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

Czechoslovak Journal of Physics

Quantum—classical correspondence of the time-dependent linear potential

Mai-Lin Liang, Zhiguo Zhang, Kai-Sheng Zhong

Published Mar 20, 2004• 4 citations
Publisher
DOI
10.1023/b:cjop.0000020579.42018.d9

Related papers

Classical-quantum Interface of a Particle in a Time-dependent Linear Potential

L. Krache, Mustapha Maamache, Y. Saadi

International Journal of Theoretical Physics•Feb 23, 2010•2 citations

Quantum-mechanical effects in a linear time-dependent potential

S. V. Mousavi

arXiv:1112.6026•The European Physical Journal D•Dec 22, 2009•2 citations

The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$

Felipe Hernández, Daniel Ranard, C. Jess Riedel

In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength $D$ show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever $D \gg \hbar^{\frac43}$, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition $D \gg \hbar^{2}$ always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling $D \sim \hbar^{\frac43}$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an $\hbar$-independent quantum-classical discrepancy at the Ehrenfest time whenever $D \ll \hbar^{\frac43}$, even for $\hbar$-independent "macroscopic" smooth observables.

quant-pharXiv:2512.17623•Dec 22, 2025•0 citations

Exact solutions of time-dependent Dirac equations and the quantum–classical correspondence

Zhiguo Zhang

Exact solutions to the Dirac equations with a time-dependent mass and a static magnetic field or a time-dependent linear potential are given. Matrix elements of the coordinate, momentum and velocity operator are calculated. In the large quantum number limit, these matrix elements give the classical solution.

Physica Scripta•Jul 19, 2006•3 citations

Interpretation of the Quantum Potential

Halil Güveniş

Physics Essays•Dec 1, 2000•0 citations

The Kostin Equation, the Deceleration of a Quantum Particle and Coherent Control

Harald Losert, Freyja Ullinger, Matthias Zimmermann +3 more

Abstract Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812 ) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.

Journal of Low Temperature Physics•Sep 19, 2022•2 citations
The threshold for quantum-classical correspondence is D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​
In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time tE∼log⁡(ℏ−1)t_{\mathrm{E}} \sim \log(\hbar^{-1})tE​∼log(ℏ−1), even in the limit ℏ→0\hbar \to 0ℏ→0. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength DDD show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever D≫ℏ43D \gg \hbar^{\frac43}D≫ℏ34​, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D≫ℏ2D \gg \hbar^{2}D≫ℏ2 always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an ℏ\hbarℏ-independent quantum-classical discrepancy at the Ehrenfest time whenever D≪ℏ43D \ll \hbar^{\frac43}D≪ℏ34​, even for ℏ\hbarℏ-independent "macroscopic" smooth observables.
Abstract Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812 ) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.