PHY 745 Group Theory

MWF 11-11:50 AM OPL 102 http://www.wfu.edu/~natalie/s17phy745/

Instructor: Natalie Holzwarth Phone:758-5510Office:300 OPL e-mail:natalie@wfu.edu

Course schedule for Spring 2017

(Preliminary schedule -- subject to frequent adjustment.)
Lecture date
DDJ Reading
Topic
HW
Due date
1 Wed: 01/11/2017 Chap. 1 Definition and properties of groups #1 01/20/2017
2 Fri: 01/13/2017 Chap. 1 Theory of representations
Mon: 01/16/2017 MLK Holiday - no class
3 Wed: 01/18/2017 Chap. 2 Theory of representations
4 Fri: 01/20/2017 Chap. 2 Proof of the Great Orthonality Theorem #2 01/23/2017
5 Mon: 01/23/2017 Chap. 3 Notion of character of a representation #3 01/25/2017
6 Wed: 01/25/2017 Chap. 3 Examples of point groups #4 01/27/2017
7 Fri: 01/27/2017 Chap. 4 & 8 Symmetry of vibrational modes #5 01/30/2017
8 Mon: 01/30/2017 Chap. 4 & 8 Symmetry of vibrational modes #6 02/01/2017
9 Wed: 02/01/2017 Chap. 8 Vibrational excitations #7 02/03/2017
10 Fri: 02/03/2017 Notes Continuous groups #8 02/06/2017
11 Mon: 02/06/2017 Notes Group of three-dimensional rotations #9 02/08/2017
12 Wed: 02/08/2017 Notes Continuous groups #10 02/10/2017
13 Fri: 02/10/2017 Chap. 5 Atomic orbitals #11 02/13/2017
14 Mon: 02/13/2017 Chap. 6 Direct product groups #12 02/15/2017
15 Wed: 02/15/2017 Chap. 7 Molecular orbital #13 02/17/2017
16 Fri: 02/17/2017 Chap. 9 Introduction to Space Groups #14 02/20/2017
17 Mon: 02/20/2017 Chap. 10 Group theory for the periodic lattice
18 Wed: 02/22/2017 Chap. 10 Group theory for the periodic lattice
19 Fri: 02/24/2017 Chap. 1-10 Review -- Distribute take-home exam
20 Mon: 02/27/2017 Chap. 10 Space group representations Exam
21 Wed: 03/01/2017 Chap. 11 Symmetry of vibrations Exam
22 Fri: 03/03/2017 Chap. 11 Symmetry of vibrations Exam Due
Mon: 03/06/2017 Spring break - no class
Wed: 03/08/2017 Spring break - no class
Fri: 03/10/2017 Spring break - no class
Mon: 03/13/2017 APS Meeting - no class
Wed: 03/15/2017 APS Meeting - no class
Fri: 03/17/2017 APS Meeting - no class
23 Mon: 03/20/2017 Chap. 7.7 Jahn-Teller Effect #15 03/24/2017
24 Wed: 03/22/2017 Chap. 7.7 Jahn-Teller Effect
25 Fri: 03/24/2017 Spin 1/2 #16 03/27/2017
26 Mon: 03/27/2017 Dirac equation for H-like atoms #17 03/29/2017
27 Wed: 03/29/2017 Chap. 14 Angular momenta #18 03/31/2017
28 Fri: 03/31/2017 Chap. 16 Time reversal symmetry #19 04/05/2017
29 Mon: 04/03/2017 Chap. 16 Magnetic point groups
30 Wed: 04/05/2017 Literature Topology and group theory in Bloch states #20 04/07/2017
31 Fri: 04/07/2017 Introduction to Lie groups #21 04/10/2017
32 Mon: 04/10/2017 Introduction to Lie groups
33 Wed: 04/12/2017 Introduction to Lie groups
Fri: 04/14/2017 Good Friday Holiday -- no class
34 Mon: 04/17/2017 Introduction to Lie groups
35 Wed: 04/19/2017 Introduction to Lie groups
36 Fri: 04/21/2017 Introduction to Lie groups
Mon: 04/24/2017 Presentations I
Wed: 04/26/2017 Presentations II

PHY 745 -- Assignment #1

January 11, 2017

Read Chapter 1 in DDJ.

