Periodic quantum systems often exhibit energy spectra with well-defined energy bands separated by band gaps. The formation of band structure in such periodic systems is usually presented in the context of Bloch's theorem or through other specialized techniques. Here, we present a simple model of a finite one-dimensional periodic quantum system that can be used to explore the formation of band structure in a straightforward way. Our model consists of an infinite square well containing several evenly spaced identical Dirac delta wells, both attractive and repulsive. We solve for the energy eigenvalues of this system directly and show the formation of band structure as the number of delta wells increases as well as how the size of the bands and gaps depends on the strength of the delta wells. These results are compared to the predictions from Bloch's theorem. In addition, we use this model to investigate how the energy spectrum is altered by the introduction of two types of defects in the periodicity of the system.
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However, a given ISW eigenvalue does not correspond to the same eigenstate for different values of N. For example
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We chose to place the defect at well seven of nine primarily because it produces typical results, but we hope that some readers will also appreciate the nod to Star Trek.
After all, the defect well disrupts the periodicity so that it is no longer assimilated into the collective.39.
In fact, our study of this quantum model was inspired by a preliminary study of defects in an acoustical syste
m conducted by Shawn Hilbert, Scott Carr, and Raphael Paolo Francisco.© 2022 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2022
Author(s)
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