Penetration of Arbitrary Double Potential Barriers with Probability Unity: Implications for Testing the Existence of a Minimum Length

TL;DR

This paper proves that particles can perfectly tunnel through arbitrary double barriers by tuning their spacing, and discusses implications for testing the existence of a minimum length in quantum mechanics.

Contribution

It rigorously demonstrates resonant tunneling with probability one for arbitrary barriers and explores its implications for minimum length hypotheses.

Findings

Large barriers can be penetrated with probability one by tuning inter-barrier spacing.

Resonant tunneling depends critically on phase factors and barrier positioning.

The results suggest a method for testing minimum length hypotheses using atomic systems.

Abstract

Quantum tunneling across double potential barriers is studied. With the assumption that the real space is a continuum, it is rigorously proved that large barriers of arbitrary shapes can be penetrated by low-energy particles with a probability of unity, i.e., realization of resonant tunneling (RT), by simply tuning the inter-barrier spacing. The results are demonstrated by tunneling of electrons and protons, in which resonant and sequential tunneling are distinguished. The critical dependence of tunneling probabilities on the barrier positions not only demonstrates the crucial role of phase factors, but also points to the possibility of ultrahigh accuracy measurements near resonance. By contrast, the existence of a nonzero minimum length puts upper bounds on the barrier size and particle mass, beyond which effective RT ceases. A scheme is suggested for dealing with the practical difficulties arising from the delocalization of particle position due to the uncertainty principle. This work opens a possible avenue for experimental tests of the existence of a minimum length based on atomic systems.