IGNOU

1. Partial Differential Equations & Special Functions

Partial Differential Equations (PDEs)

• Partial differential equations: basic concepts

• Method of separation of variables

• Laplace equation:

  • Cartesian coordinates

  • Cylindrical coordinates

  • Spherical coordinates

• Poisson equation

• Heat diffusion equation

• Wave equation

Integral Equations

• Integral equations: basic formulation

• Fredholm integral equations

• Volterra integral equations

Special Functions

• Legendre polynomials

• Hypergeometric functions

• Bessel functions

• Spherical Bessel functions

• Spherical harmonics

• Hermite polynomials

• Laguerre polynomials

Sturm–Liouville Theory

• Sturm–Liouville problems

• Expansion in orthogonal functions

2. Vector Spaces, Matrices & Tensors

Vector Spaces

• Finite-dimensional real linear vector spaces

• Linear dependence and independence

• Basis and dimension

• Vector spaces of matrices

Linear Operators & Matrices

• Linear mappings (operators)

• Matrix representation of operators

• Similarity transformations

• Matrix diagonalization

Inner Product Spaces

• Inner product

• Orthogonality

• Gram–Schmidt orthogonalization

• Norm

• Cauchy–Schwarz inequality

Complex Vector Spaces

• Finite-dimensional complex vector spaces

• Hermitian inner product

• Adjoint of an operator

Special Operators

• Hermitian operators

• Unitary operators

• Eigenvalues and eigenvectors of:

  • Hermitian matrices

  • Unitary matrices

Dual Spaces

• Dual vector space

• Dual basis

Tensors

• Elements of tensors

• Applications in physics:

  • Moment of inertia tensor

  • Elasticity tensor

  • Stress tensor in fluids

  • Metric tensor in relativity

3. Complex Analysis

Complex Functions

• Functions of a complex variable

• Analytic functions

• Cauchy–Riemann conditions

Singularities

• Zeros and singular points

• Classification of singularities

Multivalued Functions

• Branch points

• Branch cuts

Complex Integration

• Cauchy’s integral theorem

• Cauchy’s integral formula

• Contour integration

Series Expansions

• Taylor series

• Laurent series

• Analytic continuation

Residue Theory

• Residue theorem

• Jordan’s Lemma

• Evaluation of definite integrals

• Principal value integrals

• Summation of series

Conformal Mapping

• Conformal transformations

• Applications

Special Functions

• Gamma function

4. Fourier & Laplace Transforms

Fourier Transforms

• Fourier transform: definition and examples

• Sine and cosine transforms

• Complex Fourier transforms

• Dirac delta function:

  • Definition

  • Properties

  • Representations

• Properties of Fourier transforms

• Fourier transform of derivatives

• Parseval’s theorem

• Applications to partial differential equations

Laplace Transforms

• Laplace transform: definition and properties

• Examples of Laplace transforms

• Convolution theorem

• Applications of convolution

• Solving differential equations using Laplace transforms

5. Group Theory

Basic Group Concepts

• Definition of a group

• Subgroups

• Cosets

Group Mappings

• Homomorphism

• Isomorphism

Advanced Group Structure

• Invariant (normal) subgroup

• Quotient group

Matrix Representations

• Matrix representations of groups

Continuous Groups

• Continuous groups

• Groups of transformations

Geometrical Groups

• Translation groups in 2D and 3D

• Rotation groups in 2D and 3D

Lie Groups

• One-parameter subgroups

• Generators of continuous transformations

Relativistic Groups

• Lorentz group

• Poincaré group

• ( SL(2,\mathbb{C}) ) and its relation to the Lorentz group

Special Unitary Groups

• ( SU(2) )

• ( SU(3) )

Electrostatics: Electric field, Electric potential; Gauss’s law, Poisson and Laplace equations, solutions of Laplace equation (with emphasis on applications), boundary value problems in electrostatics; Special Techniques: method of images with applications, Electrostatics of macroscopic media, polarization of a medium; Microscopic properties of dielectrics (Claussius-Mossotti relation), Simple model of a dielectric; Electrostatic energy of a dielectric. Magnetostatics: Lorentz force law, Integral and differential forms of Biot-Savart law, Magnetic vector potential and multipole expansion of the vector potential (emphasis on dipole expansion); Magnetic fields of localized current distributions, Torques and forces on magnetic dipoles, Effect of a magnetic field on atomic orbits, Magnetic fields in matter, Magnetization, Bound currents, Ampere’s law in Magnetized materials, diamagnetism and paramagnetism; Ferromagnetic media, B and H fields; Magnetic circuits. Maxwell’s equations and electromagnetic waves. 

