Sub-barrier quantum tunneling: eliminating the MacColl-Hartman paradox

TL;DR

This paper explains the MacColl-Hartman paradox in quantum tunneling by linking the saturation of group delay to the phase behavior of stationary wave functions, revealing the effect's finite nature.

Contribution

It provides a new explanation for the MacColl-Hartman effect by connecting it to wave function phase properties and clarifies that the saturation plateau is finite, not infinite.

Findings

Saturation of tunneling delay is due to wave function phase behavior.

The saturation plateau has a finite length, after which delay increases.

Tunneling time increases monotonically beyond the plateau.

Abstract

I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.