Quantum chaos - Knowledge

4980: 4972: 6314: 3253:). 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. 889: 3276:) 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 1602: 1618: 988: 1016: 1131: 1119: 2602: 1382: 2445:(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. 1613:) 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. 2373:

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.

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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

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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

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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

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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,

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

1588:, 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. 1822: 1308:

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.

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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

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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

3249:). 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 ( 1621:

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 =

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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

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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

2886:). 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 2557: 1007:), 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. 108: 2974:

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

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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.

1087:, 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 1640: 1578: 1138:

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.

2073:, 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 3337:

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

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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

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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:

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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

1080:. 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. 2651: 2609:

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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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

1110:). 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. 2157: 2209: 3215: 4539: 2281: 2969: 2864: 2827: 1472: 1046: 2007: 3335: 2239: 2101: 3353:

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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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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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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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

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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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Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g.

3438: 3357:, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely 1126:

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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Heusler, Stefan; Müller, Sebastian; Altland, Alexander; Braun, Petr; Haake, Fritz (January 2007). "Periodic-Orbit Theory of Level Correlations".

2441:) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies. 2934:

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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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

875: 6282: 2559:(tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states 5608: 1817:{\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)}.} 1627: 940:

sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

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the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by

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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

121: 5254: 4942: 2365:. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo- 1019:

Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of

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Yan, Bin; Sinitsyn, Nikolai A. (2020). "Recovery of Damaged Information and the Out-of-Time-Ordered Correlators".

2788:{\displaystyle f(w)=\sum _{k}\sum _{n=1}^{\infty }D_{\it {nk}}^{i}\sin(2\pi nw{\tilde {S_{k}}}-\phi _{\it {nk}}).} 1398:

Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems.

1313:(at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them. 975:

such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.

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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

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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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Barba, J.C.; et al. (2008). "The Berry–Tabor conjecture for spin chains of Haldane–Shastry type".

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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

1065: 924: 602: 4971: 4537:(1951). "On the statistical distribution of the widths and spacings of nuclear resonance levels". 1982: 943:

In seeking to address the basic question of quantum chaos, several approaches have been employed:

5863: 5789: 5187: 4952: 4859: 3947:"Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits" 3727:

Courtney, Michael; Kleppner, Daniel (January 1996). "Core-induced chaos in diamagnetic lithium".

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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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Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory

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Doron, Cohen (2004). "Driven chaotic mesoscopic systems, dissipation and decoherence".

4236: 4194: 4163: 4137: 4101: 4062: 4050: 4031: 3997: 3946: 3899: 3848: 3817: 3791: 3675: 3605: 3571: 3259: 3246: 3164:{\displaystyle {\frac {d^{1/2}}{dx^{1/2}}}V^{-1}(x)=2{\sqrt {\pi }}{\frac {dN(x)}{dx}}} 2917: 2869: 2362: 2286: 1962: 1942: 1834: 1481: 1351: 1000: 992: 966: 382: 307: 240: 152: 4676: 4458:

Martin C. Gutzwiller (1971). "Periodic Orbits and Classical Quantization Conditions".

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Works are also focused in the study of driven chaotic systems, where the Hamiltonian

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According to closed-orbit theory, the average oscillator strength density at constant

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coordinate system parallel to the direction of movement, as it is done in many books.

2415:) semiclassically. He found agreement with quantum computations for low lying (up to 6299: 6208: 6178: 6106: 6069: 6064: 6046: 6011: 6001: 5716: 5681: 5664: 5567: 5397: 5362: 5352: 5249: 4869: 4791: 4659: 4646: 4622: 4600: 4581: 4568: 4516: 4494: 4333: 4261: 4228: 4220: 4167: 4106: 4088: 4035: 4023: 3927: 3821: 3760: 3752: 3709: 3701: 3648: 3609: 3597: 3544: 3536: 3488: 3480: 3400: 3358: 3245:

the available phase space evenly, up to random fluctuations and energy conservation (

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The transition to chaos: conservative classical systems and quantum manifestations

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Berry, M. V.; Tabor, M. (1977-09-15). "Level clustering in the regular spectrum".

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is the density of states and V(x) is the classical potential of the particle, the

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Correlating statistical descriptions of quantum mechanics with classical behaviour

6256: 6183: 6163: 6133: 6096: 6091: 5996: 5820: 5437: 5332: 5259: 5092: 4904: 4894: 4621:. Institute for nonlinear science (2. ed.). New York Heidelberg: Springer. 4355: 3625:

Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences

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is the recurrence amplitude of a closed orbit for a given initial state (labeled

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in phase space, and neighboring trajectories wind around it. For stable orbits,

6234: 6203: 6193: 5815: 5805: 5639: 5520: 5487: 5482: 5477: 5279: 5169: 5164: 5062: 5011: 4801: 4216: 4159: 3813: 3427: 3370: 3281: 3250: 1852: 1585: 1096: 1091:). In the semiclassical approach of quantum chaos, phenomena are identified in 1057: 962: 781: 741: 721: 691: 671: 621: 587: 437: 427: 220: 4560: 4328: 4303: 3922: 3887: 3871: 3836: 3221:

of the inverse of the potential is related to the density of states as in the

1831:-like term stemming from spin or other internal degrees of freedom. The index 6332: 6153: 5897: 5731: 5701: 5654: 5515: 5472: 5462: 5457: 5357: 5337: 5197: 5119: 4824: 4599:. Stockholm, Sweden: Physica Scripta, the Royal Swedish Academy of Sciences. 4337: 4224: 4092: 3931: 3756: 3705: 3652: 3540: 3532: 3484: 3234: 2887: 2829:

is a phase that depends on the Maslov index and other details of the orbits.

2619: 1399: 841: 836: 766: 736: 706: 523: 250: 225: 103:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 4255: 3748: 6036: 5649: 5644: 5467: 5432: 5342: 5299: 5154: 5149: 4748: 4662: 4614: 4351: 4232: 4182: 4110: 4027: 3713: 3644: 3601: 3492: 3277: 2606: 2010: 1386: 1135: 1123: 1107: 1092: 912: 892:

Quantum chaos is the field of physics attempting to bridge the theories of

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Keski-Rahkonen, J.; Ruhanen, A.; Heller, E. J.; Räsänen, E. (2019-11-21).

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initial state whereas periodic-orbit theory yields the density of states.

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http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf

5510: 5407: 4854: 4534: 1088: 786: 756: 676: 651: 646: 631: 4481: 4084: 3837:"Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory" 2103:

characterizes the instability of the orbit. A stable orbit moves on a

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Keski-Rahkonen, J; Luukko, P J J; Åberg, S; Räsänen, E (2019-01-21).

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is given by a smooth background plus an oscillatory sum of the form

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of quantum mechanics, specifically in the limit as the ratio of the

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For the case of one-dimensional system with the boundary condition

1855:: the shortest period orbits of a given set of initial conditions. 656: 4387: 4183:"Effects of scarring on quantum chaos in disordered quantum wells" 3983: 1498:, the normalized distribution of spacings is well approximated by 4829: 4781: 961:

Study of probability distribution of individual eigenstates (see

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Mathematical Proceedings of the Cambridge Philosophical Society

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computing matrix elements can also be a computational burden.

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Rough sketch on how to arrive at the Gutzwiller trace formula

2104: 1617: 1601: 900:. The figure shows the main ideas running in each direction. 1015: 987: 4257:

Quantum Chaos in Disordered Two-Dimensional Nanostructures

4126:"Controllable quantum scars in semiconductor quantum dots" 3886:

Chan, Amos; De Luca, Andrea; Chalker, J. T. (2018-11-08).

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Kos, Pavel; Ljubotina, Marko; Prosen, Tomaž (2018-06-08).

2601: 1130: 1118: 3888:"Solution of a Minimal Model for Many-Body Quantum Chaos" 3665: 2950: 2845: 2813: 2773: 2710: 1381: 3005: 1573:{\displaystyle P(s)={\frac {\pi }{2}}se^{-\pi s^{2}/4}.} 4529: 4048: 3777: 3778:

Vogl, M.; Pankratov, O.; Shallcross, S. (2017-07-27).

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of an oscillator strength spectrum computed at fixed

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Gutzwiller applied the trace formula to approach the

2309: 2289: 2247: 2217: 2165: 2113: 2079: 2019: 1985: 1965: 1945: 1915: 1888: 1861: 1837: 1643: 1507: 1484: 1429: 1354: 1327: 1294: 1267: 1237: 1210: 1151: 1025: 41: 4457: 4595:Berggren, Karl-Fredrik; °Aberg, Sven, eds. (2001). 1261:is non-separable in the coordinate system in which 978:

Direct application of the correspondence principle.

