Causality, apparent 'superluminality' and reshaping in barrier penetration

D. Sokolovski

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)
186 Downloads (Pure)


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 δ 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.
Original languageEnglish
Article number042115
Number of pages7
JournalPhysical Review A
Issue number4
Publication statusPublished - 30 Apr 2010

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics


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