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

arXiv:0907.1877 · Journal of Mathematical Physics

On the Ehrenfest theorem of quantum mechanics

Gero Friesecke, Mario Koppen

Published Aug 1, 2009• 17 citations
arXivHTMLPDF

Abstract

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.
DOI
10.1063/1.3191679

Related papers

On the Ehrenfest theorem of quantum mechanics

Gero Friesecke

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values, under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions. 1

•Aug 4, 2013•14 citations

A sharp version of Ehrenfest’s theorem for general self-adjoint operators

Gero Friesecke, Bernd Schmidt

Abstract We prove the Ehrenfest theorem of quantum mechanics under sharp assumptions on the operators involved.

arXiv:1003.3372•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Feb 17, 2010•9 citations

Two ways to calculate momentum expectation values with different quantum probability and flux densities

Thomas Schaupp, Klaus Renziehausen, Ingo Barth +1 more

Using the Ehrenfest theorem, two possibilites are regarded to calculate momentum expectation values. They lead to different quantum probability and flux densities being illustrated with numerical results from a model for coupled electron-nuclear motion.

International Conference on Ultrafast Phenomena•Jan 1, 2022•0 citations

Evolution of expected values in open quantum systems

Andrés Vallejo, A. Romanelli, Virginia Feldman +1 more

We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.

arXiv:2402.18895•Physical Review A•Mar 3, 2025•2 citations

The Validity of the Ehrenfest Theorem in Quantum Gravity Theory

Claudio Cremaschini, Cooper K. Watson, Ramesh Radhakrishnan +1 more

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.

Symmetry•Jan 19, 2026•1 citation

Lecture in Quantum Mechanics -- Ehrenfest theorem (part2) and evolution operator

Emmanuel Fromager

•Nov 5, 2020•0 citations
We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.
The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.