Klein paradox for bosons, wave packets and negative tunnelling times

X Gutiérrez De La Cal,D Sokolovski,M Alkhateeb,M Pons,A Matzkin

doi:10.1038/s41598-020-76065-7

Abstract

We analyse a little known aspect of the Klein paradox. A Klein–Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein–Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier’s height is enhanced yet remains finite.

Highlights

We note that the transmission and reflection amplitudes of a smooth step, t(p, q) = 1/A(p, q) and r(p, q) = B(p, q)/A(p, q), after replacing q → −q develop an additional pole in the complex p-plane, since Ŵ(1 − ip/b + iq/b − ) → ∞ as 1 − ip + iq − b → 0
We need the second incident WP, with A1(p) such that, after being reflected ψ1 = dpA1(p) exp(ipx − iEt) leaves the barrier empty
√ the m√omentum in the barrier is given by Q = ( P2 − M2 − W)2 − M2, while for the energy we find E(P) = P2 + M2

Summary

Introduction

To obtain the MREs for a smooth potential Wsmooth(x) = V [tanh(bx) − tanh[b(x − d)] we use the connection rules for a smooth step W(x) = V tanh(bx) in[20], namely x > 0 (27) We note that the transmission and reflection amplitudes of a smooth step, t(p, q) = 1/A(p, q) and r(p, q) = B(p, q)/A(p, q) , after replacing q → −q develop an additional pole in the complex p-plane, since Ŵ(1 − ip/b + iq/b − ) → ∞ as 1 − ip + iq − b → 0 .

Results

Conclusion