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In this paper we introduce an alternative approach to studying the evolutionof a quantum harmonic oscillator subject to an arbitrary time dependent force.With the purpose of finding the evolution operator, certain unitarytransformations are applied successively to Schr\"odinger's equation reducingit to its simplest form. Therefore, instead of solving the originalSchr\"odinger's partial differential equation in time and space the problem isreplaced by a system of ordinary differential equations. From the obtainedevolution operator we workout the propagator. Even though we illustrate the useof unitary transformations on the solution of a forced harmonic oscillator, themethod presented here might be used to solve more complex systems. The presentwork addresses many aspects regarding unitary transformations and the dynamicsof a forced quantum harmonic oscillator that should be useful for students andtutors of the quantum mechanics courses at the senior undergraduate andgraduate level.

Unitary approach to the quantum forced harmonic oscillator