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American Journal of Physics

Motion of an electron wave packet in a uniform electric field

John N. Churchill

Published May 1, 1978• 11 citations
Publisher

Abstract

Energy eigenstates are superimposed in order to form the wave packet for an electron propagating in one dimension under the influence of a uniform, time-independent electric field. Flux velocity, phase velocity, and envelope velocity are considered. Asymptotic expressions are derived which describe the motion when the electric field is small and/or the wave packet is far away from the classical turning point. A graphical method is presented by which one can obtain both the position and shape of the envelope at a time t if the position and shape are known at some initial time.

DOI
10.1119/1.11256

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Journal of the Physical Society of Japan•Jul 15, 1992•4 citations
Energy eigenstates are superimposed in order to form the wave packet for an electron propagating in one dimension under the influence of a uniform, time-independent electric field. Flux velocity, phase velocity, and envelope velocity are considered. Asymptotic expressions are derived which describe the motion when the electric field is small and/or the wave packet is far away from the classical turning point. A graphical method is presented by which one can obtain both the position and shape of the envelope at a time t if the position and shape are known at some initial time.
The time evolution of a Gaussian wave packet of an electron, which is confined in a quantum dot subjected to uniform magnetic fields, is analytically studied in terms of the exact Feynman kernel. The central position of the wave packet follows the classical trajectory. Although the motion of the central position is aperiodic due to its dual period, the variance of the probability density changes periodically. If the shape of the initial wave packet is the same as that of the wave function of the ground state, its shape never alters. For more general systems in which the periodic full revival of electron wave packets is realized, the characteristic of energy eigenvalues is discussed.
Energy eigenstates are superimposed, using the Stark ladder energy spectrum, to construct exact, closed-form wave packets for an electron in a uniform dc electric field and a virtual lattice. We find that (1) perfectly periodic temporal variation at the Bloch frequency does not necessarily imply that there is a spatial oscillation and (2) perfectly periodic variation in both spatial and temporal behavior at the Bloch frequency does not imply that the motion is related in any way to the equation, d k /d t = − eE/h .
Quantum hydrodynamic equations are used to describe motion of single electron. The initial and boundary conditions are formulated from the analytic formula for the electron packet in a free space. The numerical integration of equations for the free packet is performed. Then, we investigated the wave packet propagation through the back-ground positive charge, when the electric field is self-consistent. Moreover, we studied the influence of the external harmonic electric fields on the properties of the motion. The purpose of the present work consists in study of the spatial-temporal processes including packets when the quantum wave effects are essential.
We derive a useful expression for the time-development operator for quantum wave functions, exp(−iĤt/ℏ), for the case of a particle subject to a constant force. We apply it, in both position and momentum space, to construct wave packets that exhibit uniform acceleration. We use these results to generate quantum wave packets corresponding to the motion of a charged particle in a uniform electric field in two different gauges, explicitly demonstrating the change in phase of the wave function under a gauge transformation. The more standard description of the time dependence of accelerating wave packets using the formalism of propagators is briefly reviewed in an Appendix for comparison.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.