Topics to be covered
1. Group Theory
1) Role of symmetry in Physics, Basic idea of group and their emergence in physics: Symmetries and groups, Formal definition of group, Some important discrete groups that appears in molecular and lattice structure (point group and space group); Subgroup, Normal subgroup, Lagrange’s theorem; Equivalence relation, Conjugacy classes; Permutation group, even and odd permutations and different statistics of particles; Homomorphism and isomorphism, Cayley’s theorem
2) Representation theory and their emergence in physics, Angular momentum algebra as an example, Definition, Equivalence of representations, Invariant subspace and classifications of representations, Unitary and orthogonal representations, Adjoint representations, Schur’s lemma, Orthogonality of representations, Restriction on the number of irreducible representations
2) Representation theory and their emergence in physics, Angular momentum algebra as an example, Definition, Equivalence of representations, Invariant subspace and classifications of representations, Unitary and orthogonal representations, Adjoint representations, Schur’s lemma, Orthogonality of representations, Restriction on the number of irreducible representations
3) Character of representations, Definition and normalization, Construction of the character table, Character table of C3, D3
4) Applications in physics: Wigner’s theorem in Quantum Mechanics, Application of Group Theory to Molecular Vibrations, Finding the Vibrational Normal Modes, Molecular Vibrations in H2O, Overtones and Combination Modes, Infrared Activity, Raman Effect, Vibrations of the NH3 and CH4 Molecules, The Rigid Rotator, Wigner–Eckart Theorem, Vibrational–Rotational Interaction, Role of symmetry in solving a Hamiltonian: Benzene Hamiltonian and Hydrogen atom using LRL vector.
5) Lie groups and physics, Definition of Lie groups, Generators of Lie groups; Lie algebra, Definition, From Lie algebra to Lie group, Regular representations of Lie algebras, Subalgebras, Ideals and proper ideals, Simple and semi simple Lie algebras, Cartan’s criterion for semi simple Lie algebras; Root diagrams, Dynkin diagrams for SU(2), SU(3) and their applications in quantum mechanics
Lecture notes (on request)