Introduction to Group Theory - Northern Illinois University

Introduction to Group Theory

With Applications to Quantum Mechanics and Solid State Physics

Roland Winkler rwinkler@niu.edu

August 2011 (Lecture notes version: November 3, 2015)

Please, let me know if you find misprints, errors or inaccuracies in these notes. Thank you.

Roland Winkler, NIU, Argonne, and NCTU 2011-2015

General Literature

J. F. Cornwell, Group Theory in Physics (Academic, 1987) general introduction; discrete and continuous groups

W. Ludwig and C. Falter, Symmetries in Physics (Springer, Berlin, 1988). general introduction; discrete and continuous groups

W.-K. Tung, Group Theory in Physics (World Scientific, 1985). general introduction; main focus on continuous groups

L. M. Falicov, Group Theory and Its Physical Applications (University of Chicago Press, Chicago, 1966). small paperback; compact introduction

E. P. Wigner, Group Theory (Academic, 1959). classical textbook by the master

Landau and Lifshitz, Quantum Mechanics, Ch. XII (Pergamon, 1977) brief introduction into the main aspects of group theory in physics

R. McWeeny, Symmetry (Dover, 2002) elementary, self-contained introduction

and many others

Roland Winkler, NIU, Argonne, and NCTU 2011-2015

Specialized Literature

G. L. Bir und G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974) thorough discussion of group theory and its applications in solid state physics by two pioneers C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, 1972) comprehensive discussion of group theory in solid state physics G. F. Koster et al., Properties of the Thirty-Two Point Groups (MIT Press, 1963) small, but very helpful reference book tabulating the properties of the 32 crystallographic point groups (character tables, Clebsch-Gordan coefficients, compatibility relations, etc.) A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1960) comprehensive discussion of the (group) theory of angular momentum in quantum mechanics and many others

Roland Winkler, NIU, Argonne, and NCTU 2011-2015

These notes are dedicated to Prof. Dr. h.c. Ulrich R?ossler from whom I learned group theory

R.W.

Roland Winkler, NIU, Argonne, and NCTU 2011-2015

Introduction and Overview

Definition: Group

A set G = {a, b, c, . . .} is called a group, if there exists a group multiplication connecting the elements in G in the following way

(1) a, b G : c = a b G (2) a, b, c G : (ab)c = a(bc) (3) e G : a e = a a G (4) a G b G : a b = e, i.e., b a-1

(closure) (associativity) (identity / neutral element) (inverse element)

Corollaries

(a) e-1 = e (b) a-1a = a a-1 = e

a G

(c) e a = a e = a a G (d) a, b G : c = a b c-1 = b-1a-1

Commutative (Abelian) Group

(5) a, b G : a b = b a

(left inverse = right inverse) (left neutral = right neutral)

(commutatitivity)

Order of a Group = number of group elements

Roland Winkler, NIU, Argonne, and NCTU 2011-2015

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download