Quantum Signatures of Chaos or Quantum Chaos? - Springer Link

c Pleiades Publishing, Ltd., 2016. ISSN 1063-7788, Physics of Atomic Nuclei, 2016, Vol. 79, No. 6, pp. 995–1009. c V. E. Bunakov, 2016, published in Yadernaya Fizika, 2016, Vol. 79, No. 6, pp. 679–693. Original Russian Text

NUCLEI Theory

Quantum Signatures of Chaos or Quantum Chaos? V. E. Bunakov* St. Petersburg State University, Universitetskaya naberezhnaya 7-9, St. Petersburg, 199034 Russia Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, Gatchina, 188300 Russia Received January 26, 2016

Abstract—A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory. DOI: 10.1134/S1063778816060053

1. PROBLEMS OF CHAOS IN CLASSICAL MECHANICS

1.1. Searches for the Most General Physical Reasons behind Chaos For the first time, contemporary physics came across the concept of chaos when L. Boltzmann developed the kinetic theory of gases and derived respective kinetic equations. In doing this, he had to invoke the language of statistical mechanics and to introduce the hypothesis of molecular chaos. According to this hypothesis, the system being considered forgets fast the details of its previous evolution (weakening of correlations, according to the terminology frequently used at the present time). Although the validity of kinetic theory is no longer questioned, some issues concerning the derivation of kinetic equations have been hotly debated to date (for an overview, see, for example, [1, 2]). Since then, the majority of physicists had thought that it was a large number of degrees of freedom in macroscopic systems that dictated the abandonment of the idea of solving dynamical equations and resort to a probabilistic statistical description of the evolution of such systems. The prevalent opinion was that the behavior of simple systems featuring a small number of degrees of freedom is fully deterministic—it was only necessary in that case to go over to canonical variables and to solve respective equations of motion for preset initial conditions. As far back as the end of XIX century, however, mathematicians and mechanicians began to realize *

E-mail: [email protected]

that the motion of systems featuring even a very small number of degrees of freedom may be unpredictable if their trajectories are unstable against very small variations in initial conditions. For unstable systems, the distance between initially very close trajectories in the respective phase space increases with time in proportion to exp(Λt). The rate Λ at which the trajectories diverge is referred to as Lyapunov’s exponent. If the phase space of an unstable system is bounded, its phase trajectories begin to intersect repeatedly one another with time and become entangled. Since the accuracy to which one knows initial conditions is finite in any case of practical importance, an ordinary dynamical description of such an unstable system is meaningless—by integrating the respective equations of motion, we cannot predict the position of the system in the phase space at the instant t. After the lapse of a rather long time interval, the system may find its way to any point of the accessible phase space. Such a situation is referred to as dynamical chaos. Such systems are spectacularly exemplified by chaotic billiards, a family of which was demonstrated to physicists in the 1960s and 1970s. By a billiard, one means the motion of a material point in a plane bounded by elastically reflecting walls of specific shape. Although this system has only two degrees of freedom, it turns out that its motion can be either regular or chaotic, depending on the form of reflecting walls. In the case of a round billiard (that is, that which is bounded by a circle), the motion is stable (regular). But, in the case of a stadium (bounded by two parallel straight-line segments whose ends are connected by two semicircles), the motion becomes 995

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a current vogue. In view of this, it is highly desirable to find a quantitative criterion of stochastic behavior of the systems being considered. In doing this, it is natural to seek for them a dimensionless parameter that would vanish in the case of a regular stable motion and grow in response to the development of chaos.

py

y

´ Fig. 1. Phase portrait of the Henon–Heiles system for the energy of E = 1/12.

unstable. This gave sufficient grounds to recognize that even extremely simple systems may be chaotic. Attempts were made to explain the instability of motion by the nonlinearity of the equations that describe it. However, it appeared that by no means does the nonlinearity of an equation lead to an instability of its solutions. Stable solutions are exemplified by solitons for the Korteweg-de Vries equation or solutions of the equations for Toda lattices (the set of nonlinear equations that describes the dynamics of coupled nonlinear oscillators). This entailed the abandonment of the assumption that nonlinearity is responsible for the emergence of dynamical chaos (despite this, the trend toward including studies on dynamical chaos in the realms of nonlinear dynamics has persisted to date). In searches for general reasons behind chaos in classical mechanics, it therefore seemed mandatory to address Lyapunov’s instability of trajectories against small variations in initial conditions.

1.2. Searches for Quantitative Criteria of Chaos It was of paramount importance to understand that regular stable systems are idealized models and that the majority of the systems in nature are chaotic to some extent. In fact, they differ only in the characteristic time within which their instability becomes noticeable. This means that, in all fields of physics, one has or will have to deal with the problem of chaos. Therefore, investigations into chaos form a line of research of crucial importance in many fields of fundamental and applied science rather than being

It seems that Lyapunov’s exponent Λ could serve as such a criterion, and there were attempts at employing it. However, the exponent Λ is not dimensionless; moreover, it is a function of coordinates in the phase space and therefore changes upon going over from one trajectory to another. Therefore, phase portraits of the system being considered are frequently used to describe finite motions of the system. They are especially vivid in the case of Hamiltonian systems featuring two degrees of freedom. In that case, the four-dimensional phase space of such a system can be cut by a two-dimensional surface (Poincare´ section), whereby the energy of the system and one of its coordinates (for example, x) are fixed. The trajectories of the system intersect consecutively this surface, and the set of these intersection points forms the phase portrait of the system. In the case of regular motion, the points of consecutive intersections lie on rather ´ smooth closed curves. The Henon–Heiles problem, which will be discussed in detail in Subsection 3.6, is one of the most thoroughly studied examples of the transition from regular to chaotic motion. An increase in the degree of chaos in the system is due to the growth of its energy E in relative units—the system is regular at E = 0 but becomes chaotic as E grows. Figure 1 shows the phase portrait of this system at E = 1/12. The smooth closed curves in this figure are indicative of a regular behavior of the system. In the case of fully developed chaotic motion (frequently called “hard chaos”), the intersection points cover the whole surface accessible to motion absolutely at random as in Fig. 2, which shows the phase portrait ´ of the Henon–Heiles system at E = 1/6. Such phase portraits are quite readily constructed with the aid of modern computers and are frequently used in the literature. Clearly, the concept of a “rather smooth curve" is hardly appropriate in exact science, and it is necessary to invent a means for going over from phase portraits to dimensionless parameters of chaos in order to employ, in a quantitative analysis of intermediate situations between a regular behavior and hard chaos, phase portraits obtained from numerical calculations. Therefore, the method of maps, within which attempts are made to analyze the functional dependence between the coordinates of the phasepicture points arising as the trajectories consecutively intersect the chosen Poincare´ section, is frequently used in the literature. In principle, this dependence PHYSICS OF ATOMIC NUCLEI Vol. 79

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QUANTUM SIGNATURES OF CHAOS

py

y

´ Fig. 2. Phase portrait of the Henon–Heiles system for the energy of E = 1/6.

