To illustrate the basic methodology involved in the quantum mechanics of molecules, a one-dimensional caricature of the hydrogen molecular ion (H[superscript +][subscript 2]) is presented, which is exactly solvable, in the Born-Oppenheimer approximation, in terms of elementary functions. The purpose of the exercise is to elucidate in a simple setting the standard approach to the electronic and vibrational states of molecules, a subject of interest to students of physics, chemistry and biology. (Contains 9 figures and 5 footnotes.)