A metal film that is only a few atomic layers thick, on an appropriate substrate, creates a quantum well that confines the conduction electrons in the direction perpendicular to the film. As a result, components of the wavevector in the plane of the film are quasicontinuous, but the perpendicular component is discrete. The result is a strong oscillatory dependence of the film properties on film thickness, including the Fermi energy and the density of states. We show that these quantum effects associated with the film thickness can be understood within the free electron theory of metals. Because of its simplicity, the thin-film quantum well is appropriate as a problem for senior undergraduate and graduate students and provides an introduction to quantum size effects.