is determined by the dynamical equations that describe the system, but, in the majority of cases, it is very intricate. In view of this, several versions of its simplified analytic description were proposed. Within each of them, it became possible to introduce a parameter that characterizes the degree of chaos in the system being considered, which vanishes in the case of regular motion, and which exceeds a value of about unity for hard chaos (see, for example, [1–3]). Unfortunately, it was impossible to find rather simple map versions for the majority of realistic systems. Therefore, searches for a universal chaos criterion has remained a problem of great importance. 2. PROBLEMS OF CHAOS IN QUANTUM MECHANICS Since the uncertainty relation in quantum mechanics renders the concept of a trajectory inaccurate, classical stability or chaos criteria with this concept become inapplicable. This served as a motivation for the assertion made previously that chaos is impossible in quantum systems. In the case of stationary problems for discrete bound states, the deeply rooted concept of the irremovable connection between chaos and nonlinearity gave additional support to this asser¨ tion in view of the linearity of the Schrodinger equation and a seemingly complete loss of connections with classical phase space and time. In the course of time, assertions of this type became more lenient. The present-day definition of quantum chaos in Wikipedia as the dynamics of quantum systems that are chaotic in the classical limit is quite vague. A more detailed statement says that the most successful approach to quantum chaos consists in addressing the problem of identifying those properties of quantum systems that PHYSICS OF ATOMIC NUCLEI

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correspond to chaos in classical systems, while ignoring the problem of defining chaos in quantum mechanics. Within this approach, one skips the problem of chaos in quantum systems, trying to understand instead whether the properties of quantum analogs of classically chaotic systems differ from the properties of quantum versions of classically regular systems. All this corresponds to the comment made by Berry [4] more than 20 years ago: “The incorrect term quantum chaos means quantum phenomena characteristic of classically chaotic systems, quantum ’signatures’ of classical chaos.” It is proposed to perform searches for such signatures according to the following procedur. One specifies a regular and a chaotic system in classical mechanics and, on the basis of the correspondence rules, constructs their quantum analogs ¨ (Schrodinger equations with corresponding Hamiltonians), whereupon one compares the properties of the eigenfunctions and eigenvalues of these equations, nourishing the hope for finding some distinction that could be called a quantum signature of classical chaos. Over 40 years of searches, the law of the distribution of levels proved to be the only signature of this kind recognized more or less universally (see, for example, [5-7]). For quantum analogs of chaotic systems, the distribution of level spacings closely follows Wigner’s law, which involves a characteristic repulsion of levels. In accordance with this law, the probability for finding a level neighboring a given one and differing from it in energy by ε is given by the expression πε πε2 exp − 2 , (1) P (ε) = 2D2 4D where D is the average level spacing. By repulsion, one means the vanishing of the probability for finding the neighboring level as ε → 0. No repulsion of this kind was found for regular systems. Frequently, it is stated that the distribution of levels for regular quantum systems (that is, for quantum analogs of classically regular systems) is described by Poisson’s law, ε 1 exp − . (2) P (ε) = D D Since the majority of real systems occupy an intermediate position between a regular behavior and hard chaos, it would be desirable to find a numerical criterion that would specify the degree of chaos in the system. Attempts were made to formulate such a criterion by using various kinds of purely phenomenological interpolations [5, 6] between the laws in (1) and (2). An approximate relation between the level density in a quantum system and the trajectories in the phase

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space of its classical analog could be found with the aid of Gutzwiller’s trace formula (the most detailed analysis of the approximations and hypotheses underlying its derivation was given in the excellent monograph of Gutzwiller [7]). Strictly speaking, it is valid only in the semiclassical limit of → 0, and this was also used by opponents of the concept of quantum chaos. Even the possible absence of analiticity of this formula for → 0 was mentioned [8]. This formula has yet another drawback indicated by Gutzwiller himself. It gives no way to describe a transition region between a regular behavior and hard chaos (the region of so-called soft chaos). Therefore, the only quantum signature found to date for chaotic behavior may serve as its qualitative rather than quantitative feature. Since Wigner’s law holds for compound resonances of complex structure, attempts were sometimes made [9, 10] to characterize the chaotic character of nuclear states by the degree of complexity of respective wave functions. With the aid of the expansion cik Φk (3) Ψα = k

of the eigenfunction for a complicated compound state Ψα in the wave functions Φk for the Hamiltonian of the noninteracting-particle model, one introduces the concept of an information entropy for the state Ψα , |cαk |2 ln |cαk |2 , (4) Sα = − k

or the number Nα of "main components" in this wave function, −1 4 i ck . (5) Nα = k

It is the opinion of the present author that the very idea of quantum signatures of classical chaos is incorrect, since classical mechanics is a liming case of quantum mechanics, which is more general, and not vice versa. Indeed, everybody is aware of the disappearance of a number of quantum effects (including discrete spectra and interference phenomena) upon this limiting transition. Quantum picture is depleted in going over to classical mechanics, but by no means is it enriched in some specifically classical phenomena. It follows that the concept of quantum signatures of classical chaos is as incorrect as the assertion that relativistic mechanics is a "relativistic signature" of Newton’s mechanics. It would be more natural to seek quantum criteria of chaos and then address the question of what “classical signatures” may stem from them.

It is noteworthy that, while the distribution laws in Eqs. (1) and (2) reflect tentative attempts at finding some connections with trajectories of classical chaotic and regular systems by means of Gutzwiller’s formulas (albeit marred by the aforementioned flaws), the quantum-chaos criteria in Eqs. (4) and (5) rely exclusively on semi-intuitive arguments concerning the complexity of wave functions for compound resonances distributed according Wigner’s law in Eq. (1). The aforementioned trend toward ignoring the problem of defining quantum chaos led to examining, in more recent studies on chaos in nuclear physics (see, for example, [10–12]), applications of various modifications of the random-matrix method to various nuclear phenomena without making any attempt at unearthing the connection between this method and classical chaos. Possibly, it is necessary for this to disclose more profound reasons behind chaos, common both to classical and to quantum mechanics. It is the approach that was proposed by the present author for Hamiltonian systems (see, for example, [13–18]). This article contains an outline of basic principles of this approach and a survey of the results obtained within this conceptual framework. 3. APPROACH BASED ON THE LIOUVILLE–ARNOLD THEOREM

3.1. Connection between Chaos and the Symmetry of the System Thus, not only do we have at our disposal no quantitative criterion of chaos for quantum systems, but also the very definition of quantum chaos is absent. As was indicated above, this is because searches for the most general sources of chaos in classical mechanics ended up in choosing Lyapunov’s instability of trajectories against small variations in initial conditions. A definition of both classical and quantum chaos on the basis of the Liouville–Arnold theorem well known in classical mechanics (see, for example, [2]) is proposed here instead of semi-intuitive guesses concerning the nature and features of quantum signatures of chaos. This theorem states that a system featuring N degrees of freedom is regular if it has M = N linearly independent first integrals of motion in involution. First (global) integrals of motion are those that, by Noether’s theorem, are associated with the symmetry of the system (that is, with the presence of a group of transformations under which the Hamiltonian of the system is invariant). Thus, a classical regular system possesses a symmetry that is sufficiently high for this system to have integrals of motion (conservation laws) whose number M coincides with the number N of its degrees of freedom. The systems becomes chaotic as soon as symmetry PHYSICS OF ATOMIC NUCLEI Vol. 79

