Lectures

WS 22/23

Symmetries and the many-body problem RWTH Aachen & GRS Symmetries reveal essential properties of phyiscal systems. This lecture will cover the basics of group theory and some of its application to the central problem of solid-state physics, the many-electron problem. E-mail: e.pavarini@fz-juelich.de Language: English Time and Place: Tuesdays, from 14:15 to 16:45, MBP1 015 Exam: Oral, 7 February 2022, usual room Admission: self evauation. If you can solve the exercises you are ready for the exam. Literature: M.S. Dresselhaus, G.Dresselhaus, A.Jorio, Group Theory - Application to the Physics of Condensed Matter M. Tinkham Group Theory and Quantum Mechanics E. Pavarini, in the Lecture Notes of the Autumn School on Correlated Electrons: from Models to Materials download

LECTURE CONTENT

• introduction to group theory: definitions, abelian groups, Cayley tables, generators, homomorphism, isomorphism, classes
• symmetries and conservation laws, symmetries and degeneracies. Transofrmations of functions and operators. Lie groups. Reducible and irreducible representations.
• Orthogonality theorems. Characters. Character tables.
• the hydrogen atom and its hidden symmetry
• from atoms to solid-state systems
• double groups, spinors and spinor representations
• band topology
• time-reversal and Kramers degeneracy
• the N-electron problem and the symmetric group
• many-electron states and multiplets
• tensor opertators and the Wigner-Eckart theorem

LECTURE MATERIAL WILL BE ADDED DURING THE SEMESTER

• Intro Hydrogen Atom slides (by J. Musshoff)
• How to plot spherical harmonics (by J. Musshoff) slides
• Lecture 1 TOPIC: basics of group theory. Type of symmetries.
• Lecture 2 slides TOPIC: From the definition to classes. Transformations of functions and operators. Euclidean group. Lie Algebra. Reducible and irreducible representations
• Lecture 3 EXERCISES
• Lecture 4 slides TOPIC: irreducible reps and orthogonality theorems
• Lecture 5 slides TOPIC: orthogonality theorems, character tables, partner functions
• Lecture 6 EXERCISE
• Lecture 7 slides TOPIC: classification of groups
• Lecture 8 slides TOPIC: energy levels, hidden symmetry H atom
• Lecture 9 EXERCISES
• Lecture 10 slides TOPIC: point symmetry reduction, double groups, spinors, Bloch theorem
• Lecture 11 slides TOPIC: time reversal and selection rules
• Lecture 12 EXERCISES
• Lecture 13 slides TOPIC: many-electron states
• Lecture 14 slides TOPIC: many-electron states