Group Theory for Physicists, lecture & calculations (BMTE11AF40, 2020 fall)

Lectures are given and calculation classes are lead by Titusz Fehér (Department of Physics, tif (at) this.server).

First written test: 7th week of semester, 2019 October 25, 14:00, Room CHA11

Second written test: 14th week of semester, 2019 December 12, 10:00-11:30, Room E204. See results.

Retake tests (you may try only the ones you have not sat or you failed before): 2019 December 19. Both tests at the same time: 10:00–11:30, Room K376.

Please remember to sign up by e-mail for the retake tests until 15th December!

Optional consulting for retake tests and/or exam: 17 Dec, 11:00. Please sign up by e-mail until 16th December!

Requirements

• To get the calculations course signed, one has to attend at least 70% classes, and get at least 40% at both tests.
• Those attaining the three best cumulative scores on the tests, can get a proposed mark without an oral exam, or can improve the proposed mark on a simplified exam. The proposed mark is 4 for the first, and 3 for the next two students. (Repeated tests will not contribute to the score.)
• Pass oral exam based on the exam questions. (Minor modifications are to be expected following the course of the current semester. Please do not forget to download it again a few days before the exam!)
• Oral exam details: Everyone gets a problem similar to those in calculation test #2, and will also be given to exam questions. The solution will have to be explained to me first, then the questions. The mark will be some average of the three presentations. The questions are given randomly with a few constraints (max. one can come from the Q1.–6 range, and also max. one lengthy derivation/proof will be asked, in which we agree in advance).
• Consultations: about anything based and any time we agree in advance. Try to come in teams, especially during "peak periods" (e.g. before tests).

Detailed topics, extras

#1 lecture (2019 September 11)

• Rules of the game.
• The fundamental role of group theory in physics.
• Symmetries in physics and their consequences on equations (of motion) and solutions.
• Definition of the group, fundamental properties, some examples.

#1 calculations (2019 September 13)

• Integer powers of elements.
• Order of a group and group elements.
• Operations on "complexes" (i.e. on subsets in groups).
• Multiplication table of finite groups.

#2 lecture (2019 September 18)

• Subgroup, cosets, examples.
• Lagrange's theorem, and trivial consequences.
• Normal subgroups and properties. Example: C_3v.
• Conjugation, exponential notation, and properties.

#2 calculation (2019 September 20)

• Normal subgroups revisited.
• Quotient group (N⊲G →G/N), and properties.
• Homo-, iso-, endo- and automorphism. Examples.
• Properties of homomorphisms.
• Direct product of groups (the direct product group).
• Multiplication table and subgroups of C_3v. (Problem 9.3)
• Homomorphism theorem (and its proof).
• Natural homomorphism (into a quotient group). (Properties, identity element. Injective IFF N={e}.)

#3 lecture (2019 September 25)

• Revisited: Direct product of groups. Relation to quotient group: (G₁×G₂)/G₁≅G₂ and (G₁×G₂)/G₂≅G₁ but (G/N)×N does not (in general) restore G.
• Equivalence relation and equivalence classes. (Cosets as equivalence classes.)
• Conjugate equivalents. Conjugacy classes: their properties and intuitive meaning. Examples: C3v.
• Comparison of cosets and conjugacy classes.

#3 calculations (2019 September 27)

• Conjugation: intuitive meaning for permutations, C3v.
• Permutation group, disjoint cycles, parity. (Problem 9.4.)
• Group action (definition only).

#4 lecture (2019 October 2)

— (cancelled due to "Dean's day")

#4 calculations (2019 October 4)

• What rotations are compatible with 2D lattice symmetry? (Problem 10.6)
• Point groups in 3D (i.e. subgroups of O(3)).
• Schoenflies notation of point groups in 3D and their elements.
• A great web page to practice to determine the point group of molecules.
• A flow chart to determine the point group of an object.
• Familiarizing with point groups: Problems 10.1–5.

#5 lecture (2019 October 9)

• Revision: using flowcharts to determine 3D point group of an object.
• Problems 10.1–3.
• Revisit: group action. Stabilizer, orbits. A useful page in Hungarian.
• Orbit–stabilizer theorem. Examples.
• Group multiplication and conjugation as group actions.