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breaking reduces the number of first integrals, so that M proves to be less than N . In contrast to the concept of trajectories, which is applicable only in classical mechanics, the concept of symmetry is applicable in all realms of physics— from classical mechanics to quantum field theory. A "good" quantum number (eigenvalue of an operator that commutes with the Hamiltonian of the system) is the quantum analog of a first integral of motion. Therefore, it seems natural to define a regular quantum system as that whose Hamiltonian possesses a sufficiently high symmetry guaranteeing that the number M of good quantum numbers of the system is not less than the number N of its degrees of freedom. If, in the system being considered, there arises a perturbing interaction that breaks its symmetry, thereby reducing the number of good quantum numbers, so that M < N , the system ceases to be regular. Therefore, it is natural to define a chaotic quantum system as that whose symmetry is so low that the number of its good quantum numbers is smaller than the number of its degrees of freedom. It is precisely this definition of quantum chaos that was proposed earlier in [13–16].

P(ε) 2.5

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(a)

2.0 1.5 1.0 0.5 0 1.0

(b)

0.8 0.6 0.4 0.2

3.2. Criticism of Quantum Signatures of Classical Chaos We will now see that the above definition of quantum chaos makes it possible to understand quite readily the physical meaning of quantum signatures of classical chaos that were mentioned above. The presence of degenerate levels is a typical property of a quantum regular system that possesses a high symmetry (for example, this is so for an ndimensional isotropic harmonic oscillator or for a charged particle in a central Coulomb potential). Of course, the distribution of level spacings, P (ε) , in such systems is described by the delta function δ(ε) rather than by Poisson’s law in Eq. (2). A perturbation that breaks the symmetry of such a system removes degeneracy; that is, it leads to the repulsion of levels that, in the absence of perturbations, correspond to the same energy value. In the case of moderately small perturbations, this leads to a distribution P (ε) having a sharp maximum at an ε value that is much smaller than D and which is determined by the average splitting of originally degenerate levels. An example of such a distribution for the quantum ´ Henon–Heiles problem (nonlinearly perturbed twodimensional isotropic harmonic oscillator, for which it is common practice to use the excitation energy E of the system in conventional units as a degree of perturbation—for more details, see below) considered in [14] is given in Fig. 3a. Of course, no interpolation between Poisson’s and Wigner’s laws describes PHYSICS OF ATOMIC NUCLEI

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0

1

2

3

4 ε/D

Fig. 3. Distribution of spacings between neighboring ´ levels in the Henon–Heiles system in the excitationenergy ranges of (a) 0.01–0.03 and (b) 0.05−0.07. The dashed curve corresponds to Wigner’s distribution, while the dash-dotted curve corresponds to a distribution that obeys Poisson’s law.

such a distribution. As the perturbation grows, the maximum of the distribution gradually moves to the region of ε approximately equal to D (see Fig. 3b), while the distribution itself tends to Wigner’s distribution. However, it is impossible to describe this ¨ transition by any simple analytic formula. Hoenig and Wintgen [19], who studied the energy levels of the hydrogen atom in a uniform magnetic field, arrived at similar conclusions. Thus, Wigner’s repulsion of levels is nothing but the removal of degeneracy upon the breakdown of symmetries of a regular system, whereas Wigner’s law indicates that this symmetry breaking is quite strong (occurrence of hard chaos). Wigner himself derived this law in order to describe experimentally observed distributions of spacings between identicalspin resonances of a compound nucleus, in which case residual pair interactions break the high (nearly oscillator-like) symmetry of the mean field in the

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for ε → 0. Naturally, this distribution differs substantially from Poisson’s distribution. If we do not fix values of ny , then the spectrum appears to be an infinite sum of level sequences shifted with respect to one another by the unceasingly changing quantities

P(ε) 0.8 B 0.6 0.4

2 π 2 (2ny + 1) , 4m a2y

A

0.2

0

10

20

ε/D

Fig. 4. Distribution of spacings between neighboring levels for a superposition of two independent sequences (curve A) such that, in either sequence, the levels are distributed according to Wigner’s law (curve B).

independent-particle model. Indeed, the resonances in question do not have integrals of motion, with the exception of energy and spin. It is noteworthy that this law is valid only for states of the compound nucleus that have the same fixed value of the spin J. Without this spin selection, the repulsion of levels disappears, with the result that the distribution approaches that which obeys Poisson’s law. Figure 4 shows a distribution that results from a superposition of two Wigner’s distributions (for two values of the spin J). Obviously, Wigner’s repulsion disappears in this case because there are no correlations between the two superimposed Wigner’s sequences (they are absolutely independent of each other). It is natural that, as the number of independent sequences increases, we come ever closer to Poisson’s law. Thus, Wigner’s repulsion of levels disappears even for a chaotic system if we pay no attention to intact integrals of motion (spins in the case being considered) and "add apples to lorries"—that is, independent sequences of levels associated with different values of disregarded integrals of motion. Let us now consider the case of regular systems featuring no degeneracy. These are systems that have so far attracted most attention. Such systems are exemplified by a two-dimensional rectangular billiard (motion of a particle of mass m in a well surrounded by infinitely high walls) for which the ratio of the side lengths, ax /ay , is an irrational number. For this system, the energies of motion along the x and y axes are integrals of motion, while the total energy is 2 π 2 n2x n2y + 2 . Enx ny = 4m a2x ay If we fix a value of one of the integrals of motion (for example, ny ), then we obtain a sequence of nondegenerate levels repelled in such a way that P (ε) → 0

where ny is an infinite series of integers. Upon sampling a rather large number of ny values, these sequences appear to be virtually uncorrelated. Thus, we are once again dealing with a superposition of a large number of independent sequences of levels, and the distribution P (ε) approaches Poisson’s distribution. The case of a two-dimensional oscillator whose frequencies are in an irrational ratio is an even more spectacular example of the absence of dependence. The energy of such an oscillator has the form Enx ny = (nx ωx + ny ωy ) If ny is not fixed by analogy with the preceding example, then all of the resulting sequences are shifted with respect to one another by the same constant quantity ωy . Therefore, the form of the distribution P (ε) for this system has nothing in common with Poisson’s law and always changes in response to the growth of the numbers of incorporated levels without tending to any specific limit. Thus, only in the case where one fixes all integrals of motion of the system, with the exception of energy, can Wigner’s distribution be an indication of chaos in the system. However, the widespread opinion that spectra of regular systems have the form of Poisson’s distribution is in general incorrect. For regular systems featuring degeneracy, the law of the distribution of levels differs markedly from Poisson’s law. As pronounced as that are the deviations from it exhibited by the law of the distribution of levels for nondegenerate harmonic oscillators. We have seen that only for some nondegenerate regular systems and only under the condition that we forget the existence of the integrals of motion for the system other than the total energy and dump together levels corresponding to different values of these integrals (good quantum numbers) does the distribution of levels approach Poisson’s law. In the same way, however, we can also obtain Poisson’s law for a chaotic system that has at least one integral of motion in addition to energy. While Wigner’s distribution is indeed indicative of chaos in the system, Poisson’s distribution may only be an indication of the existence of missed extra (in addition to the total energy) integrals of motion. However, their number M may be insufficient for fulfillment of the condition M = N under which he system exhibits a regular behavior. This situation is exemplified PHYSICS OF ATOMIC NUCLEI Vol. 79

