Recovering the stationary phase condition for accurately obtaining scattering and tunneling times

TL;DR

This paper critically examines the stationary phase method for calculating tunneling times, highlighting boundary effects and proposing a wave packet decomposition approach to improve accuracy in scattering and tunneling time analysis.

Contribution

It introduces a framework considering multiple wave packet decomposition and boundary effects to refine tunneling phase time calculations.

Findings

Boundary effects significantly influence tunneling phase times.

Multiple wave packet decomposition clarifies scattering dynamics.

Recomposition of wave packets improves phase time accuracy.

Abstract

The stationary phase method is often employed for computing tunneling {\em phase} times of analytically-continuous {\em gaussian} or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After reexamining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one dimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.