4697:Volume 91, Number 4, July–August, 2003 pp. 296–300 3885: 3834: 3329: 3268: 3209: 3163: 3037: 2994: 2963: 2926: 2906: 2878: 2858: 2821: 2787: 2637: 2580: 2551: 2433: 2403: 2353: 2295: 2275: 2233: 2203: 2151: 2095: 2065: 2001: 1971: 1951: 1931: 1901: 1882:is the period of the primitive periodic orbit and 1874: 1843: 1816: 1572: 1490: 1466: 1360: 1340: 1300: 1280: 1253: 1223: 1193: 1040: 102: 6330: 3726: 4594: 4253: 2241:is the winding number of the periodic orbit. 5552: 4733: 4051:"Strong quantum scarring by local impurities" 3506: 3504: 3502: 1194:{\displaystyle H=H_{s}+\varepsilon H_{ns},\,} 1114:Quantum mechanics in non-perturbative regimes 869: 4686:Notices of the American Mathematical Society 97: 71: 4693:Brian Hayes, "The Spectrum of Riemannium"; 4425: 4372: 3561: 1939:and a period which is an integral multiple 5559: 5545: 4740: 4726: 4488: 3622: 3499: 876: 862: 4707:Eigenfunctions in chaotic quantum systems 4515:. Cambridge: Cambridge university press. 4507: 4386: 4327: 4286: 4198: 4141: 4100: 4066: 4001: 3921: 3903: 3870: 3852: 3795: 3679: 3575: 3344: 3284:are considered to be prototype problems. 1754: 1190: 4491:Chaos in classical and quantum mechanics 3256:Further studies concern the parametric ( 2600: 2066:{\displaystyle 1/\sinh {(\chi _{nk}/2)}} 1616: 1600: 1596: 1591: 1380: 1231:is separable in some coordinate system, 1129: 1117: 1014: 986: 887: 4301: 3413: 3388: 2354:{\displaystyle \sin {(\chi _{nk}/2)}=0} 927:states that classical mechanics is the 6331: 4613: 3944: 2596: 2152:{\displaystyle \sinh {(\chi _{nk}/2)}} 5566: 5540: 4721: 4575: 4280: 3394: 3006:One-dimensional systems and potential 2204:{\displaystyle \sin {(\chi _{nk}/2)}} 4451: 4187:Journal of Physics: Condensed Matter 3228: 951:and where quantum numbers are large. 4880:Measure-preserving dynamical system 4762: 4672:Category:Quantum Chaos Scholarpedia 4350: 3210:{\displaystyle {\frac {dN(x)}{dx}}} 2411:potential with an anisotropic mass 1385:Nearest neighbour distribution for 13: 3938: 3424:Quantum: a guide for the perplexed 3422:, "Quantum Chaology", pp 104-5 of 2947: 2842: 2810: 2770: 2707: 2696: 2532: 2500: 1702: 408:Sum-over-histories (path integral) 94: 68: 24:Part of a series of articles about 14: 6360: 5448:Oleksandr Mykolayovych Sharkovsky 4636: 2276:{\displaystyle \chi _{nk}=2\pi m} 1605:Even parity recurrence spectrum ( 45: 6313: 6312: 4978: 4970: 4747: 2964:{\displaystyle D_{\it {nk}}^{i}} 2859:{\displaystyle D_{\it {nk}}^{i}} 2822:{\displaystyle \phi _{\it {nk}}} 2383:problem (a single particle in a 1628:Einstein–Brillouin–Keller method 4461:Journal of Mathematical Physics 4419: 4366: 4344: 4295: 4274: 4254:Keski-Rahkonen, Joonas (2020). 4247: 4174: 4117: 4042: 3977: 3879: 3828: 1122:Computed regular (non-chaotic) 6262:Relativistic quantum mechanics 5213:Rabinovich–Fabrikant equations 4513:Quantum chaos: An introduction 4020:10.1103/PhysRevLett.123.214101 3945:Heller, Eric J. (1984-10-15). 3771: 3720: 3659: 3616: 3594:10.1103/PhysRevLett.125.040605 3555: 3448: 3324: 3321: 3315: 3297: 3193: 3187: 3147: 3141: 3116: 3110: 3026: 3020: 2779: 2755: 2727: 2664: 2658: 2575: 2569: 2546: 2543: 2518: 2512: 2503: 2481: 2475: 2341: 2317: 2197: 2173: 2145: 2121: 2059: 2035: 1806: 1763: 1747: 1723: 1660: 1654: 1517: 1511: 1467:{\displaystyle P(s)=e^{-s}.\ } 1439: 1433: 1317:matching boundary conditions. 1041:{\displaystyle \epsilon =-3.0} 558:Relativistic quantum mechanics 90: 83: 64: 1: 6240:Quantum statistical mechanics 6017:Quantum differential calculus 5939:Delayed-choice quantum eraser 5722:Symmetry in quantum mechanics 4667:doi:10.4249/scholarpedia.3146 3698:10.1103/PhysRevLett.98.044103 3381: 1010: 598:Quantum statistical mechanics 4699:. Discusses relation to the 4304:"Quantum chaotic scattering" 2002:{\displaystyle \alpha _{nk}} 1851:distinguishes the primitive 7: 6042:Quantum stochastic calculus 6032:Quantum measurement problem 5954:Mach–Zehnder interferometer 4948:Poincaré recurrence theorem 4578:Quantum signatures of chaos 3971:10.1103/PhysRevLett.53.1515 3477:10.1103/PhysRevLett.74.1538 3397:Quantum signatures of chaos 3364: 3330:{\displaystyle H(x,p;R(t))} 568:Quantum information science 10: 6365: 4943:Poincaré–Bendixson theorem 4489:Gutzwiller, M. C. (1990). 4405:10.1209/0295-5075/83/27005 4357:The Berry–Tabor conjecture 4160:10.1103/PhysRevB.96.094204 3814:10.1103/PhysRevB.96.035442 2234:{\displaystyle \chi _{nk}} 2096:{\displaystyle \chi _{nk}} 982: 6308: 6270: 6222: 6102:Quantum complexity theory 6080:Quantum cellular automata 6055: 5987: 5921: 5834: 5798: 5785:Path integral formulation 5752: 5617: 5574: 5496: 5313: 5295:Swinging Atwood's machine 5240: 5178: 5048: 5035: 4987: 4968: 4938:Krylov–Bogolyubov theorem 4918: 4815: 4755: 4561:10.1017/S0305004100027237 4329:10.4249/scholarpedia.9806 3986:"Quantum Lissajous Scars" 3923:10.1103/PhysRevX.8.041019 3872:10.1103/PhysRevX.8.021062 3239:quantum many-body systems 2907:{\displaystyle \epsilon } 2638:{\displaystyle \epsilon } 1586:Wigner-Dyson distribution 1301:{\displaystyle \epsilon } 6169:Quantum machine learning 6149:Quantum key distribution 6139:Quantum image processing 6129:Quantum error correction 5979:Wheeler's delayed choice 5203:Lotka–Volterra equations 5027:Synchronization of chaos 4830:axiom A dynamical system 4677:What is... Quantum Chaos 4428:"What Is Quantum Chaos?" 4426:Rudnick, Z. (Jan 2008). 4302:Gaspard, Pierre (2014). 