Lagrangian Formulation, Hamiltonian Formulation, Canonical Transformations, Dynamics of a rigid body, Hamilton – Jacobi Theory, Mechanics of Continuous media, Theory of small oscillations, Classical Perturbation Theory, Non-linear dynamics and chaos.
Quantum Mechanics I and II
Principle of superposition, Postulates of quantum mechanics, Symmetries, Single particle formulation of non-relativistic quantum mechanics, Applications to physical systems, Quantisation scheme and classical correspondence. Path integral formulation of quantum mechanics: free particle and particle in a well (perturbative approach), Quantum theory of scattering, Approximation method in quantum mechanics, Quantum computation and Quantum information theory, Bell’s inequalities, Density matrix, Reduced density matrix, entanglement.
Mathematical Tools in Theoretical Physics I and II
Theory of complex variables. Theory of linear ordinary differential equations. Integral transforms, Special functions, Boundary value problems and Green’s function, Integral equations.
Classical Theory of fields I : Electrodynamics
Action principle formulation of relativistic particle, Electromagnetic (EM) fields: relativistic formulation. Action formulation of EM fields: Maxwell equations. The vector potential: relativistic formulation, Interaction of EM fields with currents: Noether’s theorem, Interaction of charged particle with EM fields: Lorentz force equations, examples. Energy Momentum tensor: Conservation and Poynting’s theorem, ambiguities, Vacuum EM waves: geometrical optics limit; polarisation, Stokes parameters and Poincare sphere, EM waves in media: Faraday rotation, EM potentials due to an arbitrarily moving charged particle, EM fields from the moving charges: radiation and Coulomb fields, Dipole radiator: Lamor’s formula, radiated power spectrum, Synchrotron radiation: radiated power spectrum; polarization, Classical scattering by EM waves by charges: Rayleigh and Thomson scattering, Elements of multipole radiation: E1, E2 and M1 modes, Radiation reaction and inconsistencies of the Maxwell theory.
Classical Theory of fields II : General Relativity
Preliminaries – Curvilinear coordinate systems in R3 : Eucledean metric, Invariance principles : Special Relativity and Gravity, Principle of Equivalence, Redshift: Pseudo-Newtonian derivation. Curved spacetime: geodesics, Newtonian approximation. Invariants in curved spacetime – scalar, vector and tensor fields, p-form fields, metric tensor, Parallel transport and affine connection, covariant derivative, geodesics, Lie derivative and isometries, Invariant measure, Invariant matter field, Belinfante energy-momentum tensor, External field problems – Stationarity and timelike Killing vector fields, Gravitational redshift in stationary sptm, Spherically symmetric vacuum sptm: Schwarzchild Geodesics in Schwarzchild sptm : ISCOs and bounded orbits, Light bending by a spherical star, Perihelion shift of Mercury, Coordinate time and proper time, proper distance. Curved spacetime geometry – Geodesic deviation, Riemann curvature tensor: components, invariants (Ricci and Kretchmann), Weyl tensor, Bianchi Identity, Einstein-Hilbert-Lorentz action and Einstein equation; Newtonian approximation, Schwarzchild solution and properties Gravitational waves, Introduction to relativistic cosmolgy.
Statistical Mechanics I
Review of thermodynamics; Objectives of statistical mechanics; Microstates and macrostates; Phase space and concept of an ensemble; Liouville’s theorem and concept of equilibrium; Ergodic hypothesis and postulate of equal a priori probability; Microcanonical ensemble: Boltzmann’s definition of entropy and derivation of thermodynamics; The equipartition theorem; Microcanonical ensemble calculations for a classical ideal gas; Gibbs paradox; Canonical ensemble; Energy fluctuations in the canonical ensemble; Grand canonical ensemble; Density fluctuations in the grand canonical ensemble; Quantum statistical mechanics: Postulate of equal a priori probability and postulate of random phases; Density matrix; Ensembles in quantum statistical mechanics; The ideal quantum gas: Microcanonical and grand canonical ensembles; Fermi-Dirac and Bose-Einstein statistics; Bose-Einstein condensation.
