Transmitted Wave Packet Research Articles

Misinterpretation yields supervelocities during transmission of wave packets through a barrier

This paper is concerned with the transmission time of an incident Gaussian wave packet through a symmetric rectangular barrier. Following Hartman (J. Appl. Phys. 33, 3427 (1962)), the transmission time \({\tau _{Ha}}\) is usually taken as the difference between the time at which the peak of the transmitted packet leaves the barrier of thickness \(\ell \) and the time at which the peak of the incident Gaussian wave packet arrives at the barrier. This yields a corresponding transmission velocity \({C_{Ha}} = \ell /{\tau _{Ha}}\) which appears under certain conditions as a supervelocity, i.e. becomes larger than the corresponding propagation velocity in free space which is the group velocity for electrons or the velocity of light for photons, respectively. By analysing the propagation of a broadband wave packet (which leads in free space to an extremely concentrated wave packet at a certain time) we obtain the pulse response function of the barrier and show that the insertion of the barrier is physically unable to produce a supervelocity. Therefore, the peak of an incident Gaussian wave packet and the peak of the transmitted wave packet are in no causal relationship. The shape of the transmitted wave packet is produced from the incident wave by convolution with the pulse response of the barrier. This yields a distortion of the shape of the wave packet which includes also the observed negative time shift of the peak. We demonstrate further that the phenomenon of Hartman's supervelocities is not restricted to barriers with their exponentially decaying fields but occurs for instance also in transmission lines with an inserted LCR circuit.

Analytical treatment of wave packet tunneling through a resonant double barrier heterostructure

An analytical expression is found for the transmitted wave packet through a general, asymmetric double barrier structure, as well as for its width. This analytical expression is shown to be in agreement with previous treatments of wave packet tunneling and, as such, is in agreement with previous numerical calculations.

A time-dependent study of resonant tunneling through a double barrier

The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.

On the dissipation due to wave ringing in nonelliptic elastic materials

Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.

Time-evolved numerical simulation of a two-dimensional electron wave packet through a quantum point contact

We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a quantum point contact using the finite difference method to simulate the diffraction phenomenon of a single electron. We found that the transmitted wave packet has diffracted subpeaks which show the following properties: (1) The traces of diffracted subpeaks start from the center of the slit entrance. (2) Lateral wave number of diffracted subpeaks is quantized. (3) The diffraction angles are determined by slit width and the initial electron wave packet energy. (4) The number of allowed diffracted subpeaks is determined by potential barrier thickness. (5) There is a delay in the motion of diffracted subpeaks due to quantization inside the slit.

Propagation effects in a shearing field-free plasma and pulsar microstructure

The propagation of time-dependent polarized radiation through a cold, field-free plasma with relativistic velocity shear is investigated. Numerical calculations of the transport of a linearly polarized Gaussian pulse through a model shearing medium show that the transmitted wave packet in one polarization breaks up into quasi-periodic peaks modulated by a dispersed envelope. Because the shearing plasma is birefringent, the polarization of the transmitted radiation can also show a time-dependent structure, including orthogonal mode switching and circular polarization. These polarization characteristics are similar to those of observed microstructure in pulsar radio emission. It is shown that these effects are also present in a shot-noise model for the incident radiation, suggesting that the micropulse amplitude modulations may result from propagation through magnetospheric shearing regions.