A new approach to the quantization of the damped harmonic oscillator

TL;DR

This paper introduces a novel method for quantizing the damped harmonic oscillator by redefining time and coordinates, resulting in a Lagrangian that satisfies Helmholtz conditions and aligns with classical damping behavior.

Contribution

It presents a new approach to construct Lagrangians for damped systems that satisfies Helmholtz conditions and recovers classical critical damping in quantum models.

Findings

Predicts an energy decay rate consistent with previous models

Recovers the classical critical damping condition

Applicable to driven damped harmonic oscillators

Abstract

In this paper, a new approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.