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Journal of the Physical Society of Japan

Dynamics of an Electron Wave Packet in a Quantum Dot Subjected to Magnetic Fields

Tsuyoshi Ueta

Published Mar 15, 1995• 4 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.64.877

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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 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.
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.
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.
In the present work we treat the problem of a particle in a uniform magnetic field along the symmetric gauge, so chosen since the wavefunctions present the required cylindrical symmetry. It is our understanding that by means of this work we can make a contribution to the teaching of the present subject, as well as encourage students to use computer algebra systems in solving problems of quantum mechanics. We obtained the degeneracy of the spectrum of eigenvalues in a very clear way. Through the use of a computer algebra system we show graphs of the probability density associated with different eigenvalues as well as compare such functions for some degenerate states, which helps us to visualize the physics of the problem. We also present a semiclassical model which gives a physical insight regarding the paradoxical fact that eigenfunctions associated with opposite angular momenta and different energy eigenvalues have the same probability density. Finally, by solving the time-dependent Schrodinger equation we obtain the time evolution of a wave packet that at time zero was considered to be localized in a definite region of the lattice. The centroid of such a packet performs an orbit similar to that obtained in the classical treatment of a particle in a magnetic field.