Introduction to Quantum Mechanics: Wave-particle duality, Gaussian wave packets and group velocity, probabilistic interpretation of the wave function and its normalization, Dirac delta function and Fourier transforms, Momentum and energy operators, Heisenberg's uncertainty principle, Ehrenfest’s Theorem. Time independent Schrodinger equation and stationary states, free particle solution. Applications of Quantum Mechanics: One-dimensional problems: particle-in-a-box, square well potential, linear harmonic oscillator, tunnelling through a barrier; Motion in a central potential, orbital angular momentum, the hydrogen atom problem. Quantum Mechanics in Hilbert Space: Hilbert space,linear operators Dirac’s bra-ket algebra, eigen values of self-adjoint operators, unitary operators, time evolution of state vectors, Heisenberg picture, coordinate and momentum representations, observables and measurement in quantum mechanics, collapse of the state vector (project ion postulate), expectation values of observables, generalized uncertainty principle. One-dimensional harmonic oscillator using operator algebra, angular momentum in quantum mechanics, ladder operators and their matrix representation, spin angular momentum and Pauli matrices, Stern Gerlach experiment. 

Electronic Devices: Construction, working, biasing, I-V characteristics, frequency response and applications of Diodes (Power Diode, LED, Laser Diode, Photodiode, Solar Cell, Varicap, p-i-n Diode, Tunnel Diode, Gunn Diode), Transistors (BJT, j-FET, MOSFET, CMOS, UJT ,SCR) Electronic Circuits: Amplifiers : Circuits, applications, limitations of Class A, Class B, Class AB, Class C amplifiers, Darlington pair configuration; feedback and stability concept, bandwidth of amplifiers; Oscillators: Positive feedback criterion for sustained oscillations, Audio and RF oscillators, clock generators, relaxation oscillators, microwave generators Operational Amplifiers: Characteristics of practical op-amps and their effect on applications, applications of negative and positive feedback op-amp circuits- Integrators, Differentiators, Schmitt trigger, active filters, precision rectifiers, logarithmic amplifier, sample-hold circuits. Special Electronic Circuits: Astable, monostable, bi-stable multivibrators, Phase Lock Loop (PLL) and Voltage Controlled Oscillators Power Supplies: Linear and Switch-mode power supply, current limiting and fold back operations, DC-AC inverter Digital Electronics Applications: Sampling theorem, Digital to Analog and Analog to Digital Converters, Signal to noise ratio improvement techniques. Microprocessor-Microcontrollers: Microprocessor (8085): Architecture, block diagram and introduction to assembly language programming with some simple programmes; Microcontroller (8051): Architecture, block diagram and some basic applications. 

The Hamilton Equations of Motion: Legendre transformations, Hamilton equations of motion, Principle of least action. Phase plot, fixed points and their stabilities. Page | 22 Canonical Transformations: The equations of canonical transformations, Poisson brackets, equations of motion, invariance of Poisson’s brackets under canonical transformation, Liouville’s theorem. The Hamilton Jacobi Theory: Solution to the time dependent Hamilton-Jacobi equation, Jacobi’s theorem, action angle variables, adiabatic invariants. Rigid Body Dynamics: Degrees of freedom of a rigid body, Euler’s theorem, kinetic energy of rotating rigid body, angular momentum, moment of inertia tensor, Euler’s equation of motion for rigid bodies, Euler angles, motion of a heavy symmetrical top. 

