1. Mathematical Methods of Physics
Dimensional analysis: Vector algebra and vector calculus; Linear algebra, matrices. Cayley Hamilton theorem, eigenvalue problems; Linear differential equations; Special functions (Hermite, Bessel, Laguerre and Legendre); Fourier series, Fourier and Laplace transforms; Elements of complex analysis: Laurent series- Poles, residues and evaluation of integrals; Elementary ideas about tensors; Introductory group theory, SU(2), O(3);Elements of computational techniques: roots of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, solution of first order differential equations using Runge-Kutta method; Finite difference methods; Elementary probability theory, random variables, binomial, Poisson and normal distributions.
2. Classical Mechanics
Newton’s laws; Phase space dynamics, stability analysis; Central-force motion; Two–body collisions, scattering in laboratory and
3. Electromagnetic Theory
Electrostatics: Gauss’ Law and its application; Laplace and Poisson equations, boundary value problems; Magnetostatics: Biot-Savart law, Ampere’s theorem, electromagnetic induction; Maxwell’s equations in free space and linear isotropic media; boundary conditions on fields at interfaces; Scalar and vector potentials; Gauge invariance; Electromagnetic waves in free space, dielectrics and conductors; Reflection and refraction, polarization, Fresnel’s Law, interference, coherence, and diffraction; Dispersion relation in plasma; Lorentz invariance of Maxwell’s equations; Transmission lines and waveguides; Dynamics of charged particles in static and uniform electromagnetic fields; Radiation from moving charges, dipoles and retarded potentials.
4. Quantum Mechanics
Wave-particle duality; Wave functions in coordinate and momentum representations; Commutators and Heisenberg’s uncertainty principle; Matrix representations; Dirac’s bra and ket notation; Schroedinger equation(time – independent and time-independent); Eigenvalue problems such as particle-in-a-box, harmonic oscillator, etc.; Tunneling through a barrier; Motion in a central potential; Orbital angular momentum, Angular momentum algebra; spin; Addition of angular momenta; Hydrogen atom, spin-orbit coupling, fine structure; Time-independent perturbation theory and applications; Variational method; WKB approximation; Time-dependent perturbation theory and Fermi’s Golden Rule; Selection rules; Semi-classical theory of radiation; Elementary theory of scattering phase shifts, partial waves, Born approximation; Identical Particles, Pauli’s exclusion principle,
5. Thermodynamic and Statistical Physics
Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; phase space, micro- and macrostates; Microcanonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; principle of detailed balance; Blackbody radiation and Planck’s distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to nonequilibrium processes; Diffusion equation.
Semiconductor device physics, including diodes, junctions, transistors, field effect devices, homo and heterojunction devices, device structure, device characteristics. Frequency dependence and applications; Optoelectronic devices, including solar cells. Photodetectors. And LEDs: High-frequency devices, including generators and detectors; Operational amplifiers and their applications; Digital techniques and applications (registers, counters, comparators and similar circuits): A/D and D/A converters; Microprocessor and microcontroller basics.
7. Experimental Techniques and Data Analysis
Data interpretation and analysis; precision and accuracy, error analysis propagation of errors, least squares fitting. Linear and nonlinear curve fitting chi-square test; Transducers (temperature, pressure/vacuum, magnetic field, vibration, optical, and particle detectors) measurement and control; Signal conditioning and recovery, impedance matching. amplification (Op-amp based, instrumentation amp, feedback), filtering and noise reduction, shielding and grounding; Fourier transforms; lock-in detector, box-car integrator modulation techniques.
Applications of the above experimental and analytical techniques to typical undergraduate and graduate level laboratory experiments.
8. Atomic & Molecular Physics
Quantum states of an electron in an atom; Electron spin; Stern-Gerlach experiment; Spectrum of Hydrogen, helium and alkali atoms; Relativistic corrections for energy levels of hydrogen; Hyperfine structure and isotopic shift; width of spectral lines; LS & JJ coupling; Zeeman, Paschen Back & Stark effect; X-ray spectroscopy; Electron spin resonance, Nuclear magnetic resonance, chemical shift; Rotational vibrational, electronic, and Raman spectra of diatomic molecules; Frank – Condon principle and selection rules; Spontaneous and stimulated emission, Einstein A & B coefficients; Lasers, optical pumping population inversion, rate equation; Modes of resonators and coherence length.
