ideas will be discussed in detail in the sections that follow. Briefly, however, it
might be useful to summarize these ideas so that the reader will understand
where the material is headed. Remember that the ultimate goal is to have a
theory that proposes how matter behaves, and that predicts events that agree
with observation; that is, to have theory and experiment agree. Otherwise, a
different theory is necessary to understand the experiment.
The main ideas are:- The behavior of electrons, by now known to have wavelike properties,
can be described by a mathematical expression called a wavefunction. - The wavefunction contains within it all possible information that can be
known about a system. - Wavefunctions are not arbitrary mathematical functions, but must sat-
isfy certain simple conditions. For example, they must be continuous. - The most important condition is that the wavefunction must satisfy
the time-dependent Schrödinger equation. With certain assumptions,
time can be separated from the wavefunction, and what remains is a
time-independentSchrödinger equation. We focus mainly on the time-
independent Schrödinger equation in this text. - In the application of these conditions to real systems, wavefunctions are
found that do indeed yield information that agrees with experimental
observations of these systems:quantum mechanics predicts values that
agree with experimentally determined measurements.The simplest real
system to understand, covered in the next chapter, is the hydrogen
atom, a system that Rydberg and Balmer and Bohr had studied with
different degrees of success. To the extent that quantum mechanics not
only reproduces their success but also extends it, quantum mechanics
is superior to their theories trying to describe the behavior of sub-
atomic particles.
The rest of this chapter expands on the above ideas. A proper understand-
ing of quantum mechanics requires an understanding of the principles that it
uses. An adequate familiarity with these principles is essential, even irreplace-
able. In your dealings with these principles, do not lose sight of that last state-
ment in the above synopsis: quantum mechanics properly describes the be-
havior of matter, as determined by observation.
10.2 The Wavefunction
The behavior of a wave can be expressed as a simple mathematical function.
For example,
yAsin (BxC) D (10.1)
is a general expression for the amplitude,y, of a sine-type (or sinusoidal) wave
traveling in the xdimension. The constants A,B,C, and Dhave certain values
that specify exactly what the sine wave looks like.
Since de Broglie indicated that matter should have wave properties, why not
describe the behavior of matter using an expression for a wave? The first pos-
tulate of quantum mechanics is that the state of a system can be described by
an expression called a wavefunction. Wavefunctions in quantum mechanics are
typically given the symbol or (the Greek letter psi). For various physical
and mathematical reasons, these ’s are limited, or constrained,to being func-
tions that are:274 CHAPTER 10 Introduction to Quantum Mechanics