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

The Journal of Chemical Physics

A Bohmian total potential view to quantum effects. I. Methodology and simple model systems

Javier González, Josep María Bofill, Xavier Giménez

Published May 21, 2004• 7 citations
Publisher

Abstract

The coherent-state wave packet dynamics of several model systems is analyzed in terms of Bohm's total potential. The quantum dynamics has been obtained by solving the time-dependent Schrodinger equation, and a method for obtaining the total potential from it, involving just matrix algebra, has been proposed. Contrary to what one may expect, it is shown that the time- and state-dependent features of the total potential admit a rationale, classical-like description of quantum effects, leading to a unified picture of them, which is not critically dependent, as for the key features, on the classical potential. An outstanding feature is found to be the relation of the state system's density amplitude and sharpness (in its dependence with position) with quantum effects. Sharp density profiles and low densities cause the total potential to strongly depart from the classical value, in both time regimes and position ranges, which provide a clearer, more deterministic view to quantum dynamics. Free motion as well as scattering processes by square and Eckart barriers have been analyzed by means of careful inspection of several time dependent snapshots. The result is an insightful picture of processes involving tunneling and antitunneling, including their dynamical variants, as well as resonances and quantization.

DOI
10.1063/1.1747869

Related papers

Bohmian Total Potential View to Quantum Effects III. Tunnelling in Phase Space

María F. González, Josep María Bofill, Xavier Giménez

The tunnel effect in one-dimensional collision problems is analyzed in terms of numerical wavepacket time propagations, in position, momentum, and phase space representations of the problem. Bohm's total potential is used to provide a classical-like description of tunnelling quantum dynamics, under a time- and density-dependent potential. Wigner, Husimi, as well as Bohm phase space distributions are used in the present study, with their relative performances for the present problem having been analyzed.

The Journal of Physical Chemistry A•Oct 20, 2009•1 citation

Quantum trajectories in complex space

Chia‐Chun Chou, Róbert E. Wyatt

Quantum trajectories in complex space in the framework of the quantum Hamilton-Jacobi formalism are investigated. For time-dependent problems, the complex quantum trajectories determined from the exact analytical wave function for the free Gaussian wave packet and the coherent state in the harmonic potential are used to demonstrate that the information transported by the particles in the complex space can be used to synthesize the time-dependent wave function on the real axis. For time-independent problems, the exact complex quantum trajectories for the Eckart potential are obtained by numerically integrating the equations of motion. The unusual structure of the total potential (the sum of the classical and the quantum potentials) for the stationary states for the Eckart potential is pointed out. The variations of the complex-valued kinetic energy, classical potential, and quantum potential along the complex quantum trajectories are analyzed. This paper not only analyzes complex quantum trajectories for time-dependent and time-independent systems but also provides a unified description for complex quantum trajectories for one-dimensional problems in the quantum Hamilton-Jacobi formalism.

Physical Review A•Jul 20, 2007•45 citations

Bohmian mechanics in the exact factorization of electron-nuclear wave functions

Yasumitsu Suzuki, Kazuyuki Watanabe

The exact factorization of an electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us to define the rigorous nuclear time-dependent Schr\"odinger equation (TDSE) with a time-dependent potential-energy surface (TDPES) that fully accounts for the coupling to the electronic motion and drives the nuclear wave-packet dynamics. Here, we study whether the propagation of multiple classical trajectories can reproduce the quantum nuclear motion in strong-field processes when their motions are governed by the quantum Hamilton-Jacobi equation derived by applying Bohmian mechanics to this exact nuclear TDSE. We demonstrate that multiple classical trajectories propagated by the force from the gradient of the exact TDPES plus the Bohmian quantum potential can reproduce the strong-field dissociation dynamics of a one-dimensional model of the ${{\mathrm{H}}_{2}}^{+}$ molecule. Our results show that the force from the Bohmian quantum potential plays a non-negligible role in yielding quantum nuclear dynamics in the strong-field process studied here, where ionization and/or splitting of nuclear probability density occurs.

arXiv:1607.04854•Physical Review A•Sep 29, 2016•32 citations

A trajectory-based understanding of quantum interference

Ángel S. Sanz, Salvador Miret-Artés

Interference is one of the most fundamental features which characterizes quantum systems. Here we provide an exhaustive analysis of the interfere dynamics associated with wave-packet superpositions from both the standard quantum-mechanical perspective and the Bohmian one. From this analysis, clear and insightful pictures of the physics involved in these kind of processes are obtained, which are of general validity (i.e., regardless of the type of wave packets considered) in the understanding of more complex cases where interference is crucial (e.g., scattering problems, slit diffraction, quantum control scenarios or, even, multipartite interactions). In particular, we show how problems involving wave-packet interference can be mapped onto problems of wave packets scattered off potential barriers.

arXiv:0806.2105•Journal of Physics A Mathematical and Theoretical•Sep 30, 2008•104 citations

Bohmian versus semiclassical description of interference phenomena

Yi Zhao, Nancy Makri

The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations.

