c Pleiades Publishing, Ltd., 2016. ISSN 1063-7788, Physics of Atomic Nuclei, 2016, Vol. 79, No. 4, pp. 561–566. c V.E. Bunakov, 2016, published in Yadernaya Fizika, 2016, Vol. 79, No. 4, pp. 374–379. Original Russian Text
NUCLEI Theory
Quantum Chaos in Nuclear Physics V. E. Bunakov* St. Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, Orlova Roshcha, Gatchina, Leningrad oblast, 188300 Russia Received November 25, 2015
Abstract—A definition of classical and quantum chaos on the basis of the Liouville–Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated. DOI: 10.1134/S1063778816040062
1. INTRODUCTION Investigation of quantum chaos in nuclear physics is complicated by the absence of consensus on the very concept of quantum chaos. This in turn stemmed from the fact that, after long-term searches for the main source of classical chaos (see, for example, [1]), researches decided on Lyapunov’s instability of trajectories of a chaotic system against small variations in initial conditions. Indeed, such a variation leads to a divergence of the trajectories of a chaotic system in the phase space according to the exponential law exp{Λt}, where Λ is the Lyapunov exponent, which determines the rate of this divergence. Since the concept of a trajectory becomes inaccurate in quantum mechanics because of the uncertainty relation, the classical stability or chaos criteria, which are associated with the concept of a trajectory, become inapplicable. In the past, it was often asserted flatly that chaos is impossible in quantum systems. In the case of stationary problems for discrete bound states, this assertion received an especially strong support from the deeply rooted opinion that there is an indissoluble connection between chaos and nonlinearity ¨ (Schrodinger equations are linear, and the connection with classical phase space and time seems dissolved absolutely). In the course of time, these requirements softened and transformed into the following statement of Berry [2]: “the incorrect term “quantum chaos” means quantum phenomena characteristic of classical chaotic systems—that is, quantum “signatures” of classical chaos.” It was proposed to seek these *
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signatures in the following way. We consider a regular and a chaotic system in classical mechanics and con¨ struct their quantum analogs (that is, Schrodinger equations with respective Hamiltonians) according to the correspondence rules. We then compare the properties of the eigenvalues and eigenfunctions of these equations, cherishing the hope for finding some distinction, which we would refer to as a quantum signature of classical chaos. The law of the distribution of levels proved to be the only more or less commonly recognized imprint of this type (see, for example, [3–5]) found over nearly 40 years of searches. For quantum analogs of classical chaotic systems, the distribution of level spacings was close to Wigner’s law, in which there is a characteristic repulsion between the levels. According to this law, the probability for finding a neighboring level whose energy differs by ε from the energy of a given level has the form πε2 πε exp − , (1) P (ε) = 2D2 4D2 where D is the mean level spacing between the levels. By repulsion, one means here the fact that the probability for finding the neighboring level tends to zero for ε → 0. Wigner’s law is a consequence of the application of a statistical approach to describing complicated multiparticle states of a compound nucleus. A mathematical formalism for this description was developed by Wigner, Porter, Rosenzweig, and others and was called the random-matrix method (see, for example, the monographs quoted in [6–8] and references therein). 561
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It is the opinion of the present author that the very idea of quantum imprints 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. Possibly, this is precisely the reason why, at the present time, the prevalent opinion is that the most successful approach to quantum chaos consists in ignoring the problem of defining chaos in quantum mechanics and in trying, instead of this, to specify those properties of quantum systems that correspond to chaos in classical systems. Therefore, the majority of present-day original studies and review articles devoted to quantum chaos in nuclear physics (see, for example, [9, 10]) deal with various modifications of the randommatrix method as applied to various nuclear phenomena. Of course, it is highly desirable to have, as in the case of classical chaos, some dimensionless quantity that would measure the degree of chaos, since, as far back as the 1960s, physicists became aware of the fact that almost any system in nature is chaotic (that is, there is no clear-cut boundary between a regular and a chaotic behavior). Frequently, it is stated that the distribution of levels for regular quantum systems (that is, for quantum analogs of classical 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 extremely hard chaos, it would be highly desirable to find a numerical criterion that would make it possible to determine the degree of chaos in the system being considered. An attempt at formulating such a criterion relied on various forms of purely phenomenological interpolations [11, 12] between the laws in (1) and (2). However, it turned out (see, for example, [1]) that the law in (2) does not necessarily specify a regular system and that Wigner’s law itself is valid for chaotic systems only in the case where we consider a sequence of levels characterized by fixed quantum numbers (for instance, spin and parity). Since Wigner’s law holds for compound resonances of complex structure, attempts were made [10, 13] to characterize the chaotic character of nuclear
states by the degree of complexity of respective wave functions. With the aid of the expansion Ψα = cik Φk (3) k
of the eigenfunction for a complicated compound state, Ψα , in the wave functions Φk for the Hamiltonian of 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 |cik |4 . (5) Nα = k