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by a compound nucleus. Upon dumping together different-spin levels, this chaotic system formed by a multitude of interacting particles does not become regular.

3.3. Numerical Criteria of Quantum Chaos and Approximate Integrals of Motion Thus, our definition of chaos as the absence of symmetries and integrals of motion associated with them does indeed permit understanding quite straightforwardly the physical meaning of quantum signatures of chaos. Obviously, only if all M integrals of motion are known for the system being considered can conclusions concerning its regular versus chaotic behavior be drawn from the law of the distribution of its levels. But in that case, our definition of chaos permits drawing the respective conclusion much more readily: it is sufficient to compare the number M of these global integrals of motion with the number N of degrees of freedom in the system. In order to estimate quantitatively the degree of chaos in a quantum system—recall that it was impossible to do this with the aid of quantum signatures—we can resort to the same source as that which Wigner used in deriving his law: low-energy neutron resonances of a compound nucleus and the theory of neutron strength functions (see, for example, [15, 17]). The neutron strength function is the most striking example of how the traces of broken symmetries of the mean field in the shell model remain in the wave functions for compound-nucleus states. Indeed, let us consider the Hamiltonian H for a nonintegrable system that has N degrees of freedom as the sum (6) H = H0 + λV of the Hamiltonian H0 for a regular system that has a high symmetry, (7) H0 φk = εk φk , and the perturbation λV that breaks the symmetry of H0 to such an extent that the number M of good quantum numbers becomes smaller than N , M < N . We now expand the eigenfunctions ψi of the total Hamiltonian H in the basis of “regular” states φk as φk |ψi φk = cki φk (8) ψi = k

k

2 and consider the probability Wk (Ei ) = cki for finding the original “regular ” component φk in the eigenfunctions ψi (corresponding to eigenvalues Ei ) of the nonintegrable system under study. It is well known that, at rather small values of λ, this probability is localized within quite a narrow energy interval around the original energies εk . The diagonalization of the Hamiltonian matrix for the Hamiltonian H under PHYSICS OF ATOMIC NUCLEI

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quite realistic assumptions (see, for example, Appendix 2D in the monograph of A. Bohr and Mottelson [20]) shows that the energy dependence of the strength function Sk (Ei ) = Wk (Ei )/D can be approximated by a Lorentz distribution; that is, k 2 c Γkspr 1 i , (9) Sk (Ei ) = =

D 2π (Ei − εk )2 + Γkspr 2 /4 where D is the mean spacing between the levels of the nonintegrable system and the spreading width Γkspr is characterized (see [15]) by the scale of the matrix elements of the perturbing interaction λV , which mixes the basis states φ. Taking this into account, we can define the quantity Γkspr as the minimum energy range that lies around the value of εk and within which the sum of the probabilities i Wk (Ei ) for finding the regular component φk in the eigenfunctions ψi reaches the value 0.5. If the spreading width Γkspr of the states φk is smaller that the mean spacing D0 between the neighboring maxima of the strength function (9) (that is, the mean spacing between the levels of the regular system), we can discriminate between the region of localization in energy of one basis state φk (whose principal quantum number k is fixed) and the regions of localization of neighboring basis states whose principal quantum numbers are k ± 1. Although the symmetry of the original regular system has already been broken by the perturbation λV , the traces of this symmetry are clearly seen as the maxima of the strength function [or of the probability distribution Wk (Ei )]. This situation can be called soft chaos. It can be considered as a quantum analog of the Kolmogorov– Arnold–Moser (KAM) theorem in classical mechanics. This theorem states that, in the case of small perturbations, invariant tori in the phase space that are pertinent to regular systems do not disappear but only undergo slight deformation. It is noteworthy that our condition specifying the chaos parameter Γspr is intimately related to the condition obtained in [21] that makes it possible to assign an approximate value of the integral of motion to eigenstates of the Hamiltonian H of the nonintegrable (chaotic) system. It is emphatic that the the article of Hose and Taylor [21] is called “Quantum Kolmogorov–Arnol’d–Moser-like theorem: Fundamentals of Localization in Quantum Theory.” More details on the relation between our chaos parameter and this criterion and its possible simplified estimates are given in [17]. If the spreding width exceeds D0 , the maxima of the probability distribution Wk (Ei ) undergo smearing and disappear; that is, the last traces of the original symmetries of the regular system disappear. The

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limiting situation where such traces are fully lost can be called hard chaos. Thus, the dimensionless parameter κ=

Γkspr D0

=

Γspr D0

(10)

is a natural quantitative measure of symmetry breaking for a regular system (the spreading width Γspr is averaged over the indices k of the basis states φk ). If this parameter exceeds unity, the traces of the symmetry of the Hamiltonian H0 for the original regular system disappear under the effect of perturbations. Concurrently, selection rules for the original system disappear, the distribution of energy levels becomes approximately uniform (that is, there arise Wigner’s repulsion and spectral rigidity), and the law of the distribution of levels approaches Wigner’s law. All this means the onset of chaos in the quantum system under study, and we can consider the parameter κ as a quantitative measure of chaos.

3.4. Limiting Transition to Classical Mechanics The presence of the relation between the quantities Γspr and κ, on the one hand, and Lyapunov’s exponent Λ [15, 17], on the other hand, is of paramount importance. In order to demonstrate this relation, we go over to the time-dependent formalism of quantum mechanics. Suppose that, at the initial instant t = 0, the wave packet constructed from the eigenfunctions of the total Hamiltonian H describes the eigenstate φk of the unperturbed Hamiltonian H0 ; that is, ∗ (11) cki ψi (t = 0). φk (t = 0) = i

We now determine the survival probability for this state in the perturbed system, P (t) = |φk (0)|φk (t)|2 = |A(t)|2 .

(12)

Taking into account the orthogonality of the functions ψi and Eq. (9) , we can estimate (see, for example, [17]) the amplitude A(t) as (13) A(t) = φk (0)|φk (t) 2 iEi t = cki (Ei ) exp − i

iEi t iEi t dEi k 2 exp − ≈ c (Ei ) exp − D i

k Γspr iEi t dEi exp − = D (Ei − εk )2 + (Γkspr )2 /4 Γkspr εk t −i . = exp − 2

Therefore, we have

Γkspr P (t) = exp − t .