Statistical Mechanics II
Basic introduction to phase transitions: first order and continuous; Critical phenomena: critical exponents and scaling hypothesis; Ising model: exact solution in one dimension, mean-field approximation and calculation of critical exponents, Landau theory; Review of probability theory: Law of large numbers and the central limit theorem; Random walk; Brownian motion: Langevin and Fokker-Planck descriptions; Fluctuation-Dissipation theorem; Markovian process; Master equation; Concept of steady states, detailed balance and equilibrium vs. non-equilibrium; Simple illustration using interacting random walks (simple symmetric and asymmetric exclusion processes).
Particle Physics I & II
Nuclear Physics: Charge, mass, constituents, binding energy and separation energy, level scheme, excited states, spin, parity and isospin, nuclear size and form factors, static electromagnetic moments. Two-nucleon system: a) Deuteron: ground and excited states; electric quadrupole and magnetic dipole moments; non-central force and tensor interaction. b) Scattering states: n-p and p-p scattering at low energies; effective range and scattering length; singlet and triplet states; ortho- and para-hydrogen, charge independence of nuclear forces. c) Nucleon-nucleon scattering at higher energies d) Polarization in nucleon-nucleon scattering – l.s forces e) Exchange forces and saturation f) General properties of nucleon-nucleon forces; Yukawa potential. Complex– nuclear structure: a) need for nuclear models b) Fermi Gas model c) Static Liquid Drop model d) Shell Model e) Collective Model f) Unified Model. Nuclear Reactions: a) types of reactions and conservation principles b) Compound Nuclear Reactions – Resonances and the Breit Wigner formula c) Direct Reactions, Optical Model, Nuclear Fission – Bohr – Wheeler theory, Electromagnetic Transitions – Multipole transitions and selection rules.
Particle Physics: Relativistic kinematics: Mandelstamm variables; collision and decay kinematics; reaction thresholds; phase space, cross-section and decay formulae, Types of interactions and their relative strengths, Discovery of positron, muon, pion, neutrino and other particles, Symmetry, conservation laws and Quantum numbers, Classification of elementary particles, Determination of quantum numbers of different particles, Hadrons – classification by isospin and hypercharge, Quarks, colour, Leptons and gauge bosons, Weak Interactions: a) phenomenology, conservation laws and selection rules b) Fermi theory of beta decay, V-A interaction b) non-conservation od parity c) Neutral Kaon decay – CP violation and regeneration e) Z and W+ and W- bosons, E-M interactions – the QED Lagrangian from gauge invariance principles. Group Theory: Lie Group – SU(2), SU(3), SU(n) – Discrete Symmetry – C, P, T , QED – Feynman rules – Cross section and Decay rate calculations, Hadron Structure and Quark Model, Parton model, Deep Ineasltic Scattering – QCD , Weak Interaction phenomenology – Electroweak unification , Non-Abelean Gauge Theory – Standard Model
Condensed Matter Physics I
Crystal structure — Lattice and basis, Examples of crystal structures, Direct and reciprocal lattice, Xray diffraction and crystal structure determination; Specific heat of solids — Boltzmann, Einstein and Debye theories; Elecrons in metals — Drude and Sommerfeld theories; Lattice dynamics: Normal modes, phonons, anharmonic effects, lattice thermal conductivity; Electrons in solids — Electrons in a periodic potential: Nearly free electron model, Bloch’s theorem, Insulators, semiconductors, and metals: Band structures and optical properties; Magnetism — Magnetic properties of atoms: Para and diamagnetism, Spontaneous magnetic order: Ferro-, antiferro-, and ferri-magnetism, Domains and hysteresis
Spectroscopy: General definition and terminology, Multiplet structure and designation of spectral terms, coupling of two or more electrons in equivalent shells, spin orbit interaction and alkali spectra, Relativistic mass correction, Darwin term and hydroden fine structure. Zeeman and Stark effect. Two electron systems, their wavefunctions, spectral terms. Many body theory, Hatree and Hatree Fock approximation, Configuration Interaction, Lamb shift.General structure of molecular energy levels, Born Oppenheeimer approximation. Rotational, vibrational, Rotational-vibrational and electronic spectra of diatomic molecules and their detailed structures. Franck Condon principle and its implications, Raman spectra.
