- Physics:
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The course work is divided into four or six semesters spread over a period two or three years depending on whether the student has enrolled for the regular or the integrated Ph.D. program. Each semester is of roughly four months duration. The students are taught basic as well as advanced courses in physics and they also get to work on, at least, two projects. Follow the links below for the course schedule and syllabus.

The students are expected to start working on their thesis soon after the completion of their course work and projects. Follow the links below for the various aspects related to thesis work.

- Classical Mechanics
- Quantum Mechanics I
- Mathematical Methods I
- Research Methodology and Numerical Methods

- Classical Electrodynamics
- Statistical Mechanics
- Quantum Mechanics II
- Mathematical Methods II
- Project

Choose between Atomic Molecular Physics and General Relativity.

- Quantum Field Theory I
- Atomic Molecular Physics
- Condensed Matter Physics I
- General Theory of Relativity
- Project

Choose any two out of five topics listed below. Project is mandatory.

- Astrophysics
- Condensed Matter Physics II
- Particle Physics
- Quantum Field Theory II
- Quantum Information and Computation
- Project

A brief outline of the syllabus of the various courses is given below. The courses are listed in alphabetical order.

**Relativistic quantum mechanics:**Klein-Gordon and Dirac equations and their solutions, causality and other problems, interaction with electromagnetic field, magnetic moment of electron, idea of spin and conservation of total angular momentum, helicity.**Statistical Mechanics:**Basic postulates, different ensembles, general results for fluctuations in ensembles.- Fermions: applications in electronic specific heat, Chandrasekhar limit, magnetism of the electron gas, quantum Hall effect.
- Bosons: applications in specific heat of solids, BEC (including trapped gases), superfluidity of helium, radiation field and Casimir effect.
- Interacting systems: Van der Waals gas, mean field theory, Bogoliubov-Valatin transformation, magnetic transition, Ising models 1-D transfer matrix, 2-D with a limited goal of showing mapping to a quantum system, Kac Hubbard transformations, Landau theory, critical exponents, path integrals, Gaussian model.
- Renormalization group and critical phenomena: real space in d=1 and 2, general structure, scaling laws and relation between exponents,calculation of a couple of exponents in epsilon expansion, spherical limit.
- Transport properties: Boltzmann equation and applications, dynamical response, fluctuation theorems covering Jarzynski, Crooks, etc, stochastic thermodynamics.

- Introduction to celestial objects, coordinates and the concept of time.
- Radiation transfer. Equations of radiation transfer, Black-body/thermal radiation, Opacity and optical depth, solutions of the radiation transfer equations in limiting cases, Rosseland mean opacity.
- Thermal Bremsstrahlung emission, synchrotron emission. Self absorption and the emergent spectrum. Thomson scattering. Compton and Inverse-Compton scattering. Scattering in a region with magnetic field, Faraday rotation
- Introduction to fluid dynamics. Convection instability and transfer of energy from cores of stars. Supersonic motion, shocks.
- Introduction to Magneto-hydro dynamics, flux freezing, Generation and amplification of magnetic fields in astrophysical situations.
- Stellar structure. Mass-radius relation for main sequence stars, Minimum and maximum mass for nucleosynthesis, Hertzsprung-Russell diagram, Evolution of a star on the HR diagram. Novae and Supernovae, End points of stellar evolution. Inter-stellar medium. Phases of interstellar medium. Thermal, photoionization, chemical and pressure equilibrium, Star formation, feedback and the evolution of ISM.
- Orbits around massive bodies, Tidal disruption, restricted 3 body problem, Roche limit. Orbits in external potentials, potential-density pairs. An overview of models for galaxies. Accretion of matter on to a point mass, spherical accretion, Eddington limit
- Introduction to Cosmology, Friedmann models, equations. Hubble's law. A brief overview of the thermal history of the universe.

