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

International Journal of Quantum Chemistry

Inside Cover, Volume 113, Issue 13

Published May 21, 2013• 0 citations
Publisher

Abstract

Quantum potential based theories present important insights into the quantum domain behavior of classically chaotic systems. The Review on page 1747 by Munmun Khatua, Debdutta Chakraborty, and Pratim Kumar Chattaraj discusses some representative examples of quantum signature of classical chaos through the study of quantum domain behavior of classical nonintegrable systems using quantum fluid dynamics, the quantum theory of motion and the quantum fluid density functional theory, and also the reactivity dynamics of various systems in free and confined environments. The solution of the relevant generalized non-linear Schrödinger equation provides the essential time-dependent density and current density and dynamical variants of different electronic structure principles become evident.

DOI
10.1002/qua.24477

Related papers

Density dynamics in some quantum systems

Munmun Khatua, Debdutta Chakraborty, Pratim Kumar Chattaraj

Abstract The quantum domain behavior of classical nonintegrable systems is well‐understood by the implementation of quantum fluid dynamics and quantum theory of motion. These approaches properly explain the quantum analogs of the classical Kolmogorov–Arnold–Moser type transitions from regular to chaotic domain in different anharmonic oscillators. Field‐induced tunneling and chaotic ionization in Rydberg atoms are also analyzed with the help of these theories. Quantum fluid density functional theory may be used to understand different time‐dependent processes like ion‐atom/molecule collisions, atom‐field interactions, and so forth. Regioselectivity as well as confined atomic/molecular systems and their reactivity dynamics have also been explained. © 2013 Wiley Periodicals, Inc.

International Journal of Quantum Chemistry•Feb 20, 2013•7 citations

Orbital free DFT versus single density equation: a perspective through quantum domain behavior of a classically chaotic system

Debdutta Chakraborty, Susmita Kar, Pratim Kumar Chattaraj

The orbital free density functional theory and the single density equation approach are formally equivalent. An orbital free density based quantum dynamical strategy is used to study the quantum-classical correspondence in both weakly and strongly coupled van der Pol and Duffing oscillators in the presence of an external electric field in one dimension. The resulting quantum hydrodynamic equations of motion are solved through an implicit Euler type real space method involving a moving weighted least square technique. The Lagrangian framework used here allows the numerical grid points to follow the wave packet trajectory. The associated classical equations of motion are solved using a sixth order Runge-Kutta method and the Ehrenfest dynamics is followed through the solution of the time dependent Schrodinger equation using a time dependent Fourier Grid Hamiltonian technique. Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics.

Physical Chemistry Chemical Physics•Jan 1, 2015•15 citations

Statistical methods and ‘quantum chaology’

J P Connerade

Introduction A system is considered as chaotic in classical mechanics if the orbits, instead of remaining confined to a specific region, invade the whole of available phase space. A simple example is a pendulumn with a magnet below the bob: if a sheet of paper is placed between the magnet and the bob and a pen is attached, the pendulumn will write all over the page within the range accessible to it. More exactly, if we examine phase space, it will seem completely disordered, with interwoven tracks throughout its volume. If we magnify the volume, the disorder will persist, and so on ad infinitum no matter how great the magnification, because classical mechanics imposes no limit on the resolution which can be achieved. Chaotic behaviour can arise in any system whose motion is described by a nonlinear differential equation. Whether or not it is prevalent depends on the details of the problem, but it is a general theorem that any system described by a nonlinear differential equation possesses some chaotic regime. In quantum mechanics, by contrast, chaos does not occur. We may see this in several ways. First, note that we cannot magnify ad infinitum the volume to be analysed in phase space: eventually, we reach the elementary volume ħ 3 within which trajectories lose their meaning. Another way of reaching the same conclusion is to note that any Schrödinger type equation is linear: its solutions obey the superposition theorem. Under these circumstances genuine chaos is excluded by fundamental principles.

Cambridge University Press eBooks•May 7, 1998•0 citations

Ehrenfest ‘phase-space’ trajectories and quantum chaos in He atom under strong, oscillating magnetic fields: an application of time-dependent quantum fluid density functional theory (TDQFDFT)

Mainak Sadhukhan, Bimalendu Deb

Using a dynamical signature proposed earlier from our laboratory, quantum chaos in He atom interacting with strong, oscillating magnetic fields has been studied through a comparison between the nonlinear divergence of two neighbouring Ehrenfest ‘phase-space’ (EPS) trajectories differing slightly in initial conditions and the Loschmidt echo. The dynamical EPS signature can detect quantum chaos independently of the Loschmidt echo and in agreement with the latter, even for low-lying states, in the same spirit as that of classical chaos. This time-dependent signature extends the concept of quantum chaos to systems which have no classical counterparts and brings the concept of quantum chaos closer to that of classical chaos.

Molecular Physics•Feb 22, 2017•5 citations

Quantum Chaos

H.‐J. Stöckmann

This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.

