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Physical Review A

Electron delocalization and transfer induced by a time-dependent potential: Exact treatment of a simple model—formalism

J.-M. Lopez-Castillo, A.-M. Daré, Jean‐Paul Jay‐Gerin, C. Tannous

Published Jan 1, 1991• 1 citation
Publisher

Abstract

A quantum one-particle model for electron delocalization and transfer induced by a time-dependent potential is studied with a formalism based on the Lippman-Schwinger approach and fractional as well as operational calculus methods. The potential examined consists of a line of an arbitrary number of short-range (\ensuremath{\delta} function) potentials whose amplitudes have an arbitrary functional dependence on time. The temporal evolution of the ionization probability, the energy spectrum, and the time of transfer are obtained analytically and calculated numerically through a stable and accuracy-controllable technique without any assumption regarding the strength of the perturbing potential.

DOI
10.1103/physreva.43.35

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A quantum one-particle model for electron delocalization and transfer induced by a time-dependent potential is studied with a formalism based on the Lippman-Schwinger approach and fractional as well as operational calculus methods. The potential examined consists of a line of an arbitrary number of short-range (\ensuremath{\delta} function) potentials whose amplitudes have an arbitrary functional dependence on time. The temporal evolution of the ionization probability, the energy spectrum, and the time of transfer are obtained analytically and calculated numerically through a stable and accuracy-controllable technique without any assumption regarding the strength of the perturbing potential.
This paper can serve as an illustration for a course on quantum mechanics. It gives a non-trivial example of the time evolution of a system, which includes a particle in the potential of two δ-traps: one of which is in the fixed position, and the other moves with constant velocity. The numerical solution of the Schrödinger equation represents the behaviour of the vector of state in quite a transparent way; one can see rough pictures of the processes of charge transfer and ionization. The numerical procedure is explained, and an exercise can be carried out in constructing other time-dependent systems, containing δ-potentials.
Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.
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, multi-delta-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 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 Schr\"odinger equation for an effective non-unitary 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.
The time-dependent Schrodinger equation for a particle in one dimension in the field of two colliding delta potentials can be solved exactly. Assuming an initial state which is bound about the left delta potential for large negative times, the solution has been used to calculate the probabilities of finding a state bound about one of the potentials long after the collision. These probabilities are oscillating functions of the reciprocal velocity. The ionisation probability goes to zero for high collision velocities and reaches 1 / 2 for small velocities.
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.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.