Band Gap Opening: Nearly-Free-Electron Model

This simulator diagonalizes a real plane-wave Hamiltonian for an electron in a 1D periodic crystal potential, showing exactly how the free-electron parabola splits into bands with a gap at the Brillouin-zone boundary as the potential strength V₀ grows. Probe any point on any band to read its energy, group velocity, and effective mass, and watch the mass swing negative — a hole — near the top of a band.