The question of how one can measure quantum chaos in nuclear physics remains open in view of the aforementioned proposition to ignore the problem of defining chaos in quantum mechanics. 2. ALTERNATIVE DEFINITION OF REGULAR AND CHAOTIC BEHAVIOR Thus, we have seen that, for quantum systems, not only should one yet specify a quantitative criterion of chaos, but the very concept of quantum chaos remains undefined. As was indicated above, this was because searches for the most general sources of chaos ended up in choosing Lyapunov’s instability of trajectories against a small variations in initial conditions. Instead of semi-intuitive guesses concerning the nature and features of quantum signatures of chaos, the present author proposes (see, for example, [1, 14]) a definition of both classical and quantum chaos on the basis of the Liouville–Arnold theorem well known in classical mechanics (see, for example, [15, 16]). This theorem states that a system that has N degrees of freedom is regular if it possesses M = N linearly independent first integrals of motion in involution. First (also known as global or isolating) integrals of motion are those that are related to the symmetry of the system according to Noether’s theorem (that is, to the group of transformations under which the Hamiltonian of the system is invariant). Thus, a classical regular system possesses a symmetry sufficiently high for having integrals of motion (conservation laws) whose number М coincides with the number of degrees of freedom N . Upon symmetry violation that reduces the number of the first integrals, so that M becomes smaller than N , the system in question becomes chaotic. PHYSICS OF ATOMIC NUCLEI Vol. 79
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In contrast to the concept of trajectories, which is meaningful 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 (an 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 assume that a quantum system is regular if the symmetry of its Hamiltonian is sufficiently high for guaranteeing that the number M of its good quantum numbers is not less than the number N of its degrees of freedom. If one introduces in the system a perturbing interaction that violates its symmetry, reducing the number of good quantum numbers in such a way that M < N , the system ceases to be regular. Therefore, it seems natural to assume that a quantum system whose symmetry is so low that the number of its good quantum numbers is less than the number of its degrees of freedom is chaotic. This definition of quantum chaos was proposed earlier in [1, 14, 17–19].
Further, we expand the wave function ψi of the total Hamiltonian H in the basis functions φk for regular states, φk |ψi φk ≡ cki φk , (8) ψi = k
where H0 describes the motion of noninteracting nucleons in the central mean field and the perturbation V takes into account “residual” pair interactions that cannot be included in the mean field. We represent the wave function for the system (eigenfunction ψi of the total Hamiltonian H) as a superposition of various configurations of eigenstates φk of the highsymmetry noninteracting-particle Hamiltonian H0 : (7) H0 φk = εk φk . PHYSICS OF ATOMIC NUCLEI
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k
and consider the probability Wk (Ei ) = |cki |2 for finding the initial “regular” component φk in the eigenfunctions ψi (corresponding to the eigenvalues Ei ) of our nonintegrable system. The theory of neutron strength functions shows that this probability is localized in the energy range of width Γspr around the “original” energies εk . The diagonalization of the Hamiltonian matrix H under rather realistic assumptions (see, for example, Appendix 2D in the monograph of A. Bohr and Mottelson [6]) shows that the energy dependence of the strength function Sk (Ei ) = Wk (Ei )/D admits an approximation in the form of a Lorentzian distribution; that is, Sk (Ei ) =
3. APPLICATIONS TO 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. Moreover, the theory of neutron strength functions makes it possible to introduce a quantitative measure of quantum chaos. Indeed, the use of the Hartree– Fock method, which proved to be quite successful in constructing a mean field in atomic physics makes it possible to represent the Hamiltonian H for the compound nucleus in the form (6) H = H0 + V,
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Γkspr 1 |cki |2 = , (9) D 2π (Ei − εk )2 + (Γkspr )2 /4
where D is the mean spacing between the levels of the nonintegrable system and the fragmentation width Γkspr is characterized (see [14]) by the matrix elements of the perturbing interaction V , which mixes the basis states φk . Let us assume that the fragmentation width Γkspr of the states φk is smaller than the mean spacing D0 between the neighboring maxima of the strength function in (9) (that is, the mean spacing between the levels of the regular system). We can then distinguish between the “localization region” in the energy of one basis state φk (for which the principal quantum number k is fixed) and the “localization regions” for the neighboring basis states whose principal quantum numbers are k ± 1. Although the symmetry of the original regular system is broken by the perturbation V , traces of this symmetry can clearly be seen as the maxima of the strength function (or the probability distribution Wk (Ei )). This situation can be called a “soft chaos”. We can consider it 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 peculiar to regular systems do not disappear but undergo only slight deformations. It should be noted that our definition of the chaos parameter Γspr is intimately related to the condition obtained in [20], which makes it possible to assign an approximate value of the integral of motion to the eigenstates of the Hamiltonian H describing the nonintegrable (chaotic) system. For more details on the connection between the above chaos parameter