(14)

In order to find the classical limit of the quantity Γspr , one can make use of the results obtained by ´ Heller (see, for example, [22, 23] or Section 15.6 in monograph of Gutzwiller [7]) for Gaussian wave packets |Φ. These packets are chosen in such a way that, at the initial instant t = 0, the packet center lies on a periodic trajectory of period T . After that, the recurrence probability PΦ (t) = |Φ(0)|Φ(t)|2 is calculated for the packet. It was shown that, in the semiclassical limit, this probability is a periodic sequence, with a period T , of maxima corresponding to the returns of the packet to the original phasespace region. However, the amplitudes of these maxima are modulated by the factor exp − ΛT 2 , where Λ is Lyapunov’s exponent. Over the time between two successive returns, the probability in question therefore decreases by the quantity exp (−ΛT ) ≡ exp (−χ) .

(15)

(In classical mechanics, the dimensionless quantity χ = ΛT is referred to as the stability parameter of the monodromy matrix [7].) This result is physically clear. The center of the packet moves along a stable periodic trajectory, returning to the starting point with a period T . Because of a finite size of the packet in the phase space, points at its periphery move along trajectories for which initial conditions differ somewhat from the respective conditions for the trajectory of the packet center. Fully in accord with stability theory, peripheral trajectories diverge exponentially from the stable central trajectory and therefore do not return to the starting position. This circumstance causes an exponential decrease in the recurrence probability PΦ (t) = |Φ(0)|Φ(t)|2 at a rate specified by Lyapunov’s exponent. Following [22, 23], we can construct the wave packet An |φn (t), (16) |Φ(t) = n

in which each of the wave functions φn (t) has the form (11). For the sake of simplicity, we assume that the regular Hamiltonian H0 is that for a onedimensional harmonic oscillator of frequency ω and that, at the initial instant t = 0, we have (x − a)2 1 , (17) |Φ(t = 0) = √ exp − 2σ 2 (π)1/4 σ PHYSICS OF ATOMIC NUCLEI Vol. 79

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where σ = /mω specifies the width of the localization region for the packet and a is its initial shift. Determining the coefficients An with the aid of Eqs. (16) and (17) and employing relation (9), we can obtain (see, for example, [17]) an expression for the recurrence probability in the form PΦ (t) = |Φ(0)|Φ(t)|2 (18) 2 2 ≈ exp a (cos ωt − 1)/σ exp −Γspr t/ .

The first factor on the right-hand side of Eq. (18) describes periodic returns of the packet with the frequency ω, while the second factor gives the decrease in the recurrence probability because of Lyapunov’s instability of the trajectories (in the case of nuclear physics, the instability of the single-particle motion of a neutron in the nuclear mean field is due to pair interactions coupling the single-particle mode of motion to more complicated modes). A comparison of expression (18) with the results reported in [22, 23] leads to the conclusion that, in the classical limit, the quantity Γspr / goes over to Lyapunov’s exponent Λ; that is, (Γspr /) → Λ,

(19)

while the respective classical limit of the dimensionless chaos parameter is χ ΛT = , (20) κ→ 2π 2π where T is the classical period and χ is the stability parameter of the monodromy matrix.

3.5. Approximate Criterion of Quantum Chaos Thus, we have seen that, in order to determine the chaos parameter Γspr , it is necessary to assess the minimum energy range that lies around the unperturbed eigenvalue εk and within which the sum of the coefficients squared, 2 |φk |ψi |2 ≡ cki , i

i

reaches a value of 0.5 and to perform averaging over k in this energy range. In the case of degenerate states, it is first necessary to perform similar averaging over all eigenfunctions φn belonging to a given shell (that is, forming a given degenerate state). However, a symmetry-breaking interaction leads not only to the spreading of states of a regular system but also to the removal of degeneracy and to an increasingly stronger repulsion of states of this regular system that belong to a single shell. It is clear that, as long as the splitting Δ of levels belonging to the same shell is smaller than the spacing D0 between the shells, the concept of traces of the original symmetry PHYSICS OF ATOMIC NUCLEI

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is meaningful, and we then can assign (albeit approximately) the principal quantum number of the original shell to split levels in the interval Δ. This gives reasons to hope that the quantity Γspr is approximately equal to the energy splitting Δ of shell states. The ratio of Δ to D0 may then serve as an approximate criterion of chaos in a quantum system; that is, r = Δ/D0 ≈ κ

(21)

Of course, this equality is quite approximate—below, we will see that it holds within several tens of percent. However, the quantity r is calculable much more straightforwardly than κ and has a much clearer physical meaning than the aforementioned formulas obtained as interpolations between the different laws of the distribution of levels.

3.6. Examples of Employing Numerical Criteria of Quantum Chaos The two most popular examples of the transition from a regular to a chaotic behavior in classical mechanics were considered in [14, 24] in order to demonstrate the applicability of the proposed approach. The first of them is the problem formulated and examined ´ by Henon and Heiles [25] upon reasonably simplifying the problem of the motion of a star in the galaxy ´ field (Henon–Heiles system), and its analysis is an indispensable part of almost all modern monographs devoted to chaos (see, for example, [3, 10]). The ´ classical Hamiltonian of the Henon–Heiles system system has the form 1 (22) H = (p2x + x2 + p2y + y 2 ) 2 + λ(x2 y − y 3 /3) ≡ H0 + λV. Here, the regular Hamiltonian H0 describes a twodimensional harmonic oscillator in the system of units where ω = m = 1 (which is conventional for these investigations). For the choice of λ = 1, which is as standard as this choice of units (see, for example, [3]), the energy of the system E in relative units is a perturbation measure (the system is regular at ´ E = 0 and becomes chaotic as E grows). Henon and Heiles [25], who addressed this system, studied, by means of numerically integrating the respective equations of motion, the successive intersections of its trajectories with the х = 0 plane at various values of the energy E. Figures 1 and 2 given in the present article (see above) show the phase portraits of the ´ system that were obtained by Henon and Heiles for the energy values of E = 1/12 and 1/6, respectively. Further, they estimated the variance of the phasepicture points for the situation where the number of successive intersections of two close (in initial conditions) trajectories with the aforementioned plane

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BUNAKOV

r, κ 2.0

R 1.0

κ 1.5

0.8

r

1.0 0.6 0.5 0.4 0 0.2

0.14 E

0

Fig. 6. Quantum-chaos parameter κ and parameter r versus the excitation energy E (that is, the strength of ´ interaction leading to chaos) for the quantum Henon— Heiles system.

