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I. Mathematical Methods of

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

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,

I.Mathematical Methods of

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

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

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

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

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.