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and this criterion and on its simplified estimations, the interested reader is referred to [19]. We now introduce the dimensionless parameter κ=
Γkspr Γspr = , D0 D0
(10)
which is a natural quantitative measure of symmetry violation in the regular system, since Γspr is a width that is averaged over the indices k of the basis states φk . As soon as this parameter becomes greater than unity, 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 the energy levels becomes approximately uniform (that is, Wigner’s repulsion and “spectral rigidity” arise), and the distribution of levels approaches Wigner’s form. All this is indicative of the onset of a “hard chaos” in the quantum system, so that we can adopt the parameter κ as a quantitative measure of quantum chaos. It is of paramount importance that there is a connection between the quantities Γspr and κ, on one hand, and the Lyapunov exponent Λ, on the other hand. It can be shown (see, for example, [1, 14, 19]) that, in the classical limit, the quantity Γspr / goes over to the Lyapunov exponent Λ: (Γspr /) → Λ.
(11)
The respective classical limit for the dimensionless chaos parameter is given by χ ΛT = , (12) κ→ 2π 2π where T is the classical period and χ is the parameter of stability of the monodromy matrix. It follows that, in contrast to quantum signatures of classical chaos, we now have a definition of regularity and chaos that is common to classical and quantum mechanics, and the quantum measure of chaos reduces in the classical limit to the well-known classical parameters of chaos. In principle, there are many ways to break down the total Hamiltonian H into a regular part H0 and a perturbation V . Each such partition generates a specific value of the chaos parameter κ corresponding to it. This comes as no surprise since κ measures the violation of symmetry of the original Hamiltonian H0 . If we speak about symmetry violation, it is necessary to indicate which symmetry we violate. Among all possible choices of H0 , that at which the value of the parameter κ is minimal is naturally of greatest interest to us. As was already indicated in [19], this search is equivalent in actual practice to the search for approximate quantum numbers (integrals of motion)
for our chaotic system. It was considered by Hose and Taylor [20], who named their article “Quantum Kolmogorov–Arnol’d–Moser-like Theorem: Fundamentals of Localization in Quantum Theory.” This is quite a characteristic title confirming the above statement that the situation of a soft chaos is an analog of the KAM theorem for a classical chaos. At small values of the chaos parameter, the system becomes approximately integrable, while the convergence of the expansion in (8) is the fastest. 4. NEUTRON STRENGTH FUNCTION AND OPTICAL MODEL The neutron strength function is defined as S0 (Ei ) =
|c0i |2 . D
(13)
In (13), the coefficient c0i determines the contribution to the ith compound resonance from the neutron plus target nucleus single-particle configuration in the ground state, and one can determine it from the neutron width Γn(i) of the resonance state: 2 ki 0 2 (c ) . (14) mR i Here, ki is the wave vector corresponding to the neutron-resonance energy, m is the reduced neutron mass, and R is the radius of the nucleus. It seems that the simplest way to measure experimentally the neutron strength function (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 (15) KR = (2n + 1) , 2 where the radius of the nucleus is related to its mass number A by the equation R = r0 (A)1/3 and the neutron wave vector √ reckoned from the bottom of the mean field is K = 2mE . Therefore, one can go over from one resonance to another [change the value of n in expression (15)] by fixing the neutron energy (that is, the value of K) and by changing R (that is, the target mass number). This is precisely what is done in Γn(i) =
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experiments. One chooses a target of arbitrary mass number A, measures for it the accessible set of swave neutron resonances, rescales the values found for the neutron widths to a fixed energy of E0 = 1 eV (that is, to a wave vector of ki = k0 ), and finds the strength function S(A) averaged over all resonances at a given value of A. Repeating this procedure, one then plots the dependence S(A) (see figure). It is clear that, if the fragmentation width of a single-particle state exceeded the mean spacing D0 between such states, the dependence S(A) would be linear (dashed curve in the figure) fully in accord with the prediction of the statistical model. This would correspond to the model of an absolutely black absorbing nucleus. From the point of view of chaos, this implies a hard-chaos state, in which case pair residual nucleon interactions V fully violate symmetries of the mean field. However, the experimental neutron strength function exhibits maxima at A ∼ 60 and A ∼ 160. These are mass numbers of nuclei in which levels of 3s- and 4s-wave single-particle states lie at an excitation energy approximately equal to the neutron binding energy. Thus, the experimental dependence S(A) shows that, although the symmetry of the central mean field is strongly violated by residual pair 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 suffers violations, and the 4s single-particle state fragments into several components. All this is indicative of a soft chaos, in which case the dimensionless chaos parameter satisfies the condition Γspr < 1, (16) κ= D0 which, according to [20], corresponds to the situation of quantum likeness to the KAM theorem. 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 (16), 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 PHYSICS OF ATOMIC NUCLEI
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0
Γn 4 ----- × 10 D
10 8 6 4 2 0 20
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140
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A
Experimental neutron strength function versus A.
