Unitary approach to the quantum forced harmonic oscillator

TL;DR

This paper presents a novel method using successive unitary transformations to simplify and solve the quantum forced harmonic oscillator problem, replacing PDEs with ODEs for easier analysis.

Contribution

It introduces a new approach employing unitary transformations to find the evolution operator for a forced quantum harmonic oscillator, applicable to more complex systems.

Findings

Derives the evolution operator using unitary transformations.

Reduces Schrödinger's PDE to a system of ODEs.

Provides a practical method for educational purposes.

Abstract

In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schr\"odinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schr\"odinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.