The Harmonic Oscillator in the Classical Limit of a Minimal-Length Scenario

TL;DR

This paper solves the harmonic oscillator problem in a minimal-length quantum scenario, revealing anharmonic motion and a modified period, which could indicate Planck-scale physics effects in experiments.

Contribution

It provides an explicit solution showing how minimal length alters harmonic oscillator dynamics, introducing anharmonicity and period changes.

Findings

Oscillator motion becomes anharmonic due to minimal length.

The period of oscillation depends on the minimal length.

Results have implications for spectroscopy and gravity experiments.

Abstract

In this work we explicitly solve the problem of the harmonic oscillator in the classical limit of a minimal-length scenario. We show that (i) the motion equation of the oscillator is not linear anymore because the presence of a minimal length introduces an anarmonic term and (ii) its motion is described by a Jacobi sine elliptic function. Therefore the motion is periodic with the same amplitude and with the new period depending on the minimal length. This result (the change in the period of oscillation) is very important since it enables us to find in a quite simple way the most relevant effect of the presence of a minimal length and consequently traces of the Planck-scale physics. We show applications of our results in spectroscopy and gravity.