Causality, apparent “superluminality,” and reshaping in barrier penetration

We consider tunneling of a nonrelativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backward relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac $\delta$ function, the transmission amplitude is superoscillatory for finite momenta and tunneling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analyzing apparent “superluminality” in terms of spacial displacements helps avoid contradiction associated with time parameters such as the phase time.

Figure 1

Regular part of the DAD in Eq. (8) for a rectangular barrier with $\beta \equiv \sqrt{2W}d=20$ and $p_0=0$ obtained by numerical integration of Eq. (5) with $T(p)$ given by Eq. (17).

Figure 2

High rectangular barrier: (a) $\mathit{ReT}(p)/\mathit{pB}(W)$ (solid) and $\cos (\mathit{pd})$ (dashed) for $\beta \equiv \sqrt{2W}d=20$. Also shown is $|A(p-p_0)|$ scaled to a unit height (thick solid). (b) The shape of the transmitted pulse $|G^T(x,p_0,t)/T(p_0,W)|$: exact (solid) and given by Eq. (23) (dashed); (c and d) same as (a) and (b) but for $\beta =100$; (e and f) same as (a) and (b) but for $\beta =700$.

Figure 3

Wide rectangular barrier: (a) ratio between the exact $T(p)$ and its approximation in Eq. (28) for $\beta \equiv \sqrt{2W}d=20$ (solid). Also shown is $|A(p-p_0)|$ scaled to a unit height (thick solid). (b) The shape of the transmitted pulse $|G^T(x,p_0,t)/T(p_0,W)|$: exact (solid) and given by Eq. (31) (dashed); (c and d) same as (a) and (b) but for $\beta =100$; (e and f) same as (a) and (b) but for $\beta =700$.