4217:10.1088/1361-648x/aaf9fb 3533:10.1103/PhysRevA.51.3604 2013:. The amplitude factor, 925:correspondence principle 603:Quantum machine learning 356:Wheeler's delayed-choice 6085:Quantum finite automata 5188:Double scroll attractor 4953:Stable manifold theorem 4860:False nearest neighbors 3990:Physical Review Letters 3951:Physical Review Letters 3749:10.1103/PhysRevA.53.178 3668:Physical Review Letters 3564:Physical Review Letters 3457:Physical Review Letters 3432:Weidenfeld and Nicolson 313:Leggett–Garg inequality 6189:Quantum neural network 5228:Van der Pol oscillator 5208:Mackey–Glass equations 4840:Box-counting dimension 4509:Stöckmann, Hans-Jürgen 4260:. Tampere University. 3645:10.1098/rspa.1977.0140 3345:Berry–Tabor conjecture 3331: 3270: 3211: 3165: 3039: 3038:{\displaystyle y(0)=0} 2996: 2965: 2928: 2908: 2890:in strong fields, the 2880: 2860: 2823: 2789: 2700: 2639: 2610: 2582: 2553: 2458:gives a phase factor). 2435: 2405: 2355: 2297: 2277: 2235: 2205: 2153: 2097: 2067: 2003: 1973: 1953: 1933: 1932:{\displaystyle nS_{k}} 1903: 1876: 1845: 1818: 1706: 1623: 1614: 1574: 1492: 1468: 1395: 1362: 1342: 1302: 1282: 1255: 1254:{\displaystyle H_{ns}} 1225: 1195: 1139: 1127: 1049: 1042: 996: 901: 104: 6214:Quantum teleportation 5742:Wave–particle duality 5378:Svetlana Jitomirskaya 5285:Multiscroll attractor 5130:Interval exchange map 5083:Dyadic transformation 5068:Complex quadratic map 4910:Topological conjugacy 4845:Correlation dimension 4820:Anosov diffeomorphism 4701:Riemann zeta function 4576:Haake, Fritz (2001). 3395:Haake, Fritz (2001). 3376:Statistical mechanics 3332: 3271: 3212: 3166: 3040: 2997: 2995:{\displaystyle H^{-}} 2966: 2929: 2909: 2881: 2861: 2824: 2790: 2680: 2640: 2604: 2583: 2554: 2436: 2406: 2356: 2298: 2278: 2236: 2206: 2154: 2098: 2068: 2004: 1974: 1954: 1934: 1904: 1902:{\displaystyle S_{k}} 1877: 1875:{\displaystyle T_{k}} 1846: 1819: 1686: 1620: 1604: 1597:Periodic orbit theory 1592:Semiclassical methods 1575: 1493: 1469: 1384: 1363: 1343: 1341:{\displaystyle N^{3}} 1303: 1283: 1281:{\displaystyle H_{s}} 1256: 1226: 1224:{\displaystyle H_{s}} 1196: 1133: 1121: 1043: 1018: 990: 973:Semiclassical methods 891: 298:Elitzur–Vaidman 288:Davisson–Germer 105: 6349:Quantum chaos theory 6245:Quantum field theory 6174:Quantum metamaterial 6119:Quantum cryptography 5849:Consistent histories 5388:Edward Norton Lorenz 3291: 3260: 3175: 3049: 3014: 2979: 2938: 2918: 2898: 2870: 2833: 2801: 2652: 2629: 2581:{\displaystyle d(E)} 2563: 2469: 2419: 2387: 2307: 2287: 2245: 2215: 2163: 2111: 2077: 2017: 1983: 1963: 1943: 1913: 1886: 1859: 1835: 1641: 1505: 1482: 1427: 1418:Poisson distribution 1352: 1325: 1292: 1265: 1235: 1208: 1149: 1023: 995:of classical orbits. 563:Quantum field theory 475:Consistent histories 112:Schrödinger equation 39: 6230:Quantum fluctuation 6199:Quantum programming 6159:Quantum logic gates 6144:Quantum information 6124:Quantum electronics 5599:Classical mechanics 5348:Mitchell Feigenbaum 5290:Population dynamics 5275:Hénon–Heiles system 5135:Irrational rotation 5088:Dynamical billiards 5073:Coupled map lattice 4933:Liouville's theorem 4865:Hausdorff dimension 4850:Conservative system 4835:Bifurcation diagram 4651:Scientific American 4553:1951PCPS...47..790W 4474:1971JMP....12..343G 4397:2008EL.....8327005B 4320:2014SchpJ...9.9806G 4209:2019JPCM...31j5301K 4152:2017PhRvB..96i4204K 4077:2016NatSR...637656L 4012:2019PhRvL.123u4101K 3963:1984PhRvL..53.1515H 3914:2018PhRvX...8d1019C 3863:2018PhRvX...8b1062K 3806:2017PhRvB..96c5442V 3741:1996PhRvA..53..178C 3690:2007PhRvL..98d4103H 3637:1977RSPSA.356..375B 3586:2020PhRvL.125d0605Y 3525:1995PhRvA..51.3604C 3469:1995PhRvL..74.1538C 3223:Wu–Sprung potential 2960: 2855: 2720: 2597:Closed orbit theory 2434:{\displaystyle n=6} 2404:{\displaystyle 1/r} 2367:absorption spectrum 1631:density of states. 1066:solid-state physics 1005:celestial mechanics 949:perturbation theory 898:classical mechanics 351:Stern–Gerlach 148:Classical mechanics 6283:in popular culture 6065:Quantum algorithms 5913:Von Neumann–Wigner 5893:Objective collapse 5604:Old quantum theory 5526:Santa Fe Institute 5393:Aleksandr Lyapunov 5223:Three-body problem 5110:Gingerbreadman map 4997:Bifurcation theory 4875:Lyapunov stability 4695:American Scientist 4531:Eugene Paul Wigner 4436:Notices of the AMS 4055:Scientific Reports 3441:2013-03-08 at the 3327: 3266: 3247:Quantum ergodicity 3207: 3161: 3035: 2992: 2961: 2941: 2924: 2904: 2876: 2856: 2836: 2819: 2785: 2701: 2679: 2635: 2611: 2578: 2549: 2431: 2401: 2351: 2293: 2273: 2231: 2201: 2149: 2093: 2063: 1999: 1969: 1949: 1929: 1899: 1872: 1841: 1814: 1675: 1624: 1615: 1570: 1488: 1464: 1420:of energy levels: 1396: 1358: 1338: 1298: 1288:is separated, and 1278: 1251: 1221: 1191: 1140: 1128: 1050: 1038: 1001:three-body problem 997: 967:quantum ergodicity 902: 539:Von Neumann–Wigner 519:Objective-collapse 318:Mach–Zehnder 308:Leggett inequality 303:Franck–Hertz 153:Old quantum theory 100: 6344:Quantum mechanics 6326: 6325: 6300:Quantum mysticism 6278:Schrödinger's cat 6209:Quantum simulator 6179:Quantum metrology 6107:Quantum computing 6070:Quantum amplifier 6047:Quantum spacetime 6012:Quantum cosmology 6002:Quantum chemistry 5717:Scattering theory 5665:Zero-point energy 5660:Degenerate levels 5568:Quantum mechanics 5534: 5533: 5398:Benoît Mandelbrot 5363:Martin Gutzwiller 5353:Peter Grassberger 5236: 5235: 5218:Rössler attractor 4966: 4965: 4870:Invariant measure 4792:Lyapunov exponent 4660:Martin Gutzwiller 4647:Martin Gutzwiller 4628:978-0-387-98788-0 4606:978-981-02-4711-9 4587:978-3-540-67723-9 4522:978-0-521-59284-0 4500:978-0-387-97173-5 4482:10.1063/1.1665596 4452:Further resources 4267:978-952-03-1699-0 4130:Physical Review B 4085:10.1038/srep37656 3957:(16): 1515–1518. 