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

Entropy

Quantum–Classical Hybrid Systems and Ehrenfest’s Theorem

Alessandro Sergi, Daniele Lamberto, Agostino Migliore, ANTONINO MESSINA

Published Apr 1, 2023• 12 citations
Publisher

Abstract

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.

DOI
10.3390/e25040602

Related papers

Significance of Ehrenfest theorem in quantum-classical relationship

Dipak Sen, Siddhant Das, A. N. Basu +1 more

ψ-function, the Generalized Ehrenfest Theorem re- veals some interesting exact relationships between quantum and classical expectation values. These general results seem to imply a deep-rooted unity (in spite of apparent radical differences in conceptual structures) between classical and quantum mechan- ics. Some significant consequences and important physical insights which follow from the general formulation are discussed with examples. Most im- portant is that it offers, under reasonable approx i- mations, a pure quantum mechanical description of the Stern-Gerlach experiment with realistic inhomo- geneous magnetic field (∇ ∇.B = 0). EVEN almost seventy-five years after the first formul a- tion of quantum mechanics (QM), its relation with clas - sical mechanics (CM) remains obscure. The usual belief that QM has superseded CM is also not well founded. In all cases where a new theory supersedes an old one, there is a universal prescription of recovering the mathematical structure of the old theory in the doma in of its validity, i.e. where it is empirically established. This is true for both special and general theory of rel a- tivity 1 . But for QM, no such universal approximation procedure can be offered. About the success and failure of the two mechanics we can make two positive asser- tions: (i) There are experimental results in the micr o- scopic domain (such as hydr ogen spectrum) or in the macroscopic domain (such as specific heat of so lids at low temperature) which cannot be interpreted within the framework of CM. (ii) In all such cases, the mathemat i- cal formalism of QM provides the correct empirical results. Not a single failure has yet been reported. These empirical evidences show that CM cannot be the universal mechanics and that QM has such a pote n- tiality. However, since it is not yet possible to recover CM in all the domains of its validity, we cannot assert that QM has superseded CM. A second alternative that

•Jan 1, 2001•9 citations

Statistics and Nos\\'e formalism for Ehrenfest dynamics

José L. Alonso, Castro, Alberto, Jesús Clemente-Gallardo +3 more

Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.

•May 8, 2019•30 citations

Effective nonlinear Ehrenfest hybrid quantum-classical dynamics

José L. Alonso, Carlos Bouthelier-Madre, Jesús Clemente-Gallardo +2 more

Abstract The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices $${\hat{\rho }}(\xi )$$ ρ ^ ( ξ ) , for $$\xi$$ ξ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.

arXiv:2308.14440•The European Physical Journal Plus•Jul 25, 2023•11 citations

Quantum systems that follow classical dynamics

Giovanni Manfredi, S Mola, Marc Roy Feix

For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.

European Journal of Physics•May 1, 1993•14 citations

Dynamics in the quantum/classical limit based on selective use of the quantum potential

Sophya V. Garashchuk, David Dell’Angelo, Vitaly A. Rassolov

A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.

The Journal of Chemical Physics•Dec 16, 2014•11 citations

On the quantum mechanical treatment of dissipative systems

Rainer W. Hasse

Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.

Journal of Mathematical Physics•Oct 1, 1975•310 citations
The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.
ψ-function, the Generalized Ehrenfest Theorem re- veals some interesting exact relationships between quantum and classical expectation values. These general results seem to imply a deep-rooted unity (in spite of apparent radical differences in conceptual structures) between classical and quantum mechan- ics. Some significant consequences and important physical insights which follow from the general formulation are discussed with examples. Most im- portant is that it offers, under reasonable approx i- mations, a pure quantum mechanical description of the Stern-Gerlach experiment with realistic inhomo- geneous magnetic field (∇ ∇.B = 0). EVEN almost seventy-five years after the first formul a- tion of quantum mechanics (QM), its relation with clas - sical mechanics (CM) remains obscure. The usual belief that QM has superseded CM is also not well founded. In all cases where a new theory supersedes an old one, there is a universal prescription of recovering the mathematical structure of the old theory in the doma in of its validity, i.e. where it is empirically established. This is true for both special and general theory of rel a- tivity 1 . But for QM, no such universal approximation procedure can be offered. About the success and failure of the two mechanics we can make two positive asser- tions: (i) There are experimental results in the micr o- scopic domain (such as hydr ogen spectrum) or in the macroscopic domain (such as specific heat of so lids at low temperature) which cannot be interpreted within the framework of CM. (ii) In all such cases, the mathemat i- cal formalism of QM provides the correct empirical results. Not a single failure has yet been reported. These empirical evidences show that CM cannot be the universal mechanics and that QM has such a pote n- tiality. However, since it is not yet possible to recover CM in all the domains of its validity, we cannot assert that QM has superseded CM. A second alternative that
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
Abstract The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices ρ^(ξ){\hat{\rho }}(\xi )ρ^​(ξ) ρ ^ ( ξ ) , for ξ\xiξ ξ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.
For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.
A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.
Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.