Interactive visualization of the Kronig–Penney model and extended zone scheme using Python programming

Abstract In the interest of improving pedagogical clarity, we provide an interactive way to apply Python programming to visualize the extended zone scheme and the Kronig–Penney model in the same window. To investigate how straightforward physical parameters such as lattice spacing, barrier width, and potential height affect band structure characteristics in periodic potentials, the simulation was developed by students as part of an instructional end-of-semester project employing project based learning principles. This article is intended for physics educators seeking interactive tools and for students as a guide to integrating computational methods with theoretical physics. The simulation has several features including sliding graphs in real time. The simulation aims to promote conceptual understanding in solid state physics by connecting theory into the abstract and presenting visual representations. With Python libraries such as matplotlib, numpy , and ttkbootstrap , we create an intuitive user interface appropriate for both undergraduate and graduate level instruction. We discussed how these simulations may broaden pedagogical space in a pedagogical frame in developing interactive capabilities for learning and provided a brief outline of the underlying theoretical principles, programmatic design and pedagogical features of the simulations.