiv
3860:arXiv
3829:S2CID
3803:arXiv
3687:arXiv
3617:S2CID
3583:arXiv
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3262:scars
3182:here
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793:Pauli
788:Onnes
723:Fermi
698:Debye
688:Dirac
653:Bloch
643:Bethe
511:Local
449:Pauli
439:Dirac
237:State
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5051:list
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4612:ISBN
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