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YMMETRY IS ONE OF THE MOST POWERFUL TOOLS that can be ap-

plied to quantum mechanics and wavefunctions. Most people are gener-

ally aware of the concept of symmetry: an object is round or square, or the left

side is the same as the right side, or maybe they are mirror images. All of these

statements imply a recognition of symmetry, a spatial similarity due to the

shape of an object. But more technically, symmetry is a powerful mathemati-

cal tool that can potentially simplify our study of quantum mechanics.

Consider a random quadrilateral, a plane figure having four sides. In order

to define a specific quadrilateral, one must specify not only the lengths of each

side, but their order and angles of intersection. Now consider a square. A

square is also a plane figure having four sides. But by definition, the sides are

at 90° angles and all have the same length. A square has more symmetry,and

so is simpler to define.

Such comparisons apply in quantum mechanics, too. Recognizing the sym-

metry of an atomic or molecular system allows one to simplify the quantum

mechanics, sometimes dramatically. We have already seen some aspects of

symmetry: odd and even functions, the spherical nature of the hydrogen atom’s

1 sorbital, the cylindrical shape of H 2 and H 2. All these are applications of

symmetry. In this chapter, we will develop a general understanding of symme-

try using a mathematical tool called group theory. Then, we can see how sym-

metry applies to some aspects of quantum mechanics.

13.1 Synopsis

This chapter begins with an introduction to group theory, the branch of math-

ematics that considers symmetry. We will find that each symmetry operation

has a corresponding symmetry operator, just like other quantum-mechanical

operators. Symmetry operators move objects, including molecules, in three-

dimensional space into spatially equivalent objects. Every object satisfies a

collection, or group,of symmetry operators. Understanding the characteristics

of that group of operators is an important part of symmetry. At first, there

will be little connection between symmetry and the topics of the previous

chapters, but that will change quickly. Wavefunctions also have symmetry, and

their symmetry can be used to understand their properties and to define and

13.1 Synopsis

13.2 Symmetry Operations and Point Groups

13.3 The Mathematical Basis
of Groups

13.4 Molecules and Symmetry

13.5 Character Tables

13.6 Wavefunctions and
Symmetry

13.7 The Great Orthogonality
Theorem

13.8 Using Symmetry in Integrals

13.9 Symmetry-Adapted Linear
Combinations

13.10 Valence Bond Theory

13.11 Hybrid Orbitals

13.12 Summary

Introduction to Symmetry

in Quantum Mechanics