9. Condensed Matter Physics
Bravais lattices; Reciprocal lattice, diffraction and the structure factor; Bonding of solids; Elastic properties, phonons, lattice specific heat; Free electron theory and electronic specific heat; Response and relaxation phenomena; Drude model of electrical and thermal conductivity; Hall effect and thermoelectric power; Diamagnetism; paramagnetism, and ferromagnetism; Electron motion in a periodic potential, band theory of metals, insulators and semiconductors; Superconductivity, type – I and type – II superconductors, Josephson junctions; Defects and dislocations; Ordered phases of matter, translational and orientational order, kinds of liquid crystalline order; conducting polymers; Quasicrystals.
10. Nuclear and Particle Physics
Basic nuclear properties size, shapes, charge distribution, spin and parity; Binding energy, semi-empirical mass formula; Liquid drop model; Fission and fusion; Nature of the nuclear force, form of nucleon potential; Charge-independence and charge-symmetry of nuclear forces; Isospin; Deuteron problem; Evidence of shell structure, single particle shell model, its validity and limitation; Rotational spectra; Elementary ideas of alpha, beta and gamma decays and their selection rules; Nuclear reaction mechanisms, compound nuclei and direct reactions; Classification of fundamental forces; Elementary particles (quarks, baryons, mesons, leptons); Spin and parity assignments, isospin, strangeness; Gell-Mann-Nishijima formula; C. P. and T invariance and applications of symmetry arguments to particle reactions, parity non-conservation in weak interaction; Relativistic Kinematics.
- Basic Principles and applications of quantum mechanics – hydrogen atom, angular momentum.
- Variational and perturbational methods.
- Basics of atomic structure, electronic configuration,
shapeof orbitals, hydrogen atom spectra.
- Theoretical treatment of atomic structures and chemical bonding.
- Chemical applications of groups’ theory.
- Basic principles and application of spectroscopy – rotational, vibrational, electronic, Raman, ESR, NMR.
- Chemical thermodynamics.
- Phase equilibria.
- Statistical thermodynamics.
- Chemical equilibria.
- Electrochemistry – Nernst equation, Electrode Kinetics, electrical double layer, Debye-Hückel theory.
- Chemical Kinetics – empirical rate laws, Arrhenius equation, theories of reaction rates, determination of reaction mechanisms, experimental techniques for fast reactions.
- Concepts of catalysis.
- Polymer chemistry. Molecular weights and their determinations. Kinetics of chain polymerization
- Solids – structural classification of binary and ternary compounds, diffraction techniques, bonding, thermal, electrical and magnetic properties.
Collidsand surface phenomena.
- Data analysis.
- Chemical Periodicity
- Structure and bonding in homo-and heteronuclear molecules, including shapes of molecules.
- Concepts of acids and bases.
- Chemistry of the main group elements and their compounds. Allotropy, synthesis, bonding
- Chemistry of transition elements and coordination compounds – bonding theories, spectral and magnetic properties, reaction mechanisms.
- Inner transition elements – spectral and magnetic properties, analytical applications.
- Organometallic compounds – synthesis, bonding and structure, and reactivity. Organometallics in homogeneous catalysis.
- Cages and metal clusters.
- Analytical chemistry – separation techniques. Spectroscopic electro- and thermoanalytical methods.
- Bioinorganic chemistry – Photosystems, Porphyrines, metalloenzymes, Oxygen transport, electron- transfer reactions, nitrogen fixation.
characterisationof inorganic compounds by IR, Raman, NMR, EPR, Mössbauer, UV-, NQR, MS, electron spectroscopy and microscopic techniques.
- Nuclear chemistry – nuclear reactions, fission and fusion, radioanalytical techniques and activation analysis.
- IUPAC nomenclature of organic compounds.
- Principles of stereochemistry, conformational analysis, isomerism
- Reactive intermediates and organic reaction mechanisms.
- Concepts of aromaticity.
- Pericyclic reactions.
- Named reactions.
- Transformations and rearrangements.
- Principles and applications of organic photochemistry, Free radical reactions.
- Reactions involving
- Oxidation and reduction of functional groups.
- Common reagents (organic, inorganic and organometallic) in organic synthesis.
- Chemistry of natural products such as steroids, alkaloids, terpenes, peptides, carbohydrates, nucleic acids lipids.
