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Japanese Journal of Applied Physics

Time-Dependent Ballistic Phenomena of Electron Injected into Half-Ellipse Confined Room

Takuji Koiso, M. Muraguchi, Kyozaburo Takeda, Naoki Watanabe

Published Jun 1, 2005• 9 citations
Publisher

Abstract

We theoretically studied the time-developing ballistic phenomena of a single-electron confined in a half-ellipse infinite-potential wall by solving the time-dependent Schrödinger equation numerically. We also solved the corresponding Newton equation in order to compare the classical results with the quantum ones, and extracted the quantum features. The ellipse-shaped potential wall completely reflects an electron and causes the focusing ratio of unity in the classical limit. The dispersion of the wave packet of an electron, however, weakens this characteristic nature, and reduces the focusing ratio from unity. Because the dispersion also lets an electron arrive at the collector indistinctly, we define the effective arrival time by finding inflections in the time-dependent profile of the probability density at the collector. Based on the second-derivation technique, we further determine the quantum arrival time (QAT) at which the intrusion of the wave packet occurs dominantly. The comparison of this QAT with the classical arrival time (CAT) determines whether the corresponding ballistic propagation can be discussed on the basis of the quantum consideration or the classical prediction. We further studied how the change in the half-ellipse potential wall shape affects the ballistic phenomena through the change in the ellipticity γ, the system size L and the dispersion degree σ of the wave packet. Using the ellipse-shaped infinite-potential wall, the application of the magnetic field causes irrational cyclotron motion assisted by the ellipse potential, in addition to the rational cyclotron motions. The numerical solution of the time-dependent Schrödinger equation determines the unique cyclotron motion whose peculiarity is caused by the dispersion of the wave packet and is rarely predicted by the classical limit.

DOI
10.1143/jjap.44.4252

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We theoretically study the time-developed progress of resonant tunneling for an electron wave packet injected into a two-dimensional quantum ring (QR) by solving the time-dependent (TD) Schr\"odinger equation numerically. Focusing on an extraction of the angular momentum lz{l}_{z}lz​, we examine the TD features in the resonant tunneling electron by projection analysis in which the resulting TD wave function at the QR is decomposed into the (static) resonant states by calculating the inner products among them. This analysis reveals that the two-states approach is well applicable for the QR system and the cross terms between these two states are crucial for the TD vacillation of both the expectation value of the angular momentum $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{l}}_{z}⟩$ and the electron density $\ensuremath{\rho}$. The quasidegeneracy of the resonant states causes a characteristic beating whose frequency is determined by the difference between the eigenenergies. We further study the corresponding TD phenomena under a magnetic field and find that the rotational direction in $\ensuremath{\rho}$ changes in accordance with the strength of the magnetic field. This feature seems to be very different from the classical prospect of the cyclotron motion where the application of the magnetic field determines the rotational direction uniquely.
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Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.
A propagator for the one-dimensional time-dependent Schr¨odinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the non-classical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schr¨odinger equation (defines the wave function w(x, t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → −ih/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |w(x, t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
Dispersion associated with the complex band structure of a periodic potential can be used to explain single-electron wave packet tunneling through a finite-sized tunnel barrier. The imaginary k-space dispersion in a tunnel barrier may be viewed as a filter that controls the arrival time, shape, and group velocity of a transmitted wave packet. This k-space filter can be designed to increase or decrease the group velocity of the transmitted pulse and a transmitted peak-pulse probability arrival time can be engineered to occur before or after that of an electron wave packet propagating in the absence of the tunnel barrier.
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.