  1. Work problem 1.4 at the end of Chapter 1.
No Title PDF VERSION
January 20, 2017
PHY 745 - Problem Set #2
Finish reading Chaper 2 in Dresselhaus2 and Jorio
  1. Consider the following non-unitary representation of the P(3) group.
    Γ(E)=⎛
    ⎜
    ⎜
    ⎝
    1
    0
    0
    1
    		⎞
    ⎟
    ⎟
    ⎠     		    Γ(A)=⎛
    ⎜
    ⎜
    ⎝
    −1
    0
    0
    1
    		⎞
    ⎟
    ⎟
    ⎠     		    Γ(B)=⎛
    ⎜
    ⎜
    ⎝
    1/2
    √3/4
    √3
    −1/2
    		⎞
    ⎟
    ⎟
    ⎠     		    

    Γ(C)=⎛
    ⎜
    ⎜
    ⎝
    1/2
    −√3/4
    −√3
    −1/2
    		⎞
    ⎟
    ⎟
    ⎠     		  Γ(D)=⎛
    ⎜
    ⎜
    ⎝
    −1/2
    −√3/4
    √3
    −1/2
    		⎞
    ⎟
    ⎟
    ⎠     		  Γ(F)=⎛
    ⎜
    ⎜
    ⎝
    −1/2
    √3/4
    −√3
    −1/2
    		⎞
    ⎟
    ⎟
    ⎠     		  
    Transform this representation into a unitary representation using the procedure discussed in your textbook and the lecture notes.


File translated from TEX by TTH, version 4.01.
On 19 Jan 2017, 18:23.
No Title PDF VERSION
January 23, 2017
PHY 745 - Problem Set #3
Continue reading Chapter 3 in Dresselhaus2 and Jorio
(Note: You may wish to consult with an updated version of Lecture 2, where a typo has been corrected.)
  1. Consider a group of order 4 with the following multiplication table:
    EABC
    E EABE
    A AECB
    B B CEA
    C CBA E
    Find the classes of this group and construct a possible character table.


File translated from TEX by TTH, version 4.01.
On 21 Jan 2017, 23:41.

PHY 745 -- Assignment #4

January 25, 2017

Continue reading Chapter 3 in DDJ.

  1. Consider the following point group of order 8 based on operations on a general spacial point (x,y,z): (x,y,z), (-x,-y,z), (-y,x,z), (y,-x,z), (-x,y,-z), (x,-y,-z), (y,x,-z), (-y,-x,-z) Find the classes of this group and determine the number and dimensions of the irreducible representations.

PHY 745 -- Assignment #5

January 27, 2017

Read Chapter 4 and the beginning of Chapter 8 in DDJ.

  1. Following the analysis of the coordinate transformations discussed in class (especially on slide 10 of Lecture 7), show that the characters for the transformation matrices for the two reflections are χ(σv) =3 and χ(σv')=1.

PHY 745 -- Assignment #6

January 30, 2017

Continue reading Chapter 4 and the beginning of Chapter 8 in DDJ.

  1. Consider the motions of a NH3 molecule which has C3v symmetry. In class, we discussed its character table and the characters of the general coordinate transformations. Decompose the character of the general coordinate transformations into the irreducible representations and determine the numbers and types of irreducible representations of the molecular vibrations.

PHY 745 -- Assignment #7

February 1, 2017

Continue reading Chapter 8 in DDJ.

  1. Using the character table for the Td group, determine the possible final state symmetries for infrared excitation of a CH4 molecule.

