The time-dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave-packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for various orbital angular momentum quantum numbers can be obtained from: (1) Fourier transform of the amplitude density from the time to the energy domain, and (2) direct analysis of the scattered wave packet. The approach is illustrated by an application to a standard potential scattering model problem.
The time-dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave-packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for various orbital angular momentum quantum numbers can be obtained from: (1) Fourier transform of the amplitude density from the time to the energy domain, and (2) direct analysis of the scattered wave packet. The approach is illustrated by an application to a standard potential scattering model problem.
Time-independent wave-packet forms of the Schr\"odinger equation (TIWSE) and Lippmann-Schwinger equation (TIWLSB) have been derived by a partial time-to-energy Fourier transform of L2 wave-packet solutions to the time-dependent Schr\"odinger equation. The new equations retain the initial wave packet \ensuremath{\chi}(t0) as a ``universal source'' of scattered waves, which applies for all collision energies E contained in the initial wave packet. The relationship between the solution ${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\ensuremath{\Psi}}}^{(+)}$(E) of the standard time-independent Lippmann-Schwinger or Schr\"odinger equation is given and the method illustrated by a computational application.
In this paper we reexamine, and analyze solutions to, the recently derived time-independent wave packet-Schrödinger (TIWS) and time-independent wave packet-Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time-dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrödinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninteracting to the interacting region. The method is illustrated by calculations for two 1-D scattering problems, namely the transmission of an electron through a single barrier and through a double barrier. The latter is especially challenging because of the occurrence of long-lived resonances with the electron trapped inside the double barrier. In addition, we show that the method can still make use of absorbing potentials to decrease grid size, and we show how the formalism can be used to treat inelastic and reactive scattering, and radial scattering variables, thereby constituting a general approach to time-independent wave packet quantum scattering.
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
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.
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t→−iℏ/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time (t→−it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density ∣ψ(x,t)∣2, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.