Exploring one‐dimensional quantum mechanics with transfer matrices

Walker, James S.; Gathright, J.

doi:doi:10.1119/1.17541

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American Journal of Physics -- May 1994 -- Volume 62, Issue 5, pp. 408

Exploring one‐dimensional quantum mechanics with transfer matrices

James S. Walker¹ and J. Gathright²

¹Department of Physics, Washington State University, Pullman, Washington 99164‐2814
²Space Science Center, University of Hawaii at Hilo, Hilo, Hawaii 96720

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An exact transfer‐matrix formalism is developed for analyzing and solving problems in one‐dimensional quantum mechanics. We show that with only three general‐purpose matrices—one to propagate a wave function over a region of constant potential, one to take a wave function over a discontinuity in a potential, and one to connect a wave function across a delta function—a rich and intriguing variety of behavior is revealed. Not only are standard results recovered with this technique, in ways suitable for presentation in the classroom, but new findings and applications are discussed as well. A primary advantage of the transfer‐matrix approach is that it facilitates wide‐ranging explorations of one‐dimensional quantum mechanics by both students and researchers, especially when implemented with Mathematica. For those interested in pursuing independent explorations, an electronic, interactive version of this paper, complete with the figures given here and the code that generates them, is available over Internet as a Mathematica notebook.