Accessing Position Space Wave Functions in Band Structure Calculations of Periodic Systems─A Generalized, Adapted Numerov Implementation for One-, Two-, and Three-Dimensional Quantum Problems.

In this work, a generalized, adapted Numerov implementation capable of determining band structures of periodic quantum systems is outlined. Based on the input potential, the presented approach numerically solves the Schrödinger equation in position space at each momentum space point. Thus, in addition to the band structure, the method inherently provides information about the state functions and probability densities in position space at each momentum space point considered. The generalized, adapted Numerov framework provided reliable estimates for a variety of increasingly complex test suites in one, two, and three dimensions. The accuracy of the proposed methodology was benchmarked against results obtained for the analytically solvable Kronig-Penney model. Furthermore, the presented numerical solver was applied to a model potential representing a 2D optical lattice being a challenging application relevant, for example, in the field of quantum computing.