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

Journal of Physics B Atomic Molecular and Optical Physics

On the quantum foundations of some classical approximations for ion-atom collisions

Carlos O Reinhold, C A Falcon

Published May 28, 1988• 23 citations
Publisher

Abstract

A connection between quantum mechanics and classical methods used in ion-atom collisions is developed. The connection is based on the fluid dynamical interpretation of the one-electron quantum wavefunction. The classical trajectory Monte Carlo method and the Thomson approximation are obtained as quantum models and the wavefunctions associated with these approximations are defined. A discussion is given of the procedure to be used to calculate scattering cross sections. Scaling laws derivable from the classical Liouville equation are deduced in the impact parameter formalism.

DOI
10.1088/0953-4075/21/10/015

Related papers

Trajectory method for the solution of the time-dependent Schrodinger equation in atomic physics and application to H+-H scattering

Georg Terlecki, NORBERT GRÜN, W. Scheid

A trajectory method has been developed for the solution of the one-electron Schrodinger equation in atomic scattering problems. The method is based on the fluid dynamical interpretation of the quantum mechanical probability density and current. The time evolution is described by trajectories of particles, which represent the probability distribution. The trajectories are calculated by solving classical equations of motion with forces depending on the probability density. Results are obtained for the differential and total ionisation and charge exchange cross sections for the scattering of H + on H.

Journal of Physics B Atomic and Molecular Physics•Sep 28, 1984•20 citations

A tutorial on quantum scattering theory and coupled-channel calculations of bimolecular collisions

Karl P. Horn, Christiane P. Koch

Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.

quant-pharXiv:2610.20882•Oct 16, 2026

Superoperator methods for calculating cross sections: I. Correlation functions and the Born approximation

Ralph Eric Turner, Robert F. Snider

Quantum mechanical scattering theory has recently been formulated exclusively in terms of density operators. The present work interprets the resulting generalized collision cross sections as integrals overtime correlation functions of relative coordinate, two body operators. It is shown that a straight line trajectory approximation to these time correlation functions is identical to the Born approximation. A phase space representation of the quantum collision dynamics is formulated. In these terms, the quantum cross section can be reduced in a natural way to the corresponding classical cross section provided there is no 'singular' behaviour. It would appear that this formulation would be very appropriate for making semiclassical approximations for the differential cross sections.

Canadian Journal of Physics•Jul 1, 1976•7 citations

Using the wavelet transform to separate scales in the Schrödinger equation and subsequently derive the Boltzmann equation

A. P. Meilakhs

Here we continue our investigation aimed at the derivation of the Boltzmann equation from the first principles of quantum mechanics. The key difficulty that we overcome in this manuscript is the incompatibility between traditional methods employed in the derivation of classical mechanics from quantum mechanics and methods used to describe transitions between quantum states due to interaction. Here we use a novel technique that is similar to the wavelet transform method that originally appeared in the context of signal processing. With this method, we can simultaneously work with slowly varying potentials that give rise to classical-like wave-function behaviour, motion along a prescribed trajectory, and sharp potentials that lead to behaviour that is best understood in terms of quantum transitions. Working with smooth potentials in this framework, we derive the Liouville equation from the Schrödinger equation. When using this method for sharp potentials, we naturally obtain the description of system dynamics in terms of a transition-rate matrix. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the transition-rate matrix, we can deduce the collision integral. We combine the results to derive the Boltzmann equation, the main formula of physical kinetics.

quant-pharXiv:2511.21601•Nov 27, 2025•0 citations

Analysis of an ion–atom collision at intermediate velocities by means of quantum trajectories

Lino Ponce

We present the results of a quantum calculation designed to discuss the dynamics of an ion–atom collision at intermediate velocities. The description is based on a quantum fluid picture. The time-dependent Schrodinger equation is integrated numerically for a model proton–hydrogen collision and quantum trajectories are computed. By these means, we obtain a pictural view of the scattering process. Analysis of the resulting trajectories yields insights into the dynamics of the physical phenomena involved, with special emphasis on ionization.

