  The ability to induce a sizable gap in the excitation spectrum of a normal
layer placed in contact with a conventional superconductor has become
increasingly important in recent years in the context of engineering a
topological superconductor. The quasiclassical theory of the proximity effect
shows that Andreev reflection at the superconductor/normal interface induces a
nonzero pairing amplitude in the metal but does not endow it with a gap.
Conversely, when the normal layer is atomically thin, the tunneling of Cooper
pairs induces an excitation gap that can be as large as the bulk gap of the
superconductor. We study how these two seemingly different views of the
proximity effect evolve into one another as the thickness of the normal layer
is changed. We show that a fully quantum-mechanical treatment of the problem
predicts that the induced gap is always finite but falls off with the thickness
of the normal layer, $d$. If $d$ is less than a certain crossover scale, which
is much larger than the Fermi wavelength, the induced gap is comparable to the
bulk gap. As a result, a sizable excitation gap can be induced in normal layers
that are much thicker than the Fermi wavelength.
