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Japanese Journal of Applied Physics

Confinement and Transfer of a Two-Dimensional Wave Packet under Crossed Electric and Magnetic Fields

Hidetoshi Takagi, Mitsuru Ishida Mitsuru Ishida, Nobuhiko Sawaki

Published Mar 1, 2001• 1 citation
Publisher

Abstract

The motion of a two-dimensional Gaussian wave packet was analyzed under crossed electric and magnetic fields by solving the time-dependent Schrödinger equation numerically. It was found that the perpendicular magnetic field prevents the expansion of the wave packet and the in-plane electric field transfers the wave packet. The possibility of trapping and transfer of the wave packet is discussed by introducing a phenomenological decay term into the partial waves.

DOI
10.1143/jjap.40.1973

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The motion of a two-dimensional Gaussian wave packet was analyzed under crossed electric and magnetic fields by solving the time-dependent Schrödinger equation numerically. It was found that the perpendicular magnetic field prevents the expansion of the wave packet and the in-plane electric field transfers the wave packet. The possibility of trapping and transfer of the wave packet is discussed by introducing a phenomenological decay term into the partial waves.
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.
The eigensolutions of the Schrödinger equation for the motion of a charged particle in a uniform magnetic field depend on the choice of electromagnetic gauge. In the Landau gauge it is hard to discern a relationship between the eigensolutions and the circular or helical orbits of classical theory. In this paper a wave packet is constructed from the Landau gauge eigensolutions which simulates the classical motion in the plane perpendicular to the magnetic field. Some properties of the wave packet are discussed.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the Schrödinger equation is approximated by Gaussian wave packets via the time-dependent variational formulation by Dirac and Frenkel. For the numerical approximation we derive ordinary differential equations for the parameters of the variational solution. Moreover, we prove L2L^2L2-error bounds and observable error bounds for the approximating Gaussian wave packet.
Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.