BMETE11AF40

Course data
Course name: Group Theory for Physicists
Neptun ID: BMETE11AF40
Responsible teacher: Titusz Fehér
Department: Department of Physics
Programme: BSc Physics
Course data sheet: BMETE11AF40
Requirements, Informations

Further information

Lectures are given on Wednesdays between 12:15 and 13:45, calculations are on Fridays between 10:15 and 11:45, both online until week #2. Afterwards we probably meet on the campus. COVID-special: Those who cannot come to the campus, will get the material in MS Teams to follow the course

Lectures are given and practicals are lead by Titusz Fehér (Department of Physics, tif (at) esr.phy.bme... ).

 

First written test: 6th week of semester.

Second written test: 14th week of semester.

Retake tests: on demand. (You may try only the ones you have not sat or you failed before and both will be organized on the same day.Please remember to sign up by e-mail for the retake tests at least two days in advance, indicating whether you intend to take the first, the second one or both of them!

Requirements

  • To get the calculations course signed, you have to submit at least 70% of the assignments, and get at least 40% at both tests.
  • The TOP3 of the tests (cumulative points, but maybe only that of the 2nd, depending on covid situation), 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, tételsor. (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 1-3 problems similar to those in test #2, and will also be given two 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

Test #1 of the 2019/fall semester

Topics: fundamental group definitions and properties.

  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.

 

Test #2 of the 2019/fall semester

Topic: linear representation theory of finite groups and its applications.

  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.

You have to be able to state and apply definitions, lemmas, theorems etc. but reproducing their proofs will not be asked. During the test you may not use anything besides the following hand-outs: character tables, and flow charts to identify point groups. Good luck.

 

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.

Offered benefits for the results on the written tests
Name Exam benefint
?? Mark 4 offered, for 5: I will ask only 1-2 questions selected in advance (please contact me by e-mail for details).
?? 3 offered, for 4 or 5: you will have to solve a QM problem on the exam + answer exam questions.

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).

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