Journal of Physics B Atomic Molecular and Optical Physics•Jan 24, 2007•1 citation

Quantum-classical computing for time-dependent ion-atom collision dynamics: Applications to charge-transfer cross-section simulations

Minchen Qiao, Yu-xi Liu

The simulation of ion-atom collisions remains a formidable challenge due to the complex interplay between electronic and nuclear degrees of freedom. We present a hybrid quantum-classical computing framework for simulating time-dependent ion-atom collision dynamics, within which two variational quantum time evolution algorithms are implemented. To validate our framework, we simulate the charge-transfer dynamics and compute the corresponding cross sections for the proton-hydrogen collision system across an energy range 1--25 keV. Our results accurately reproduce the charge-transfer dynamics with high fidelity and exhibit very good agreement with available experimental and theoretical cross-section data across the entire energy range. These results highlight the accuracy and applicability of our hybrid quantum-classical framework for scattering cross-section calculations. Our work demonstrates an effective approach for mapping time-dependent many-body collision problems onto near-term quantum computing devices, and also provides promising directions for practical applications of universal quantum computing in the noisy intermediate-scale quantum era.

arXiv:2506.19374•Physical Review A•Nov 10, 2025•0 citations
A connection between quantum mechanics and classical methods used in ion-atom collisions is developed. The connection is based on the fluid dynamical interpretation of the one-electron quantum wavefunction. The classical trajectory Monte Carlo method and the Thomson approximation are obtained as quantum models and the wavefunctions associated with these approximations are defined. A discussion is given of the procedure to be used to calculate scattering cross sections. Scaling laws derivable from the classical Liouville equation are deduced in the impact parameter formalism.
A trajectory method has been developed for the solution of the one-electron Schrodinger equation in atomic scattering problems. The method is based on the fluid dynamical interpretation of the quantum mechanical probability density and current. The time evolution is described by trajectories of particles, which represent the probability distribution. The trajectories are calculated by solving classical equations of motion with forces depending on the probability density. Results are obtained for the differential and total ionisation and charge exchange cross sections for the scattering of H + on H.
Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.
Quantum mechanical scattering theory has recently been formulated exclusively in terms of density operators. The present work interprets the resulting generalized collision cross sections as integrals overtime correlation functions of relative coordinate, two body operators. It is shown that a straight line trajectory approximation to these time correlation functions is identical to the Born approximation. A phase space representation of the quantum collision dynamics is formulated. In these terms, the quantum cross section can be reduced in a natural way to the corresponding classical cross section provided there is no 'singular' behaviour. It would appear that this formulation would be very appropriate for making semiclassical approximations for the differential cross sections.
Here we continue our investigation aimed at the derivation of the Boltzmann equation from the first principles of quantum mechanics. The key difficulty that we overcome in this manuscript is the incompatibility between traditional methods employed in the derivation of classical mechanics from quantum mechanics and methods used to describe transitions between quantum states due to interaction. Here we use a novel technique that is similar to the wavelet transform method that originally appeared in the context of signal processing. With this method, we can simultaneously work with slowly varying potentials that give rise to classical-like wave-function behaviour, motion along a prescribed trajectory, and sharp potentials that lead to behaviour that is best understood in terms of quantum transitions. Working with smooth potentials in this framework, we derive the Liouville equation from the Schrödinger equation. When using this method for sharp potentials, we naturally obtain the description of system dynamics in terms of a transition-rate matrix. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the transition-rate matrix, we can deduce the collision integral. We combine the results to derive the Boltzmann equation, the main formula of physical kinetics.
We present the results of a quantum calculation designed to discuss the dynamics of an ion–atom collision at intermediate velocities. The description is based on a quantum fluid picture. The time-dependent Schrodinger equation is integrated numerically for a model proton–hydrogen collision and quantum trajectories are computed. By these means, we obtain a pictural view of the scattering process. Analysis of the resulting trajectories yields insights into the dynamics of the physical phenomena involved, with special emphasis on ionization.
The simulation of ion-atom collisions remains a formidable challenge due to the complex interplay between electronic and nuclear degrees of freedom. We present a hybrid quantum-classical computing framework for simulating time-dependent ion-atom collision dynamics, within which two variational quantum time evolution algorithms are implemented. To validate our framework, we simulate the charge-transfer dynamics and compute the corresponding cross sections for the proton-hydrogen collision system across an energy range 1--25 keV. Our results accurately reproduce the charge-transfer dynamics with high fidelity and exhibit very good agreement with available experimental and theoretical cross-section data across the entire energy range. These results highlight the accuracy and applicability of our hybrid quantum-classical framework for scattering cross-section calculations. Our work demonstrates an effective approach for mapping time-dependent many-body collision problems onto near-term quantum computing devices, and also provides promising directions for practical applications of universal quantum computing in the noisy intermediate-scale quantum era.