- Energy quantization, transition probabilities, linewidth, Lamb shift
- Coupling of angular momenta, multiplet structure. gyromagnetic effects. Hyperfine and nuclear quadrupole interactions.
- Hartree Fock and correlation effects.
- Molecular spectra. Harmonic and anharmonic vibrations, Vibrational-rotational coupling, Coupling of rotational and electronic motion.
- Molecular orbital and valence bond theories.
- Nuclear binding, energy and forces. Weiszacker mass formula
- Two nucleon problem. Alpha, beta and gamma decay, Nuclear reactions
- Nucleon-nucleon scattering, effective range theory.
- Nuclear shell model, collective model and deformed nuclei, BCS model
- Nuclear reaction theory, interacting boson model, heavy ion collision, quark-gluon plasma
- Topics in Cold Atoms

- Special theory of relativity and electrodynamics: Lorentz transformation of electromagnetic fields, Lorentz covariant formulation of electrodynamics; gauge invariance, Maxwell's equations from action principle.
- The electrostatic limit of Maxwell's equations, multipole expansion, uniqueness, boundary value problems, solution of Poisson's equation.
- The magnetostatic limit of Maxwell's equations, applications.
- Electrodynamics: motion of charges in external fields; electromagnetic waves in vacua and propagation through continuous media; energy-momentum of electromagnetic field and Poynting's theorem.
- Advanced and retarded Green's functions; Lienard-Wiechert potentials; dipole radiation and Larmor's formula; spectral resolution and angular distribution of radiation from a relativistic point charge; synchrotron radiation; Rayleigh and Thomson scattering; collision problems; Bremsstrahlung and Cerenkov radiation.
- Scattering of electromagnetic waves: Rayleigh and Thomson scattering, radiation damping.

- A rapid review/summary of Newtonian mechanics
- Calculus of variations: Concept of variation - Euler equation - Applications - Variation subject to constraints and Lagrange multipliers.
- The Lagrangian formulation: Generalized coordinates and velocities - The principle of least action and the Lagrange equations of motion - Extension to constrained systems.
- Conservation laws: Symmetries and Noether's theorem - Conservation of energy, momentum and angular momentum.
- Integration of the Lagrange equations of motion: Motion in one dimension - The two body problem, reduced mass and the equivalent one-dimensional problem - Motion in a central field - Kepler's problem - Scattering.
- Small oscillations: Free, damped and forced oscillations in one dimension - Resonance - Damped and forced oscillations - Parametric resonance.
- Rigid body motion: Angular velocity - The inertia tensor and angular momentum of a rigid body - The equations of motion of a rigid body - Eulerian angles - Euler's equations - Motion of tops - Motion in a rotating frame - Coriolis force.
- The Hamiltonian formulation: Conjugate momentum - Hamiltonian and Hamilton's equations - Poisson brackets - Canonical transformations - Dynamics in the phase space - Hamilton-Jacobi equation - Separation of variables and solutions - Action-angle variables - Adiabatic invariants.
- Elements of non-linear dynamics: Differential equations as dynamical systems - Lyapunov exponents - Sensitive dependence on initial conditions - Discrete time dynamics and maps - Period doubling and chaos.

- Atoms to solids: atomic physics: Coulomb effects, `crystal fields' in solids, `local moments' and `band electrons', vibrations: phonons
- Structure: characterizing structures - crystalline/amorphous/liquids, periodic structures: classification, reciprocal space, x-ray/neutron diffraction.
- Electronic spectrum: free electrons: states, spectrum, thermodynamics. Bloch states: `nearly free electron' and tight binding limits, consequences for thermodynamics; transport. `Metal physics': Sp. heat, susceptibility, transport, magnetic fields: Landau quantization.
- Phonons: Debye and Einstein model, spectrum of a real lattice, thermodynamics of phonons, anharmonic effects, Debye-Waller factor.
- Magnetism: spin paramagnetism, Itinerant -vs- localized electrons, Stoner and Heisenberg models, `mean-field' theory, spin waves.
- Superconductivity: phenomenology, `pairing' interaction, BCS theory, type II superconductors.
- Possible advanced topics: electron localization/Anderson transition, Mott transition, Bose condensation/superfluids, the Kondo effect, low dimensional systems.

The course will consist of any two of A-E.

- Foundation: Low dimensional systems: Quantum Wells, Wires and Quantum Dots, one and two dimensional heterostructures, coupled wells and super- lattices. Density of states in low dimensional systems
- Charge Transport: Transmission and its relation to conductance, Landauer theory of coherent charge transport - phenomenology as well scattering theory. Transmission function and its relation to S matrix and Greens function. Non-equilibrium Greens function and its relation to Landauer-Buttiker theory.
- Noise in Charge transport: Thermal and Shot Noise in Mesoscopic conductors. Scattering theory of Shot noise and its application.
- Spintronics: 1: Introduction to spintronics.(Datta-Das spin transistor) 2: Spin currents: Equilibrium and non-equilibrium spin currents and their measurement and its relation to spin-Hall effect, generalized Landauer-Buttiker theory for coupled charge and spin transport. 3: Tunnel Magneto-resistance and spin currents 4: Spin Shot Noise, Entanglement generation and its detection.

