Relativistic tunneling through opaque barriers

TL;DR

This paper presents an analytical approach to relativistic tunneling through opaque barriers, providing a corrected phase time formula that aligns with numerical results and reveals a linear relationship with barrier width under specific conditions.

Contribution

It introduces a new analytical formula for phase time in relativistic tunneling that improves upon the standard stationary phase method.

Findings

The new phase time formula matches numerical simulations.

Phase time is proportional to barrier width when energy limits coincide.

Standard formulas are corrected for better accuracy.

Abstract

We propose an analytical study of relativistic tunneling through opaque barriers. We obtain a closed formula for the phase time. This formula is in excellent agreement with the numerical simulations and corrects the standard formula obtained by the stationary phase method. An important result is found when the upper limit of the incoming energy distribution coincides with the upper limit of the tunneling zone. In this case, the phase time is proportional to the barrier width.