MWF 12-12:50 PM | OPL 107 |
http://www.wfu.edu/~natalie/s09phy745/ |

http://www.wfu.edu/~ecarlson/groups/ |

Instructors:
Natalie Holzwarth
Eric Carlson |
Phone:758-5510 | Office:300 OPL |
e-mail:natalie@wfu.edu |

Phone:758-4994 | Office:306 OPL |
e-mail:ecarlson@wfu.edu |

- Problem Set #1 (1/15/2009)
- Problem Set #2 (1/21/2009)
- Problem Set #3 (1/23/2009)
- Problem Set #4 (1/26/2009)
- Problem Set #5 (1/28/2009)
- Problem Set #6 (1/30/2009)
- Problem Set #7 (2/02/2009)
- Problem Set #8 (2/04/2009)
- Problem Set #9 (2/06/2009)
- Problem Set #10 (2/09/2009)
- Problem Set #11 (2/11/2009)
- Problem Set #12 (2/13/2009)
- Problem Set #13 (2/16/2009)
- Problem Set #14 (2/23/2009)
- Problem Set #15 (2/25/2009)
- Problem Set #16 (2/27/2009)

January 14, 2009

Briefly skim Chapter 1 and read Chapter 2 in **Tinkham**.
This problem has 6 parts and covers material from Lectures 1 & 2. It
is due Wed. Jan. 21, 2009.

- Tinkham Problem #2-1.

January 21, 2009

Continue reading Chapter 3 in **Tinkham**. For the following matrices ,
find the similar transformation which creates the related diagonal matrix
:

choosing to be

- In this example, represents a real number.

- In this example, represents a real number.

- In this example, you may wish to ask Maple to help.

January 23, 2009

Continue reading Chapter 3 in **Tinkham**.

- Consider the following ``normal" matrix
in terms of
real constants , , , and .

- Find the eigenvalues and eigenvectors

- Show that

- Find the relationships between the constants for the case that N is Hermitian.
- Find the relationships between the constants for the case that N is unitary.

- Find the eigenvalues and eigenvectors

January 22, 2009

Continue reading Chapter 3 in **Tinkham**.

- On page 8 of
**Tinkham**, you will find an example of 2-dimension representation of the triangular group described by Fig. 2-1 and the multiplication table in the previous page. Consider the following alternative 2-dimension representation:

- Show that this alternative representation satisfies the group multiplication table.
- If the alternative representation is not unitary, use the procedure described in Section 3-2 of your text to transform it into a unitary transformation.

## PHY 745 -- Assignment #5

January 28, 2009Continue reading Chapter 3 in

**Tinkham**. This problem is due Fri. Jan. 30, 2009.- Tinkham Problem #3-1. (To save time, you may wish to consult the back of your textbook and verify that the appropriate character table satisfies the necessary relations.)

## PHY 745 -- Assignment #6

January 30, 2009Finish reading Chapter 3 in

**Tinkham**. This problem is due Mon. Feb. 2, 2009.- Find the Character table for a group "
*i*" composed of the identity*E*and inversion*i*( x,y,z -> -x,-y,-z ). - Using the results from HW 5, find the Character table for the
direct product group D
_{4}×*i*.

hw7 February 2, 2009**PHY 745 - Problem Set #7**Finish reading Chapter 3 and start Chapter 4 in

**Tinkham**.- Consider the following 3-dimensional transformation matrix

(1) - Find Euler angles , , and that correspond to that transformation (with or without inversion).
- Consider the transformation of the spherical harmonic functions, using your Euler angles and Eq. 5-36 of your text.
- Check that

(2)

## PHY 745 -- Assignment #8

February 4, 2009Start reading Chapter 4 in

**Tinkham**. This problem is due Fri. Feb. 6, 2009.A perfect cube has the following 48 symmetry elements designated as transformations on a general point with cartesian coordinates x,y,z with respect the origin at the center of the cube.

- x,y,z
- -x,y,z
- x,-y,z
- x,y,-z
- -x,-y,z
- -x,y,-z
- x,-y,-z
- -x,-y,-z
- y,x,z
- -y,x,z
- y,-x,z
- y,x,-z
- -y,-x,z
- -y,x,-z
- y,-x,-z
- -y,-x,-z
- x,z,y
- -x,z,y
- x,-z,y
- x,z,-y
- -x,-z,y
- -x,z,-y
- x,-z,-y
- -x,-z,-y
- z,y,x
- -z,y,x
- z,-y,x
- z,y,-x
- -z,-y,x
- -z,y,-x
- z,-y,-x
- -z,-y,-x
- y,z,x
- -y,z,x
- y,-z,x
- y,z,-x
- -y,-z,x
- -y,z,-x
- y,-z,-x
- -y,-z,-x
- z,x,y
- -z,x,y
- z,-x,y
- z,x,-y
- -z,-x,y
- -z,x,-y
- z,-x,-y
- -z,-x,-y

_{h}group on page 329 of your text.

## PHY 745 -- Assignment #9

February 6, 2009Continue reading Chapter 4 in

**Tinkham**. This problem is due Mon. Feb. 9, 2009.-
The figure shows an object which could have D
_{3}(32) symmetry. If the red and green balls were identical, the object would have D_{6}(622) symmetry. On page 327 of your text, you will find character tables for the D_{6}(622) and D_{3}(32) groups.- Check whether D
_{3}(32) is a subgroup of D_{6}(622). - Find the compatability relations between the representations of
D
_{3}(32) and of D_{6}(622). (Please define a reasonable notation to distinguish the two group representations.)

