Tunneling Through a One-Dimensional Square Potential Barrier Under Fluctuations in an Observer’s Frame of Reference

Yun-Sok Shin

doi:10.11648/j.ajpa.20200803.12

Abstract

This study reports tunneling through a one-dimensional (1D) square potential barrier (SPB) under fluctuations in an observer’s frame of reference (OFR). To date, tunneling through an SPB has been studied under the assumption that the OFR remains constant throughout the tunneling measurements; therefore, the change of the tunneling probability when the OFR is assumed to fluctuate remains unanswered. In this paper, a 1D SPB is considered under fluctuations of an OFR. The average transmission probability of a particle through an SBP for two types of OFR fluctuations (periodic-square-wave and periodic-sawtooth-wave fluctuations) is formulated in time representations. Under these types of fluctuations, the average transmission probability gradually increases with a particle’s energy, which is saturated to the transmission probability in the case of the stationary OFR at a much greater energy than the amplitude of the fluctuations. The average transmission probability is much higher at the amplitude of the fluctuations in the case of periodic-square-wave fluctuations. Therefore, the average transmission probability with a particle’s energy has the potential to reveal the distribution of OFR fluctuations.

Highlights

Tunneling is the quantum mechanical phenomenon in which a particle passes through a potential barrier
Tunneling can be estimated in terms of Heisenberg’s uncertainty principle [3], which holds that measuring the momentum of a particle can be conducted with precision, whereas the resulting position disturbance of the particle is limited by Planck’s constant divided by the uncertainty of the momentum of the particle
In this paper, tunneling through an square potential barrier (SPB) was studied under the periodic fluctuations of an OFR

Summary

Introduction

Tunneling is the quantum mechanical phenomenon in which a particle passes through a potential barrier. The average current in it exhibited an asymmetric zero-bias Coulomb peak as a function of the energy of the channel under periodic-square-wave and periodic-sawtooth-wave fluctuations—the amplitude of which gradually increased as the amplitude of the fluctuations increased. The average transmission probability of a particle to transmit an SPB [32] was investigated under OFR fluctuations. The average transmission probability rapidly increases just above the energy corresponding to the amplitude of fluctuations of the OFR in the case of periodic-square-wave fluctuations. The average transmission probability with a particle’s energy may be able to reveal the distribution of OFR fluctuations. SPB, where and are the potential energy and width of the barrier, respectively, the energy of a particle, and the energy of an OFR at time. Where εN and aN are the amplitude and period of the periodic-square-wave fluctuations, respectively

Periodic-Square-Wave Fluctuations of One Period

Periodic-Square-Wave Fluctuations at High-Frequency Limits

Periodic-Sawtooth-Wave Fluctuations of One Period

Conclusions