0.04

0.08

0.12

0.16 E

Fig. 5. Relative phase-space fraction covered by regular ´ trajectories in the Henon—Heiles problem [25].

becomes great. If the variance in question did not exceed a preset value μ, the trajectories were thought to be regular. Otherwise, they were assumed to be chaotic. In this way, they estimated the fraction R of the phase-portrait area covered by regular trajectories as a function of energy (Fig. 5) and arrived at the conclusion that, up to the energy of E = 0.11, this fraction is equal to unity (in which case the motion is regular). This means that the transition to chaos occurs at a critical energy of Ecr ≈ 0.11. They wrote, "the situation can be very roughly described by saying that the second integral exists for orbits below critical energy and does not exist for orbits above that ´ energy." It should be noted that Henon and Heiles associated the onset of chaos in the system with the disappearance of a integral of motion. Undoubtedly, the curve in Fig. 5 reflects the stability of invariant tori against small perturbations in accordance with the KAM theorem. It is possible, however, that an absolute regularity of the system in the region of E < Ecr is a result of an insufficiently high accuracy ´ of the method used by Henon and Heiles in the case of small perturbations [26]. The respective quantum Hamiltonian was constructed in [14] by replacing the momenta p in expression (22) by the momentum operators. The diagonalization of the resulting Hamiltonian matrix led 2 to finding the probability distribution Wk (Ei ) = cki [see Eq. (9)] and to constructing the dependence of the quantities Γspr and κ = Γspr /D0 on the energy E (see curve κ at Fig. 6). One can see that the chaos parameter κ reaches unity at E ≈ 0.11 in agreement

0.02

0.06

0.10

with the classical results. Figure 6 also shows the energy dependence of the parameter r(E). This parameter reaches unity at E ≈ 0.13, which is a reasonably good result in view of quite an approximate character of this parameter. The so-called diamagnetic Kepler’s problem, which is merely a classical spinless analog of the hydrogen atom in a uniform magnetic field, is yet another classical system for which a transition from a regular to a chaotic behavior was studied in detail in classical mechanics [27, 28]. The respective Hamiltonian has the form 1 H = p2 /2m − e2 /r + ωlz + mω 2 (x2 + y 2 ), (23) 2 where the frequency ω = eB/2mc is one-half of the cyclotron frequency and B is the strength of the magnetic field directed along z axis. The dimensionless field-strength parameter γ = ω/ (here, is the Rydberg energy) is usually combined with the electron energy E to produce the scaled energy ε = Eγ −2/3 . As the scaled energy ε grows from −∞ (for B = 0) to 0 (for B = 1), the regular motion of the system becomes more and more chaotic. By anal´ ogy with the procedure used by Henon and Heiles, the fraction R of the phase-portrait area occupied by regular trajectories was obtained numerically in [27, 28] at lz = 0 (Fig. 7) as a function of ε. One can see that a transition to chaos in this system occurs at the critical scaled-energy value of εcr = −0.48. Following the same line of reasoning as in the ´ case of the Henon–Heiles system, Bunakov and Ivanov [16, 17] replaced the momenta in Eq. (23) by the momentum operators and examined the way in which a uniform magnetic field breaks O(4) symmetry, which is characteristic of unperturbed motion in a PHYSICS OF ATOMIC NUCLEI Vol. 79

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κ 2.0

E –1.5

–1.0

–0.5

1005

0

1.0 1.5

R

0.8 0.6

1.0

0.4

0.5

0.2 0 –1.5

–1.0

ε

–0.5

0 –2.5

0

Fig. 7. Relative phase-space fraction covered by regular trajectories in the diamagnetic Kepler problem [27, 28].

Coulomb potential. After diagonalizing the resulting Hamiltonian matrix in parabolic coordinates in the basis of Coulomb wave functions at the magneticnumber value of m = 0, the quantity Γspr and the chaos parameter κ were calculated in those studies for various values of ε (see Fig. 8). One can see that the chaos parameter reaches a value of unity at εcr = −0.45 and that the curve representing its behavior is a nearly mirror reflection of the curve in Fig. 7 (we recall that the parameter R measures the degree of regularity in the system, while the parameter κ measures the degree of chaos in it).

3.7. Quantum Chaos in Nuclear Physics For any three-dimensional system containing more than two interacting particles, the number of first integrals of motion (conservation laws) is less than the number of degrees of freedom. Therefore, all nuclei heavier than the deuteron are chaotic systems. Neutron resonances in medium-mass and heavy nuclei [this was the case for which Wigner derived the law of distribution in (1)] possess only three good quantum numbers. These are energy, spin, and parity (yet, we now know that parity violation is possible for these resonances). At the same time, the number of quasiparticles that determine the structure of the resonances in question is about eight to twelve. Therefore, neutron resonances provide a typical example of quantum chaotic systems. As was already indicated, the numerical quantumchaos parameter proposed above was constructed by analogy with the theory of neutron strength functions (see, for example, [20]). The neutron strength 2 function is defined as [see Eq. (9)] S0 (Ei ) = c0i /D. Here, the coefficient c0i determines the contribution to the ith compound resonance from the neutron PHYSICS OF ATOMIC NUCLEI

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–2.0

–1.5

–1.0

–0.5

0 ε

Fig. 8. Quantum-chaos parameter κ as a function of the scaled energy ε for the quantum diamagnetic Kepler problem.

plus + target nucleus single-particle configuration in the ground state, and one can determine it from the neutron width Γn(i) of the ith resonance state. It seems that the simplest way to measure experimentally the neutron strength function in (9) is to choose any target nucleus and to measure cross sections for neutron resonances at various gradually growing energies of projectile neutrons. However, the density of neutron resonances grows exponentially with increasing energy (that is, the spacing between the resonances decreases exponentially), whereas the resonance widths grow as new resonance-decay channels open. Very soon, we therefore enter the energy region where resonances overlap ever more strongly, so that it becomes impossible to measure the function in (9). In view of this, this function is determined by using the fact that the positions of single-particle s-wave resonances in a nucleus are approximately given by the expression π (24) KR = (2n + 1) , 2 where the radius of the nucleus R is related to its mass number A by the equation R = r0 (A)1/3 , the neutron momentum√reckoned from the bottom of the 2mE . Therefore, one can go mean field is K = over from one resonance to another [change the value of n in Eq. (24)] by fixing the neutron energy (that is, the value of К) and by changing R (that is, the target mass number). This is precisely what is done in experiments. One chooses a target nucleus of arbitrary mass number А, measures for it the accessible set of s-wave neutron resonances, rescales the values found for the neutron width to the energy fixed at E0 = 1 eV (that is, to the momentum value of ki = k0 ), and finds the strength function S(A) averaged

1006

BUNAKOV 0

Γn 4 ----- × 10 D

10

8

6

4

2 0 20

60

100

140

180

220

A

Fig. 9. Experimental neutron strength function.

over all resonances at a given value of A. Repeating this procedure, one than plots the dependence S(A) (see Fig. 9).

