# MSc Physics Syllabus

### Classical Mechanics

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, entanglement entropy (von Neumann and Renyi).

### 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

Equilibrium Statistical Mechanics: Introduction: Macrostates, microstates, phase space and ensembles, Ergodic hypothesis, postulate of equal apriori probabilities, Boltzmann's postulate of entropy. Counting the number of microstates in phase space. Entropy of ideal gas: Sackur-Tetrode equation and Gibb's paradox, Liouville's theorem, Canonical Ensemble, Grand Canonical ensemble, Mean field theory and Van der Waal's equation of state, Cluster integrals and Mayer-Ursell expansion, Qutantum Statistical Mechanics, Density Matrix; Quantum Liouville theorem, Identical particles – BE and FD distributions. Ideal Bose and Fermi gas, Bose condensation. Ising Model: partition function for one dimensional case; Chemical equilibrium and Saha ionisation formula. Phase transitions: first order and continuous, critical exponents and scaling relations. Calculation of exponents from Mean field theory and Landau's theory, upper critical dimension.

### Statistical Mechanics II : Non-equlibrium Statistical Mechanics

Basic Introduction: equlibrium states, steady states, dynamics, repsonse functions, fluctuation dissipation theorem, Kramers-Kronig relation, Fluctuation relations (work theorem). Langevein equation, Diffusion, Fokker-Planck equation, Master equation, detailed balance. Specific cases: metals, Nyquist theorem, Kobo formula, Specific cases: ASEP (asymmetric exclusion process), SOC (self-organised criticality), growth problems, random walk, hydrodynamics.

### Atomic and Molecular Physics

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.

### Nuclear and Particle Physics

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**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.

### Condensed Matter Physics I

Crystal Structure: Lattice and Basis; Examples of crystal structures; Direct and Reciprocal Lattice; Xray diffraction and crystal structure determination; Symmetry operations – Point and Space Groups. Band Theory: Electrons in Periodic Potential – Bloch’s theorem and Energy Bands; Plane Wave expansions and Tight Binding; OPW, APW and KKR methods of Band Structure Calculations; Fermi Surface – examples of bands structure of metals. 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; Boltzmann transport equation – conductivity in Relaxation time approximation. Lattice Dynamics: Normal modes of a three dimensional lattice; Phonons – measuring phonon dispersion relation; Anharmonic effects – Lattice Thermal Conductivity. 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. Magnetism: Diamagnetism and Paramagnetism; Magnetic order – Ferro , antiferro and ferrimagnetism; Heisenberg model – mean field theory of ferromagnetic and antiferromagnetic transitions; 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.

### 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.

### Cosmology

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.

### Condensed Matter Physics II