PHY 745 -- Assignment #8

February 3, 2017

Review the lecture notes for Lecture 10.

  1. Verify the algebra on slide 8 of the lecture notes for the transformation of the l=1 spherical harmonic functions. (Extra credit for considering the corresponding transformation of the l=2 spherical harmonics.)

PHY 745 -- Assignment #9

February 6, 2017

Review the lecture notes for Lecture 11.

  1. From the character of the representation χl(α) of the three-dimensional rotation group for angular momentum l, determine the character of χl(α+2 π).

PHY 745 -- Assignment #10

February 8, 2017

Review the lecture notes for Lecture 12.

  1. Consider the example of the "multiplication" of the two rotations described on slides 5 and 6. Extend the analysis on slide 6 to third order in the spin-angular operators Ji. Note that this "Cambell-Baker-Hausdorff" formula is only correct to second order in the spin-angular operators or for special values of the computator relationship. By expanding to third order, you will get an idea of the more general expression.

PHY 745 -- Assignment #11

February 10, 2017

Start reading Chapter 5 of DDJ.

  1. Using the character table for the Td group found in appendix A of your textbook,, find the decomposition of the l=1 and l=2 spherical harmonic states into the representations of the tetrahedral group.

PHY 745 -- Assignment #12

February 13, 2017

Start reading Chapter 6 of DDJ.

  1. Consider electronic states described by the direct product point group D3d discussed on slide of 14 of the lecture notes. For each polarization of the vector potential and for each initial state representation, determine whether transitions are possible and if so what are the representation(s) of the final state.

PHY 745 -- Assignment #13

February 15, 2017

Start reading Chapter 7 of DDJ.

  1. Work problem 7.2 in your textbook.

PHY 745 -- Assignment #14

February 17, 2017

Start reading Chapter 9 of DDJ.

  1. Work out the group multiplication table for a general point xy in the 2-dimensional hexagonal space group p6 discussed in class and in your textbook.

PHY 745 -- Assignment #15

March 20, 2017

Read Section 7.7 of DDJ, and Appendix 8 of Born and Huang.

  1. Work out the details of the Born-Oppenheimer approximation using the approach of Born and Huang. Show how a degeneracy in the electronic system effects the potential energy of the nuclei.

PHY 745 -- Assignment #16

March 24, 2017

  1. Consider the three Pauli spin matrices
    1. For each of the Pauli spin matrices, find the eigenvalues and eigenvectors.
    2. For each of the Pauli spin matrices, find the unitary matrix which diagonalizes it.

PHY 745 -- Assignment #17

March 27, 2017

  1. Consider a hydrogen-like atom with one electron and a nuclear charge of Z protons. Supply the missing steps in the lecture notes to derive the ground state energy and wavefunction.

PHY 745 -- Assignment #18

March 29, 2017

  1. Consider an atom having total spin quantum number 5/2, placed into an octahedral field. Find the characters of the atomic angular momentum representations of the octahedral double group and express them in terms of the irreducible representations of the octahedral double group representations.

PHY 745 -- Assignment #19

March 31, 2017

  1. Work problem #16.4 at the end of Chap. 16 in DDJ.

PHY 745 -- Assignment #20

April 5, 2017

Read the first 5 pages of the paper by David DiVincenzo and Eugene Mele, PRB 29 1685 (1984) which analyzes a simple model of the states of two dimensional graphene near its Fermi level (taken at E=0).

  1. Starting from Eq. (7), "derive" the solutions given by equation (11).
  2. Evaluate the expression for the case U2(r) = 0.
  3. Discuss how these equations are similar/different with respect to the Dirac equation.

PHY 745 -- Assignment #21

April 7, 2017

  1. Slide 10 of the lecture notes for Lecture 31 gives some results for the 3-dimensional representation of a SU(2) group. Check the algebraic steps that lead to these results.
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Last modfied: Saturday, 07-Jan-2017 21:58:16 EST