- Selective organic transformations – chemoselectivity, regioselectivity, stereoselectivity, enantioselectivity, Protecting groups.
- Chemistry of aromatic and aliphatic heterocyclic compounds.
- Physical characterization of organic compounds by IR, UV-, MS, and NMR.
- Chemistry in nanoscience and technology.
- Catalysis and green chemistry.
- Medicinal chemistry.
- Supramolecular chemistry.
- Environmental chemistry.
UNIT – 1
- Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean Property, supremum, infimum.
- Sequences and series, convergence, limsup, liminf.
- Bolzano Weierstrass theorem, Heine Borel theorem.
- Continuity, uniform continuity, differentiability, mean value theorem.
- Sequences and series of functions, uniform convergence.
- Riemann sums and Riemann integral, improper Integrals.
- Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral.
- Matric spaces, compactness, connectedness. Normed Linear Spaces. Spaces of Continuous functions as examples.
- Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations.
- Algebra of matrices, rank
anddeterminant of matrices, linear equations.
- Eigenvalues and eigenvectors, Cayley- Hamilton theorem.
- Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.
- Inner product spaces, orthonormal basis.
- Quadratic forms, reduction
andclassification of quadratic forms.
UNIT – 2
- Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions.
- Analytic functions, Cauchy- Riemann equations.
- Contour integral, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem.
- Taylor series, Laurent series, calculus of residues.
- Conformal mappings, Mobius transformations.
- Permutations, combinations, pigeon-hole principal, inclusion-exclusion principal, derangements.
Fundamentaltheorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s O- function, primitive roots.
- Groups, subgroups, normal subgroups, quotient groups, homomorphisms.
- Cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems.
- Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain.
- Polynomial rings and irreducibility criteria.
- Fields, finite fields, field extensions.
UNIT – 3
Ordinary Differential Equations (ODEs):
- Existence and Uniqueness of
solutionof initial value problems for first orderordinary differential equations, singular solutions of first orderODEs, systemof first orderODEs. Generaltheory of homogeneous and non-homogeneous linear ODEs, variationof parameters, Sturm- Liouville boundary value problem, Green’s function.
Partial Differential Equations (PDEs):
- Lagrange and Charpit methods for solving
first orderPDEs, Cauchyproblem for first orderPDEs.
- Classification of
second orderPDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat andWave equations.
- Numerical solutions of algebraic equations, Method of iteration and Newton- Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.
Calculus of Variations:
- Variation of a functional, Euler- Lagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations.
Linear Integral Equations:
Linearintegral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.
- Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s principle and
principleof least action. Two- dimensional motion of rigid bodies, Euler’s dynamical
UNIT – 4
Descriptive statistics, exploratory data analysis.
- Sample space, discrete probability, independent events, Bayes theorem. Random variables and distribution functions (univariate and multivariate): expectation and moments. Independent random variables, marginal and conditional distributions. Characteristic functions. Probability inequalities (
Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central limit theorems (i.i.d. case).
- Markov chains with finite and countable state space, classification of states, limiting behavior of n-step transition probabilities, stationary distribution.
- Standard discrete and continuous univariate distributions. Sampling Distributions. Standard errors and asymptotic distributions, distribution of order statistics and range.
- Methods of estimation. Properties of estimators. Confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, Likelihood ratio tests. Analysis of discrete data and chi-square test of goodness of fit. Large sample tests.
- Simple nonparametric tests for one and two sample problems, rank correlation and test for independence. Elementary Bayesian inference.
- Gauss-Markov models, estimability of parameters, Best linear unbiased estimators, tests for linear hypotheses and confidence intervals. Analysis of variance and covariance. Fixed, random and mixed effects models. Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression.
- Multivariate normal distribution, Wishart distribution
andtheir properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation.
- Simple random sampling, Stratified sampling
andSystematic Sampling. Probability Proportional to size sampling. Ratio and regression methods.
- Completely randomized, randomized blocks and Latin-square designs. Connected, complete and orthogonal blocks designs, BIBD. 2k factorial experiments: confounding and construction.
- Series and parallel systems, hazard function and failure rates, censoring and life testing.
- Linear programming problem. Simplex methods, duality. Elementary queuing and inventory models. Steady-state solutions of Markovian queuing models: M/M/I, M/M/I with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/I.