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WORLD SCIENTIFIC eBooks

Quantum Non-integrability

D.H. Feng, Jian‐Min Yuan

Published Sep 1, 1992• 18 citations
Publisher

Abstract

Quantum chaotic scattering, R. Blumel microwave ionization of Rydberg atoms, P. Koch and L. Moorman geometry and quantum mechanics, L.N. Chang and Y.G. Liang where is the chaos in two-electron atoms?, R. Blumel and W.P. Reinhardt quantum classical correspondence and quantum chaos, W-M. Chang et al phase space structure and fractal dissociation boundary of a kicked molecular system, J.M Yuan et al from classical to quantum chaos, G. Zaslavsky quantum chaos in nuclear physics, F. Sakata and M. Marumori.

DOI
10.1142/1376

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Introduction

R. Blümel, William P. Reinhardt

By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.

Cambridge University Press eBooks•Jul 24, 1997•0 citations

Quantum chaos: diffusion photoeffect in hydrogen

Dima L. Shepelyansky

Some properties of quantum chaos G P Berman Stochastic web and structure generation G M Zaslavski, Renad Z Sagdeev, O A Usikov et al. Dynamical chaos in magnetic systems K N Alekseev, G P Berman, V I Tsifrinovich et al. Quantum chaos in interactions of multilevel quantum systems with a coherent radiation field G P Berman and A R Kolovski Minimal chaos, stochastic webs, and structures of quasicrystal symmetry G M Zaslavski, Roal'd Z Sagdeev, D A Usikov et al. Ionization of excited hydrogen atoms by microwave fields: a test case for quantum chaos R Blumel and U Smilansky Quantum effects in the long time behavior of the dissipative standard map T Dittrich and R Graham Quantum chaos: diffusion photoeffect in hydrogen

Soviet Physics Uspekhi•May 31, 1987•0 citations

Quantum Mechanics, Diffusion and Chaotic Fractals

Mohamed Saladin El Naschie, Otto E. Rössler, Ilya Prigogine

Quantum mechanics and chaotic fields quantum chaos, complex spectral representation and time-symmetry breaking, T. Petrosky and I. Prigogine scale relativity, fractal space-time and quantum mechanics, L. Nottale introduction to wave phenomena and uncertainty in a fractal space - I, A. Le Mehaute et al young double-slit experiment, Heisenberg uncertainty principle and cantoria space-time, M.S. El Naschie the quantum dimension of space-time, E. Alvarez et al intra-observer chaos - hidden root of quantum mechanics?, O.E. Rossler amplification of superpositional effects through electronic - conformational interactions, M. Conrad quantum cellular automation in 1-D, D.W. Belousek et al an energy-barrier model of biased transport in disordered systems, M. Giona chaos in the Einstein equations, S.E. Rugh.

•May 1, 1995•210 citations

Quantum Chaos in Quantum Optics - Lectures on the Quantum Dynamics of Classically Chaotic Systems

Robert Graham

NATO ASI series. Series B : Physics•Jan 1, 1988•0 citations

Introduction

Giulio Casati, Boris Chirikov

The present collection of papers is devoted to a rather new and very controversial topic, the so-called “quantum chaos”. Some researchers see nothing essentially new at all in this phenomenon (apart from a number of various examples and models), and they have good reason to believe so. Indeed, the problems in this field all belong to the traditional, “old-fashion”, and rather “simple” quantum mechanics of finite-dimensional systems with a given interaction and no quantized fields. Nevertheless, many, including ourselves, consider quantum chaos to be a new discovery, though in an old field, of a great importance for fundamental physics. To understand this, the phenomenon of quantum chaos should be put into its proper perspective in recent developments in physics. The central point of this perspective is the conception of dynamical chaos (also a rather new topic) in classical mechanics (for a good review see, e.g., Refs.[l-3]). Thus before discussing the current understanding of quantum chaos we need briefly to describe classical dynamical chaos

Cambridge University Press eBooks•Apr 27, 1995•0 citations

Classical Fractals and Quantum Chaos in Ultracold Dipolar Collisions

Bing Yang, Jesús Pérez‐Ríos, F. Robicheaux

We examine a dipolar-gas model to address fundamental issues regarding the correspondence between classical chaos and quantum observations in ultracold dipolar collisions. The theoretical model consists of a short-range Lennard-Jones potential well with an anisotropic, long-range dipole-dipole interaction between two atoms. Both the classical and quantum dynamics are explored for the same Hamiltonian of the system. The classical chaotic scattering is revealed by the fractals developed in the scattering function (defined as the final atom separation as a function of initial conditions), while the quantum chaotic features lead to the repulsion of the eigenphases from the corresponding quantum S matrix. The nearest-eigenphase-spacing statistics have an intermediate behavior between the Poisson and the Wigner-Dyson distributions. The character of the distribution can be controlled by changing an effective Planck constant or the dipole moment. The degree of quantum chaos shows a good correspondence with the overall average of the classical scattering function. The results presented here also provide helpful insights for understanding the role of the inherent dipole-dipole interaction in the currently ongoing experiments on ultracold collisions of highly magnetic atoms.