The Journal of Chemical Physics•Jun 19, 2003•36 citations

Quantum Dynamics

Uri Peskin

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Cambridge University Press eBooks•May 11, 2023•0 citations
The coherent-state wave packet dynamics of several model systems is analyzed in terms of Bohm's total potential. The quantum dynamics has been obtained by solving the time-dependent Schrodinger equation, and a method for obtaining the total potential from it, involving just matrix algebra, has been proposed. Contrary to what one may expect, it is shown that the time- and state-dependent features of the total potential admit a rationale, classical-like description of quantum effects, leading to a unified picture of them, which is not critically dependent, as for the key features, on the classical potential. An outstanding feature is found to be the relation of the state system's density amplitude and sharpness (in its dependence with position) with quantum effects. Sharp density profiles and low densities cause the total potential to strongly depart from the classical value, in both time regimes and position ranges, which provide a clearer, more deterministic view to quantum dynamics. Free motion as well as scattering processes by square and Eckart barriers have been analyzed by means of careful inspection of several time dependent snapshots. The result is an insightful picture of processes involving tunneling and antitunneling, including their dynamical variants, as well as resonances and quantization.
The tunnel effect in one-dimensional collision problems is analyzed in terms of numerical wavepacket time propagations, in position, momentum, and phase space representations of the problem. Bohm's total potential is used to provide a classical-like description of tunnelling quantum dynamics, under a time- and density-dependent potential. Wigner, Husimi, as well as Bohm phase space distributions are used in the present study, with their relative performances for the present problem having been analyzed.
Quantum trajectories in complex space in the framework of the quantum Hamilton-Jacobi formalism are investigated. For time-dependent problems, the complex quantum trajectories determined from the exact analytical wave function for the free Gaussian wave packet and the coherent state in the harmonic potential are used to demonstrate that the information transported by the particles in the complex space can be used to synthesize the time-dependent wave function on the real axis. For time-independent problems, the exact complex quantum trajectories for the Eckart potential are obtained by numerically integrating the equations of motion. The unusual structure of the total potential (the sum of the classical and the quantum potentials) for the stationary states for the Eckart potential is pointed out. The variations of the complex-valued kinetic energy, classical potential, and quantum potential along the complex quantum trajectories are analyzed. This paper not only analyzes complex quantum trajectories for time-dependent and time-independent systems but also provides a unified description for complex quantum trajectories for one-dimensional problems in the quantum Hamilton-Jacobi formalism.
The exact factorization of an electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us to define the rigorous nuclear time-dependent Schr\"odinger equation (TDSE) with a time-dependent potential-energy surface (TDPES) that fully accounts for the coupling to the electronic motion and drives the nuclear wave-packet dynamics. Here, we study whether the propagation of multiple classical trajectories can reproduce the quantum nuclear motion in strong-field processes when their motions are governed by the quantum Hamilton-Jacobi equation derived by applying Bohmian mechanics to this exact nuclear TDSE. We demonstrate that multiple classical trajectories propagated by the force from the gradient of the exact TDPES plus the Bohmian quantum potential can reproduce the strong-field dissociation dynamics of a one-dimensional model of the H2+{{\mathrm{H}}_{2}}^{+}H2​+ molecule. Our results show that the force from the Bohmian quantum potential plays a non-negligible role in yielding quantum nuclear dynamics in the strong-field process studied here, where ionization and/or splitting of nuclear probability density occurs.
Interference is one of the most fundamental features which characterizes quantum systems. Here we provide an exhaustive analysis of the interfere dynamics associated with wave-packet superpositions from both the standard quantum-mechanical perspective and the Bohmian one. From this analysis, clear and insightful pictures of the physics involved in these kind of processes are obtained, which are of general validity (i.e., regardless of the type of wave packets considered) in the understanding of more complex cases where interference is crucial (e.g., scattering problems, slit diffraction, quantum control scenarios or, even, multipartite interactions). In particular, we show how problems involving wave-packet interference can be mapped onto problems of wave packets scattered off potential barriers.
The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.