On Close Relationship between Classical Time-Dependent Harmonic Oscillator and Non-Relativistic Quantum Mechanics in One Dimension

TL;DR

This paper establishes a novel mapping between classical time-dependent harmonic oscillators and quantum phenomena in one dimension, demonstrating that quantum tunneling and related effects can be described using classical physics without energy conservation violations.

Contribution

It introduces a new classical-quantum mapping and provides formulas for potential barriers with specific reflection and transmission properties, supported by numerical simulations.

Findings

Quantum tunneling described by classical oscillators

Exact solutions for potential barriers with desired properties

Classical oscillators mimic quantum uncertainty relations

Abstract

In this paper, I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be described in terms of classical physics without invoking violations of the energy conservation law at any time instance. A formula is presented that generates a wide class of potential barrier shapes with the desirable reflection (transmission) coefficient and transmission phase shift along with the corresponding exact solutions of the time-independent Schr\"odinger's equation. These results, with support from numerical simulations, strongly suggest that two uncoupled classical harmonic oscillators, which initially have a 90\degree relative phase shift and then are simultaneously disturbed by the same parametric perturbation of a finite duration, manifest behavior which can be mapped to that of a single quantum particle, with classical 'range relations' analogous to the uncertainty relations of quantum physics.