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 transmission 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 thoroughly explicated and several coding examples are provided.

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