Analytical and Numerical Study of the Kronig–Penney Model in One Dimension
DOI:
https://doi.org/10.70882/josrar.2026.v3i4.231Keywords:
Kronig–Penney model, Energy band structure, Bloch’s theorem, Transfer matrix method, Band gap, Periodic potentialAbstract
The Kronig–Penney model is a foundational exactly solvable problem in solid state physics that illustrates the emergence of energy band structure in periodic potentials. This paper presents both an analytical solution of the model for a one-dimensional periodic delta-function potential and a numerical validation of the resulting band gaps. The analytical approach solves the time-independent Schrödinger equation by applying Bloch’s theorem and continuity conditions, leading to a transcendental equation relating the energy and the Bloch wavenumber. Numerically, the band structure is computed using a transfer matrix method, and the results are compared with the analytical dispersion relation. The study confirms the existence of alternating allowed and forbidden energy bands and demonstrates how the model potential strength affects the band gap widths.
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Copyright (c) 2026 Abdullahi Tanimu, Shamsuddeen Sani Alhassan (Author)
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