3.1 Introduction

The purpose of this chapter is to get some practice calculating what happens to an electron moving in a potential according to Schr dinger s equation. There is a remarkable richness in the type and variety of the predictions. In fact, to the uninitiated, specific solutions to Schr dinger s equations can be quite unexpected. For this reason alone, one should be motivated to explore the possibilities. Getting used to the behavior of waves can take some time, so in this chapter we want to carefully reveal some of the key features of Schr dinger s equation that describe the waviness of matter.

Let us start by considering some basics. According to our approach, a particle of mass m moves in space as a function of time in the presence of a potential. Time and space are assumed to be smooth and continuous. The potential can cause the electron motion to be localized to one region of space, forming what is called a bound state. The alternative one may consider is an electron able to move anywhere in space, in which case the electron is in an unbound state (sometimes called a scattering state).

In Section 2.2 we introduced the time-independent Schr dinger equation for a particle of mass m in a potential V( r), which is a function of space only. The second-order differential equation is

where ? is the Hamiltonian operator, E _n = ? ? _n are energy eigenvalues, and ? _n(