Solution of the quantum finite square well problem using the Lambert W function

We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping $w \to z = w e^w$ between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. There are many opportunities to follow up, and we present the method in a pedagogical manner to stimulate further research in this and related avenues.