Applied Group Theory

For Physicists and Chemists

George H. Duffey

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486783147
• eBook ISBN: 9780486801322
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

This text introduces advanced undergraduates and graduate students to symmetry relations by means of group theory. Key relationships are derived in detail from first principles. Rather than matrix theory, the treatment employs algebraic theory in deriving the properties of characters and projection operators. This approach is customarily employed in quantum mechanics courses and makes the connection to group structure clearer. Cayley diagrams illustrate the structure of finite groups. Permutation groups are considered in some detail, and the special methods needed for continuous groups are developed.
The treatment's broad range of applications offers students assistance in analyzing the modes of motion of symmetric classical systems; the constitutive relations in crystalline systems; the modes of vibration in molecules; the molecular orbitals of molecules; the electronic structures of atoms; the attendant spectra; and fundamental particle multiplets. Each chapter concludes with a concise review, discussion questions, problems, and references.
1992 edition.

CHAPTER 1

Symmetry Operations

1.1

Causality and Symmetry

The universe is not a single indivisible whole. Instead, it consists of parts that can act independently in spite of interactions binding the parts together. This feature allows observers and observing instruments to exist. It also permits both analysis and synthesis to proceed.

An observer first notes that events in his life fall in order; he experiences time locally. Second, whatever he observes can be located at points or small regions in a 3-dimensional space based on his own position at the time of observation. Similar relationships presumably prevail for an observing instrument.

Time is not observed as a global entity but as an independent property of the observing point or small region. It behaves as a directed coordinate orthogonal to the three spatial coordinates of the point. Furthermore, an interval of time can be measured by the distance traveled by a photon in the interval. Consequently, the arena in which phenomena occur is a 4-dimensional continuum in which a displacement may be oriented to be either timelike or spacelike.

In constructing science, one seeks out the patterns that exist among the observations. One presumes that the material world is not capricious or lawless—that it is not governed by spirits as primitive man believed. If certain events appear to follow as a consequence of particular conditions, these events are said to be caused by the conditions. Thus in Newtonian mechanics, one says that the acceleration of a body is caused by the net force acting on it.

In principle, uniqueness need not prevail. A given set of conditions, a given cause, may lead to various possible results rather than to a single result. Then degeneracy is said to obtain. For instance, the radioactive nuclei in a sample may be shown to be identical by statistical tests. Nevertheless, they will disintegrate at random times with a definite half life.

In general, we will call the part of the universe under study a system. The system may be subdivided in various ways. And the resulting parts maybe further subdivided. Each of the subsystems maybe considered a system in its own right in the approximation that it behaves as an entity.

Now, an operation performed on a system may yield an equivalent system with the same spectrum of properties. The entity under study is then said to possess symmetry. The operation is called a symmetry operation.

When these conditions are only approximately satisfied, one says that a near-symmetry exists. The system may then be considered as a pertubation of a corresponding symmetric entity. When the perturbation is small, it may be neglected.

Symmetry operations may act in position space, or in the space-time continuum. They may act in momentum space, or in phase space. Alternatively, they may act in a more general space or plot.

Symmetry operations may also involve other attributes besides position and momentum. Examples of these include partic...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter 1/Symmetry Operations
• Chapter 2/Classes and Characters
• Chapter 3/Expedient Arrangements of Displacements
• Chapter 4/Symmetry-Adapted Stresses and Strains
• Chapter 5/Matrix Representations
• Chapter 6/Symmetry-Adapted Kets and Bras
• Chapter 7/Combinations of Products of Bases
• Chapter 8/Permutation Groups
• Chapter 9/Continuous Groups
• Chapter 10/Rotation Groups
• Chapter 11/Physical Lie Algebras
• Appendix/Characters and Bases for the Primitive Symmetry Species of Important Groups
• Index