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American Journal of Physics

Simulations in quantum tunneling

Kevin Richard Smith, Guy Blaylock

Published Sep 18, 2017• 16 citations

Abstract

We study the timing effects of nonrelativistic wave packet tunneling through a barrier using a numerical simulation readily accessible to an undergraduate audience. We demonstrate that the peak of the transmitted packet can sometimes emerge from the barrier ahead of the peak of an undisturbed wave packet that does not encounter a barrier. Under the right circumstances, this effect can give the appearance that transmission through the barrier occurs at superluminal speeds. We demonstrate that this seemingly paradoxical effect is not all that puzzling. Rather, components from the front of the incoming wave packet are preferentially transmitted, forming a transmitted packet ahead of the average of the incoming wave packet but not ahead of the leading edge of that packet. Our studies also show how the timing depends on barrier height and width, consistent with expectations based on the different energy components of the wave packet.

DOI: 10.1119/1.4991367

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We study the timing effects of nonrelativistic wave packet tunneling through a barrier using a numerical simulation readily accessible to an undergraduate audience. We demonstrate that the peak of the transmitted packet can sometimes emerge from the barrier ahead of the peak of an undisturbed wave packet that does not encounter a barrier. Under the right circumstances, this effect can give the appearance that transmission through the barrier occurs at superluminal speeds. We demonstrate that this seemingly paradoxical effect is not all that puzzling. Rather, components from the front of the incoming wave packet are preferentially transmitted, forming a transmitted packet ahead of the average of the incoming wave packet but not ahead of the leading edge of that packet. Our studies also show how the timing depends on barrier height and width, consistent with expectations based on the different energy components of the wave packet.

We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schr\"odinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schr\"odinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.

We have modeled the transmission of an electron wave packet through a resonant tunneling semiconductor nanostructure by solving the time-dependent Schr\"odinger equation using the finite-difference method. We have found in all cases that the passage of the electron wave packet through the tunneling barrier is accompanied by a propagation delay relative to the propagation of an undisturbed wave packet. Tunneling transport is shown to be causal, and no evidence of superluminal behavior is seen, either for resonant or for off-resonant tunneling. In the case of off-resonant tunneling, the peak of the transmitted wave packet is observed to exit a double resonant tunneling barrier before the peak of the incident wave packet enters the structure. However, these two peaks are not directly related, and the appearance of a well-formed peak in the transmitted intensity is shown to be a result of the transient behavior of the tunneling event.

The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space.

An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.

Based on the Dirac equation, the behavior of relativistic electrons which tunnel a potential barrier of height V 0 for incoming energies between V 0 and V 0 + m is studied by using the wave packet formalism. The choice of this incoming energy zone guarantees that only electrons participate in the tunneling process. In the opaque limit, as shown in a previous analysis, the transit velocity is proportional to the barrier width and for relativistic potential tends to a constant value greater than c . In this paper, a new numerical study shows a very surprising result: superluminal transmissions are also evident for thin barriers.