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 A Mathematical Nuclear and General

Scattering theory in a time-dependent external field. I. General theory

Eduard Prugovečki, A. Tip

Published Mar 21, 1974• 13 citations
Publisher

Abstract

A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H 0 (t)=H 0 +H 1 (t) and the full hamiltonian H(t)=H 0 (t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega +or- (s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H 1 (t) of the type H 1 (t)=H 1 cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H 1 (t) is absent.

DOI
10.1088/0305-4470/7/5/007

Related papers

The Quantum-Mechanical Scattering Problem

Thomas F. Jordan

The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.

Journal of Mathematical Physics•May 1, 1962•11 citations

Some results in algebraic scattering theory

E. Mourre

Some results of the the dependent formalism in scattering theory are presented. The asymptotic condition with an observable algebra A acting on a small Hilbert space, h, is imposed. H is given by H = H/sub 0/ + V, where H/sub 0/ is the free Hamiltonian on h and V the potential present. The demonstrations given are based on the convergence properties of the spectral measurements associated with the vectors, exp(-iHt) psi, psi being the scattering state. It is shown thnt the OMEGA /sup plus or minus / operators plus or minus infinity), where g belongs to a group of dense h vectors and where T/sub t/ is a family of limited operators in A. A necessary and sufficient condition is given for OMEGA /sup plus or minus / to exist and to permit a choice of T/sub t/ unitary operators to be made. (auth)

•Jan 1, 1973•0 citations

Quantum Dynamics Using The Time-Dependent Schrödinger Equation

Abraham Nitzan

This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.

Oxford University Press eBooks•Apr 6, 2006•0 citations

Configuration space properties of the scattering operator and time delay for potentials decaying like $\vert \chi \vert^{-\alpha}, \, \alpha > 1$

Abhisek Mohapatra, Kalyan B. Sinha, Werner O. Amrein

The time-delay in a simple quantum mechanical scattering process is defined to be the limit as r → ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x)∼|x| #75α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1#75δ (δ > 0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation

Annales De L Institut Henri Poincare-physique Theorique•Jan 1, 1992•9 citations

Theory of the scattering operator. II, Multichannel scattering

J. M. Jauch

The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.

•Dec 30, 1958•8 citations

Cyclic states, Berry phases and the Schrodinger operator

Alla N. Seleznyova

The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.

Journal of Physics A Mathematical and General•Feb 21, 1993•17 citations
A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H 0 (t)=H 0 +H 1 (t) and the full hamiltonian H(t)=H 0 (t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega +or- (s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H 1 (t) of the type H 1 (t)=H 1 cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H 1 (t) is absent.
The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.
Some results of the the dependent formalism in scattering theory are presented. The asymptotic condition with an observable algebra A acting on a small Hilbert space, h, is imposed. H is given by H = H/sub 0/ + V, where H/sub 0/ is the free Hamiltonian on h and V the potential present. The demonstrations given are based on the convergence properties of the spectral measurements associated with the vectors, exp(-iHt) psi, psi being the scattering state. It is shown thnt the OMEGA /sup plus or minus / operators plus or minus infinity), where g belongs to a group of dense h vectors and where T/sub t/ is a family of limited operators in A. A necessary and sufficient condition is given for OMEGA /sup plus or minus / to exist and to permit a choice of T/sub t/ unitary operators to be made. (auth)
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
Configuration space properties of the scattering operator and time delay for potentials decaying like ∣χ∣−α, α>1\vert \chi \vert^{-\alpha}, \, \alpha > 1∣χ∣−α,α>1
The time-delay in a simple quantum mechanical scattering process is defined to be the limit as r → ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x)∼|x| #75α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1#75δ (δ > 0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation
The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.
The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.