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

A unified description of classical and quantal scattering

Ralph Eric Turner

Published Jan 1, 2010• 0 citations
Publisher

Abstract

A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.

DOI
10.14288/1.0060984

Related papers

Superoperator methods of calculating cross sections: III. Unified description of classical and quantal scattering

Ralph Eric Turner, Robert F. Snider

It is shown how differential cross sections can be obtained from the time dependence of phase space packets. This procedure is valid both for classical and quantum mechanics. Two methods are described. In one the trajectory of the packet is emphasized, while in the second the packet is appropriately spread to infinite size. Both methods are applicable to either mechanics. It is shown how the quantal results agree with those of the stationary state approach as formulated in terms of the density operator. The description is also used to elucidate the difference between the scattered flux and the generalized flux that arises naturally in the superoperator formulation.

Canadian Journal of Physics•Aug 1, 1980•13 citations

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

Semiclassical mechanics of periodic motion: I. General scheme

Peter Kasperkovitz, Momtchil Peev

A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.

Journal of Physics A Mathematical and General•Mar 6, 1998•2 citations

Deformation quantization and semiclassical expansion in many-body potential scattering problem

M. I. Krivoruchenko, Christopher A. Fuchs, Amand Faessler

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in $\\hbar$. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into semiclassical transport models.

•May 6, 2006•4 citations

General time-dependent formulation of quantum scattering theory

Stuart C. Althorpe

We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.

Physical Review A•Apr 2, 2004•18 citations

Superoperator methods of calculating cross sections: II. Distorted wave Born approximation

Robert F. Snider, Ralph Eric Turner

The superoperator form of the collision cross section is evaluated within the distorted wave Born approximation. It is first verified that the obvious expansion methods give a result identical to that obtained by standard methods. Formalistically, the algebraic expansions of the transition operator and superoperator are shown to have parallel structures. The distorted wave Born approximation for the cross section also has a structure parallel to the structure of the Born approximation cross section. This is especially brought out by formulating the results in terms of time correlation functions. Certain simplifying features are found for cross sections averaged over initial and final velocity directions. These cross sections for internal state transitions are further simplified by averaging over a Maxwellian distribution of initial velocities in such a way as to obtain 'kinetic cross sections' appropriate for gas kinetic phenomena. Connection is also made with the 'constant acceleration approximation' used to estimate correlation functions in gas phase NMR.

Canadian Journal of Physics•Jul 1, 1976•2 citations
A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.
It is shown how differential cross sections can be obtained from the time dependence of phase space packets. This procedure is valid both for classical and quantum mechanics. Two methods are described. In one the trajectory of the packet is emphasized, while in the second the packet is appropriately spread to infinite size. Both methods are applicable to either mechanics. It is shown how the quantal results agree with those of the stationary state approach as formulated in terms of the density operator. The description is also used to elucidate the difference between the scattered flux and the generalized flux that arises naturally in the superoperator formulation.
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.
A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in hbar\\hbarhbar. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into semiclassical transport models.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.
The superoperator form of the collision cross section is evaluated within the distorted wave Born approximation. It is first verified that the obvious expansion methods give a result identical to that obtained by standard methods. Formalistically, the algebraic expansions of the transition operator and superoperator are shown to have parallel structures. The distorted wave Born approximation for the cross section also has a structure parallel to the structure of the Born approximation cross section. This is especially brought out by formulating the results in terms of time correlation functions. Certain simplifying features are found for cross sections averaged over initial and final velocity directions. These cross sections for internal state transitions are further simplified by averaging over a Maxwellian distribution of initial velocities in such a way as to obtain 'kinetic cross sections' appropriate for gas kinetic phenomena. Connection is also made with the 'constant acceleration approximation' used to estimate correlation functions in gas phase NMR.