TABLE OF CONTENTS Our objective for the next few chapters is to learn how the Hamiltonian matrix [ H] for a given device structure (see Fig. 1.6.5) is written down. We start in this chapter with (1) the hydrogen atom (Section 2.1) and how it led scientists to the Schr dinger equation, (2) a simple approach called the finite difference method (Section 2.2) that can be used to convert this differential equation into a matrix equation, and (3) a few numerical examples (Section 2.3) showing how energy levels are calculated by diagonalizing the resulting Hamiltonian matrix.
Early in the twentieth century scientists were trying to build a model for atoms which were known to consist of negative particles called electrons surrounding a positive nucleus. A simple model pictures the electron (of mass m and charge ? q) as orbiting the nucleus (with charge Zq) at a radius r (Fig. 2.1.1) kept in place by electrostatic attraction, in much the same way that gravitational attraction keeps the planets in orbit around the Sun.
| (2.1.1) |  |
A faster electron describes an orbit with a smaller radius. The total energy of the electron is related to the radius of its orbit by the relation
| (2.1.2) |  |
However, it was soon realized that this simple viewpoint was inadequate since, according to classical electrodynamics, an orbiting electron...
