Dirac equation studies in the tunneling energy zone

Abstract

Abstract. We investigate the tunnelling zone V0 −m < E < V0 +m for a one-dimensional potential within the Dirac equation. We find the appearance of superluminal transit times akin to the Hartman effect.

Relativistic tunneling through opaque barriers

“…from which we observe that energy components with E < V 0 correspond to below potential states, the interpretation of which is still subject to discussions [26,27,31]. In Sec.…”

“…This avoids Klein and diffusion phenomena, as well the presence of below potential evanescent solutions. By solving the Dirac equation (2) and imposing the continuity of the wave function at 0 and L, we find, for the transmitted wave packet [26][27][28],…”

“…Although the majority of the discussions about tunneling times have been based on the nonrelativistic Schrödinger equation, some recent works have attended to particular features of relativistic tunneling particles, such as superluminality [19][20][21][22] and existence of the Hartman effect for relativistic potentials [23][24][25][26]. The solutions of the Dirac equation are oscillatory in the diffusion E > V 0 + m [27,28] and Klein E < V 0 − m [29] energy zones, where V 0 is the barrier height and m is the particle mass.…”

“…The Klein region, in its turn, involves pair production and dynamical localized states [31]. The tunneling energy zone max{m,V 0 − m} < E < V 0 + m is characterized by evanescent solutions [26]. In this paper, * deleo@ime.unicamp.br † leonnardi@bol.com.br we aim to present an analytic and numerical study of the phase time for one-dimensional relativistic tunneling in the opaque barrier limit.…”

A Probability Distribution for Quantum Tunneling Times

“…Still, the intrinsic experimental difficulties associated both with the measurements and the interpretation of the results have, so far, prevented an elucidation of the problem and, in fact, contradictory results persist, with some experiments obtaining a finite non-zero result [3,6] and others compatible with instantaneous tunneling [4]. It should be noticed that the similarity between Schrödinger and Helmholtz equations allows for analogies between quantum tunneling of massive particles and photons [7], and a noninstantaneous tunneling time is supported by this analogy and experiments measuring photonic tunneling times [8], as well as by many theoretical calculations based on both the Schrödinger (for reviews see, e.g., [1,2]) and the Dirac equations (e.g., [9][10][11][12][13][14][15][16]).…”