Quantum Field Theory I
Relativistic quantum mechanics and the Dirac equation and its solutions, Canonical quantisation: Free scalar field, electromagntic field, Dirac field, Wick’s Theorem, Correlation functions, Propagators for the scalar, Dirac and electromagnetic field. Simple introduction to interacting theories and Feynman diagrams.
Quantum Field Theory II
Interacting Quantum Field theories, Quantum electrodynamics (QED), Calculation techniques for Feynman diagrams of all major processes in QED, Divergences in Quantum Field Theory, Removal of divergences, radiative corrections, explicit calculation of Lamb shift, Renormalisation theory, Wilson renormalisation group. Statistical field theory and applications to condensed matter physics, Two dimensional Ising model and gauge theories.
Advanced General Relativity and Astrophysics
Gravitational waves – Linearized General Relativity – Graviational waves in linearised GR – Energy radiated by gravitational waves – Detection of gravitational waves. White dwarfs – Astronomy basics – theormodynamics preliminaries – Degenerate electron gas – Equations of state – Chandrasekhar limit – Thomas-Fermi approximation approach to white dwarf – white dwarf cooling. Neutron stars – Histroy and formation – Structure and stability – Interior – Equations of state – Maximum mass – rotating neutron stars, pulsars. Black Holes – Penrose-Carter diagram of Minkowski and Schwarzschild spacetime – Reissner – Nordstrom blackhole – Majumdar-Papapetrou solutions – Kerr black hole – Kerr-Newman black holes – Geodesic congruences and the Raychaudhuri equation – Hamiltonian formulation of GR – Laws of black hole mechanics.
Comological observations, The expansion of the universe, Spacetime geometry, Comoving coordinates, Friedmann-Roberson-Walker (FRW) metric, Proper distances, Dynamics of a photon moving in FRW background, particle and event horizons. The cosmological redshift. Hubble’s law, Luminosity distances. Dynamics of expansion: Einstein field equations, Friedmann equation, Critical density, Matter dominated and radiation dominated expansion. Galaxy Rotation curves, Indirect evidence for dark-matter, Discovery of accelerated expansion. Dynamics of dark energy, consmological constant. The Cosmic Mircrowave Background Radiation (CMBR), The equilibrium era, recombination and last scattering, the dipole aniotropy, The Synyaev Zel’dovich effect, Primary fluctuations in CMBR, Scahs-Wolfe effect, Harrison – Zel’dovich spectrum, Doppler fluctuations, Intrinsic temperature fluctuations, Integrated Scahs – Wolfe effect. Thermal History of early universe, Cosmological nucleosynthesis, Baryosysthesis and Leptosynthesis, cold dark matter. Comic inflation: flatness, horizon, monopole problem, Slow-roll inflation, Reheating. Comological perturbation theory, Origin of large scale structure.
Two-dimensional conformal field theory
Conformal Group in D> 2 dimensions, Quasi primary fields, Conformal group in D=2 dimensions, Quasi primary and primary fields, secondary fields, 2-pt, 3-pt, f-pt correlation functions. Conformal ward identities, Stress energy tensor and conformal invariance, Mode expansion of Stress energy tensor, Virasoro Algebra, Conformal anomalies and Central charge, Operator product expansions. Kac determinants and Virasoro modules, briefly mentioned the minimal models, Crossing symmetry and conformal bootstrap method.
Advanced Condensed Matter Physics
Electron transport — Semi-classical equations, Bloch electrons in magnetic and electric fields, Hall effect and magneto-resistance, de Haas-van Alphen effect and Fermi surface determination; Semiconductors — Homogeneous semiconductors: carrier density, inhomogeneous semiconductors, carrier densities in a p-n junction, rectification; Dielectric properties — Screening, Thomas-Fermi and Lindhard expressions for dielectric constants, local field, optical properties, ferroelectrics; Mean field theory of ferromagnetic and antiferromagnetic transitions — Heisenberg model, spin waves; Superconductiviry — Persistent current, Meissner effect and critical fields – type I and II superconductors, specific heat, Electron-Phonon interaction and BCS theory, Ginzburg-Landau theory, Superconducting tunneling-Josephson effect, high temperature superconductivity – brief discussion.