Physical Review Letters•Apr 10, 2017•22 citations
Quantum chaotic scattering, R. Blumel microwave ionization of Rydberg atoms, P. Koch and L. Moorman geometry and quantum mechanics, L.N. Chang and Y.G. Liang where is the chaos in two-electron atoms?, R. Blumel and W.P. Reinhardt quantum classical correspondence and quantum chaos, W-M. Chang et al phase space structure and fractal dissociation boundary of a kicked molecular system, J.M Yuan et al from classical to quantum chaos, G. Zaslavsky quantum chaos in nuclear physics, F. Sakata and M. Marumori.
By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.
Some properties of quantum chaos G P Berman Stochastic web and structure generation G M Zaslavski, Renad Z Sagdeev, O A Usikov et al. Dynamical chaos in magnetic systems K N Alekseev, G P Berman, V I Tsifrinovich et al. Quantum chaos in interactions of multilevel quantum systems with a coherent radiation field G P Berman and A R Kolovski Minimal chaos, stochastic webs, and structures of quasicrystal symmetry G M Zaslavski, Roal'd Z Sagdeev, D A Usikov et al. Ionization of excited hydrogen atoms by microwave fields: a test case for quantum chaos R Blumel and U Smilansky Quantum effects in the long time behavior of the dissipative standard map T Dittrich and R Graham Quantum chaos: diffusion photoeffect in hydrogen
Quantum mechanics and chaotic fields quantum chaos, complex spectral representation and time-symmetry breaking, T. Petrosky and I. Prigogine scale relativity, fractal space-time and quantum mechanics, L. Nottale introduction to wave phenomena and uncertainty in a fractal space - I, A. Le Mehaute et al young double-slit experiment, Heisenberg uncertainty principle and cantoria space-time, M.S. El Naschie the quantum dimension of space-time, E. Alvarez et al intra-observer chaos - hidden root of quantum mechanics?, O.E. Rossler amplification of superpositional effects through electronic - conformational interactions, M. Conrad quantum cellular automation in 1-D, D.W. Belousek et al an energy-barrier model of biased transport in disordered systems, M. Giona chaos in the Einstein equations, S.E. Rugh.
The present collection of papers is devoted to a rather new and very controversial topic, the so-called “quantum chaos”. Some researchers see nothing essentially new at all in this phenomenon (apart from a number of various examples and models), and they have good reason to believe so. Indeed, the problems in this field all belong to the traditional, “old-fashion”, and rather “simple” quantum mechanics of finite-dimensional systems with a given interaction and no quantized fields. Nevertheless, many, including ourselves, consider quantum chaos to be a new discovery, though in an old field, of a great importance for fundamental physics. To understand this, the phenomenon of quantum chaos should be put into its proper perspective in recent developments in physics. The central point of this perspective is the conception of dynamical chaos (also a rather new topic) in classical mechanics (for a good review see, e.g., Refs.[l-3]). Thus before discussing the current understanding of quantum chaos we need briefly to describe classical dynamical chaos
We examine a dipolar-gas model to address fundamental issues regarding the correspondence between classical chaos and quantum observations in ultracold dipolar collisions. The theoretical model consists of a short-range Lennard-Jones potential well with an anisotropic, long-range dipole-dipole interaction between two atoms. Both the classical and quantum dynamics are explored for the same Hamiltonian of the system. The classical chaotic scattering is revealed by the fractals developed in the scattering function (defined as the final atom separation as a function of initial conditions), while the quantum chaotic features lead to the repulsion of the eigenphases from the corresponding quantum S matrix. The nearest-eigenphase-spacing statistics have an intermediate behavior between the Poisson and the Wigner-Dyson distributions. The character of the distribution can be controlled by changing an effective Planck constant or the dipole moment. The degree of quantum chaos shows a good correspondence with the overall average of the classical scattering function. The results presented here also provide helpful insights for understanding the role of the inherent dipole-dipole interaction in the currently ongoing experiments on ultracold collisions of highly magnetic atoms.