Applications of Group Theory in Quantum Mechanics

M. I. Petrashen, J. L. Trifonov

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486472232
• eBook ISBN: 9780486172729
• Topic: Physical Sciences
• Subtopic: Quantum Theory

Geared toward postgraduate students, theoretical physicists, and researchers, this advanced text explores the role of modern group-theoretical methods in quantum theory. The authors based their text on a physics course they taught at a prominent Soviet university. Readers will find it a lucid guide to group theory and matrix representations that develops concepts to the level required for applications.
The text's main focus rests upon point and space groups, with applications to electronic and vibrational states. Additional topics include continuous rotation groups, permutation groups, and Lorentz groups. A number of problems involve studies of the symmetry properties of the Schroedinger wave function, as well as the explanation of "additional" degeneracy in the Coulomb field and certain subjects in solid-state physics. The text concludes with an instructive account of problems related to the conditions for relativistic invariance in quantum theory.

Chapter 1

Introduction

In the first chapter of this monograph we shall try, in so far as it is possible at the beginning of a book, to show how one can naturally and advantageously apply the theory of groups to the solution of physical problems. We hope that this will help the reader who is mainly interested in the applications of group theory to physics to become familiar with the general ideas of abstract groups which are necessary for applications.

1.1 Symmetry properties of physical systems

It is frequently possible to establish the properties of physical systems in the form of symmetry laws. These laws are expressed by the invariance (invariant form) of the equations of motion under certain definite transformations. If, for example, the equations of motion are invariant under orthogonal transformations of Cartesian coordinates in three-dimensional space, it may be concluded that reference frames oriented in a definite way relative to each other are equivalent for the description of the motion of the physical system under consideration. Equivalent reference frames are usually defined as frames in which identical phenomena occur in the same way when identical initial conditions are set up for them. Conversely, if in a physical theory it is postulated that certain reference frames are equivalent, then the equations of motion should be invariant under the transformations relating the coordinates in these systems. For example, the postulate of the theory of relativity which demands the equivalence of all reference frames moving with uniform velocity relative to one another is expressed by the invariance of the equations of motion under the Lorentz transformation. The class of equivalent reference frames for a given problem is frequently determined from simple geometrical considerations applied to a model of the physical system. This is done, for example, in the case of symmetric molecules, crystals and so on. However, not all transformations under which the equations of motion are invariant can be interpreted as transformations to a new reference frame. The symmetry of a physical system may not have an immediate geometrical interpretation. For example, V. A. Fock has shown that the Schroedinger equation for the hydrogen atom is invariant under rotations in a four-dimensional space connected with the momentum space.

The symmetry properties of a physical system are general and very important features. Their generality usually ensures that they remain valid while our knowledge of a given physical system grows. They must not, however, be regarded as absolute properties; like any other descriptions of physical systems they are essentially approximate. The approximate nature of some symmetry properties is connected with the current state of our knowledge, while in other cases it is due to the use of simplified models of physical systems which facilitate the solution of practical problems.

Thus, by the symmetry of a system we shall not always understand the invariance of its equations of motion under a certain set of transformations. The following important property must always be remembered: if an equation is invariant under transformations A and B, it is also invariant under a transformation C which is the result of the successive application of the transformations A and B. The transformation C is usually called the product of the transformations A and B. A set of symmetry transformations for a given physical system is therefore closed with respect to the operation of multiplication which we have just defined. Such a set of transformations is called a group of symmetry transformations for the given physical system. A rigorous definition of a group is given below.

1.2 Definition of a group

A group G is defined as a set of objects or operations (elements of the group) having the following properties.

1. The set is subject to a definite ‘multiplication’ rule, i.e. a rule by which to any two elements A and B of the set G, taken in a definite order, there corresponds a unique element C of this set which is called the product of A and B. The product is written C = AB.

2. The product is associative, i.e. the equation (AB) D = A (BD) is satisfied by any elements A, B and D of the set. The product may not be commutative, i.e. in general AB ≠ BA. Groups for which multiplication is commutative are Abelian.

3. The set contains a unique element E (the identity or unit element) such that the equation

AE = EA = A

is satisfied by any element A in the set.

4. The set G always includes an element F (the inverse) such that for any element A

AF = E

The inverse is usually denoted by A^-1.

The above four properties define a group. We see that a group is a set which is closed with respect to the given rule of multiplication. The following are consequences of the above properties.

a. The group contains only one unit element. Thus, for example, if we suppose that there are two unit elements E and E’ in the group G, then in view of property 3 we have

EE′ = E = E′E = E′

i.e. E = E′.

b. If F is the inverse of A, the element A will be the inverse of F, i.e. if AF = E, then FA = E. In fact, multiplying the first of these equations on the left by F, we have

FÁF = F

The element F (like any other element of the set G) has an inverse F^–1. Multiplying the last equation on the right by F^–1 we obtain FAFF^–1 = FF^–1, i.e. FA = E.

c. For each element in the set there is only one inverse element. Let us suppose that an element A in G has two inverse elements F and D, i.e. AF = E and AD = E. If this is so, then by multiplying the equation AF = AD on the left by A^–1 we obtain F = D.

d. If C = AB then C^–1 = B^–1A^–1, because of the associative property of the product of two elements in the group.

We note also that if the number of elements in a group is finite, then the group is called a finite group; if the number of elements is infinite, the group is called an infinite group. The number of elements in a finite group is the order of the group.

The following are examples of groups.

1. The set of all integers, including zero, forms an infinite group if addition is taken as group multiplication. The unit element in this group is 0, the inverse element of a number A is − A, and the group is clearly Abelian.

2. The set of all rational numbers, excluding zero, forms a group for which the multiplication rule is the same as the familiar multiplication rule used in arithmetic. The unit element is 1. This is again an infinite Abelian group. The positive rational numbers also form a group, but the negative rational numbers do not.

3. The set of vectors in n-dimensional linear space forms a group. The group multiplication rule is the vector addition; the unit element is t...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• Foreword
• Chapter 1 - Introduction
• Chapter 2 - Abstract Groups
• Chapter 3 - Representations of Point Groups
• Chapter 4 - Composition of Representations and the Direct Products of Groups
• Chapter 5 - Wigner’s Theorem
• Chapter 6 - Point Groups
• Chapter 7 - Decomposition of a Reducible Representation into an Irreducible Representation
• Chapter 8 - Space Groups and Their Irreducible Representations
• Chapter 9 - Classification of the Vibrational and Electronic States of a Crystal
• Chapter 10 - Continuous Groups
• Chapter 11 - Irreducible Representations of the Three—Dimensional Rotation Group
• Chapter 12 - The Properties of Irreducible Representations of the Rotation Group
• Chapter 13 - Some Applications of the Theory of Representation of the Rotation Group in Quantum Mechanics
• Chapter 14 - Additional Degeneracy in a Spherically Symmetric Field
• Chapter 15 - Permutation Groups
• Chapter 16 - Symmetrized Powers of Representations
• Chapter 17 - Symmetry Properties of Multi-Electron Wave Functions
• Chapter 18 - Symmetry Properties of Wave Functions for a System of Identical Particles with Arbitrary Spins
• Chapter 19 - Classification of the States of a Multi-Electron Atom
• Chapter 20 - Applications of Group Theory To Problems Connected With the Perturbation Theory
• Chapter 21 - Selection Rules
• Chapter 22 - The Lorentz Group and its Irreducible Representations
• Chapter 23 - The Dirac Equation
• Appendix to Chapter 7
• Bibliography
• Index