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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 exploration of one-dimensional quantum mechanics by both students and researchers, expecially 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.
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