Small Corrections to the Tunneling Phase Time Formulation

TL;DR

This paper revisits tunneling phase time calculations by analyzing wave packet collisions, proposing a symmetrical collision setup to clarify the stationary phase method's application and address previous analytical issues.

Contribution

It introduces a symmetrical collision framework that allows proper application of the stationary phase method for tunneling phase time calculations, correcting prior analytical inconsistencies.

Findings

Revealed multiple reflected and transmitted wave packets in tunneling.

Demonstrated a symmetrical collision setup to recover stationary phase conditions.

Provided insights into correct application of the stationary phase method.

Abstract

After reexamining the above barrier diffusion problem where we notice that the wave packet collision implies the existence of {\em multiple} reflected and transmitted wave packets, we analyze the way of obtaining phase times for tunneling/reflecting particles in a particular colliding configuration where the idea of multiple peak decomposition is recovered. To partially overcome the analytical incongruities which frequently rise up when the stationary phase method is adopted for computing the (tunneling) phase time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a unidimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted wave components so that the conditions for applying the stationary phase principle are totally recovered. Lessons concerning the use of the stationary phase method are drawn.