3892:Physical Review X 3841:Physical Review X 3784:Physical Review B 3729:Physical Review A 3631:(1686): 375–394. 3513:Physical Review A 3406:978-3-540-67723-9 3269:{\displaystyle R} 3229:Recent directions 3205: 3159: 3130: 3095: 2927:{\displaystyle w} 2914:as a function of 2892:Fourier transform 2879:{\displaystyle i} 2758: 2670: 2498: 2296:{\displaystyle m} 1972:{\displaystyle n} 1952:{\displaystyle n} 1844:{\displaystyle k} 1752: 1666: 1611:density of states 1607:Fourier transform 1531: 1491:{\displaystyle s} 1463: 1361:{\displaystyle N} 1134:Computed chaotic 917:dynamical systems 894:quantum mechanics 886: 885: 593:Scattering theory 573:Quantum computing 346:Schrödinger's cat 278:Bell's inequality 86: 61: 30:Quantum mechanics 6356: 6316: 6315: 6027:Quantum geometry 6022:Quantum dynamics 5879:Superdeterminism 5775:Matrix mechanics 5630:Bra–ket notation 5561: 5554: 5547: 5538: 5537: 5506:Butterfly effect 5418:Itamar Procaccia 5368:Brosl Hasslacher 5265:Elastic pendulum 5193:Duffing equation 5140:Kaplan–Yorke map 5058:Arnold's cat map 5046: 5045: 5022:Stability theory 5007:Dynamical system 5002:Control of chaos 4982: 4974: 4958:Takens's theorem 4890:Poincaré section 4760: 4759: 4742: 4735: 4728: 4719: 4718: 4649:(1992 and 2008, 4632: 4615:Reichl, Linda E. 4610: 4591: 4572: 4526: 4504: 4485: 4445: 4444: 4432: 4423: 4417: 4416: 4390: 4370: 4364: 4363: 4362: 4348: 4342: 4341: 4331: 4299: 4293: 4292: 4290: 4288:quant-ph/0403061 4278: 4272: 4271: 4251: 4245: 4244: 4202: 4178: 4172: 4171: 4145: 4121: 4115: 4114: 4104: 4070: 4046: 4040: 4039: 4005: 3981: 3975: 3974: 3942: 3936: 3935: 3925: 3907: 3883: 3877: 3876: 3874: 3856: 3832: 3826: 3825: 3799: 3775: 3769: 3768: 3724: 3718: 3717: 3683: 3663: 3657: 3656: 3620: 3614: 3613: 3579: 3559: 3553: 3552: 3519:(5): 3604–3620. 3508: 3497: 3496: 3463:(9): 1538–1541. 3452: 3446: 3417: 3411: 3410: 3392: 3336: 3334: 3333: 3328: 3275: 3273: 3272: 3267: 3216: 3214: 3213: 3208: 3206: 3204: 3196: 3179: 3170: 3168: 3167: 3162: 3160: 3158: 3150: 3133: 3131: 3126: 3109: 3108: 3096: 3094: 3093: 3092: 3088: 3071: 3070: 3066: 3053: 3044: 3042: 3041: 3036: 3001: 2999: 2998: 2993: 2991: 2990: 2970: 2968: 2967: 2962: 2959: 2954: 2953: 2933: 2931: 2930: 2925: 2913: 2911: 2910: 2905: 2885: 2883: 2882: 2877: 2865: 2863: 2862: 2857: 2854: 2849: 2848: 2828: 2826: 2825: 2820: 2818: 2817: 2816: 2794: 2792: 2791: 2786: 2778: 2777: 2776: 2760: 2759: 2754: 2753: 2744: 2719: 2714: 2713: 2699: 2694: 2678: 2644: 2642: 2641: 2636: 2587: 2585: 2584: 2579: 2558: 2556: 2555: 2550: 2536: 2535: 2499: 2491: 2440: 2438: 2437: 2432: 2410: 2408: 2407: 2402: 2397: 2360: 2358: 2357: 2352: 2344: 2337: 2332: 2331: 2302: 2300: 2299: 2294: 2282: 2280: 2279: 2274: 2260: 2259: 2240: 2238: 2237: 2232: 2230: 2229: 2210: 2208: 2207: 2202: 2200: 2193: 2188: 2187: 2158: 2156: 2155: 2150: 2148: 2141: 2136: 2135: 2102: 2100: 2099: 2094: 2092: 2091: 2072: 2070: 2069: 2064: 2062: 2055: 2050: 2049: 2027: 2008: 2006: 2005: 2000: 1998: 1997: 1978: 1976: 1975: 1970: 1958: 1956: 1955: 1950: 1938: 1936: 1935: 1930: 1928: 1927: 1908: 1906: 1905: 1900: 1898: 1897: 1881: 1879: 1878: 1873: 1871: 1870: 1850: 1848: 1847: 1842: 1823: 1821: 1820: 1815: 1810: 1809: 1802: 1794: 1793: 1778: 1777: 1753: 1751: 1750: 1743: 1738: 1737: 1708: 1705: 1700: 1685: 1684: 1674: 1653: 1652: 1579: 1577: 1576: 1571: 1566: 1565: 1561: 1556: 1555: 1532: 1524: 1497: 1495: 1494: 1489: 1473: 1471: 1470: 1465: 1461: 1457: 1456: 1392:Wigner's surmise 1367: 1365: 1364: 1359: 1347: 1345: 1344: 1339: 1337: 1336: 1307: 1305: 1304: 1299: 1287: 1285: 1284: 1279: 1277: 1276: 1260: 1258: 1257: 1252: 1250: 1249: 1230: 1228: 1227: 1222: 1220: 1219: 1200: 1198: 1197: 1192: 1186: 1185: 1167: 1166: 1047: 1045: 1044: 1039: 878: 871: 864: 505:Superdeterminism 158:Bra–ket notation 109: 107: 106: 101: 93: 88: 87: 79: 67: 62: 60: 49: 21: 20: 6364: 6363: 6359: 6358: 6357: 6355: 6354: 6353: 6329: 6328: 6327: 6322: 6304: 6290:Wigner's friend 6266: 6257:Quantum gravity 6218: 6204:Quantum sensing 6184:Quantum network 6164:Quantum machine 6134:Quantum imaging 6097:Quantum circuit 6092:Quantum channel 6051: 5997:Quantum biology 5983: 5959:Elitzur–Vaidman 5934:Davisson–Germer 5917: 5869:Hidden-variable 5859:de Broglie–Bohm 5836:Interpretations 5830: 5794: 5748: 5635:Complementarity 5613: 5570: 5565: 5535: 5530: 5498: 5492: 5438:Caroline Series 5333:Mary Cartwright 5315: 5309: 5260:Double pendulum 5242: 5232: 5181: 5174: 5100:Exponential map 5051: 5037: 5031: 4989: 4983: 4976: 4962: 4928:Ergodic theorem 4921: 4914: 4905:Stable manifold 4895:Recurrence plot 4811: 4765: 4751: 4746: 4709:by Arnd Bäcker. 4683:(January 2008, 4639: 4629: 4607: 4588: 4535:Dirac, P. A. M. 4523: 4501: 4454: 4449: 4448: 4430: 4424: 4420: 4371: 4367: 4360: 4349: 4345: 4300: 4296: 4279: 4275: 4268: 4252: 4248: 4179: 4175: 4122: 4118: 4047: 4043: 3982: 3978: 3943: 3939: 3884: 3880: 3833: 3829: 3776: 3772: 3725: 3721: 3664: 3660: 3621: 3617: 3560: 3556: 3509: 3500: 3453: 3449: 3443:Wayback Machine 3418: 3414: 3407: 3393: 3389: 3384: 3367: 3355:Riemann surface 3347: 3292: 3289: 3288: 3261: 3258: 3257: 3231: 3219:half derivative 3197: 3180: 3178: 3176: 3173: 3172: 3151: 3134: 3132: 3125: 3101: 3097: 3084: 3080: 3076: 3072: 3062: 3058: 3054: 3052: 3050: 3047: 3046: 3015: 3012: 3011: 3008: 2986: 2982: 2980: 2977: 2976: 2955: 2946: 2945: 2939: 2936: 2935: 2919: 2916: 2915: 2899: 2896: 2895: 2871: 2868: 2867: 2850: 2841: 2840: 2834: 2831: 2830: 2809: 2808: 2804: 2802: 2799: 2798: 2769: 2768: 2764: 2749: 2745: 2743: 2742: 2715: 2706: 2705: 2695: 2684: 2674: 2653: 2650: 2649: 2630: 2627: 2626: 2599: 2564: 2561: 2560: 2531: 2527: 2490: 2470: 2467: 2466: 2451: 