One-dimensional (1D) systems in which metastable states exist are physically important, but they are usually not discussed quantitatively in textbooks. This paper presents a relatively simple 1D system involving a piecewise-constant potential for which metastable states can be easily calculated using a computer algebraic system. The metastable-state wave functions are computed and plotted for various particle energies. The Breit–Wigner approximation is used to fit the resulting resonant structure in the transmission coefficient. Connections are made between the initial analysis and poles of the scattering matrix that corresponds to the potential.

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The difference in the real parts of the energy is based on an extremely subtle aspect of where the pole in the S matrix is in the complex plane, a topic we plan to explore in a follow-up publication.

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