Syllabus

Module 1: (8 hours)
Basic mathematical background, symmetry and group concepts general theorems, conjugation and class, factor groups, homomorphic and isomorphic mappings.
Discrete symmetry, groups proper point groups, improper point groups, colour and magnetic groups, double groups, lattice translation, and crystallographic space groups.

Module 2: (6 hours)
Representations of groups, reducible and irreducible representations, orthogonality theorems, character of a representation, decomposition of reducible representations.

Module 3: (6 hours)
Symmetry operations and basis functions, basis functions and irreducible representations, representation theory in quantum mechanics, direct-product groups.

Module 4: (8 hours)
Applying symmetry in quantum mechanics, full rotation group and angular momentum, homomorphism between rotation and unitary groups, irreducible representations of the rotation group, addition of angular momentum, tensors and spinors, Wigner-Eckart theorem, group theoretical matrix-element theorems, and quantum mechanical selection rules.

Module 6: (8 hours)
Applying symmetry in condensed matter, atomic orbitals in a cubic crystal field, spin-orbit coupling, and d electrons in octahedral crystal fields, the effect of a magnetic field. Electronic and vibrational states of molecules, selection rules for infrared and Raman activity.

Prerequisite
PH 4005: Quantum Mechanics – I, and PH 4006: Quantum Mechanics – II

To introduce the principles of group theory and its mathematical structures relevant to physical symmetries.

To classify and analyze discrete symmetry operations, including point groups, space groups, and their applications in crystal structures.

To develop competence in the representation theory of groups and its applications in quantum mechanics.

To apply symmetry principles in solving problems related to angular momentum, spin, and spectroscopic transitions in condensed matter systems.

At the end of course, students will be able to:
CO1: classify symmetry groups, understand their algebraic properties, and apply group theoretical concepts to physical systems.
CO2: demonstrate proficiency in constructing and analyzing reducible and irreducible representations of group in physical contexts.
CO3: gain a strong foundation in the application of symmetry principles in quantum mechanics,
CO4: apply group theoretical methods to problems in condensed matter physics.

M. S. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory: Application to the Physics of Condensed Matter, Springer, 2008

A. W. Joshi, Elements of Group Theory for Physicists, New Age International Private Limited (2023)

J. F. Cornwell, Group theory in Physics: An Introduction, Academic Press Abridged edition (1997)

W Ludwig and C. Falter, Symmetries in Physics, Springer (1996)