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In this course, students become familiar with basic notions and results of the group theory and the representation
theory for both finite and continuous (Lie) groups and learn how to use them to solve problems in physics. For the
1st and 2nd year of the TF and JSF studies.
Last update: Podolský Jiří, prof. RNDr., CSc., DSc. (29.04.2019)
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The condition for granting the credit is the fulfillment of take-home problems. This credit is not a condition for participation in the exam. Last update: Houfek Karel, doc. RNDr., Ph.D. (18.02.2022)
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Cornwell J. F.: Group Theory in Physics, Volumes I and II (Academic Press, London 1984)
Morton Hamermesh: Group Theory and Its Application to Physical Problems, Dover Publications, 1989
Shlomo Sternberg: Group theory and physics, Cambridge University Press, Cambridge 1994
Otto Litzman, Milan Sekanina: Užití grup ve fyzice, Academia, Praha 1982
Ma, Z.-Qi: Group Theory for Physicists (World Scientific, New Jersey 2007)
Marián Fecko: Diferenciálna geometria a Lieove grupy pre fyzikov, IRIS, Bratislava 2004, chapt. 10-12
Isham, C. J.: Modern Differential Geometry for Physicists, 2nd Ed. (World Scientific, Singapore 1999) Last update: Kolorenč Přemysl, doc. RNDr., Ph.D. (28.09.2021)
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The exam is oral. The requirements correspond to the syllabus of the course to the extent that was presented at the lectures. Last update: Houfek Karel, doc. RNDr., Ph.D. (18.02.2022)
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Fundamentals of the theory of finite and Lie groups
Groups and their subgroups (basic properties and theorems), group homomorphism and isomorphism, group action on a set, Lie groups and its algebra (geometrical and matrix approach), one-parameter subgroups of the Lie group and exponential map, summary of matrix groups and their properties (double cover of SO(3) by SU(2)) Fundamentals of the representation theory of groups Representation as a group action on linear spaces, invariant subspaces, equivalent, unitary, irreducible, and (completely) reducible representations, basic theorems for finite and compact Lie groups (Schur's lemma, orthogonality relations, characters and their properties, Peter-Weyl theorem, Casimir operators, Racah theorem), summary of results of the representation theory of the symmetric group and the group SU(n) Applications in quantum theory Classification of eigenvalues and eigenstates of an operator by irreducible representations of a symmetry group, coupled systems and decomposition of reducible representations (Clebch-Gordan series and coefficients), evaluation of matrix elements using group-theoretical methods (irreducible tensor operators, general Wigner-Eckart theorem, selection rules)
All notions and theorems will be illustrated by examples of point groups (which describe molecular and crystal symmetries and which play important role in quantum chemistry, molecular spectroscopy and solid state physics) and selected Lie groups such as SO(3), SU(2), and SU(3) (which are important in atomic, nuclear and particle physics).
Previous knowledge of groups is not assumed, but working knowledge of linear algebra and basic quantum mechanics is necessary for application. Last update: Kolorenč Přemysl, doc. RNDr., Ph.D. (30.09.2019)
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