provides a spectacular example of simple and viable calculations for quantum chaotic systems. There is a simple relation between the imaginary part of the optical potential, W (E), and the fragmentation width in (9); that is, Γspr = 2W (E).
(17)
In the region of neutron resonances (see solid curve in the figure with maxima), W ≈ 3 MeV, while the chaos parameter for medium-mass and heavy nuclei takes a value of κ ≈ 0.5. The systematics presented in the monograph of A. Bohr and Mottelson [6] makes it possible to establish an empirical dependence of W on the projectilenucleon energy in the form W (E) ∼ (3 + 0.1E) MeV. (18) Therefore, it is clear that the chaos parameter grows with energy, but it remains less than unity even for energies of about 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 (10) increases. As was indicated in [14, 17, 18], 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
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optical model in calculating observables of nuclear reactions. In those cases where k ≥ 1, it is reasonable to employ random-matrix methods or Fokker– Planck equations. 5. CONCLUSIONS 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 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. V. E. Bunakov, Phys. At. Nucl. 77, 1550 (2014). 2. M. Berry, in Proceedings of the Adriatico Research Conference and Miniworkshop on Quantum Chaos, Trieste, 1990, Ed. by H. A. Cerdeira, R. Ramaswamy, M. C. Gutzwiller, and G. Casati (World Sci., Singapore, 1991), p. VII. 3. S. W. McDonald and A. N. Kaufman, Phys. Rev. 42, 1189 (1979). 4. G. Casati, F. Valz-Gris, and I. Guarneri, Lett. Nuovo Cimento 28, 279 (1980).
5. O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984). 6. A. Bohr and B. R. Mottelson, Nuclear Structure, vol. 1: Single-Particle Motion(Benjamin, New York, 1969; Mir, Moscow, 1971). 7. M. L. Mehta, Random Matrices (Academic, New York, 1967). 8. C. E. Porter and N. Rozenzweig, Ann. Acad. Sci. Finland A 6, 44 (1960). ¨ 9. H. A. Weidenmuller and G. E. Mitchell, Rev. Mod. Phys. 81, 539 (2009). 10. V. Zelevinsky and A. Volya, Phys. Scripta T 125, 147 (2006). 11. T. Brody, Lett. Nuovo Cimento 7, 482 (1973). 12. M. V. Berry and M. Robnik, J. Phys. A 17, 2413 (1984). 13. V. Zelevinsky, B. A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276, 85 (1996). 14. V. E. Bunakov, Phys. At. Nucl. 62, 1 (1999). 15. G. M. Zaslavsky and R. Z. Sagdeev, Introduction to Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Nauka, Moscow, 1988) [in Russian]. 16. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988). 17. V. E. Bunakov, in Proceedings of the IV International Conference on Selected Topics in Nuclear Structure, Dubna, Russia, 1994, E4-94-370 (JINR, Dubna, 1994), p. 310. 18. V. E. Bunakov, in Lecture Notes of the 30th Winter School of SPb Nuclear Physics Institute (PNPI, St. Petersburg, 1996), p. 136. 19. V. E. Bunakov and I. B. Ivanov, J. Phys. A 35, 1907 (2002). 20. G. Hose and H. S. Taylor, Phys. Rev. Lett. 51, 947 (1983).
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