Chapter 3 - Particles in Boxes

Abstract

Because of the mathematical complexity, many systems that contain multiple particles must be described by wave equations that cannot be solved in an exact manner. Among the simple models that can be solved in an exact manner are those that describe particles in one-dimensional and three-dimensional boxes. This chapter illustrates the solution of equations for these models, and in order to reduce the mathematical complexity of the three-dimensional model, the separation of variables technique is applied to the differential equation. An electron in a site normally occupied by an anion behaves as a particle in a three-dimensional box. The result is the difference between allowed energy levels that correspond to the energy of visible light (producing an F-center). This leads to the absorbance of light so that the crystal appears colored. A similar situation exists for an electron that is contained in a solvent “cage.” These models are discussed as if the systems are particles in boxes.