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Journal of the Physical Society of Japan

Wave Packet Motion in Magnetic Field

Hideo Tsuru

Published Jul 15, 1992• 4 citations
Publisher

Abstract

A motion of a Gaussian wave packet of a single electron in a uniform magnetic field is obtained by solving the time dependent Schrödinger equation analytically. The wave packet moves changing its width and central position periodically. The period of the change of the width is the same as that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.

DOI
10.1143/jpsj.61.2246

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Physical Review•Aug 1, 1931•37 citations
A motion of a Gaussian wave packet of a single electron in a uniform magnetic field is obtained by solving the time dependent Schrödinger equation analytically. The wave packet moves changing its width and central position periodically. The period of the change of the width is the same as that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.
A motion of a Gaussian wave packet of a single electron in a harmonic potential is obtained by solving the time dependent Schrödinger equation analytically. The analytic formula of the phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the wave packet moves changing its width and central position periodically. The period of the change of the width is half of that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.
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.
A Gaussian wave packet confined to move on a plane perpendicular to a magnetic field remains a Gaussian wave packet in its time evolution. The average position and momentum follow the Ehrenfest equations which are identical to the classical Hamilton equations. A set of nonlinear equations decoupled from the Ehrenfest equation is derived for the parameters describing the time evolution of the density distribution and phases of a wave packet. Explicit solutions are then obtained when the “internal” angular momentum of the wave packet vanishes. In this case it is shown that the motion of the wave packet is a superposition of a translational motion, a rotation and a vibration.
Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr\"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order ${10}^{\ensuremath{-}8}$ to ${10}^{\ensuremath{-}10}$. Hence it is concluded that the difference between magnetic deflection measurements of em\frac{e}{m}me​ and other determinations cannot be explained as a quantum effect.