chaos parameter satisfies the condition Γspr κ= 160. These are mass numbers of nuclei in which levels of 3s and 4s single-particle states lie at an excitation energy approximately equal to the neutron binding energy. Thus, the picture observed experimentallly implies that, although the symmetry of the central mean field is strongly violated by residual two-body nucleon interactions V , traces of this symmetry can still clearly be seen in our chaotic system. Moreover, it is well known that, in the region of 140 < A < 200, nuclei are strongly deformed. Therefore, the spherical symmetry of the mean field is broken, and the 4s single-particle state is spread over several components. All this is indicative of a soft chaos, in which traces of the symmetry of the original regular system still survive, so that the dimensionless

An experimental proof of the fact that the nucleus is not an absolutely black body paved the way to creating the optical model in the period spanning the 1950s and 1960s. In this model, a nucleon moves in a mean field specified by a real potential. At the same time, losses in the single-particle mode of motion by collisions with other nucleons are taken into account via the imaginary part of the potential. Of course, individual resonances in complex nuclei cannot be described within this statistical approach. Owing to the smallness of the parameter in (25), however, the optical model describes fairly well energy-averaged elastic and total cross sections and angular distributions for nucleon–nucleus reactions. Even in describing inelastic processes by the distorted-wave method, the optical model, which is a workhorse of nuclear physics, proves to be quite successful. This provides a spectacular example of simple and viable calculations for quantum chaotic systems. There exists a simple relation between the imaginary part of the optical potential, W (E), and the spreading width in Eq. (9); that is,

(25)

Γspr = 2W (E). In the region of neutron resonances (see the solid curve in Fig. 9 with maxima), W ≈ 3 MeV, while the chaos parameter for medium-mass and heavy nuclei takes a value of κ ≈ 0.5. PHYSICS OF ATOMIC NUCLEI Vol. 79

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The systematics presented in the monograph of A. Bohr and Mottelson [20] makes it possible to establish an empirical dependence of W on the projectile nucleon energy in the form W (E) ≈ (3 + 0.1E) [MeV]. Therefore, it is clear that the chaos parameter grows with energy, but it remains less than unity even for energies of about 50 to 100 MeV and for the deepest hole states in light and medium-mass nuclei. In the harmonic-oscillator approximation, which describes fairly well the mean field, the spacing between the neighboring shells is given by 40 ω = 1/3 . A It follows that, as the mass number A grows, the spacing between the mean-field levels, D0 ω, decreases, while the chaos parameter in Eq. (25) increases. As was indicated in [13, 15], the chaos parameter is in fact the only small parameter in nuclear physics. It is the smallness of this parameter that justifies the use of the basis of mean-field states in calculations, which provides a rather fast convergence. The same smallness validates the use of the optical model in calculating observables of nuclear reactions. In nuclear physics, one says that the nucleus is semitransparent rather than absolutely black, as was thought earlier. In terms of chaos, this means that soft rather than hard chaos is dominant in nuclei. It should be noted that the random-matrix method developed by Wigner, Porter, and other theorists is based on the statement that the matrix elements of the nuclear Hamiltonian Н are uncorrelated and are distributed according to a normal law. It is of paramount importance that, in order to obtain this random distribution law, it was necessary to consider a statistical ensemble of Hamiltonian matrices and to require [29] that the distribution law be invariant under any unitary transformations of basis states. This ensemble of matrices is referred to as a Gaussian orthogonal ensemble (GOE). Since the choice of basis states is determined by the choice of “basis” Hamiltonian H0 in breaking down the total Hamiltonian H into parts, (26)

H = H0 + V,

this requirement means that the distribution law should not depend on the way of the partition in (26). This may be so only if all possible choices of “basis” Hamiltonian H0 yield equally “bad” zero-order approximations to the exact Hamiltonian of the problem. This means that, for all of them, the quantumchaos parameter exceeds unity. This is the situation that is referred to as hard chaos. Clearly, this is a limiting (“ideal”) regime that occurs for a chaotic PHYSICS OF ATOMIC NUCLEI

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system as rarely as the cases of perfectly regular systems do. In all other cases of soft chaos, the chaos parameter depends on the choice of “basis” Hamiltonian H0 . In practice, we will of course be interested in that choice of H0 , among all possible ones, which guarantees the best convergence of the method for diagonalizing the total Hamiltonian matrix—that is, the minimum value of the parameter κ. This dependence of the parameter κ on the choice of basis comes as no surprise since κ measures the degree of violation of a symmetry of the Hamiltonian H0 from which we start. In discussing symmetry breaking, it is natural to indicate which kind of symmetry we break. In the case of classical mechanics, the set of Lyapunov’s exponents and the form of the phase portrait in the case of soft chaos similarly depend on the choice of regular system whose symmetry undergoes breakdown. In the example of the diamagnetic Kepler problem, one usually considers the situation where a uniform magnetic field is a perturbation that breaks the symmetry of motion in a Coulomb field. If, however, the strength of the magnetic field grows substantially, it becomes natural to consider a regular motion of a particle in a uniform magnetic field (Landau levels) perturbed by Coulomb interaction, in which case H0 and V are interchanged in Eq. (26). Of course, it is then meaningless to assess the degree of chaos in the system by employing the Coulomb basis, but, in classical mechanics, this only corresponds to considering the destruction of the phase portrait of a totally different regular system. The fact that, in contrast to other systems in which physicists and mathematicians studied dynamical chaos, the hydrogen atom in a uniform magnetic field is not an abstract theoretical model but a real physical system that admits experimental studies, which did indeed start in the late 1980s, numbered among the main motivations [30] to analyze the transition to chaos in the diamagnetic Kepler problem. Basically, the experimental results obtained for this system were interpreted in classical terms of trajectories of periodic orbits. It is noteworthy that atomic nuclei are also real physical systems not less accessible to laboratory investigations. These investigations have been performed for nearly 70 years, and the accumulated data on the subject and the range of reliable theoretical approaches are incommensurably richer than those in the diamagnetic Kepler problem. In nuclear physics, we first encountered the phenomenon of quantum chaos without having any idea of this notion and had to find practicable methods for describing experimental data. All of them reduced to searches for approximate symmetries and integrals of motion. If the integrals of motion found in this way make it possible to construct an approximately regular system (model), the respective calculations