#5 calculations (2019 October 11)

• Revisit group action, stabilizer, orbits and their properties. Examples, incl. D_6h and C_4v.
• Problem 10.4–5.
• Problems for practicing:
  • Liboff-01
  • Jones-01
  • Jones-02

#6 lecture (2019 October 16)

• Problem 11.1.
• Problem 2.2
• On the level of story-telling: subnormal/composition series, Jordan–Hölder theorem. Simple groups and (the history of) their classification. The Monster group and its relation to physics.
• Representations, and related basic definitions: faithful, equivalent.
• Characters of representations.

#6 calculations (2019 October 18)

• Representations and characters, further definitions: (ir)reducible, (in)decomposable, fully decomposable, unitary representations.
• Every finite dimensional unitary reducible representation is decomposable.
• Maschke's theorem: every finite representation of a finite group is unitary-equivalent. Corollary: every finite representation of finite groups is equivalent to the direct sum of irreducible representations.
• Examples for representations of groups, their transformations and decomposition of representations.

Test #1 (Week #7, 2019 October 25, 14:00, room CHA11)

1. Group axioms.
2. Subgroups, cosets (Lagrange's theorem).
3. Normal subgroup and properties.
4. Quotient group, direct product of groups. Their relation.
5. Homomorphism, isomorphism, endomorphism, automorphism.
6. Properties of homomorphisms, homomorphism theorem.
7. Conjugation and its properties. Conjugacy classes.
8. Permutation group and operations, disjoint cycles.
9. Lagrange's theorem.
10. Point groups, and their identification (using a flow chart).
11. Group action, orbit, stabilizer. Orbit-stabilizer theorem.

You will have to reproduce definitions, statements and theorems, and you will have to apply them to solve problems but proofs/derivations will not be asked. To identify point groups, you can use one or all the three flow charts linked above. You may also use the hand-out with the table "Correspondance between different notations of point groups" and the graph "Subgroup relations of the 32 crystallographic point groups".

Evaluation will be weighted as theory:application = 40%:60%.

You have to collect at least 40% to pass. The 3 best students (based on the total points collected in the two test) will enjoy some benefits during the final exam.

Results of test one.

#8 lecture (2019 October 30)

• The regular representation.
• Grand/Fundamantal orthogonality theorem (i.e. Schur orthogonality relations).
• The Φ_G space, the C_Gcentral space.
• A long list of consequences of GOT, including orthonormality of irrep characters.

#8 calculations (2019 November 1)

— (cancelled due to National holiday)

#9 lecture (2019 November 6)

• Continued: A long list of consequences of GOT, including orthonormality of irrep characters, number of irreps and their dimensions, characters as "fingerprints".
• Simple physical example of a representation and its decomposition: three bodies on a ring with springs.

#9 calculations (2019 November 8)

• Example: decomposition of the regular representation.
• Completeness of irrep mátrix elements. Orthogonality of irrep matrix elements "in the other direction".
• Completeness of irrep characters in the central space. Orthogonality of irrep characters "in the other direction".

#10 lecture (2019 November 13)

• GOT, and its manifestation in character tables: character tables and their properties.
• Warning: the character table does not define the group! (Not even up to an isomorphism, see D₄ vs Q!)
• Character tables of point groups, Mulliken symbols.
• Interactive character tables by Gernot Katzer and a collection by György Kriza.
• Real-life physics examples: representations, their decomposition.
• Projection into irrep subspaces: the conventional and a less conventional formula.
• The representation generated (induced) by a group action on a linear space.
• The consequencies of symmetries of a mechanical system on its properties.
• Small vibrations: relation between the point group and normal modes.
• Detailed discussion of the one-dimensional problem of three masses connected by springs: |~~k~~•~~K~~⬤~~K~~•~~k~~|.
• Consequences of commuting time development and symmetry operations.