- Check whether D

hw10 February 8, 2009**PHY 745 - Problem Set #10**Continue reading Chapter 4 in

**Tinkham**.- Consider a quantum mechanical free particle of mass confined within
a rectangular box of dimensions
,
, and
.
- Check that
the eigenstates of the particle all vanish on the 6 planes
that bound the box and take the form:

where

- Now consider these states with reference to the point group discussed in your text book and in the Notes for Lecture 9 - http://www.wfu.edu/~natalie/s09phy745/lecturenote/ . From the character table for this point group, for each irreducible representation, find at least one example of a basis function from the eigenstates.

- Check that
the eigenstates of the particle all vanish on the 6 planes
that bound the box and take the form:

## PHY 745 -- Assignment #11

February 11, 2009Continue reading Chapter 4 in

**Tinkham**. This problem is due Fri. Feb. 13, 2009.-
An Nd
^{+2}ion in its ground state in a spherical environment has total orbital angular momentum L=6. Suppose that this ion is introduced into a crystal at a site that has O_{h}symmetry. What are the compabible irreducible representations of the ion?

Homework Set 1 - Using the Web Physics 745 - Group Theory

Homework Set 12 Due Monday, February 16

Note: the lecture and homework set was prepared by Prof. Carlson.

1. Diamond is a version of carbon. The position of the carbon atoms takes the form

_{}where

*d*= 356.683 pm,_{}are arbitrary integers, and_{}takes on the following eight values:_{}Thus there are eight carbon atoms per cell of size

*d*^{3}.(a) For what values of

_{}will_{}be a translation vector;*i.e.*, if there is a carbon atom at**r**, there will always be a carbon atom at_{}? To make your answer finite, only include values with_{}.(b) Find primitive vectors

**a**,**b**and**c**such that*all*translation vectors take the form_{}, where_{}are integers. Demonstrate it explicitly for those vectors you found in part (a) (which will probably be trivial), and also for the three vectors_{},_{}and_{}.(c) What are the lengths of these vectors

_{}and the angles between them,_{}?

## PHY 745 -- Assignment #13

February 16, 2009Continue reading Chapter 4 in

**Tinkham**. This problem is due Wed. Feb. 18, 2009.-
Consider a monoclinic lattice in the "first setting" with non-orthogonal angle
γ. The conventional translation vectors are given by

**a**= a**x**

**b**= b ( cos γ**x**+ sin γ**y**)

**c**= c**z**

and the lattice positions are

**τ**_{1}= 0

**τ**_{2}= ½**a**+ ½**c**.

- Show that the same lattice can be generated using the primitive lattice
translations

**T**_{1}= ½**a**+ ½**c**

**T**_{2}=**b**

**T**_{3}= ½**a**- ½**c**

- What are the volumes of the conventional and primitive unit cells?

- Show that the same lattice can be generated using the primitive lattice
translations

## PHY 745 -- Assignment #14

February 23, 2009Continue reading Chapters 4 and 8 in

**Tinkham**. This problem is due Wed. Feb. 25, 2009.Consider the following two examples of crystals which have the fcc (O

_{h}^{5}) structure. For each, find an expression for their structure factors S(Δk). For evaluating the atomic structure factors, you may use the expressionF where the index "a" and the related parameters will depend on the kind of atom. For both crystals, the primitive lattice and reciprocal lattice vectors are given by_{a}(Δk) = Z_{a}exp(-|Δk|^{2}&Lambda_{a}^{2}/4),Real space lattice Reciprocal space lattica **T**_{1}=(a/2)*(**x**+**y**)**G**_{1}=(2π/a)*(**x**+**y**-**z**)**T**_{2}=(a/2)*(**x**+**z**)**G**_{2}=(2π/a)*(**x**-**y**+**z**)**T**_{3}=(a/2)*(**y**+**z**)**G**_{3}=(2π/a)*(-**x**+**y**+**z**)- Consider the NaCl lattice where within the reference cell a Na is located
at
**σ**_{Na}=0 and a Cl is located at**σ**_{Cl}=(a/2)*(**x**+**y**+**z**). - Consider the diamond lattice where within the reference cell
one C is located
at
**σ**_{C1}=(a/8)*(**x**+**y**+**z**) and the other is located at**σ**_{C2}=-(a/8)*(**x**+**y**+**z**).

## PHY 745 -- Assignment #15

February 25, 2009Continue reading Chapters 4 and 8 in

**Tinkham**. This problem is due Fri. Feb. 27, 2009.-
Consider the above diagram of the Brillouin Zone of a cube taken from the
paper of
Bouckaert, Smoluchowski, and Wigner (Phys. Rev.
**50**, 58 (1936) The Γ point has the point group symmetry ({R|0}) of cubic space group -- ({R|t}) -- O_{h}.- For at least 3 lines within the Brillouin Zone (such as Σ, Δ, &Lambda) find the point group symmetries which describe Bloch states of the crystal with those wave vectors.
- For at least 3 points within the Brillouin Zone (such as X, M, R) find the point group symmetries which describe Bloch states of the crystal with those wave vectors.

## PHY 745 -- Assignment #16

February 27, 2009Continue reading Chapters 4 and 8 in

**Tinkham**. This problem is due Mon. Mar. 2, 2009.-
Consider a molecule C
_{3}which takes the form of an equilateral triangle. (I am not sure such a molecule is stable...) In equilibrium this molecule has D_{3h}symmetry. Since there are 3 atoms, there are 3 × 3 = 9 normal modes of vibration, 3 of which have non-zero frequency. (The other 6 modes involve uniform translation or rotation.)- Using the D
_{3h}character table on page 328 of your text, find the number of times each irreducible representation characterizes a general vibrational mode. - Determine which of these irreducible representations correspond to the non-trivial (non zero-frequency) modes.

- Using the D

Last modfied: Friday, 27-Feb-2009 09:07:25 EST