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

AIP conference proceedings

Time-dependent approach to transport and scattering in atomic and mesoscopic physics

Tobias Kramer, R. Bijker, Octavio Castaños, Rocío Jáuregui, Óscar Rosas-Ortiz

Published Jan 1, 2011• 1 citation
Publisher

Abstract

Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.

DOI
10.1063/1.3575532

Related papers

Time-dependent approach to transport and scattering in atomic and mesoscopic physics

Tobias Kramer

Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.

arXiv:1011.3194•AIP conference proceedings•Jan 1, 2011•8 citations

Time evolution of quantum systems

Dževad Belkić

In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.

CRC Press eBooks•Jan 15, 2020•0 citations

The Schrödinger Equation

Uri Peskin

The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.

Cambridge University Press eBooks•May 11, 2023•0 citations

A strategy for time dependent quantum mechanical calculations using a Gaussian wave packet representation of the wave function

Shin‐ichi Sawada, Robert Heather, Bret Jackson +1 more

We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.

The Journal of Chemical Physics•Sep 15, 1985•205 citations

The Time-Dependent Schrödinger Equation

Reinhold A. Bertlmann, Nicolai Friis

Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description

•Sep 7, 2023•0 citations

Hamilton‐jacobi dynamics for the solution of time dependent quantum problems I. Formalism and wave packet propagation in one dimension

Ersin Yurtsever, Jürgen Brickmann

Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.

Berichte der Bunsengesellschaft für physikalische Chemie•Apr 1, 1994•4 citations
Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.
Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.
In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.