Tutorial: The quantum finite square well and the Lambert W function

Authors

Sree Ram Valluri, University of Western OntarioFollow
Ken Roberts, University of Western Ontario

Document Type

Article

Publication Date

2-2017

Journal

CANADIAN JOURNAL OF PHYSICS

Volume

2

Issue

95

First Page

105

Last Page

110

URL with Digital Object Identifier

http://dx.doi.org/10.1139/cjp-2016-0602

Abstract

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 -> z = we(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. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.

Notes

http://dx.doi.org/10.1139/cjp-2016-0602