  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.
