Dirac Equation Studies in the Tunnelling Energy Zone

In several recent articles, we have investigated in some detail one-dimensional electrostatic potentials by means of both the Schrödinger [1–3] and the Dirac equation [4–6]. Several original phenomena have been observed, such as the transition from resonance phenomena to multiple (infinite) peak formation [5] equivalent to a shift from wave-like to particle behavior in the barrier diffusion zone E > V0 + m, where V0 is the barrier height, E one of the wave packet energies and m the particle mass. We have also investigated the compatibility of the barrier results with the Klein paradox [7–9], when m < E < V0 −m. In this latter case, we have noted the existence of dynamic localized states with a continuous spectrum [6]. These states are the nearest approximation to the bound states of Schrödinger or of Dirac in the evanescent energy zone considered in this paper. The evanescent zone is the last energy zone we need to consider to complete our analysis. It is given by V0 −m < E < V0 +m (V0 > 2m) or m < E < V0 +m (V0 < 2m). It has evanescent (real exponential) space forms in the barrier region. For a well potential, it is just such forms that give rise to discrete bound states. In this paper, we shall concentrate upon the single barrier potential and hence complete our analysis for this elementary structure. The evanescent stationary solutions become dynamic if instead of plane-waves we work with incoming wave packets. Then, the particles within the classically forbidden region are measurable only for a finite time, the time of transition from an incoming wave packet to reflected/transmitted wave packets. Even for the step potential in this energy zone there will exist, during this transitory time, current flow both into and then subsequently out of the step. Since, the stationary solution has zero current flow within the step, this feature is not always recognized. It is however obvious when one admits that there will be a non zero transitory dρ/dt for any space interval within the step. The most important barrier feature (both theoretically and experimentally) of this energy zone is tunnelling. A part of the incoming wave packet will continue its course beyond the barrier region. Its magnitude will be modulated by the barrier. For ”large” barriers (compared to the wave packet size) an exponential reduction in amplitude ∝ exp[−ql] occurs, where l is the barrier length and q is √