Group Theory in Solid State Physics
Delve into the world of group theory in solid state physics with topics like group multiplication tables, representations of groups, and the great orthogonality theorem. Explore concepts such as subgroup properties, unit elements, and abstract group theory. Discover the importance of representations in describing symmetry properties and examples of one and two-dimensional representations.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 3: Reading: Chapter 1 & 2 in MPM; Continued brief introduction to group theory 1. Group multiplication tables 2. Representations of groups 3. The great orthogonality theorem 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 1
1/21/2015 PHY 752 Spring 2015 -- Lecture 3 2
1/21/2015 PHY 752 Spring 2015 -- Lecture 3 3
Short digression on abstract group theory What is group theory ? 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 4
Example of a 6-member group E,A,B,C,D,F,G 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 5
Check on group properties: 1. Closed; multiplication table uniquely generates group members. 2. Unit element included. 3. Each element has inverse. 4. Multiplication process is associative. Definitions Subgroup: members of larger group which have the property of a group Class: members of a group which are generated by the construction wh ere i i X YX C = 1 and are group eleme Y n t s X i 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 6
Group theory some comments The elements of the group may be abstract; in general, we will use them to describe symmetry properties of our system Representations of a group A representation of a group is a for each group element) -- set of matrices (on e ( ), ( )... that satisfies A B the multip of the ma representation. lication table of the group. The trices is called the dimension dimension of the 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 7
Example: Note that the one-dimensional "identical representa ( ) ( ) ( ) A B C = = = Another one-dimensional representation is ( ) ( ) ( ) ( ) E D = = tion" = = = 1 1 1 1 1 1 ( ) D ( ) E ( ) F 1 is always possible = = = = 2 2 2 ( ) ( ) F 1 A B C 2 2 2 1 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 8
Example: A two-dimensional representation is 1 0 0 1 = = 1 0 0 ( ) E ( ) A = = 3 3 1 3 3 1 2 1 2 ( ) B ( ) C = 2 2 3 3 3 3 1 2 1 2 2 2 3 3 1 2 1 2 ( ) D ( ) F = 2 2 3 3 3 3 1 2 1 2 2 2 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 9
What about 3 or 4 dimensional representations for this group? 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 ( ) E ( ) A ( ) B = = = 3 0 1 2 2 3 0 1 2 2 1 0 0 1 0 0 1 0 0 ( ) C ( ) D ( ) F = = = 3 3 3 0 0 0 1 2 1 2 1 2 2 2 2 3 3 3 0 0 0 1 2 1 2 1 2 2 2 2 The only irreducible representations for this group are 2 one-dimensional and 1 two-dimensional 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 10
Comment about representation matrices A representation is not fundamentally altered by a similarity transformatio = n 1 ( ) A ( ) A S S Check: ( = Typically, unitary matrices are chosen for representations Typically representations are reduced to block diagonal form and the irreducible blocks are considered in the representation theory = = 1 1 ) ( ) ( ) ( ) ( ) A SS A AB S AB S S A B S 1 1 ( ) B S S = ( ) ( ) B 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 11
The great orthogonality theorem Notation: R order of the group element of the group ( ) th representation of R i h i R denote matrix i ndices ension of the representation dim l i h ( ) * = i j ( ) R ( ) R ij l R i 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 12
Great orthogonality theorem continued ( ( ) R h ) * = i j ( ) R R ij l i Analysis shows that = 2 i l h i 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 13
Simplified analysis in terms of the characters of the representations jl j j R = ( ) ( ) R 1 Character orthogonality theorem ( R ) * = i j ( ) R ( ) R h ij Note that all members of a class have the same character for any given representation i. 1/21/2015 PHY 752 Spring 2015 -- Lecture 3 14
