SYLLABUS OF PHYSICAL SCIENCES

 

 

Dimensional analysis. Vector algebra and vector calculus. Linear algebra, matrices, Cayley-Hamilton

Theorem. Eigenvalues and eigenvectors. Linear ordinary differential equations of first & second order,

Special functions (Hermite, Bessel, Laguerre and Legendre functions). Fourier series, Fourier and Laplace

transforms. Elements of complex analysis, analytic functions; Taylor & Laurent series; poles, residues

and evaluation of integrals. Elementary probability theory, random variables, binomial, Poisson and

normal distributions. Central limit theorem.

 

II. Classical Mechanics

Newton’s laws.  Dynamical systems, Phase space dynamics, stability analysis. Central force motions.

Two body Collisions  - scattering in laboratory and Centre of mass frames.  Rigid body dynamicsmoment of inertia tensor. Non-inertial frames and pseudoforces. Variational principle. Generalized

coordinates. Lagrangian and Hamiltonian formalism and equations of motion. Conservation laws and

cyclic coordinates. Periodic motion:  small oscillations, normal modes. Special theory of relativityLorentz transformations, relativistic kinematics and mass–energy equivalence.

 

III. Electromagnetic Theory  

Electrostatics: Gauss’s law and its applications,  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 the 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. Dynamics

of charged particles in static and uniform electromagnetic fields.

 

IV. Quantum Mechanics   

Wave-particle duality. Schrödinger equation (time-dependent and time-independent). Eigenvalue

problems (particle in a box, harmonic oscillator, etc.). Tunneling through a barrier. Wave-function in

coordinate and momentum representations. Commutators and Heisenberg uncertainty principle. Dirac

notation for state vectors. Motion in a central potential: orbital angular momentum, angular momentum

algebra, spin, addition of angular momenta; Hydrogen atom. Stern-Gerlach experiment. Timeindependent perturbation theory and applications. Variational method. Time dependent perturbation

theory and Fermi’s golden rule, selection rules. Identical particles, Pauli exclusion principle, spin-statistics

connection.

 

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their consequences. Thermodynamic potentials, Maxwell relations,

chemical potential, phase equilibria. Phase space, micro- and macro-states. Micro-canonical, canonical and grand-canonical ensembles and partition functions. Free energy and its connection with

thermodynamic quantities. Classical and quantum statistics. Ideal  Bose and Fermi gases. Principle of

detailed balance. Blackbody radiation and Planck’s distribution law.

 

VI. Electronics and Experimental Methods

Semiconductor devices (diodes, junctions, transistors, field effect devices, homo- and hetero-junction

devices), device structure, device characteristics, frequency dependence and applications. Opto-electronic

devices (solar cells, photo-detectors, LEDs).  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.

Data interpretation and analysis. Precision and accuracy. Error analysis, propagation of errors. Least

squares fitting,

 

PART ‘B’ ADVANCED

I. Mathematical Methods of Physics 

Green’s function. Partial differential equations (Laplace, wave and heat equations in two and three

dimensions). Elements of computational techniques: root of functions, interpolation, extrapolation,

integration by trapezoid and Simpson’s rule, Solution of first order differential equation using RungeKutta method. Finite difference methods. Tensors. Introductory group theory: SU(2), O(3).

 

II. Classical Mechanics

Dynamical systems, Phase space dynamics, stability analysis.    Poisson brackets and canonical

transformations. Symmetry, invariance and Noether’s theorem. Hamilton-Jacobi theory.

 

III. Electromagnetic Theory  

Dispersion relations in plasma. Lorentz invariance of Maxwell’s equation. Transmission lines and wave

guides. Radiation- from moving charges and dipoles and retarded potentials.

 

IV. Quantum Mechanics 

Spin-orbit coupling, fine structure. WKB approximation. Elementary theory of scattering: phase shifts,

partial waves, Born approximation. Relativistic quantum mechanics: Klein-Gordon and Dirac equations.

Semi-classical theory of radiation.

 

V. Thermodynamic and Statistical Physics

First- and second-order phase transitions. Diamagnetism, paramagnetism, and ferromagnetism. Ising

model. Bose-Einstein condensation. Diffusion equation. Random walk and Brownian motion.

Introduction to nonequilibrium processes.

 

VI. Electronics and Experimental Methods

Linear and nonlinear curve fitting, chi-square test. Transducers (temperature, pressure/vacuum, magnetic

fields,  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.

High frequency devices (including generators and detectors).