2420: 2417: 2416: 2393: 2388: 2385: 2384: 2361:at a classical 2333: 2324: 2320: 2316: 2308: 2305: 2304: 2288: 2285: 2284: 2252: 2248: 2246: 2243: 2242: 2222: 2218: 2216: 2213: 2212: 2189: 2180: 2176: 2172: 2164: 2161: 2160: 2137: 2128: 2124: 2120: 2112: 2109: 2108: 2084: 2080: 2078: 2075: 2074: 2051: 2042: 2038: 2034: 2023: 2018: 2015: 2014: 2009:is the orbit's 1990: 1986: 1984: 1981: 1980: 1964: 1961: 1960: 1944: 1941: 1940: 1923: 1919: 1914: 1911: 1910: 1893: 1889: 1887: 1884: 1883: 1866: 1862: 1860: 1857: 1856: 1853:periodic orbits 1836: 1833: 1832: 1798: 1786: 1782: 1773: 1769: 1759: 1755: 1739: 1730: 1726: 1722: 1712: 1707: 1701: 1690: 1680: 1676: 1670: 1648: 1644: 1642: 1639: 1638: 1599: 1594: 1557: 1551: 1547: 1540: 1536: 1523: 1506: 1503: 1502: 1483: 1480: 1479: 1449: 1445: 1428: 1425: 1424: 1379: 1353: 1350: 1349: 1332: 1328: 1326: 1323: 1322: 1293: 1290: 1289: 1272: 1268: 1266: 1263: 1262: 1242: 1238: 1236: 1233: 1232: 1215: 1211: 1209: 1206: 1205: 1178: 1174: 1162: 1158: 1150: 1147: 1146: 1116: 1103:distributions. 1085:level repulsion 1024: 1021: 1020: 1013: 985: 933:Planck constant 929:classical limit 921:classical chaos 911:focused on how 907:is a branch of 882: 853: 852: 851: 616: 608: 607: 553: 552:Advanced topics 545: 544: 543: 495:Hidden-variable 485:de Broglie–Bohm 464: 462:Interpretations 454: 453: 452: 422: 414: 413: 412: 370: 362: 361: 360: 327: 283:CHSH inequality 272: 264: 263: 262: 191:Complementarity 185: 177: 176: 175: 143: 114: 89: 78: 77: 63: 53: 48: 40: 37: 36: 17: 12: 11: 5: 6362: 6352: 6351: 6346: 6341: 6324: 6323: 6321: 6320: 6309: 6306: 6305: 6303: 6302: 6297: 6292: 6287: 6286: 6285: 6274: 6272: 6268: 6267: 6265: 6264: 6259: 6254: 6253: 6252: 6242: 6237: 6235:Casimir effect 6232: 6226: 6224: 6220: 6219: 6217: 6216: 6211: 6206: 6201: 6196: 6194:Quantum optics 6191: 6186: 6181: 6176: 6171: 6166: 6161: 6156: 6151: 6146: 6141: 6136: 6131: 6126: 6121: 6116: 6115: 6114: 6104: 6099: 6094: 6089: 6088: 6087: 6077: 6072: 6067: 6061: 6059: 6053: 6052: 6050: 6049: 6044: 6039: 6034: 6029: 6024: 6019: 6014: 6009: 6004: 5999: 5993: 5991: 5985: 5984: 5982: 5981: 5976: 5971: 5969:Quantum eraser 5966: 5961: 5956: 5951: 5946: 5941: 5936: 5931: 5925: 5923: 5919: 5918: 5916: 5915: 5910: 5905: 5900: 5895: 5890: 5885: 5884: 5883: 5882: 5881: 5866: 5861: 5856: 5851: 5846: 5840: 5838: 5832: 5831: 5829: 5828: 5823: 5818: 5813: 5808: 5802: 5800: 5796: 5795: 5793: 5792: 5787: 5782: 5777: 5772: 5767: 5762: 5756: 5754: 5750: 5749: 5747: 5746: 5745: 5744: 5739: 5729: 5724: 5719: 5714: 5709: 5704: 5699: 5694: 5689: 5684: 5679: 5674: 5669: 5668: 5667: 5662: 5657: 5652: 5642: 5640:Density matrix 5637: 5632: 5627: 5621: 5619: 5615: 5614: 5612: 5611: 5606: 5601: 5596: 5595: 5594: 5584: 5578: 5576: 5572: 5571: 5564: 5563: 5556: 5549: 5541: 5532: 5531: 5529: 5528: 5523: 5521:Predictability 5518: 5513: 5508: 5502: 5500: 5494: 5493: 5491: 5490: 5488:Lai-Sang Young 5485: 5483:James A. Yorke 5480: 5478:Amie Wilkinson 5475: 5470: 5465: 5460: 5455: 5450: 5445: 5440: 5435: 5430: 5425: 5420: 5415: 5413:Henri Poincaré 5410: 5405: 5400: 5395: 5390: 5385: 5380: 5375: 5370: 5365: 5360: 5355: 5350: 5345: 5340: 5335: 5330: 5325: 5319: 5317: 5311: 5310: 5308: 5307: 5302: 5297: 5292: 5287: 5282: 5280:Kicked rotator 5277: 5272: 5267: 5262: 5257: 5252: 5250:Chua's circuit 5246: 5244: 5238: 5237: 5234: 5233: 5231: 5230: 5225: 5220: 5215: 5210: 5205: 5200: 5195: 5190: 5184: 5182: 5179: 5176: 5175: 5173: 5172: 5170:Zaslavskii map 5167: 5165:Tinkerbell map 5162: 5157: 5152: 5147: 5142: 5137: 5132: 5127: 5122: 5117: 5112: 5107: 5102: 5097: 5096: 5095: 5085: 5080: 5075: 5070: 5065: 5060: 5054: 5052: 5049: 5043: 5033: 5032: 5030: 5029: 5024: 5019: 5014: 5012:Ergodic theory 5009: 5004: 4999: 4993: 4991: 4985: 4984: 4969: 4967: 4964: 4963: 4961: 4960: 4955: 4950: 4945: 4940: 4935: 4930: 4924: 4922: 4919: 4916: 4915: 4913: 4912: 4907: 4902: 4897: 4892: 4887: 4882: 4877: 4872: 4867: 4862: 4857: 4852: 4847: 4842: 4837: 4832: 4827: 4822: 4816: 4813: 4812: 4810: 4809: 4804: 4802:Periodic point 4799: 4794: 4789: 4784: 4779: 4774: 4768: 4766: 4763: 4757: 4753: 4752: 4745: 4744: 4737: 4730: 4722: 4716: 4715: 4710: 4704: 4690: 4674: 4669: 4654: 4638: 4637:External links 4635: 4634: 4633: 4627: 4611: 4605: 4592: 4586: 4573: 4527: 4521: 4505: 4499: 4486: 4468:(3): 343–358. 4453: 4450: 4447: 4446: 4418: 4365: 4343: 4294: 4273: 4266: 4246: 4193:(10): 105301. 4173: 4116: 4041: 3996:(21): 214101. 3976: 3937: 3878: 3827: 3770: 3735:(1): 178–191. 3719: 3658: 3615: 3554: 3498: 3447: 3428:Jim Al-Khalili 3412: 3405: 3386: 3385: 3383: 3380: 3379: 3378: 3373: 3371:Scar (physics) 3366: 3363: 3346: 3343: 3326: 3323: 3320: 3317: 3314: 3311: 3308: 3305: 3302: 3299: 3296: 3282:kicked rotator 3265: 3235:Hilbert spaces 3230: 3227: 3203: 3200: 3195: 3192: 3189: 3186: 3183: 3157: 3154: 3149: 3146: 3143: 3140: 3137: 3129: 3124: 3121: 3118: 3115: 3112: 3107: 3104: 3100: 3091: 3087: 3083: 3079: 3075: 3069: 3065: 3061: 3057: 3034: 3031: 3028: 3025: 3022: 3019: 3007: 3004: 2989: 2985: 2958: 2952: 2949: 2944: 2923: 2903: 2875: 2853: 2847: 2844: 2839: 2815: 2812: 2807: 2796: 2795: 2784: 2781: 2775: 2772: 2767: 2763: 2757: 2752: 2748: 2741: 2738: 2735: 2732: 2729: 2726: 2723: 2718: 2712: 2709: 2704: 2698: 2693: 2690: 2687: 2683: 2677: 2673: 2669: 2666: 2663: 2660: 2657: 2634: 2598: 2595: 2590: 2589: 2577: 2574: 2571: 2568: 2548: 2545: 2542: 2539: 2534: 2530: 2526: 2523: 2520: 2517: 2514: 2511: 2508: 2505: 2502: 2497: 2494: 2489: 2486: 2483: 2480: 2477: 2474: 2463: 2459: 2455: 2450: 2447: 2430: 2427: 2424: 2400: 2396: 2392: 2350: 2347: 2343: 2340: 2336: 2330: 2327: 