  A contemporary physicist would be hard put to agree entirely with the author
of a 1959 textbook on quantum mechanics, who wrote: "A second simple,
one-dimensional system, somewhat divorced from reality but illustrative of the
principles of the theory, is a particle in a box with finite walls." Interest
in this prototypic system has not diminished over the years, and is not likely
to do so in the near future, because it has now been wedded to reality and seen
as a paradigm for quantum wells and other nanosystems. The treatment of this
topic in standard textbooks has changed little over the last few decades, and
such changes as have come to our notice leave, in our opinion, much to be
desired. The purpose of this article is to enable, even encourage, other
teachers to give the finite square well the attention that it deserves; to this
end, the authors provide a tutorial review that is more instructive and
comprehensive than the accounts presented in numerous textbooks and dispensed
in still more numerous online resources, but does not make any additional
demands on the mathematical abilities of the student. Given the elemental
nature of the topic, the authors claim no originality; with nothing more than
the winnowing fork in their hands, they can only clear the threshing floor,
gather the wheat into the barn, and let others decide whether or no the chaff
should be burnt with unquenchable fire.
