PART 'A' : CORE

## Mathematical Methods of Physics

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 Laplac 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

## 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.

## 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

## 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

## 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

## 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

Leastsquares fitting

PART 'B' : ADVANCED

## 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

Simpson's rule, Solution of first order differential equation using Runge kutta method.

Finite difference methods.

Tensor.

Introductory group theory: SU(2), O(3)

## Classical Mechanics

Dynamical systems

Phase space dynamics

stability analysis.

Poisson brackets and canonical transformations.

Symmetry, invariance and Noether's theorem.

Hamilton-Jacobi theory.

## 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.

## 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.

## 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.

## 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), filteringand noise reduction, shielding and grounding.

Fourier transforms, lock-in detector, box-car integrator, modulation techniques.

High frequency devices (including generators and detectors).

## 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.

## 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.

## 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.