2323: 2319: 2315: 2312: 2292: 2272: 2269: 2266: 2263: 2258: 2255: 2251: 2228: 2225: 2221: 2199: 2196: 2192: 2186: 2183: 2179: 2175: 2171: 2168: 2147: 2144: 2140: 2134: 2131: 2127: 2123: 2119: 2116: 2090: 2087: 2083: 2061: 2058: 2054: 2048: 2045: 2041: 2037: 2033: 2030: 2026: 2022: 1996: 1993: 1989: 1968: 1948: 1926: 1922: 1918: 1896: 1892: 1869: 1865: 1840: 1825: 1824: 1813: 1808: 1805: 1801: 1797: 1792: 1789: 1785: 1781: 1776: 1772: 1768: 1765: 1762: 1758: 1749: 1746: 1742: 1736: 1733: 1729: 1725: 1721: 1718: 1715: 1711: 1704: 1699: 1696: 1693: 1689: 1683: 1679: 1673: 1669: 1665: 1662: 1659: 1656: 1651: 1647: 1598: 1595: 1593: 1590: 1581: 1580: 1569: 1564: 1560: 1554: 1550: 1546: 1543: 1539: 1535: 1530: 1527: 1522: 1519: 1516: 1513: 1510: 1487: 1475: 1474: 1460: 1455: 1452: 1448: 1444: 1441: 1438: 1435: 1432: 1378: 1375: 1357: 1335: 1331: 1297: 1275: 1271: 1248: 1245: 1241: 1218: 1214: 1202: 1201: 1189: 1184: 1181: 1177: 1173: 1170: 1165: 1161: 1157: 1154: 1115: 1112: 1097:time evolution 1068:, and even to 1037: 1034: 1031: 1028: 1012: 1009: 984: 981: 980: 979: 976: 970: 959: 952: 884: 883: 881: 880: 873: 866: 858: 855: 854: 850: 849: 844: 839: 834: 829: 824: 819: 814: 809: 804: 799: 794: 789: 784: 779: 774: 769: 764: 759: 754: 749: 744: 739: 734: 729: 724: 719: 714: 709: 704: 699: 694: 689: 684: 679: 674: 669: 664: 659: 654: 649: 644: 639: 634: 629: 624: 618: 617: 614: 613: 610: 609: 606: 605: 600: 595: 590: 588:Density matrix 585: 580: 575: 570: 565: 560: 554: 551: 550: 547: 546: 542: 541: 536: 531: 526: 521: 516: 511: 510: 509: 508: 507: 492: 487: 482: 477: 472: 466: 465: 460: 459: 456: 455: 451: 450: 445: 440: 435: 430: 424: 423: 420: 419: 416: 415: 411: 410: 405: 400: 395: 390: 385: 379: 378: 377: 371: 368: 367: 364: 363: 359: 358: 353: 348: 342: 341: 340: 339: 338: 336:Delayed-choice 331:Quantum eraser 326: 325: 320: 315: 310: 305: 300: 295: 290: 285: 280: 274: 273: 270: 269: 266: 265: 261: 260: 259: 258: 248: 243: 238: 233: 228: 223: 221:Quantum number 218: 213: 208: 203: 198: 193: 187: 186: 183: 182: 179: 178: 174: 173: 168: 162: 161: 160: 155: 150: 144: 141: 140: 137: 136: 135: 134: 129: 124: 116: 115: 110: 99: 96: 92: 85: 82: 76: 73: 70: 66: 59: 56: 52: 47: 44: 33: 32: 26: 25: 15: 9: 6: 4: 3: 2: 6361: 6350: 6347: 6345: 6342: 6340: 6337: 6336: 6334: 6319: 6311: 6310: 6307: 6301: 6298: 6296: 6293: 6291: 6288: 6284: 6281: 6280: 6279: 6276: 6275: 6273: 6269: 6263: 6260: 6258: 6255: 6251: 6248: 6247: 6246: 6243: 6241: 6238: 6236: 6233: 6231: 6228: 6227: 6225: 6221: 6215: 6212: 6210: 6207: 6205: 6202: 6200: 6197: 6195: 6192: 6190: 6187: 6185: 6182: 6180: 6177: 6175: 6172: 6170: 6167: 6165: 6162: 6160: 6157: 6155: 6154:Quantum logic 6152: 6150: 6147: 6145: 6142: 6140: 6137: 6135: 6132: 6130: 6127: 6125: 6122: 6120: 6117: 6113: 6110: 6109: 6108: 6105: 6103: 6100: 6098: 6095: 6093: 6090: 6086: 6083: 6082: 6081: 6078: 6076: 6073: 6071: 6068: 6066: 6063: 6062: 6060: 6058: 6054: 6048: 6045: 6043: 6040: 6038: 6035: 6033: 6030: 6028: 6025: 6023: 6020: 6018: 6015: 6013: 6010: 6008: 6007:Quantum chaos 6005: 6003: 6000: 5998: 5995: 5994: 5992: 5990: 5986: 5980: 5977: 5975: 5974:Stern–Gerlach 5972: 5970: 5967: 5965: 5962: 5960: 5957: 5955: 5952: 5950: 5947: 5945: 5942: 5940: 5937: 5935: 5932: 5930: 5927: 5926: 5924: 5920: 5914: 5911: 5909: 5908:Transactional 5906: 5904: 5901: 5899: 5898:Quantum logic 5896: 5894: 5891: 5889: 5886: 5880: 5877: 5876: 5875: 5872: 5871: 5870: 5867: 5865: 5862: 5860: 5857: 5855: 5852: 5850: 5847: 5845: 5842: 5841: 5839: 5837: 5833: 5827: 5824: 5822: 5819: 5817: 5814: 5812: 5809: 5807: 5804: 5803: 5801: 5797: 5791: 5788: 5786: 5783: 5781: 5778: 5776: 5773: 5771: 5768: 5766: 5763: 5761: 5758: 5757: 5755: 5751: 5743: 5740: 5738: 5735: 5734: 5733: 5732:Wave function 5730: 5728: 5725: 5723: 5720: 5718: 5715: 5713: 5710: 5708: 5707:Superposition 5705: 5703: 5702:Quantum state 5700: 5698: 5695: 5693: 5690: 5688: 5685: 5683: 5680: 5678: 5675: 5673: 5670: 5666: 5663: 5661: 5658: 5656: 5655:Excited state 5653: 5651: 5648: 5647: 5646: 5643: 5641: 5638: 5636: 5633: 5631: 5628: 5626: 5623: 5622: 5620: 5616: 5610: 5607: 5605: 5602: 5600: 5597: 5593: 5590: 5589: 5588: 5585: 5583: 5580: 5579: 5577: 5573: 5569: 5562: 5557: 5555: 5550: 5548: 5543: 5542: 5539: 5527: 5524: 5522: 5519: 5517: 5516:Edge of chaos 5514: 5512: 5509: 5507: 5504: 5503: 5501: 5495: 5489: 5486: 5484: 5481: 5479: 5476: 5474: 5473:Marcelo Viana 5471: 5469: 5466: 5464: 5463:Audrey Terras 5461: 5459: 5458:Floris Takens 5456: 5454: 5451: 5449: 5446: 5444: 5441: 5439: 5436: 5434: 5431: 5429: 5426: 5424: 5421: 5419: 5416: 5414: 5411: 5409: 5406: 5404: 5401: 5399: 5396: 5394: 5391: 5389: 5386: 5384: 5381: 5379: 5376: 5374: 5371: 5369: 5366: 5364: 5361: 5359: 5358:Celso Grebogi 5356: 5354: 5351: 5349: 5346: 5344: 5341: 5339: 5338:Chen Guanrong 5336: 5334: 5331: 5329: 5326: 5324: 5323:Michael Berry 5321: 5320: 5318: 5312: 5306: 5303: 5301: 5298: 5296: 5293: 5291: 5288: 5286: 5283: 5281: 5278: 5276: 5273: 5271: 5268: 5266: 5263: 5261: 5258: 5256: 5253: 5251: 5248: 5247: 5245: 5239: 5229: 5226: 5224: 5221: 5219: 5216: 5214: 5211: 5209: 5206: 5204: 5201: 5199: 5198:Lorenz system 5196: 5194: 5191: 5189: 5186: 5185: 5183: 5177: 5171: 5168: 5166: 5163: 5161: 5158: 5156: 5153: 5151: 5148: 5146: 5145:Langton's ant 5143: 5141: 5138: 5136: 5133: 5131: 5128: 5126: 5123: 5121: 5120:Horseshoe map 5118: 5116: 5113: 5111: 5108: 5106: 5103: 5101: 5098: 5094: 5091: 5090: 5089: 5086: 5084: 5081: 5079: 5076: 5074: 5071: 5069: 5066: 5064: 5061: 5059: 5056: 5055: 5053: 5047: 5044: 5041: 5034: 5028: 5025: 5023: 5020: 5018: 5017:Quantum chaos 5015: 