Review of topics in group theory

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Presentation transcript:

Review of topics in group theory PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 19: Review of topics in group theory Chapters 1-10 in DDJ General concepts and definitions in group theory Representations of groups; great orthogonality theorem Point groups; space groups 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

2/24/2017 PHY 745 Spring 2017 -- Lecture 19

An abstract algebraic construction in mathematics Group theory An abstract algebraic construction in mathematics Definition of a group: 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Class members of a group generated by the conjugate construction Some definitions: Order of the group  number of elements (members) in the group (positive integer for finite group, ∞ for infinite group) Subgroup  collection of elements within a group which by themselves form a group Coset  Given a subgroup gi of a group a right coset can be formed by multiply an element of g with each element of gi Class members of a group generated by the conjugate construction 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Example of a 6-member group E,A,B,C,D,F,G 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

For our example: 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Representations of a group 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Example: 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

The great orthogonality theorem on unitary irreducible representations 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Great orthogonality theorem for characters 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Character table for P(3): 1 c2 -1 c3 2 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

The number of characters=the number of classes The characters ci behave as a vector space with the dimension equal to the number of classes. The number of characters=the number of classes 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

“Standard” notation for representations of C2v Example of H2O sv sv’ “Standard” notation for representations of C2v 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Symmetry analysis Dy1 Dy2 Dy3 Dx1 Dy1 Dz1 Dx2 R Dy2 Dz2 Dx3 Dy3 Dz3 C2 sv sv’ 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Dx1 1 0 0 0 0 0 0 0 0 Dx1 Dy1 0 1 0 0 0 0 0 0 0 Dy1 Dz1 0 0 1 0 0 0 0 0 0 Dz1 Dx2 0 0 0 1 0 0 0 0 0 Dx2 E Dy2 = 0 0 0 0 1 0 0 0 0 Dy2 Dz2 0 0 0 0 0 1 0 0 0 Dz2 Dx3 0 0 0 0 0 0 1 0 0 Dx3 Dy3 0 0 0 0 0 0 0 1 0 Dy3 Dz3 0 0 0 0 0 0 0 0 1 Dz3 c(E)=9 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Dx1 -1 0 0 0 0 0 0 0 0 Dx1 Dy1 0 1 0 0 0 0 0 0 0 Dy1 Dz1 0 0 -1 0 0 0 0 0 0 Dz1 Dx2 0 0 0 0 0 0 -1 0 0 Dx2 C2 Dy2 = 0 0 0 0 0 0 0 1 0 Dy2 Dz2 0 0 0 0 0 0 0 0 -1 Dz2 Dx3 0 0 0 -1 0 0 0 0 0 Dx3 Dy3 0 0 0 0 1 0 0 0 0 Dy3 Dz3 0 0 0 0 0 -1 0 0 0 Dz3 c(C2)=-1 Similarly: c(sv)=3 c(sv’)=1 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

2/24/2017 PHY 745 Spring 2017 -- Lecture 19

2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Some properties of continuous group SO(3) 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Irreducible representations in terms of angular momentum eigenfunctions 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Group of all unitary matrices of dimension 2 – SU(2) 2/24/2017 PHY 745 Spring 2017 -- Lecture 19

Group of all unitary matrices of dimension 2 – SU(2) --continued 2/24/2017 PHY 745 Spring 2017 -- Lecture 19