#10 calculations (2019 November 15)

• Application of group theory on a non-trivial system: five point masses connected by springs, of C_4v symmetry: equation of motion, and the induced 10-dimensional representation.
• More coming soon...
• The Hermann–Mauguin ("international") notation of point groups and symmetries. Determine the point group of an object with Hermann–Mauguin notation (WARNING! This flowchart is faulty!): G. L. Breneman, J. Chem. Educ. 64 (3), p 216 (1987): Crystallographic symmetry point group notation flow chart.
• This Wikipedia page compares the two most used notation of point groups.

#11 lecture (2019 November 20)

#11 calculations (2019 November 22)

#12 lecture (2019 November 27)

#12 calculations (2019 November 29)

— (cancelled due to "Open university")

#13 lecture (2019 December 4)

#13 calculations (2019 December 6)

#13 calculations-extra (2019 December 7)

#14 lecture (2019 December 11)

#14 calculations (2019 December 13)

Test #2 (2019 December 12, 10:00–11:30, room E204)

1. Representations, their classification. Decomposition of representations (def).
2. Schur's lemmas.
3. Grand/Fundamantal orthogonality theorem. A Φ space. Completeness of irrep mátrix elements.
4. Character of representations, and its properties. Central space. Orthonormality of irrep characters.
5. Completeness of irrep characters. Orthogonality of irrep characters "in the other direction" (i.e. "vertically").
6. Character tables and their properties. Mulliken symbols.
7. Reduction of characters/representations (method).
8. Consequence of symmetries of a (classical) harmonic mechanical system on the normal modes. Relation between irrep and normal modes.
9. Projection into irrep subspaces by characters. Projection into irrep subspaces by irrep matrix elements. How these methods compare?
10. Effect of symmetry breaking on normal modes.
11. Product representation, its character and decomposition. Multiplication table of irreps.
12. Application of group theory in quantum mechanical eigenstate problems.
13. Application of group theory for degenerate and non-degenerate perturbation theory.
14. Selection rules.
15. Neumann's principle.

Evaluation

Evaluation will be weighted as theory:application = 40%:60%. Application means solving problems similar to those in your course notes.

You have to collect at least 40% to pass. The 3 best students (based on the total points collected in the two test) will enjoy some benefits during the final exam.

Preparation

Besides your course notes, you may want to check Section 4 in Jones' book, which covers our most important topics. Sec. 5.2 and 5.3 show, from a slightly different point of view, similar physical problems we visited during the course, and similar to those you may see in the test. You will find relevant problems at the end of both sections, but those that were not part of this course will not be in the test, either.

You may also check Burns' book (see Suggested Reading), you may have a look at problems at the end of sections 3, 5 and 6, and you may find problems 7.1, 7.2 and 7.4 useful as well.

Results

See detailed results.

Note: Since noone was able to cope with QM problems, one of the fundamental aims of this course, I cannot let you pass with a 5 without it.

Suggested Reading

1. H. F. Jones: Groups, Representations and Physics (IOP Publishing, 1998)
2. R. L. Liboff: Primer for Point and Space Groups (Springer, 2003).
3. M.S. Dresselhaus, G. Dresselhaus, A. Jorio: Group Theory – Application to the Physics of Condensed Matter (Springer, 2008).
4. G. Burns: Introduction to Group Theory with Applications (Academic Press, 1977, ISBN 0-12-145750-8).

Downloads

• Oral exam questions in English: Exam questions (v2009104) (2nd page in Hungarian).
• Interactive character tables by Gernot Katzer.
• M. S. Dresselhaus: Application of Group Theory to the Phsyics of Solids.
• Only in Hungarian: Csoportelméleti alapfogalmak (helyi másolat)
• Only in Hungarian: A kvantumgenerátorról és e pillanatban innen is letölthető az általam ismeretlen szerzőtől származó Bevezetés a csoportelmélet alapjaiba fizikus hallgatóknak (H.F. Jones nyomán) c. magyar nyelvű jegyzet. (Figyelem, vannak benne hibák, pl. a 11. oldalon *"ha Im f⊂B a leképezés injektív".)
• György Kriza's page for his group theory course contains a nice collection of supplementary documents. We will use his character tables in this course. Kriza György csoportelmélet specijéhez tartozó kiegészítő anyagok. (Egy-két módosítással az itt található karaktertáblákat használjuk az órán.)