1. 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
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PHY 745 Spring Lecture 19

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PHY 745 Spring Lecture 19

3. An abstract algebraic construction in mathematics
Group theory
An abstract algebraic construction in mathematics
Definition of a group:
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PHY 745 Spring Lecture 19

4. 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
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PHY 745 Spring Lecture 19

5. Example of a 6-member group E,A,B,C,D,F,G
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PHY 745 Spring Lecture 19

6. For our example:
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PHY 745 Spring Lecture 19

7. Representations of a group
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PHY 745 Spring Lecture 19

8. Example:
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PHY 745 Spring Lecture 19

9. The great orthogonality theorem on unitary irreducible representations
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PHY 745 Spring Lecture 19

10. Great orthogonality theorem for characters
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PHY 745 Spring Lecture 19

11. Character table for P(3):1 c2 -1 c3 2
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PHY 745 Spring Lecture 19

12. 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
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PHY 745 Spring Lecture 19

13. “Standard” notation for representations of C2v
Example of H2O sv
sv’
“Standard” notation for representations of C2v
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PHY 745 Spring Lecture 19

14. Symmetry analysis Dy1 Dy2 Dy3 Dx1 Dy1 Dz1 Dx2 R Dy2 Dz2 Dx3 Dy3 Dz3C2 sv
sv’
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PHY 745 Spring Lecture 19

15. Dx Dx1
Dy Dy1
Dz Dz1
Dx Dx2
E Dy2 = Dy2
Dz Dz2
Dx Dx3
Dy Dy3
Dz Dz3
c(E)=9
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PHY 745 Spring Lecture 19

16. Dx Dx1
Dy Dy1
Dz Dz1
Dx Dx2
C2 Dy2 = Dy2
Dz Dz2
Dx Dx3
Dy Dy3
Dz Dz3
c(C2)=-1
Similarly: c(sv)=3
c(sv’)=1
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PHY 745 Spring Lecture 19

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PHY 745 Spring Lecture 19

19. Some properties of continuous group SO(3)
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PHY 745 Spring Lecture 19

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21. Irreducible representations in terms of angular momentum eigenfunctions
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PHY 745 Spring Lecture 19

22. Group of all unitary matrices of dimension 2 – SU(2)
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PHY 745 Spring Lecture 19

23. Group of all unitary matrices of dimension 2 – SU(2) --continued
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PHY 745 Spring Lecture 19