Maxwell’s equations and electromagnetic waves, Faraday’s law, generalized Ampere’s law, Displacement current, Maxwell’s equations, Gauge symmetry, Coulomb and Lorentz gauges, Electromagnetic energy and momentum (mention only), Poynting’s theorem and conservation of energy and momentum, Electromagnetic waves in vacuum. Dynamics of charged particles, Motion of charged particles in uniform static electric field, uniform static magnetic field, crossed electric and magnetic fields, spatially non-uniform magnetic field, B  drift and time varying electric fields.Electromagnetic waves in different media, Derivation of ac conductivity, ac dielectric susceptibility, effective relative permittivity for a plasma, Wave propagation in dielectrics, conductors and plasmas, phase and group velocity; Reflection and refraction (including oblique incidence); Dispersion relation and energy propagation; Guided wave propagation, Waveguides, Applications (radio communication and other applications). Radiating systems, Inhomogeneous wave equation and Green’s function solution, wave equation for vector and scalar potential, electric dipole fields and radiation, magnetic dipole, short antenna. Special theory of relativity and electrodynamics, Michelson Morley experiment, Lorentz transformations, four vectors, Transformation of electric and magnetic fields, Invariance of Maxwell’s equations, Lorentz invariants; Four vector potential, Electromagnetic field tensor, Lorentz force on a charged particle, invariance of electric charge and Lorentz covariant formulation of electrodynamics; Electromagnetic field tensor in four dimensions and Maxwell’s equations, Dual field tensor. Page | 23 Radiation from Moving Charges, Liénard-Wiechert potentials, Electric and magnetic fields due to an accelerated charge and a uniformly moving charge. 

Symmetries and Conservation Laws in Quantum Mechanics: General concept of symmetry in Q. M. (unitary operators and symmetry), space translation and conservation of linear momentum, time translation and conservation of energy, rotation and conservation of angular momentum, parity. Multi-Particle Systems: Identical particles, Exchange symmetry for Fermions and Bosons, addition of angular momenta, CG coefficients. Approximation Methods for Stationary States: Time-independent perturbation theory and applications, degenerate case, Stark effect. spin-orbit coupling and fine structure, Zeeman effect without spin, WKB approximation, alpha particle lifetime calculations; the double oscillator, variational methods.Approximation Methods for Time-dependent Problems: Time- dependent potential, Interaction picture, time-dependent two-state problems with sinusoidal oscillating potential, application to NMR, time-dependent perturbation theory, Dyson series, transition probability, Fermi Golden Rule, constant and harmonic perturbation, semi-classical theory of radiation. Scattering Theory: scattering cross sections, integral equation formulation, partial wave analysis, Born approximation. Relativistic Wave Equation: Klein Gordonequation, Dirac’s relativistic wave equation, position probability density; expectation values, Dirac matrices, plane wave solutions of the Dirac equation; the spin of the Dirac particle, significance of negative energy states and antiparticles. 

Fundamentals of Statistical Mechanics: Macroscopic and microscopic states of a physical system, contact between statistical physics and thermodynamics, basic postulates of statistical mechanics, concept of ergodicity (time and ensemble average). Classical Ensemble Theory: Phase space, classical Liouville’s equation, Ensembles: microcanonical, canonical and grand canonical; classical ideal gas, the entropy of mixing and Gibbs paradox; partition functions, Equipartition and virial theorems; calculation of statistical quantities (examples of ideal gas and harmonic oscillator); energy and density fluctuations. Quantum Mechanical Ensemble Theory: Quantum micro and macro states, the density matrix/operator, quantum Liouville’s equation, quantum micro-canonical, canonical and grand canonical ensembles, examples of particle in a box; partition functions and distribution functions of ideal quantum gases, Fermi-Dirac statistics, and Bose-Einstein statistics, Maxwell- Boltzmann statistics as a limiting case, Black-body radiation, Bose- Einstein Condensation, electronic specific heat, Fermi energy, Pauli paramagnetism. Statistical Mechanics of Interacting Systems: Cumulant expansion and Cluster expansionfora classical gas, Virial expansion of equation of state, Pair correlation function, Radial distribution function, structure factor, relation to thermodynamic functions. Phase Transitions: Basics of phase transitions (first order and continuous phase transitions), Ising model, mean field theory, Landau’s theory of phase transitions, critical exponents.