Cambridge University Press eBooks•Oct 13, 1999•959 citations

Evaluation of theapplicability of Schrödinger-variant equations for classical fluid dynamics

Y. S. Ciou

The artificial fluid model known as "Schrödinger flow" (SF) can represent rotational flow with dissipative effects, and has attracted attention despite its gap from real-world fluid behavior. To address the structural discrepancy arising from the incomplete transition from quantum hydrodynamics to classical fluid dynamics, we propose a variant of the hydrodynamic Schrödinger equation (HSE) that better aligns with classical formulations. We then revisit an alternative method for eliminating the quantum potential (referred to in this work as the "cancellation approach") in light of the known non-universality of the classical limit. Beyond examining conservation laws via Lagrangian field theory, we identify its deeper connection to a generalized Lagrangian field theory, which offers a more straightforward route to constructing a Lagrangian density in terms of the components of a two-component wave function for rotational inviscid flow. Finally, we discuss the completeness of Clebsch parameterization, concluding our evaluation of the applicability of the proposed SF-variant formulation.

arXiv:2505.19265•May 25, 2025•0 citations
Quantum potential based theories present important insights into the quantum domain behavior of classically chaotic systems. The Review on page 1747 by Munmun Khatua, Debdutta Chakraborty, and Pratim Kumar Chattaraj discusses some representative examples of quantum signature of classical chaos through the study of quantum domain behavior of classical nonintegrable systems using quantum fluid dynamics, the quantum theory of motion and the quantum fluid density functional theory, and also the reactivity dynamics of various systems in free and confined environments. The solution of the relevant generalized non-linear Schrödinger equation provides the essential time-dependent density and current density and dynamical variants of different electronic structure principles become evident.
Abstract The quantum domain behavior of classical nonintegrable systems is well‐understood by the implementation of quantum fluid dynamics and quantum theory of motion. These approaches properly explain the quantum analogs of the classical Kolmogorov–Arnold–Moser type transitions from regular to chaotic domain in different anharmonic oscillators. Field‐induced tunneling and chaotic ionization in Rydberg atoms are also analyzed with the help of these theories. Quantum fluid density functional theory may be used to understand different time‐dependent processes like ion‐atom/molecule collisions, atom‐field interactions, and so forth. Regioselectivity as well as confined atomic/molecular systems and their reactivity dynamics have also been explained. © 2013 Wiley Periodicals, Inc.
The orbital free density functional theory and the single density equation approach are formally equivalent. An orbital free density based quantum dynamical strategy is used to study the quantum-classical correspondence in both weakly and strongly coupled van der Pol and Duffing oscillators in the presence of an external electric field in one dimension. The resulting quantum hydrodynamic equations of motion are solved through an implicit Euler type real space method involving a moving weighted least square technique. The Lagrangian framework used here allows the numerical grid points to follow the wave packet trajectory. The associated classical equations of motion are solved using a sixth order Runge-Kutta method and the Ehrenfest dynamics is followed through the solution of the time dependent Schrodinger equation using a time dependent Fourier Grid Hamiltonian technique. Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics.
Introduction A system is considered as chaotic in classical mechanics if the orbits, instead of remaining confined to a specific region, invade the whole of available phase space. A simple example is a pendulumn with a magnet below the bob: if a sheet of paper is placed between the magnet and the bob and a pen is attached, the pendulumn will write all over the page within the range accessible to it. More exactly, if we examine phase space, it will seem completely disordered, with interwoven tracks throughout its volume. If we magnify the volume, the disorder will persist, and so on ad infinitum no matter how great the magnification, because classical mechanics imposes no limit on the resolution which can be achieved. Chaotic behaviour can arise in any system whose motion is described by a nonlinear differential equation. Whether or not it is prevalent depends on the details of the problem, but it is a general theorem that any system described by a nonlinear differential equation possesses some chaotic regime. In quantum mechanics, by contrast, chaos does not occur. We may see this in several ways. First, note that we cannot magnify ad infinitum the volume to be analysed in phase space: eventually, we reach the elementary volume ħ 3 within which trajectories lose their meaning. Another way of reaching the same conclusion is to note that any Schrödinger type equation is linear: its solutions obey the superposition theorem. Under these circumstances genuine chaos is excluded by fundamental principles.
Using a dynamical signature proposed earlier from our laboratory, quantum chaos in He atom interacting with strong, oscillating magnetic fields has been studied through a comparison between the nonlinear divergence of two neighbouring Ehrenfest ‘phase-space’ (EPS) trajectories differing slightly in initial conditions and the Loschmidt echo. The dynamical EPS signature can detect quantum chaos independently of the Loschmidt echo and in agreement with the latter, even for low-lying states, in the same spirit as that of classical chaos. This time-dependent signature extends the concept of quantum chaos to systems which have no classical counterparts and brings the concept of quantum chaos closer to that of classical chaos.
This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.
The artificial fluid model known as "Schrödinger flow" (SF) can represent rotational flow with dissipative effects, and has attracted attention despite its gap from real-world fluid behavior. To address the structural discrepancy arising from the incomplete transition from quantum hydrodynamics to classical fluid dynamics, we propose a variant of the hydrodynamic Schrödinger equation (HSE) that better aligns with classical formulations. We then revisit an alternative method for eliminating the quantum potential (referred to in this work as the "cancellation approach") in light of the known non-universality of the classical limit. Beyond examining conservation laws via Lagrangian field theory, we identify its deeper connection to a generalized Lagrangian field theory, which offers a more straightforward route to constructing a Lagrangian density in terms of the components of a two-component wave function for rotational inviscid flow. Finally, we discuss the completeness of Clebsch parameterization, concluding our evaluation of the applicability of the proposed SF-variant formulation.