© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Comments on eikonal and sudden approximations

Ralph Eric Turner

Published Jun 1, 1980• 7 citations

Abstract

An approximation scheme is presented which leads systematically from the exact Moller (or scattering) operator to the eikonal approximation and then to the sudden approximation. In terms of the Moller (or scattering) operator, the eikonal approximation is local in position space and is parametrised by the average incoming free momentum. Carrying the procedure a step further decouples the internal and translational motions to produce the sudden approximation. Distorted as well as straight-line approximations are considered. To do so distorted scattering conditions are presented which define distorted M Moller, transition and scattering operators. Some properties of these operators are discussed. Relations with the usual Moller, transition and scattering operators are presented. Finally, an eikonal approximation to the Moller super-operator is presented which does not depend explicitly upon the form of the dynamics that is used. Quantally, the eikonal Moller super-operator is equivalent to the eikonal Moller operator and, in the limit of small h, reduces to the classical eikonal Moller super-operator. This classical Moller super-operator gives a particle picture for the eikonal approximation in contrast to the traditional wave picture.

DOI: 10.1088/0305-4470/13/6/030

Related papers

A unified description of classical and quantal scattering

Ralph Eric Turner

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.

•Jan 1, 2010•0 citations

Tutorial: Perturbation Approximations to Asymptotic Quantum Transition Rates

Steven Kenneth Kauffmann

The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.

•Aug 1, 2019•0 citations

Eikonal approximation and the spreading of a wave packet

J. Lerner

Standard eikonal methods, as asymptotic perturbation expansions in 1/\ensuremath{\Elzxh}, are incapable of describing fully the dispersive spreading of a wave packet. By simple models, we distinguish the two types of dispersion associated with a packet having initially uniform and nonuniform distributions of wave number. A nonperturbative extension of the eikonal approximation is applied to two initial-value problems for the free-particle Schr\"odinger equation and compared with exact solutions. The method maintains the ray picture fully. Applications of the method to more general classes of problems are discussed.

Physical Review A•Feb 1, 1986•1 citation

The relationship between formal scattering theory and the density matrix

L. Stanton

A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.

Molecular Physics•Jan 1, 1971•7 citations

Time Delay in the Eikonal Approximation

Takako Suzuki

The time delay or advance in the eikonal approximation is discussed in terms of the wave packet motion in the interaction region as well as the scattering amplitude. The result of the time delay or advance obtained from the wave packet motion does not strictly coincide with that given by the derivative of the phase of the amplitude.

Europhysics Letters (EPL)•Jul 15, 1989•17 citations

Potential Scattering in One Dimension

T. Ishihara

General solution is derived in operator form on the one-dimensional potential scattering problem without any approximation. Backscattering is included explicitly in this formulation. Coordinate representation is presented and its relation to the eikonal approximation is discussed. Effects of backscattering are examined numerically in the case of a square-well potential.

Progress of Theoretical Physics•Oct 1, 1975•2 citations

An approximation scheme is presented which leads systematically from the exact Moller (or scattering) operator to the eikonal approximation and then to the sudden approximation. In terms of the Moller (or scattering) operator, the eikonal approximation is local in position space and is parametrised by the average incoming free momentum. Carrying the procedure a step further decouples the internal and translational motions to produce the sudden approximation. Distorted as well as straight-line approximations are considered. To do so distorted scattering conditions are presented which define distorted M Moller, transition and scattering operators. Some properties of these operators are discussed. Relations with the usual Moller, transition and scattering operators are presented. Finally, an eikonal approximation to the Moller super-operator is presented which does not depend explicitly upon the form of the dynamics that is used. Quantally, the eikonal Moller super-operator is equivalent to the eikonal Moller operator and, in the limit of small h, reduces to the classical eikonal Moller super-operator. This classical Moller super-operator gives a particle picture for the eikonal approximation in contrast to the traditional wave picture.

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.

The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.

Standard eikonal methods, as asymptotic perturbation expansions in 1/\ensuremath{\Elzxh}, are incapable of describing fully the dispersive spreading of a wave packet. By simple models, we distinguish the two types of dispersion associated with a packet having initially uniform and nonuniform distributions of wave number. A nonperturbative extension of the eikonal approximation is applied to two initial-value problems for the free-particle Schr\"odinger equation and compared with exact solutions. The method maintains the ray picture fully. Applications of the method to more general classes of problems are discussed.

A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.