- Physics at low dimensions: surface physics--surface states, reconstructions, adsorption on surfaces; atomic wires and clusters
- Treating electron-electron interactions: Hartree-Fock approximation, the electron gas; Density functional theory: as a theory of interacting electrons, and a framework to do calculations for systems discussed 1.
- Anharmonic effects in crystals: Thermal expansion, lattice thermal conductivity, Umklapp precesses.
- Phonons in Metals: Kohn anomaly, dielectric constant, temperature dependence of electrical resistivity.
- Dielectric properties of insulators.
- Elementary exciations in solids: plasmons, magnons etc.

- Quantum Hall effect
- Quantum dots and quantum wires
- Kondo effect
- Spin systems
- Fermi liquid theory
- Non-Fermi liquids
- Bosonisation and Luttinger liquids
- Quantum phase transitions

- Dynamic critical phenomena and their classification, dynamic scaling concepts, introduction to dynamical renormalization group
- Statistical mechanics of non-equilibrium systems: Kinetics of phase-ordering, Dynamic and static scaling concepts in surface growth and morphology, reaction-diffusion systems, phenomenological theories.
- Basic concepts in stochastic processes, Langevin and Fokker-Planck equations, Markov processes.
- Random walk in 1,2 and 3 dimensions and their various applications, anomalous diffusion.

- Mott physics: electron localisation, magnetic order, doped phase, physics in the cuprates.
- Kondo systems: physics of the single impurity, dense systems - Kondo and Anderson lattice, heavy fermions, quantum criticality.
- Metallic magnets: ferromagnetism in strongly repulsive systems, the transition metals, spin-fermion systems, the double exchange model, the classical Kondo lattice.
- Electron-phonon coupling: the classical theory, polaron formation, many electron systems, polaron ordering, physics in the manganites.
- Disorder in interacting systems: illustrative examples.

- On facilities arranged at other DAE institutions.

- Review of special theory of relativity.
- Riemannian geometry of Euclidean signature manifolds: tensors on Euclidean manifolds and their transformation laws; Christoffel symbol and Riemann tensor; geodesics; parallel transport along open lines and closed curves; general properties of the Riemann tensor.
- Generalization to manifolds with Lorenzian signature: Comparison of geodesic motion and motion under Newtonian gravitational potential in the non-relativistic limit.
- Equivalence principle and its application: gravity as a curvature of space-time; geodesics as trajectories under the influence of gravitational field; generalization to massless particles; gravitational red-shift; utility of equivalence principle for studying gravity plus other interactions; motion of a charged particle in curved space-time in the presence of an electric field; Maxwell's equation in curved space-time
- Einstein's equation
- Schwarzschild solution: construction of the metric and its symmetries; Motion of a particle in the Schwarzschild metric; precession of the perihelion; bending of light; horizon, its properties and significance.
- Cosmological models: principles of homogeneity and isotropy; FRW metric; open, closed and flat universes; relation between distance, red-shift and scale factor; role of equation of state; equations of state for matter, radiation, and cosmological constant, and their effect on scale factor evolution; explicit solutions for matter, radiation or cosmological constant dominated universes.

- Vector Analysis: operations with vectors, scalar and vector fields, gradient, curl and divergence. Line, surface, and volume integrals, Curvilinear coordinate systems, Elements of tensors.
- Vector Spaces, linear transformations, scalar product and dual space, bases, linear operators, eigenvalues and eigenfunctions, unitary and hermitian operators
- Complex Analysis: functions of a complex variable, analytic functions, integral calculus, contour integrals, Taylor and Laurent series, singularities, residues, principal values, Riemann surfaces, conformal mapping, analytic continuation
- Ordinary differential equations: linear ODE's, Green's functions, second order differential equations: series solution, special functions
- Elements of statistics: probability, random walk. Probability distributions
- Calculus of variations

