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A variety of one-dimensional quantum systems is studied using a transfer-matrix formalism. In particular, only two matrices are required: one to propagate a wave function over a region of constant potential, the other to connect wave functions at a discontinuity in the potential. Using these simple matrices as building blocks, we constructed complex transfer matrices with which to study energy bands in one-dimensional crystals, as well as the intriguing characteristics of tranmission through multiple barriers. All systems studied here are modeled with finite-step potentials, not delta functions. The solutions obtained are exact and can be generated with relative ease using Mathematica. The method is throroughly explicated and several coding examples are provided.
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