5013: 5010: 5008: 5005: 5003: 5000: 4998: 4995: 4994: 4992: 4986: 4981: 4977: 4973: 4959: 4956: 4954: 4951: 4949: 4946: 4944: 4941: 4939: 4936: 4934: 4931: 4929: 4926: 4925: 4923: 4917: 4911: 4908: 4906: 4903: 4901: 4898: 4896: 4893: 4891: 4888: 4886: 4883: 4881: 4878: 4876: 4873: 4871: 4868: 4866: 4863: 4861: 4858: 4856: 4853: 4851: 4848: 4846: 4843: 4841: 4838: 4836: 4833: 4831: 4828: 4826: 4825:Arnold tongue 4823: 4821: 4818: 4817: 4814: 4808: 4805: 4803: 4800: 4798: 4795: 4793: 4790: 4788: 4785: 4783: 4780: 4778: 4775: 4773: 4770: 4769: 4767: 4761: 4758: 4754: 4750: 4743: 4738: 4736: 4731: 4729: 4724: 4723: 4720: 4714: 4713:ChaosBook.org 4711: 4708: 4705: 4702: 4698: 4696: 4691: 4688: 4687: 4682: 4681:Ze'ev Rudnick 4678: 4675: 4673: 4670: 4668: 4664: 4661: 4658: 4657:Quantum Chaos 4655: 4652: 4648: 4644: 4643:Quantum Chaos 4641: 4640: 4630: 4624: 4620: 4616: 4612: 4608: 4602: 4598: 4593: 4589: 4583: 4579: 4574: 4570: 4566: 4562: 4558: 4554: 4550: 4546: 4542: 4541: 4536: 4532: 4528: 4524: 4518: 4514: 4510: 4506: 4502: 4496: 4492: 4487: 4483: 4479: 4475: 4471: 4467: 4463: 4462: 4456: 4455: 4442: 4438: 4437: 4429: 4422: 4414: 4410: 4406: 4402: 4398: 4394: 4389: 4384: 4380: 4376: 4369: 4359: 4358: 4353: 4352:Marklof, Jens 4347: 4339: 4335: 4330: 4325: 4321: 4317: 4313: 4309: 4305: 4298: 4289: 4284: 4277: 4269: 4263: 4259: 4258: 4250: 4242: 4238: 4234: 4230: 4226: 4222: 4218: 4214: 4210: 4206: 4201: 4196: 4192: 4188: 4184: 4177: 4169: 4165: 4161: 4157: 4153: 4149: 4144: 4139: 4136:(9): 094204. 4135: 4131: 4127: 4120: 4112: 4108: 4103: 4098: 4094: 4090: 4086: 4082: 4078: 4074: 4069: 4064: 4060: 4056: 4052: 4045: 4037: 4033: 4029: 4025: 4021: 4017: 4013: 4009: 4004: 3999: 3995: 3991: 3987: 3980: 3972: 3968: 3964: 3960: 3956: 3952: 3948: 3941: 3933: 3929: 3924: 3919: 3915: 3911: 3906: 3901: 3898:(4): 041019. 3897: 3893: 3889: 3882: 3873: 3868: 3864: 3860: 3855: 3850: 3847:(2): 021062. 3846: 3842: 3838: 3831: 3823: 3819: 3815: 3811: 3807: 3803: 3798: 3793: 3790:(3): 035442. 3789: 3785: 3781: 3774: 3766: 3762: 3758: 3754: 3750: 3746: 3742: 3738: 3734: 3730: 3723: 3715: 3711: 3707: 3703: 3699: 3695: 3691: 3687: 3682: 3677: 3674:(4): 044103. 3673: 3669: 3662: 3654: 3650: 3646: 3642: 3638: 3634: 3630: 3626: 3619: 3611: 3607: 3603: 3599: 3595: 3591: 3587: 3583: 3578: 3573: 3570:(4): 040605. 3569: 3565: 3558: 3550: 3546: 3542: 3538: 3534: 3530: 3526: 3522: 3518: 3514: 3507: 3505: 3503: 3494: 3490: 3486: 3482: 3478: 3474: 3470: 3466: 3462: 3458: 3451: 3444: 3440: 3437: 3433: 3429: 3425: 3421: 3420:Michael Berry 3416: 3408: 3402: 3398: 3391: 3387: 3377: 3374: 3372: 3369: 3368: 3362: 3360: 3356: 3352: 3342: 3340: 3318: 3312: 3309: 3306: 3303: 3300: 3294: 3285: 3283: 3279: 3263: 3254: 3252: 3248: 3242: 3240: 3236: 3226: 3224: 3220: 3201: 3198: 3190: 3184: 3181: 3155: 3152: 3144: 3138: 3135: 3127: 3122: 3119: 3113: 3105: 3102: 3098: 3089: 3085: 3081: 3077: 3073: 3067: 3063: 3059: 3055: 3032: 3029: 3023: 3017: 3003: 2987: 2983: 2972: 2956: 2942: 2921: 2901: 2893: 2889: 2888:Rydberg atoms 2873: 2851: 2837: 2805: 2782: 2765: 2761: 2750: 2746: 2739: 2736: 2733: 2730: 2724: 2721: 2716: 2702: 2691: 2688: 2685: 2681: 2675: 2671: 2667: 2661: 2655: 2648: 2647: 2646: 2632: 2623: 2621: 2620:Rydberg atoms 2615: 2608: 2607:Rydberg atoms 2603: 2594: 2572: 2566: 2540: 2537: 2528: 2524: 2521: 2515: 2509: 2506: 2495: 2492: 2487: 2484: 2478: 2472: 2464: 2460: 2456: 2453: 2452: 2446: 2442: 2428: 2425: 2422: 2414: 2398: 2394: 2390: 2382: 2379: 2374: 2370: 2368: 2364: 2348: 2345: 2338: 2334: 2328: 2325: 2321: 2313: 2310: 2290: 2270: 2267: 2264: 2261: 2256: 2253: 2249: 2226: 2223: 2219: 2194: 2190: 2184: 2181: 2177: 2169: 2166: 2142: 2138: 2132: 2129: 2125: 2117: 2114: 2106: 2088: 2085: 2081: 2056: 2052: 2046: 2043: 2039: 2031: 2028: 2024: 2020: 2012: 1994: 1991: 1987: 1966: 1946: 1924: 1920: 1916: 1894: 1890: 1867: 1863: 1854: 1838: 1830: 1811: 1803: 1799: 1795: 1790: 1787: 1783: 1779: 1774: 1770: 1766: 1760: 1756: 1744: 1740: 1734: 1731: 1727: 1719: 1716: 1713: 1709: 1697: 1694: 1691: 1687: 1681: 1677: 1671: 1667: 1663: 1657: 1649: 1645: 1637: 1636: 1635: 1632: 1629: 1619: 1612: 1608: 1603: 1589: 1587: 1567: 1562: 1558: 1552: 1548: 1544: 1541: 1537: 1533: 1528: 1525: 1520: 1514: 1508: 1501: 1500: 1499: 1485: 1458: 1453: 1450: 1446: 1442: 1436: 1430: 1423: 1422: 1421: 1419: 1413: 1410: 1404: 1401: 1400:Random matrix 1393: 1388: 1383: 1374: 1370: 1355: 1333: 1329: 1318: 1314: 1310: 1295: 1273: 1269: 1246: 1243: 1239: 1216: 1212: 1187: 1182: 1179: 1175: 1171: 1168: 1163: 1159: 1155: 1152: 1145: 1144: 1143: 1137: 1132: 1125: 1120: 1111: 1109: 1104: 1102: 1098: 1094: 1090: 1086: 1081: 1079: 1075: 1071: 1067: 1063: 1059: 1055: 1035: 1032: 1029: 1026: 1017: 1008: 1006: 1002: 994: 989: 977: 974: 971: 968: 964: 960: 957: 953: 950: 946: 945: 944: 941: 938: 934: 930: 926: 922: 918: 914: 910: 906: 905:Quantum chaos 899: 895: 890: 879: 874: 872: 867: 865: 860: 859: 857: 856: 848: 845: 843: 840: 838: 835: 833: 830: 828: 825: 823: 820: 818: 815: 813: 810: 808: 805: 803: 800: 798: 795: 793: 790: 788: 785: 783: 780: 778: 775: 773: 770: 768: 765: 763: 760: 758: 755: 753: 750: 748: 745: 743: 740: 738: 735: 733: 730: 728: 725: 723: 720: 718: 715: 713: 710: 708: 705: 703: 700: 698: 695: 693: 690: 688: 685: 683: 680: 678: 675: 673: 670: 668: 665: 663: 660: 658: 655: 653: 650: 648: 645: 643: 640: 638: 635: 633: 630: 628: 625: 623: 620: 619: 612: 611: 604: 601: 599: 596: 594: 591: 589: 586: 584: 581: 579: 578:Quantum chaos 576: 574: 571: 569: 566: 564: 561: 559: 556: 555: 549: 548: 540: 537: 535: 534:Transactional 532: 530: 527: 525: 524:Quantum logic 