Vector fields: Orthogonal curvilinear co-ordinate systems – Expressions for gradient, divergence, curl and Laplacian – Linear vector spaces: Linear independence, basis, dimension, inner product – Schwartz inequality – Orthonormal basis – Gram – Schmidt orthogonalization process – Linear operators – Representation of vectors and operators in a basis – Matrix theory –
Cayley-Hamilton theorem – Inverse of a matrix – Diagonalisation of matrices – Operational methods: Laplace transforms – Solution of linear differential equations with constant coefficients – Fourier integral – Fourier transforms – Convolution theorems – Applications – Complex variables: Analytic function – Cauchy-Riemann conditions – Singular points – Multivalued function and branch points – Cauchy’s integral theorem and formula – Taylor’s and Laurentz’s expansions – Residue theorem and its applications.
Lagrangian and Hamiltonian formulations – Newton’s equations and conservation laws for a system of particles – D’Alembert’s principle and Lagrange’s equations of motion – Hamiltonian and Hamilton’s equation of motion – Application: Two-body central force problem – Scattering by central potential, two particle scattering – Cross section in Lab system-Small oscillations – Transformation to normal coordinates and frequencies of normal modes – Mechanics of rigid bodies: Angular momentum and kinetic energy – Moment of inertia tensor – Euler angles – Euler’s equation of motion – Torque free motion – Symmetric top - Wave motion – Phase velocity – Group velocity – Dispersion - Relativity: Special theory of relativity – Lorentz transformation – Addition of velocities – Mass-energy equivalence.
Basic principles: Wave - particle duality – Heisenberg’s uncertainty principle – Postulates of quantum mechanics – Interpretation of wave function – Schrodinger’s wave equation and its application to particle in a box – Harmonic oscillator – tunneling through a barrier – Motion in central field potential: Hydrogen atom – angular momentum and spherical harmonics – Addition of two angular momenta – Approximate methods: Time independent perturbation theory for non-degenerate case – application to anharmonic oscillator – time dependent perturbation theory – Fermi’s golden rule – Scattering theory: Scattering amplitude – cross section – Born approximation – Partial wave analysis – Identical particles and spin – Symmetric and antisymmetric wave functions – Representation theory – Coordinate and momentum representations.
Electrostatics – Laplace and Poisson equations – Boundary value problems – Magnetostatics – Ampere’s theorem - Biot- Savart law – Electromagnetic induction - Maxwell’s equations in free space and in linear isotropic media – Boundary conditions on the fields at interfaces – Scalar and vector potentials – Gauge invariance – Electromagnetic waves – Reflection, refraction, dispersion, interference, diffraction and polarization – Electrodynamics of a charged particle in electric and magnetic fields – Radiation from moving charges and from a dipole – Retarded potential.
Laws of thermodynamics and their consequences – Thermodynamic potentials and Maxwell’s relations – Chemical potential and phase equilibria – Phase space, microstates and macrostates – Partition function – Free energy and its connection with thermodynamic quantities – Classical and quantum statistics – Degenerate electron gas – Black body radiation and Planck’s distribution law – Bose – Einstein condensation – Einstein and Debye models for lattice specific heat.
Quantum states of an electron in an atom - Hydrogen atom spectrum – Electron spin - Stern-Gerlach experiment – Spinorbit coupling – Fine structure – Relativistic correction – Spectroscopic terms and selection rules – Hyperfine structure – Exchange symmetry of wave functions _ Pauli’s exclusion principle – Periodic table – Alkali-type spectra – LS and JJ coupling – Zeeman, Paschen-Back and Stark effects – X-rays and Auger transitions – Compton effect – Principles of ESR, NMR – Molecular Physics: Covalent, ionic and Vander Waal’s interactions – Rotation/vibration spectra – Raman spectra – Selection rules – Nuclear spin and intensity alternation – Isotopic effects – Electronic states of diatomic molecules – Frank-Condon principle – Lasers: Spontaneous and stimulated emission – Optical pumping – Population inversion – Coherence (temporal and spatial) – Simple description of ammonia maser - CO2 and He-Ne lasers.
Crystal classes and systems – 2d and 3d lattices – Bonding in common crystal structures – Reciprocal lattice – Diffraction and structure factor – Elementary ideas about point defects and dislocations – Lattice vibrations – Phonons – Specific heat of solids – Free electron theory – Fermi statistics – Heat capacity – Electron motion in periodic potential – Energy bands in metals, insulators and semiconductors – Tight binding approximation – Impurity level in doped semiconductors – Electronic transport from classical kinetic theory – Electrical and thermal conductivities – Hall effect and thermoelectric power -transport in semiconductors – Dielectrics – Polarization mechanism – Clausius-Mossotti equation – Piezo, pyro and ferroelectricity – Dia and paramagnetism – Exchange interactions – Magnetic ordering: ferro, antiferro and ferrimagnetism – Superconductivity: Basic phenomenology – Meissner effect – Type 1 and Type 2 superconductors – BCS pairing mechanism.
Basic nuclear properties – Size, shape, charge distribution, spin and parity - binding energy - empirical mass formula – liquid drop model – Nuclear forces – Elements of two-body problem – Charge independence and charge symmetry of nuclear forces – Evidence of nuclear shell structure – Single particle shell model – Its validity and limitations – Collective model – Interactions of charged particles and e. m. rays with matter – Basic principles of particle detectors – Ionization chamber – Proportional counter – GM counter – Scintillation and semiconductor detectors – Radioactive decays: Basic theoretical understanding - Nuclear reactions – Elementary ideas of reaction mechanism – Compound nucleus and direct reactions – Elementary ideas of fission and fusion – Particle Physics: Symmetries and Conservation laws – Classification of fundamental forces and elementary particles – Iso-spin – Strangeness – Gell-Mann Nishijima formula – Quark model – C. P. T invariance in different interactions – Parity nonconservation in weak interaction.
Physics of p-n junction – Diode as a circuit element – Clipping – Clamping – Rectification – Zener regulated power supply – Transistor as a circuit element – CC, CB and CE configuration – Transistor as a switch, OR, AND, NOT gates – Feedback amplifiers – Operational amplifier and its applications - Inverting, non-inverting amplifier – Adder – Subtractor – Integrator -
Differentiator – Waveform generator – Comparator – Schmidt trigger – Digital integrated circuits – NAND and NOR gates as building blocks – X - OR gate – Simple combinational circuits – Half and full adder – Flip-flop – Shift register – Counters – Basic principles of A/D and D/A converters – Simple applications of A/D and D/A converters – Microprocessor 8085: Architecture – Addressing modes – Instruction sets – Simple programming.
Measurement of fundamental constants: e, h, c – Measurement of high and low resistances, L and C – Detection of X – rays, gamma rays, charged particles, neutrons etc – Ionization chamber - proportional counter – GM counter – Scintillation detectors – Solid state detectors – Emission and absorption spectroscopy – Measurement of magnetic field – Hall effect – Magnetoresistance – X – ray and neutron diffraction – Vacuum techniques – Basic idea of conductance – Pumping speed etc – Pumps: Mechanical pump – Diffusion pump – Gauges: Thermocouple – Penning – Pirani – Hot cathode – Low temperature: Cooling a sample over a range upto 4 K and measurement of temperature – Error analysis and hypothesis testing – Propagation of errors – Plotting of graph - Distributions - Least squares fitting - Criteria for goodness of fits – Chi square test.