 

VII. Atomic & Molecular Physics

Quantum states of an electron in an atom. Electron spin. Spectrum of helium  and alkali atom. Relativistic

corrections for energy levels of hydrogen atom,  hyperfine structure and isotopic shift, width of spectrum

lines, LS & JJ couplings. Zeeman, Paschen-Bach & Stark effects. Electron spin resonance. Nuclear

magnetic resonance, chemical shift. Frank-Condon principle. Born-Oppenheimer approximation.

Electronic, rotational, vibrational and Raman spectra of diatomic molecules, selection rules.  Lasers:

spontaneous and stimulated emission, Einstein A & B coefficients.  Optical pumping, population

inversion, rate equation. Modes of resonators and coherence length.

 

VIII. 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. Electron motion in a periodic potential, band theory of solids: metals, insulators

and semiconductors. Superconductivity: type-I and type-II superconductors. Josephson junctions.

Superfluidity. Defects and dislocations.  Ordered phases of matter: translational and orientational order,

kinds of liquid crystalline order. Quasi crystals.

 

IX. Nuclear and Particle Physics

Basic nuclear properties: size, shape and charge distribution, spin and parity. Binding energy, semiempirical mass formula, liquid drop model. Nature of the nuclear force, form of nucleon-nucleon

potential, charge-independence and charge-symmetry of nuclear forces. Deuteron problem. Evidence of

shell structure, single-particle shell model, its validity and limitations. Rotational spectra. Elementary

ideas of alpha, beta and gamma decays and their selection rules. Fission and fusion. Nuclear reactions,

reaction mechanism, compound nuclei and direct reactions.

 

Classification of fundamental forces. Elementary particles and their quantum numbers (charge, spin,

parity, isospin, strangeness, etc.). Gellmann-Nishijima formula. Quark model, baryons and mesons. C, P,

and T invariance. Application of symmetry arguments to particle reactions. Parity non-conservation in

weak interaction.  Relativistic kinematics.

EXAM SCHEME OF PHYSICAL SCIENCES

CSIR-UGC (NET) EXAM FOR AWARD OF JUNIOR RESEARCH FELLOWSHIP AND ELIGIBILITY FOR LECTURERSHIP

PHYSICAL SCIENCES

EXAM SCHEME

TIME: 3 HOURS  

MAXIMUM MARKS: 200

SYLLABUS OF MATHEMATICAL SCIENCES

 

CSIR-UGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship

COMMON SYLLABUS FOR PART ‘B’ AND ‘C’ 

MATHEMATICAL SCIENCES

UNIT – 1 

Analysis: 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.

 

Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.

 

Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions

as examples.

 

Linear Algebra: Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear

transformations.

 

Algebra of matrices, rank and determinant 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 and classification of quadratic forms

 

UNIT – 2 

Complex Analysis: 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 theorem, 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.

 

Algebra: Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle, derangements.

 

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem,

Euler’s Ø- 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, Galois Theory.

 

Topology: basis, dense sets, subspace and product topology, separation axioms, connectedness and

compactness.

 

UNIT – 3 

Ordinary Differential Equations (ODEs): 

Existence and uniqueness of solutions of initial value problems for first order ordinary differential

equations, singular solutions of first order ODEs, system of first order ODEs.

 

General theory of homogenous and non-homogeneous linear ODEs, variation of parameters,

Sturm-Liouville boundary value problem, Green’s function.

 

Partial Differential Equations (PDEs): 

Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.

 

Classification of second order PDEs, General solution of higher order PDEs with constant

coefficients, Method of separation of variables for Laplace, Heat and Wave equations.

 

Numerical Analysis : 

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: 

Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with

separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.

 

Classical Mechanics: 

Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s

principle and principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical

equations for the motion of a rigid body about an axis, theory of small oscillations.

 

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 behaviour of n-step

transition probabilities, stationary distribution, Poisson and birth-and-death processes.

 

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, confidence intervals,

tests for linear hypotheses. 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 and their 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 and systematic sampling. Probability proportional to size

sampling. Ratio and regression methods.

 

Completely randomized designs, randomized block designs and Latin-square designs. Connectedness and

orthogonality of block designs, BIBD. 2K factorial experiments: confounding and construction.

 

Hazard function and failure rates, censoring and life testing, series and parallel systems.

 

Linear programming problem, simplex methods, duality. Elementary queuing and inventory models.

Steady-state solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C,

M/M/C with limited waiting space, M/G/1.

 

All students are expected to answer questions from Unit I. Students in mathematics are expected to answer additional question from Unit II and III.  Students with in statistics are expected to answer additional question from Unit IV.