Group Theory In Solid State Physics I.pdf

Group Theory in Solid State

Physics I

Preface

This lecture introduces group theoretical concepts and methods with the

aim of showing how to use them for solving problems in atomic, molecular

and solid state physics. Both ﬁnite and continuos groups will be discussed

in this lectures. Finite groups are important because the symmetry elements

in molecular and solid state physics consist of discrete rotations and trans-

lations. Continuos groups are crucial in problems containing the spin. The

relevant literature for the topics presented in this lectures is:

L.D. Landau, E.M. Lifshitz, Lehrbuch der Theor. Pyhsik, Band III, ”Quan-

tenmechanik”, Akademie-Verlag Berlin, 1979, Kap. XII and Band V, ”Statis-

tische Physik”, Teil 1, Akademie-Verlag 1987, Kap. XIII.

Zurich, March 2003

D. Pescia

Contents

Preface

1 Groups 1

1.1 Exercises.............................. 12

2 Group Representations 14

2.1 UnitaryRepresentations ..................... 14

2.2 Reducibleandirreduciblerepresentations............ 18

2.3 Characters of a representation and theorems involving them . 21

2.4 Exercises.............................. 25

3 Group theory and Quantum Mechanics 27

3.1 The symmetry group of the Schr¨odingerequation ....... 27

3.2 Construction of Basis functions and matrix representations . . 30

3.3 Applicationstoquantummechanicalproblems......... 32

3.4 Exercises.............................. 35

4 SO (3) and SU (2) 44

4.1 LieAlgebra ............................ 44

4.2 The group SO (3) ......................... 46

4.3 The j =1 / 2 irreducible representation: SU (2).......... 51

4.4 Doubleﬁnitegroups ....................... 54

4.4.1 The group C 1 ....................... 55

4.4.2 The group C i ....................... 55

4.4.3 The double group C 3 v .................. 56

4.5 Exercises.............................. 57

iii

Chapter 1

Groups

Aset G of elements g 1 ,g 2 ... is said to form a group if

1. a law of composition (or multiplication)

◦

of two elements g 1 and g 2 is

deﬁned, so that g 1 ◦ g 2 ∈ G

2. a identity element e exists such that e

◦

g = g

◦

e = g

3. the associative law is fullﬁled, i.e. g 1 ◦ ( g 2 ◦ g 3 )=( g 1 ◦ g 2 ) ◦ g 3

4. there exists an inverse element g − 1 such that g

◦

g − 1 = g − 1

◦

g = e

g 1 is said to be a commutative or Abelian

group. The number of elements in a group might be ﬁnite, in which case it is

denoted by h and called the order of the group. It might also be inﬁnite, in

which case the group is called an inﬁnite group. A special class of groups are

transformation groups. We consider a set M containing, as elements, objects

x, y, .. such as e.g. vectors and numbers. A transformation g is a one-to one

map from M to M , i.e

◦

g 2 = g 2

◦

g : x

−→

a ( x )

∈

∀

∈

Of particular importance in physics are a such transformations which trans-

form a physical body into itself: such a transformation is called symmetry

transformations .

Theorem: The set G of all symmetry transformations form a group .

The group is called the symmetry group of the system.

Proof.

First, we deﬁne a composition law

◦

a means that

In general, changing the order of multiplication produces a diﬀerent result.

A group for which g 1

consisting of performing two symmetry

transformations of the system successively: by convention, b

CHAPTER 1. GROUPS

a is performed ﬁrst, followed by b . The result of performing two symmetry

transformations successively is to transform the system into itself: i.e.

•

a and b are symmetry operations, b

◦

a is also a symmetry operation

This means that the set G is closed under the law of successive transfor-

mations. We can also deﬁne an identity transformation e which leaves the

system invariant:

•

e belongs obviously to G

Given a symmetry transformation a we see that there exists an inverse trans-

formation a − 1 which transform back the system into itself:

•

a − 1 also belongs to G

Finally, successive symmetry transformations of the system obeys the asso-

ciative laws, i.e

•

◦

( b

◦

c )=( a

◦

b )

◦

QED

Exercise . Show that the following sets are groups

•

the set consisting of (1 ,

−

•

the set complex numbers (1 ,i,

−

1 ,

−

i )

•

the set of all integers Z =( ..., − 2 , − 1 , 0 , 1 , 2 , ... ) with the addition as

law of composition and the set of rational numbers

under ordinary

multiplication (but not N =1 , 2 , 3 , ... )

•

the set of real numbers

under addition or under multiplication (if

zero is excluded)

•

the set of all matrices of order m × n under addition

Symmetry transformations can be divided into three types:

1. Rotations by a certain angle about some axis and reﬂections at a certain

plane

2. translations by some vector and

3. combination of the type 1. and type 2. transformations.

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