# 402-0516-10L Group Theory and its Applications

Semester | Spring Semester 2021 |

Lecturers | D. Pescia |

Periodicity | yearly recurring course |

Language of instruction | English |

Abstract | This lecture introduces the use of group theory to solve problems of quantum mechanics, condensed matter physics and particle physics. Symmetry is at the roots of quantum mechanics: this lecture is also a tutorial for students that would like to understand the practical side of the (often difficult) mathematical exposition of regular courses on quantum mechanics. |

Objective | The aim of this lecture is to give a practical knowledge on the application of symmetry in atomic-, molecular-, condensed matter- and particle physics. The lecture is intended for students at the master and Phd. level in Physics that would like to have a practical and comprehensive view of the role of symmetry in physics. Students in their third year of Bachelor will be perfectly able to follow the lecture and can use it for their future master curriculuum. Students from other Departements are welcome, as the lecture is designed to be (almost) self-contained. As symmetry is omnipresent in science and in particular quantum mechanics, this lecture is also a tutorial on quantum mechanics for students that would like to understand what is behind the often difficult mathematical exposition of regular courses on quantum mechanics. |

Content | 1. Abstract Group Theory and representation theory of groups (Fundamentals of groups, Groups and geometry, Point and space groups, Representation theory of groups (H. Weyl, 1885-1955), Reducible and irreducible representations , Properties of irreducible representations, Characters of a representation and theorems involving them, Symmetry adapted vectors) 2. Group theory and eigenvalue problems (General introduction and practical examples) 3. Representations of continuous groups (the circle group, The full rotation group, atomic physics, the translation group and the Schrödinger representation of quantum mechanics, Cristal field splitting, The Peter-Weyl theorem, The Stone-von Neumann theorem, The Harisch-Chandra character) 4. Space groups and their representations (Elements of crystallography, irreducible representations of the space groups, non-symmorphic space groups) 5. Topological properties of groups and half integer spins: tensor products, applications of tensor products, an introduction to the universal covering group, the universal covering group of SO3, SU(2), how to deal with the spin of the electron, Clebsch-Gordan coefficients, double point groups, the Clebsch-Gordan coefficients for point groups, the Wigner-Eckart-Koster theorem and its applications 6 The application of symmetry to phase transitions (Landau). 7. Young tableaus: many electron and particle physics (SU_3). |

Lecture notes | A manuscript is made available. |

Literature | -B.L. van der Waerden, Group Theory and Quantum Mechanics, Springer Verlag. ("Old" but still modern). - L.D. Landau, E.M. Lifshitz, Lehrbuch der Theor. Pyhsik, Band III, "Quantenmechanik", Akademie-Verlag Berlin, 1979, Kap. XII and Ibidem, Band V, "Statistische Physik", Teil 1, Akademie-Verlag 1987, Kap. XIII and XIV. (Very concise and practical) -A. Fässler, E. Stiefel, Group Theoretical Methods and Their applications, Birkhäuser. (A classical book on practical group theory, from a strong ETHZ school). - C. Isham, Lectures on group and vector spaces for physicists, World Scientific. (More mathematical but very didactical) |