Different Facets of Chaos in Quantum Mechanics

Nowadays there is no universally accepted definition of quantum chaos. In this paper we review and critically discuss different approaches to the subject, such as Quantum Chaology and the Random Matrix Theory. Then we analyze the problem of dynamical chaos and the time scales associated with chaos suppression in quantum mechanics. Summary: 1. Introduction 2. Quantum Chaology and Spectral Statistics 3. From Poisson to GOE Transition: Comparison with Experimental Data 3.1 Atomic Nuclei 3.2 The Hydrogen Atom in the Strong Magnetic Field 4. Quantum Chaos and Field Theory 5. Alternative Approaches to Quantum Chaos 6. Dynamical Quantum Chaos and Time Scales 6.1 Mean-Field Approximation and Dynamical Chaos 7. ConclusionsComment: RevTex, 25 pages, 7 postscript figures, to be published in Int. J. Mod. Phys.

Order and Chaos in Roto--Vibrational States of Atomic Nuclei

Using a classical analytical criterion (that of curvature) and numerical results (Poincar\`e sections and spectral statistics), a transition order--chaos--order in the roto--vibrational model of atomic nuclei has been shown. Numerical calculations were performed for some deformed nuclei.Comment: LaTex, 6 figures (available on request), accepted for publication in Int. J. Mod. Phys.

Quantum Corrections to the Semiclassical Quantization of the SU(3) Shell Model

We apply the canonical perturbation theory to the semi--quantal hamiltonian of the SU(3) shell model. Then, we use the Einstein--Brillowin--Keller quantization rule to obtain an analytical semi--quantal formula for the energy levels, which is the usual semi--classical one plus quantum corrections. Finally, a test on the numerical accuracy of the semiclassical approximation and of its quantum corrections is performed.Comment: LaTex, 17 pages, 1 figure (available upon request to the authors), to be published in Modern Physics Letters

Anomalous Spectral Statistics of the Asymmetric Rotor Model

We investigate the spectral statistics of the asymmetric rotor model (triaxial rigid rotator). The asymmetric top is classically integrable and, according to the Berry-Tabor theory, its spectral statistics should be Poissonian. Surprisingly, our numerical results show that the nearest neighbor spacing distribution $P(s)$ and the spectral rigidity $\Delta_3(L)$ do not follow Poisson statistics.Comment: 9 pages, 4 figures, presented to the 7th International Spring Seminar on Nuclear Physics, Maiori, May 27-31 (2001), to be published in the proceedings, Ed. A. Covello (World Scientific, 2002

Coexistence of Ordered and Chaotic States in Nuclear Structure

In atomic nuclei, as in other many-body systems, the classical phase space is mixed, so ordered and chaotic states generally coexist. In this contribution we discuss some models, showing the transition from order to chaos. In several cases a clear correspondence between classical and quantum chaos has been established. In particular the transition from ordered to chaotic states will be discussed in the framework of the shell and roto-vibrational models.Comment: Latex, 3 pages. Talk of Luca Salasnich to the International Conference on Nuclear Data for Science and Technology, May 19-24, ICTP, Trieste, Italy. To be published in the Proceeding

New Results on Quantum Chaos in Atomic Nuclei

In atomic nuclei, ordered and chaotic states generally coexist. In this paper the transition from ordered to chaotic states will be discussed in the framework of roto-vibrational and shell models. In particular for $^{160}Gd$, in the roto-vibrational model, the Poincar\`e sections clearly show the transition from order to chaos for different values of rotational frequency. Furthermore, the spectral statistics of low-lying states of several $fp$ shell nuclei are studied with realistic shell-model calculations.Comment: Latex, 10 pages, invited talk of Prof. V.R. Manfredi to the International Symposium on Large-Scale Collective Motion of Atomic Nuclei, October 1996, Brolo (Messina), to be published in the Proceedings (World Scientific, Singapore

A Note on the Toda Criterion for Interacting Dipole-Quadrupole Vibrations

The Toda criterion of the Gaussian curvature is applied to calculate analytically the transition energy from regular to chaotic motion in a schematic model describing the interaction between collective dipole and quadrupole modes in atomic nuclei.Comment: Latex, 9 pages, 2 figures (available upon request), to be published in Modern Physics Letters

Energy Level Quasi-Crossings: Accidental Degeneracies or Signature of Quantum Chaos?

In the field of quantum chaos, the study of energy levels plays an important role. The aim of this review paper is to critically discuss some of the main contributions regarding the connection between classical dynamics, semi-classical quantization and spectral statistics of energy levels. In particular, we analyze in detail degeneracies and quasi-crossings in the eigenvalues of quantum Hamiltonians which are classically non-integrable. Summary: 1. Introduction; 2. Quasi-Crossing and Chaos; 3. Molecular Spectroscopy; 4. Nuclear Models; 4.1 Zirnbauer-Verbaashot-Weidenmuller Model; 4.2 Lipkin-Meshow-Glick Model; 5. Particle Physics and Field Theory; 6. Conclusions.Comment: 26 pages, Latex, 9 figures, to be published in International Journal of Modern Physics

Spectral Statistics of the Triaxial Rigid Rotator: Semiclassical Origin of their Pathological Behavior

In this paper we investigate the local and global spectral properties of the triaxial rigid rotator. We demonstrate that, for a fixed value of the total angular momentum, the energy spectrum can be divided into two sets of energy levels, whose classical analog are librational and rotational motions. By using diagonalization, semiclassical and algebric methods, we show that the energy levels follow the anomalous spectral statistics of the one-dimensional harmonic oscillator.Comment: 14 pages with 5 figures, to be published in Int. J. Mod. Phys.

Quantal Overlapping Resonance Criterion in the Pullen Edmonds Model

In order to highlight the onset of chaos in the Pullen-Edmonds model a quantal analog of the resonance overlap criterion has been examined. A quite good agreement between analytical and numerical results is obtained.Comment: 12 pages, LATEX, 2 figures available upon request to the Authors, submitted to Mod. Phys. Lett.