Modification of Heisenberg’s Uncertainty Relation for a Damped Harmonic Oscillator under the Generalized Uncertainty Principle

Abstract

We investigate the effects of the Generalized Uncertainty Principle (GUP), which intro- duces a fundamental minimal observable length, on the quantum dynamics of a damped harmonic oscillator. By incorporating a modified Heisenberg algebra into the framework, we derive the altered uncertainty relations for damped quantum systems and examine their dependence on the discrete energy spectrum. Our analysis reveals significant devia- tions from the standard Heisenberg uncertainty relation, demonstrating that the presence of a minimal length scale fundamentally modifies the quantum-classical correspondence in dissipative systems. We present explicit expressions for the modified uncertainty relations and discuss their physical implications for quantum measurements in damped oscillatory systems. These results contribute to our understanding of quantum mechanics at length scales approaching the Planck regime and provide testable predictions for experimental verification of GUP effects in dissipative quantum systems.