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arXiv:quant-ph/0008084 · Journal of Physics A Mathematical and General

Exact relativistic time evolution for a step potential barrier

Jorge Villavicencio

Published Aug 16, 2000• 11 citations
arXivHTMLPDF

Abstract

We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth 2 x p inside the potential, where x p is the penetration length of the stationary solution. Beyond the characteristic distance 2 x p , a Sommerfeld-type precursor travels along the potential at the speed of light, c . However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.

DOI
10.1088/0305-4470/33/34/311

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We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth 2 x p inside the potential, where x p is the penetration length of the stationary solution. Beyond the characteristic distance 2 x p , a Sommerfeld-type precursor travels along the potential at the speed of light, c . However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.
We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point-source initial condition. We show that in both the propagation and Klein-tunneling regimes, the zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the zitterbewegung frequency and that the amplitude of these oscillations decays as ${t}^{\ensuremath{-}3/2}$. We show that the beating effect also manifests itself in the free Klein-Gordon and Dirac equations within a quantum shutter setup, which involve the dynamics of cutoff quantum states. We also find a time domain where the particle density of the point source is governed by the propagation of a main wavefront, exhibiting an oscillating pattern similar to the diffraction-in-time phenomenon observed in nonrelativistic systems. The relative positions of these wavefronts are used to investigate the time delay of quantum waves in the Klein-tunneling regime. We show that depending on the energy difference E\mathcal{E}E between the source and the potential step, the time delay can be positive, negative, or zero. The latter case corresponds to a super-Klein-tunneling configuration, where E\mathcal{E}E equals half the energy of the potential step.
By using an exact solution to the time-dependent Schr\"odinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a forerunner associated with a nontunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the forerunner at a given fixed position inside the potential is exactly the traversal time $\ensuremath{\tau}.$ We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the forerunner, and to analyze the role played by the time of arrival ${3}^{\ensuremath{-}1/2}\ensuremath{\tau}$ found recently by Muga and B\"uttiker [Phys. Rev. A 62, 023808 (2000)].
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr\"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different ``shallow'' and ``deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B\"uttiker-Landauer) time $\ensuremath{\tau}.$ The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.
We consider exact time-dependent analytic solutions to the Schr\"odinger equation for tunneling in one dimension with cutoff wave initial conditions at t=0.t=0.t=0. We obtain that as soon as $t\ensuremath{\ne}0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance x0{x}_{0}x0​ from the potential the probability density rises after a time t0=x0/c{t}_{0}{=x}_{0}/ct0​=x0​/c restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.
Wavepackets representing relativistic quantum particles injected into a half-space, from a source that is switched on at a definite time, are represented by superpositions of plane waves that must include negative frequencies. Propagation is causal: it is a consequence of analyticity that at time t no part of the wave has travelled farther than ct , corresponding to the front of the signal. Nevertheless, interference fringes behind the front travel superluminally. For Klein–Gordon and Dirac wavepackets, the spatially integrated density increases because current is injected at the boundary. Even in the simplest causal model, understanding the shape of the wave after long times is an instructive exercise in the asymptotics of integrals, illustrating several techniques at a level suitable for graduate students; different spatial features involve contributions from a pole and from two saddle points, the uniform asymptotics for the pole close to a saddle, and the coalescence of two saddles into the Sommerfeld precursor immediately behind the front.