- Integral transforms, Fourier transforms, inversion and convolution, Laplace transforms
- Partial differential equations: classification of second order PDE's, Laplace and Poisson equations, applications to electrostatics, Heat equation, Wave equation
- Group theory, definitions and examples of groups. Homomorphism, isomorphism and automorphism, Permutation groups, Group representation: reducibility, equivalence, Schur's lemma. Lie groups and Lie algebra's. Root system and Dynkin diagram. Representations of simple Lie algebras, SO(n), Lorentz group. Symmetries in physical systems

- Experimental methods of particle physics: fixed target and collider experiments, particle detectors.
- Symmetries in particle physics: charge conjugation, parity, time reversal, isospin and SU(2), motivation for the quark model and SU(3)
- Introduction to relativistic kinematics: Mandelstam variables, phase space, calculation of cross-sections and decay widths
- Basics of quantum electrodynamics: electron-positron annihilation, electron-muon scattering, Bhabha scattering, Compton scattering.
- Deep inelastic scattering: Bjorken scaling, parton model, scaling violation, introduction to quantum chromodynamics.
- Introduction to weak interactions: parity violation, (V-A) theory, pion and muon decay, neutrino scattering.
- The gauge theory of electroweak interactions: Glashow-Salam-Weinberg model, applications of the model, neutral current phenomena, The physics of W-and Z-bosons, physics of the Higgs boson.
- Selected topics in flavour physics and neutrino physics, neutrino oscillation.

All regular as well as the integrated Ph.D. students are expected to do two projects with the faculty members. The students are advised to choose projects so that, at least one of them will eventually lead to the topic of their thesis.

- Non-relativistic quantum field theory: quantum mechanics of many particle systems; second quantization; Schroedinger equation as a classical field equation and its quantization; inclusion of interparticle interactions in the first and second quantized formalism
- Free Klein-Gordon equation: classical action and its quantization; spectrum; Feynman rules for computing n-point Green's function of elementary and composite operators.
- Symmetries and conservation laws: examples in non-relativistic and relativistic field theories; translation, rotation, Lorentz boost/Galilean transformation and internal symmetry transformations; associated conserved charges
- Interacting Klein-Gordon field; Feynman rules for computing Greens functions; physical mass of the particle from the analysis of two point Greens functions; S-matrix and its computation from n-point Greens functions; relating S-matrix to cross-section.
- Quantization of free Dirac fields: spectrum; Feynman rules
- Quantization of free electromagnetic field: role of gauge invariance; gauge fixing; physical state condition; spectrum; Feynman rules
- Quantum electrodynamics: coupling Dirac field to electromagnetic field; gauge invariance; quantization; Feynman rules for computing Green's function; Spectrum and S-matrix from the Greens function.

- Path integral method: inclusion of fermions, path integral formulation of Abelian gauge theories, gauge fixing and Fadeev-Popov ghosts.
- Non-Abelian gauge theories: classical theory, quantization via path integral method, BRST symmetry.
- Renormalization: scalar field theory and fermions, gauge theories, use of Ward identities.
- Renormalization group equation
- Spontaneous symmetry breaking
- Standard model

- Quantum formalism: states, evolution, measurements
- Multipartite quantum systems: description and manipulation of bipartite systems and beyond
- Entanglement: quantification and detection in bipartite and multipartite systems
- Quantum communication: no-cloning theorem, quantum teleportation, quantum dense coding, multipartite communication protocols
- Quantum cryptography: essential classical cryptography, BB84, B92, Ekert, and secret sharing protocols
- Quantum computation: quantum algorithms, universal gates
- Interface of quantum information with other sciences
- Experimental realizations

- Basic notions, states and operators
- Schrodinger equation, one-dimensional problems, harmonic oscillator, periodic potential, Kronig-Penny model
- Three-dimensional problems, central force potential, the hydrogen atom
- Symmetries and conservation laws in QM, Degeneracies, Bergman's theorem; Discrete Symmetries
- Angular momentum in quantum mechanics raising and lowering operators, angular momentum addition, Clebsch-Gordon coefficients; Schwinger oscillator Model, tensor operators and Wigner-Eckart theorem
- Time-independent perturbation theory, non-degenerate and degenerate cases, Fine structure and Zeeman effect
- Time-dependent perturbation theory.
- Approximation methods: WKB approximation, variational methods

- Scattering theory and its applications, Heisenberg, Schrodinger and interaction pictures
- Charged particle in a magnetic field, Landau levels, Quantum Hall effect
- Relativistic quantum mechanics, Klein-Gordon and Dirac equations and their solutions, Causality and other problems, Interaction with electromagnetic field, magnetic moment of electron, idea of spin and conservation of total angular momentum, helicity, Hydrogen atom, fine structure of spectral lines
- Path integrals, propagators, solution using path integral methods, Semiclassical methods revisited
- Quantum mechanics of many particles, identical particles and symmetries of wavefunction.
- Possible additional topics: supersymmetric quantum mechanics and exactly solvable potential problems, EPR paradox and Bells inequalities.

