Classical and quantum harmonic oscillators subject to a time dependent force

TL;DR

This paper presents a method for quantizing a harmonic oscillator influenced by a time-dependent force by employing canonical transformations and their unitary equivalents, simplifying the analysis of such perturbed quantum systems.

Contribution

It introduces a canonical transformation approach to remove time-dependent forces in quantum harmonic oscillators, linking classical and quantum formalisms for simplified quantization.

Findings

Canonical transformations effectively remove time-dependent forces

Unitary transformations map perturbed and unperturbed quantum states

Method simplifies quantization of perturbed harmonic oscillators

Abstract

In this work we address the problem of the quantization of a simple harmonic oscillator that is perturbed by a time dependent force. The approach consists of removing the perturbation by a canonical change of coordinates. Since the quantization procedure uses the classical Hamiltonian formalism as staring point, the change of variables is carried out using canonical transformations, and to transform between the quantized systems the canonical transformation is implemented as a unitary transformation mapping the states of the perturbed and unperturbed system onto each other.