Infinite chain of atoms and Coulomb chain: tight binding method

Abstract

The solution in quantum mechanics for systems in which the Hamilton operator is not time-dependent (stationary states) focuses on the solution of the Schrodinger equation independent of time. However, when the system becomes one with many particles, the solution of the equation cannot be addressed by analytical means. Therefore, there are several methods of approximate solution; these fall into two broad categories: the numerical and the self-consistent. The essential difference lies in the process that must be followed to find those solutions. Among these methods, one that is widely applied to problems with many particles is the so-called Tight Binding (Hold Strong) as it allows some freedom of programming and tracking algorithm besides having very good approximation margins. In this article, an overview of the method and its application to three specific problems (set of linear loads and Coulomb potential) will be provided. These two systems will be developed for three main reasons. The first is associated with the simplicity of the process and clarification of the method, the second is related to the utility of practical application of the physical model in engineering such as integrated circuits, resistance alternators and other complex systems, and the third is because in some references a comparison is made with alternative (usually self-consistent) methods to very similar problems.