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BUNAKOV

take advantage of a rather fast convergence of the expansion in this basis (that is, the smallness of the parameter κ against unity). Such approaches underlay almost all of the spectroscopic calculations of low-lying states. The remarkable potential of the optical model, which is also based on the smallness of the parameter κ in relation to unity, has already been mentioned. Statistical approaches based on the random-matrix method or Boltzmann or Fokker– Plank equations work fairly well as κ approaches unity. In this connection, the following comment is in order: according to our analysis of neutron strength functions, the commonly accepted quantum signature of chaos in the form of Wigner’s distribution is a much less sensitive feature of quantum chaos than the proposed parameter κ. Wigner’s law, whose derivation relied on the theory of random matrices (see, for example, [20]) and which suggests the occurrence of hard chaos (which is equivalent to the statement that nuclei are absolutely black) holds fairly well for neutron resonances in medium-mass and heavy nuclei, but distinct maxima of the strength functions for these resonances are indicative of soft chaos for which κ ≈ 0.5. 4. CONCLUSIONS The statement advocated in the present article is that the term "quantum signatures of classical chaos" is as incorrect as the statement that relativistic mechanics is a relativistic signature of Newtonian mechanics would be. The definition proposed here for a regular and a chaotic behavior is based, fully in accord with the Liouville–Arnold theorem, on the concept of symmetry that the system being considered possesses and which ensures the existence of global integrals of motion (good quantum numbers). If the symmetry of the system is so high that the number M of the integrals of motion in question is equal to the number N of degrees of freedom in this system, the system is regular. If additional perturbations break the symmetry of the system to such an extent that M proves to be smaller than N , the system becomes chaotic. Since, in contrast to the concept of trajectories, the concept of symmetry is common to all branches of physics, the symmetry-inspired concepts of a regular and a chaotic behavior are equally applicable in classical and in quantum mechanics. The width Γspr of the energy interval in the spectrum of states of the chaotic system being considered such that the spreading states of the respective regular system for a fixed principal quantum number are concentrated within this interval is proposed as a

measure of quantum chaos. In contrast to the previously proposed semi-intuitive criteria of quantum chaos, the quantity Γspr / reduces in the classical limit to Lyapunov’s exponent Λ. For a dimensionless parameter of quantum chaos, we choose the ratio Γspr κ= , D0 where D0 is the mean spacing between the levels of the regular system that correspond to fixed quantum numbers. If Γspr D0 , we can discriminate between the region of localization in energy of one basis state φk (corresponding to a fixed value of the principal quantum number k) and the regions of localization of neighboring basis states corresponding to the principal quantum numbers k ± 1. Although the symmetry of the original regular system has already been broken by the perturbation, the traces of this symmetry can clearly be seen in that case as maxima of the strength function [or the probability distribution Wk (Ei )]. This situation can be termed as soft chaos and can be considered as a quantum analog of the KAM theorem. If Γspr > D0 , then the traces of the broken symmetry of the regular system disappear completely and a transition to hard chaos occurs. The classical limit for the dimensionless chaos parameter is given by χ ΛT = , κ→ 2π 2π where T is the classical period and χ is the parameter of stability of the monodromy matrix. We have considered quantum analogs of those “standard” classical systems for which a transition from a regular to a chaotic behavior could be traced in detail within classical mechanics. These are the ´ Henon—Heiles problem and the diamagnetic Kepler problem (classical spinless hydrogen atom in a uniform magnetic field). The use of the methods proposed here revealed that, even quantitatively, transitions from a quantum regular behavior to quantum chaos are similar to the respective transitions in classical mechanics. It is the opinion of the present author that study of quantum chaos in nuclear physics is substantially less promising than the use of nuclear-physics methods in an analysis of quantum chaotic systems in different fields of physics, since we now know that chaos is a universal phenomenon and that all systems in nature are chaotic to some extent. Nuclear physicists were the first who had to deal with a quantum chaotic system—the atomic nucleus—and have had over nearly 70 years to develop methods of approximate calculations for this system. As a matter of PHYSICS OF ATOMIC NUCLEI Vol. 79

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fact, almost each nuclear model results from more or less successful searches for approximate integrals of motion, and we have accumulated over the past 70 years a rather large number of successful methods that could be used in those fields of physics that have already come across quantum chaos or will come across it in the near future. REFERENCES 1. G. M. Zaslavsky, Chaos in Dynamical Systems (Nauka, Moscow, 1984; Harwood Academic, New York, 1985). 2. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988). 3. A. Lichtenberg and M. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983). 4. M. Berry, in Quantum Chaos, Proceedings of the Adriatico Research Conference and Miniworkshop, Trieste, 1990, Ed. by H. Cerdeira, R. Ramaswamy, M. Gutzwiller, and G. Casati (World Sci., Singapore, 1991), p. VII. 5. T. Brody, Lett. Nuovo Cimento 7, 482 (1973). 6. M. V. Berry and M. Robnik, J. Phys. A 17, 2413 (1984). 7. M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990). 8. F. Steiner, in Festschrift Universitaet Hamburg, 1994, Schlaglichter der Forschung zum 75, Jahrestag, Ed. by R. Ansorge (Dietrich Reimer Verlag, Hamburg, 1994), p. 543 [in German]; F. Shtainer, Nelin. Dinam. 2, 214 (2006). 9. V. Zelevinsky and A. Volya, Phys. Scr. T 125, 147 (2006). 10. V. Zelevinsky, B. A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276 , 85 (1996). 11. G. Mitchel, A. Richter, and H. A. Weidenmueller, Rev. Mod. Phys. 81, 539 (2009).

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12. V. Zelevinsky, Phys. At. Nucl. 65, 1188 (2002). 13. V. E. Bunakov, in Proceedings of International Conference on Selected Topics in Nuclear Structure, JINR Report No. E3-94-370 (Joint Inst. Nucl. Res., Dubna, 1994), p. 310. 14. V. E. Bunakov, F. F. Valiev, and Yu. M. Tchuvilsky, Phys. Lett. A 243, 288 (1998). 15. V. E. Bunakov, Phys. At. Nucl. 62, 1 (1999). 16. V. E. Bunakov and I. B. Ivanov, Phys. At. Nucl. 62, 1099 (1999). 17. V. E. Bunakov and I. B. Ivanov, J. Phys. A 35, 1907 (2002). 18. V. E. Bunakov, Phys. At. Nucl. 77, 1550 (2014). ¨ 19. A. Hoenig and D. Wintgen, Phys. Rev. 39, 5642 (1989). 20. A. Bohr and B. Mottelson, Nuclear Structure, Vol. 1: Single-Particle Motion (Benjamin, New York, 1969). 21. G. Hose and H. Taylor, Phys. Rev. Lett. 51, 947 (1983). 22. E. Heller, in Chaos and Quantum Physics, Proceedings of the Les Houches Summer School on Quantum Chaos, Sec. 52, 1989 (Elsevier, Amsterdam, 1989), p. 548. 23. S. Tomsovich and E. Heller, Phys. Rev. E 47, 282 (1993). 24. V. E. Bunakov, I. B.Ivanov, and R. B. Panin, Bull. Russ. Acad. Sci.: Phys. 64, 20 (2000). ´ 25. M. Henon and C. Heiles, Astron. J. 69, 73 (1983). 26. G. Benettin, L. Galgani, and J. M. Strelcyn, Phys. Rev. A 14, 2338 (1976). 27. A. Harada and J. Hasegawa, J. Phys. A 16, L259 (1983). 28. J. Hasegawa, M. Robnik, and G. Wunner, Prog. Theor. Phys. Suppl. 98, 198 (1989). 29. C. Porter and N. Rosenzweig, Ann. Acad. Sci. Finland. A 6, 44 (1960). 30. H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989).