Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Yuanzihe wuli pinglun

Quantum-classical Correspondence of Regular Motion of Integrable Systems and Related Problems

Gong-ou Xu

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Based on the dynamical symmetry group characterizing the integrability of classical as well as quantum mechanics, quantum dynamics with proper initial conditions was genuinely formulated, and analytical solutions in the form of soliton-like state evolving around a certain invariant torus were obtained. It has been shown that, in case the intrinsic size of the evolving quantum state is significantly smaller than the extent of its evolving orbit, the motion can be satisfactorily treated with classical approximation. Having demonstrated the quantum-classical correspondence of regular motion, the possibility to study quantum chaos on the basis dynamical symmetry breaking was briefly discussed.

Related papers

General Character of Quantum Regular Motion and Quantum-Classical Correspondence

Gong-ou Xu, Yatian Yang, Xing Yong-Zhong

In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.

Chinese Physics Letters•May 1, 1999•4 citations

Dynamical symmetry, integrability of quantum systems, and general character of quantum regular motion

Gong-ou Xu, Yatian Yang, Xing Yong-Zhong

The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ${\ensuremath{\rho}}_{\ensuremath{\gamma}}(t)$ of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ${\ensuremath{\rho}}_{\ensuremath{\gamma}}^{0}$ varying smoothly with $\ensuremath{\gamma}$ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.

Physical Review A•Mar 17, 2000•4 citations

Integrability and chaos in quantum systems (as viewed from geometry and dynamical symmetry)

Wei-Min Zhang, D.H. Feng

It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.

•Nov 1, 1989•2 citations

Dynamical Symmetry Breaking and Quantum Nonintegrability

Wei-Min Zhang, Craig C. Martens, D.H. Feng +1 more

In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.

Physical Review Letters•Nov 7, 1988•41 citations

Quantum Chaos and Quantum-Classical Correspondence

Joseph Emerson

The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.

arXiv:quant-ph/0211035•Nov 7, 2002•10 citations

Quantization of integrable systems

M. Brack, Rajat K. Bhaduri

This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.

CRC Press eBooks•Mar 5, 2018•0 citations
Based on the dynamical symmetry group characterizing the integrability of classical as well as quantum mechanics, quantum dynamics with proper initial conditions was genuinely formulated, and analytical solutions in the form of soliton-like state evolving around a certain invariant torus were obtained. It has been shown that, in case the intrinsic size of the evolving quantum state is significantly smaller than the extent of its evolving orbit, the motion can be satisfactorily treated with classical approximation. Having demonstrated the quantum-classical correspondence of regular motion, the possibility to study quantum chaos on the basis dynamical symmetry breaking was briefly discussed.
In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ${\ensuremath{\rho}}_{\ensuremath{\gamma}}(t)$ of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ${\ensuremath{\rho}}_{\ensuremath{\gamma}}^{0}$ varying smoothly with $\ensuremath{\gamma}$ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.
It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.
In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.
The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.