522: 520: 517: 515: 512: 506: 503: 502: 501: 498: 497: 496: 493: 491: 488: 486: 483: 481: 478: 476: 473: 471: 468: 467: 463: 458: 457: 449: 446: 444: 441: 439: 436: 434: 431: 429: 426: 425: 418: 417: 409: 406: 404: 401: 399: 396: 394: 391: 389: 386: 384: 381: 380: 376: 373: 372: 366: 365: 357: 354: 352: 349: 347: 344: 343: 337: 334: 333: 332: 329: 328: 324: 321: 319: 316: 314: 311: 309: 306: 304: 301: 299: 296: 294: 291: 289: 286: 284: 281: 279: 276: 275: 268: 267: 257: 254: 253: 252: 251:Wave function 249: 247: 244: 242: 239: 237: 234: 232: 231:Superposition 229: 227: 224: 222: 219: 217: 214: 212: 209: 207: 204: 202: 199: 197: 194: 192: 189: 188: 181: 180: 172: 169: 167: 164: 163: 159: 156: 154: 151: 149: 146: 145: 139: 138: 133: 130: 128: 125: 123: 120: 119: 118: 117: 113: 80: 74: 57: 54: 50: 42: 35: 34: 31: 28: 27: 23: 22: 19: 6339:Chaos theory 6037:Quantum mind 6006: 5949:Franck–Hertz 5811:Klein–Gordon 5760:Formulations 5753:Formulations 5682:Interference 5672:Entanglement 5650:Ground state 5645:Energy level 5618:Fundamentals 5582:Introduction 5468:Mary Tsingou 5433:David Ruelle 5428:Otto Rössler 5373:Michel Hénon 5343:Leon O. Chua 5300:Tilt-A-Whirl 5270:FPUT problem 5155:Standard map 5150:Logistic map 5016: 4975: 4749:Chaos theory 4694: 4684: 4665:2(12):3146. 4663:Scholarpedia 4650: 4618: 4596: 4577: 4544: 4538: 4512: 4490: 4465: 4459: 4440: 4434: 4421: 4381:(2): 27005. 4378: 4374: 4368: 4356: 4346: 4311: 4308:Scholarpedia 4307: 4297: 4276: 4256: 4249: 4190: 4186: 4176: 4133: 4129: 4119: 4061:(1): 37656. 4058: 4054: 4044: 3993: 3989: 3979: 3954: 3950: 3940: 3895: 3891: 3881: 3844: 3840: 3830: 3787: 3783: 3773: 3732: 3728: 3722: 3681:nlin/0610053 3671: 3667: 3661: 3628: 3624: 3618: 3567: 3563: 3557: 3516: 3512: 3460: 3456: 3450: 3423: 3415: 3396: 3390: 3348: 3338: 3286: 3278:standard map 3255: 3243: 3232: 3009: 2973: 2797: 2624: 2616: 2612: 2591: 2443: 2375: 2371: 2011:Maslov index 1826: 1633: 1625: 1582: 1476: 1414: 1407:of freedom ( 1405: 1397: 1387:Rydberg atom 1371: 1319: 1315: 1311: 1203: 1141: 1136:Rydberg atom 1124:Rydberg atom 1108:Dyson series 1105: 1093:spectroscopy 1082: 1051: 998: 993:bifurcations 942: 904: 903: 577: 433:Klein–Gordon 369:Formulations 206:Energy level 201:Entanglement 184:Fundamentals 171:Interference 122:Introduction 18: 6295:EPR paradox 6075:Quantum bus 5944:Double-slit 5922:Experiments 5888:Many-worlds 5826:Schrödinger 5790:Phase space 5780:Schrödinger 5770:Interaction 5727:Uncertainty 5697:Nonlocality 5692:Measurement 5687:Decoherence 5677:Hamiltonian 5453:Nina Snaith 5443:Yakov Sinai 5328:Rufus Bowen 5078:Duffing map 5063:Baker's map 4988:Theoretical 4900:SRB measure 4807:Phase space 4777:Bifurcation 4443:(1): 32–34. 4314:(6): 9806. 2378:anisotropic 2363:bifurcation 1829:Berry phase 956:Hamiltonian 822:von Neumann 807:Schrödinger 583:EPR paradox 514:Many-worlds 448:Schrödinger 403:Schrödinger 398:Phase-space 388:Interaction 293:Double-slit 271:Experiments 246:Uncertainty 216:Nonlocality 211:Measurement 196:Decoherence 166:Hamiltonian 6333:Categories 6223:Extensions 6057:Technology 5903:Relational 5854:Copenhagen 5765:Heisenberg 5712:Tunnelling 5575:Background 5511:Complexity 5408:Edward Ott 5255:Convection 5180:Continuous 4855:Ergodicity 4547:(4): 790. 4200:1806.02598 4143:1710.00585 4068:1511.04198 4003:1911.09729 3905:1712.06836 3854:1712.02665 3797:1611.08879 3577:2003.07267 3382:References 3359:integrable 1089:scattering 1074:microwaves 1011:Approaches 915:classical 817:Sommerfeld 732:Heisenberg 727:Gutzwiller 667:de Broglie 615:Scientists 529:Relational 480:Copenhagen 383:Heisenberg 241:Tunnelling 142:Background 5929:Bell test 5799:Equations 5625:Born rule 5423:Mary Rees 5383:Bryna Kra 5316:theorists 5125:Ikeda map 5115:Hénon map 5105:Gauss map 4787:Limit set 4772:Attractor 4569:120852535 4388:0804.3685 4338:1941-6016 4225:0953-8984 4168:119083672 4093:2045-2322 4036:208248295 3932:2160-3308 3822:119028983 3757:1050-2947 3706:0031-9007 3653:0080-4630 3610:212725801 3541:1050-2947 3485:0031-9007 3349:In 1977, 3339:many-body 3128:π 3103:− 2988:− 2902:ϵ 2806:ϕ 2766:ϕ 2762:− 2756:~ 2734:π 2725:⁡ 2697:∞ 2682:∑ 2672:∑ 2633:ϵ 2533:′ 2510:⁡ 2501:ℑ 2496:π 2488:− 2322:χ 2314:⁡ 2268:π 2250:χ 2220:χ 2178:χ 2170:⁡ 2126:χ 2118:⁡ 2082:χ 2040:χ 2032:⁡ 1988:α 1796:π 1784:α 1780:− 1728:χ 1720:⁡ 1703:∞ 1688:∑ 1668:∑ 1545:π 1542:− 1526:π 1451:− 1296:ϵ 1172:ε 1101:amplitude 1070:acoustics 1062:molecular 1033:− 1027:ϵ 958:(system). 847:Zeilinger 692:Ehrenfest 421:Equations 98:⟩ 95:Ψ 84:^ 72:⟩ 69:Ψ 46:ℏ 6318:Category 6112:Timeline 5864:Ensemble 5844:Bayesian 5737:Collapse 5609:Glossary 5592:Timeline 5499:articles 5241:Physical 5160:Tent map 5050:Discrete 4990:branches 4920:Theorems 4756:Concepts 4617:(2004). 4511:(1999). 4413:53550992 4241:51693305 4233:30566927 4111:27892510 4028:31809168 3714:17358777 3602:32794812 3493:10059054 3439:Archived 3365:See also 3280:and the 2283:, where 2211:, where 2159:becomes 1348:, where 772:Millikan 697:Einstein 682:Davisson 637:Blackett 622:Aharonov 490:Ensemble 470:Bayesian 375:Overview 256:Collapse 236:Symmetry 127:Glossary 6271:Related 6250:History 5989:Science 5821:Rydberg 5587:History 5497:Related 5305:Weather 5243:systems 5036:Chaotic 4782:Fractal 4549:Bibcode 4470:Bibcode 4393:Bibcode 4316:Bibcode 4205:Bibcode 4148:Bibcode 4102:5124902 4073:Bibcode 4008:Bibcode 3959:Bibcode 3910:Bibcode 3859:Bibcode 3802:Bibcode 3765:9912872 3737:Bibcode 3686:Bibcode 3633:Bibcode 3582:Bibcode 3549:9912027 3521:Bibcode 3465:Bibcode 3434:2003), 1609:of the 1054:nuclear 983:History 935:to the 923:?" 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