ABOUT THE COURSE:This is an advanced core course for the students who want a career in physics. It consists of three parts. First 4 weeks will be devoted to some advanced topics in usual Quantum mechanics, in next 4 weeks we will learn details of relativistic quantum mechanics with more emphasis to Dirac equations and the last 4 weeks is to introduce the concept field quantization. This course is the most essential to the students who want to carry out research in high energy physics, condensed matter physics and other branches of theoretical physics. In this course difficult concepts will be introduced through examples in very simple language and is pre-requisites for the advanced level courses like, Particle Physics, Quantum Field theory Quantum many body systems, advanced statistical mechanics and advanced condensed matter theory etc. This course will be helpful to the students who will be writing national level examinations like JEST, NET, GATE etc.INTENDED AUDIENCE:B.Sc. 4th year, M.Sc. Students in Physics,PhD students in High Energy Physics/Condensed matter physicsCollege Teachers and other ProfessionalsPREREQUISITES:Basic Quantum MechanicsBasic Classical MechanicsSpecial Theory of RelativityMathematical physicsINDUSTRY SUPPORT:NIL

Syllabus

Week 1:Basic features of Classical Mechanics, Postulates of Quantum Mechanics, Concept of Vector space, Essential features of Special Theory of Relativity and Concept of four vectors.
Week 2:Essential features of Angular momentum, Addition of Angular Momentum, Properties of CG coefficients, Calculation of CG coefficients in Simple systems.

Week 3:Application of CG coefficients, Identical Particles, Symmetrisation and Anti-symmetrisation of wave functions.
Week 4:Slater determinant, Scattering of identical particles, permutation symmetry, Hartree-Fock method for self-consistent solutions.
Week 5:Need for relativistic QM, KG equations, Probability density and probability current density, Solution of KG equation. Limitations of KG equation. Introduction of Dirac Equation. Continuity Equation, Probability density and Probability Current density, Solution for Dirac equation for particle at rest.
Week 6:Dirac Equation in a Magnetic field and its non-relativistic limit, Negative energy solutions, Plane wave solutions. Dirac Equation in four vector notation. Properties of Dirac Gamma Matrices
Week 7:Covariance of Dirac Equation. Properties of Dirac Gamma matrices, Bilinear Covariants. Symmetries (Parity, Time Reversal, Charge Conjugation) of Dirac Equation.
Week 8:Hole Theory, Feynman-Stueckelberg interpretation.Solution for massless particle (Neutrinos), Projection operator, Helicity, Chirality, Zitterbewegung, Klein Paradox, Foldy-Wouthuysen transformation. H-atom Spectrum ( without derivation).
Week 9:Discrete to Continuous systems and infinite degrees of freedom, Concept of Lagrangian density, Construction of Lagrangian density for simple systems. Euler-Lagrange Equation for simple systems. Introduction to 2nd quantization, Quantisation of Scalar field
Week 10:Hamiltonian density, Fock space, Normal Ordering. Propagator for Scalar field, Vacuum fluctuation, Microscopic Causality.
Week 11:Symmetries and Conservation laws, Noether theorem, Charged Scalar field, Electro-Magnetic (EM) fields, Problems in quantisation of EM fields
Week 12:Gauge fixing and Quantization of EM field. Propagator, Gupta-Bleuler method. Dirac field quantization and anti-commutation relations.