Relativistic analysis of a wave packet interacting with a quantum-mechanical barrier (original) (raw)
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Superluminal Tunneling of a Relativistic Half-Integer Spin Particle Through a Potential Barrier
This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling is always superluminal. For antiparticle states, the tunneling may be either subluminal or superluminal instead, depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For particle states these are always negatives while for antiparticle states they are always positives, whatever the height and width of the barrier. The scattering also leads to an anomalous distortion of the Dirac spinor that tends to disappear as the particle velocity approaches the speed of light. Moreover, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study's results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement. 1 Introduction Several theoretical and experimental studies in the past decades have examined phenomena involving superluminal waves and objects because of their implication in quantum and cosmological physics [1-6]. Among them, the study of the tunneling time problem is one of the topics that has most attracted the interest of quantum physicists [7-12]. Researchers have approached this issue both from the perspective of non-relativistic [13] and relativistic [14] quantum theory. In both cases the tunneling time does not depend on the barrier width (at least for large enough barriers), thus proving superluminal behaviour of the quantum object (wave or particle). However, the tunneling time problem remains a controversial one in quantum physics. A comprehensive and clear theory to explain how long does it take a particle to tunnel through a barrier still does not exist [15]. As is well known, classical quantum mechanics does not treat time as an Hermitian operator but rather as a parameter [16]. Time does not appear in the commutation relationships typical of the Hermitian operators, even if it appears in one of the forms of the Heisenberg uncertainty principle, being a physical variable conjugated to the energy. For this reason we have to give up directly knowing the tunneling time. We may bypass the obstacle by assuming that the wave packet inside the barrier is stationary, with an imaginary wave vector. We can then interpret the tunneling time as the phase variation of the evanescent stationary wave that crossing
Superluminal effects for quantum tunneling through two successive barriers
2000
We study the phenomenon of one-dimensional non-resonant tunnelling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time is INDEPENDENT not only of the barrier widths (the so-called "Hartman effect"), but also of the R-width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered because of the formal identity between the Schroedinger and the Helmholtz equation [PACS numbers: 73.40.Gk; 03.65.-w; 03.30.+p; 41.20.Jb; 84.40.Az].
Transmission times of wave packets tunneling through barriers
Journal of Experimental and Theoretical Physics, 1999
The transmission of wave packets through tunneling barriers is studied in detail by the method of quantum molecular dynamics. The distribution function of the times describing the arrival of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The behavior of the average coordinate of a packet, the average momentum, and their variances is investigated. It is found that under the barrier a part of the packet is reflected and a Gaussian barrier increases the average momentum of the transmitted packet and its variance in momentum space.
New Journal of Physics, 2020
Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the time-dependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for above-barrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that post-selected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a phot...
Quantum temporal probabilities in tunneling systems
Annals of Physics, 2013
In this article, we propose a resolution to the paradox of apparent superluminal velocities for tunneling particles, by a careful treatment of temporal observables in quantum theory and through a precise application of the duality between particles and waves. To this end, we employ a new method for constructing probabilities associated to quantum time measurements that provides an explicit link between the tunneling time of particles and the associated quantum fields. We demonstrate that the idea of faster-than-light speeds in tunneling follows from an inadmissible use of classical reasoning in the description of quantum systems. Our results suggest that direct measurements of the transit time in tunneling could provide a new testing ground for the predictions of quantum theory versus local hidden-variables theories.
Relativistic tunneling through two successive barriers
Physical Review A, 2007
We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called Generalized Hartman Effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts, and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the Generalized Hartman Effect. We compare our results with the nonrelativistic ones and discuss their interpretation.
Superluminal tunnelling through successive barriers: Does QM predict infinite group-velocities?
The phenomenon of one-dimensional non-resonant tunnelling is analyzed through two or more successive (opaque) potential barriers, separated by intermediate free regions R, just by exploiting the relevant solutions to the Schroedinger equation. The total traversal time has been shown by us to be independent not only of the barrier widths (the so-called 'Hartman effect'), but also of the R-widths: so that the effective group velocity in the regions R, between two successive barriers, can be regarded as practically infinite. Such a prediction has been theoretically confirmed and generalized (as well as interpreted in terms of 'super-oscillations') by Aharonov et al. A recent experiment by Longhi et al. supported the predictions by considering two successive gratings in an optical fibre, that is, by having recourse to two 'classical barriers' (which allow simulating the tunnelling, due to the known formal identity between the Schro¨dinger and the Helmholtz equation).
Unified time analysis of photon and particle tunnelling
2004
A uniÿed time analysis of photon and nonrelativistic particle tunnellings is presented, in which time is regarded as a quantum observable, canonically conjugated to energy. Within this approach, one can introduce self-consistent deÿnitions of the tunnelling times, on the basis of conventional quantum mechanics (or one-dimensional quantum electrodynamics) only. The validity of the Hartman e ect [which states the tunnelling duration to be independent of the (opaque) barrier width, with superluminal group velocities of the tunnelling packet as a consequence] is veriÿed for all the known expressions of the mean tunnelling time. However, some noticeable generalizations of (and deviations from) the Hartman e ect are, as well, brie y investigated. Moreover, the analogy between particle and photon tunnelling is suitably exploited; on the basis of such an analogy, an explanation of some recent interesting microwave and optical experimental results on tunnelling times is proposed. Attention is devoted, at last, to some aspects of the causality problem for particle and photon tunnelling.