- Research Methodology including quantitative methods, communication skills, seminar presentation and review of research papers
- Introduction to programming languages: F77, F90 or C
- Errors in numerical calculations.
- Numerical linear algebra, eigenvalue and eigenvectors.
- Interpolation techniques.
- Generation and use of random numbers.
- Sorting and searching.
- Differentiation and Integration (including Monte Carlo techniques)
- Root finding algorithms
- Optimization, extrema of many variable functions.
- ODE's and PDE's: including FFT and finite difference methods, integral equations.

- Review of thermodynamics: free energy, thermodynamic potential, thermodynamic relations, chemical potential, Nernst's Theorem.
- Statistical Distributions, Liouville's Theorem, the BBGKY hierarchy, Boltzmann equation and approximations.
- Calculation of averages; microcanonical, canonical and grand canonical ensembles; Partition function.
- Equilibrium physics of classical systems: Ideal gases, internal degrees of freedom. Non-ideal gases: van der Waals force. Cluster expansions. Liquids: constructing a theory. Magnetism: Ising and Heisenberg models.
- Non interacting quantum systems: method of second quantizationon, electrons in metals, relativistic electron systems, electrons in a strong magnetic field.
- Lattice vibrations: phonon physics. Photons: blackbody radiation, Bose condensation.
- Interacting quantum systems: magnetism, interacting bosons in a trap.
- Phase transitions: mean field approach; Landau theory; Universality and scaling; Landau-Ginzburg functional; Importance of fluctuations; Infrared divergence; Basic idea of RG
- Possible additional topics: Computational methods, Kosterlitz-Thouless transition, Quantum phase transitions.

The following courses will be offered either as a special topic or as a reading course, depending on the number of students opting for them.

- Cosmology
- String theory
- Supersymmetry
- Galactic dynamics
- Quantum many body theory
- Non-equilibrium statistical mechanics
- Special topics in quantum field theory
- Field theory in curved spacetime
- Mesoscopic physics and spintronics

The grading of the courses will be based on continuous assessment and an end-of-semester exam.

- A student getting less than 50% fails the course, and would have to repeat the course and pass on his second attempt.
- To continue through the course work, a student should not fail in two or more courses at any given time.
- A student completing the course and the project requirements satisfactorily is said to clear the course work successfully.

Students who are very confident of their mastery of the course material can ask the course instructor to give them a test at the start of the course. If they pass this test, they can be exempt from attending the course, further examinations, etc.. The request for an exemption test is not automatically granted, it depends on prior impression about the ability of the student and interaction between the student and course instructor. Even if an instructor does not allow test based exemption in a particular case, he/she may allow the student to be absent from the regular lectures, but submit all assignments/projects, and take all the tests.

At HRI, research in physics is conducted in the following five areas:

- Astrophysics
- Condensed Matter Physics
- High Energy Phenomenology
- String Theory
- Quantum Information and Computation

For further details about the research carried out in physics at HRI, click here.

The students are expected to choose their advisor by/before the end of their course work. Though it need not necessarily be so, it would be advisable for students to choose projects such that at least one of them will lead to their thesis topic.

The students are expected to register for their Ph.D. soon after they have chosen their thesis advisor.

Students pursuing Ph.D. have to give a seminar before July 31 every year until they have submitted their thesis.

All students are expected to submit their thesis within five years of joining HRI.

- Physics:
- Research
- People
- Graduate programs
- Activities
- Timetable

Harish-Chandra Research
Institute

Chhatnag Road, Jhusi

Allahabad 211 019,
India

Phone: +91 (532) 2667 510,
2667 511, 2668 311, 2668 313, 2668 314

Fax: +91 (532) 2567 748,
2567 444

Physics Graduate Programme <physgradp at hri dot res dot in>

Updated: September 03, 2016 T22:25:33Z