Errors and uncertainties in computations: Types of errors, error in functions, errors in algorithms. Matrix computing and scientific libraries.

Zero-finding and matching: Newton's rule for finding roots. Quantum eigenvalues, particle in a box. Fields due to moving charges.

Integration: Trapezoid rule, Simpson's rule, Gaussian quadrature, multi-dimensional integrals. Monte-Carlo integrations.

Phase Transitions and Critical Phenomena
General Introduction: Origin of phase transition, thermodynamic instabilites, Maxwell construction. Classification of phase transitions: first order and second order. Phase transitions in different systems (e.g. liquid-gas and paramagnet-ferromagnetic transition), order parameter, critical exponents, concept of long-range order.

Lattice models: Ising Model, exact solution in one dimension, high-temperature and low-temperature expansions. Phase transitions in X-Y and Heisenberg Models.

Essential techniques: Probability distributions and statiscs, error analysis and error propagation, covariance, least-square fitting. Vacuum technology: gas flow equations, flow regimes, types of pumps, gauges and seals. Sensors and analog instrumentation: analog signal processing. Lock in amplifiers and applications: measurements in noise prone environments. Digital electronics: microprocessors and micro-controllers, ADC/DAC, PLCs, computer interfaces.

Semiconductor in equilibrium: Equilibrium distribution of electrons and holes, qualitative description of dopant atoms and energy levels, equilibrium distribution of electrons and holes in extrinsic semiconductor, degenerate and non-degenerate semiconductors, statistics of donors and acceptors, probability function, compensated semiconductors, Fermi energy levels and its variation with doping concentration and temperature, relevance of Fermi energy.

Approximation methods: Variational methods, WKB approximation; time-indepdndent perturbation theory; time-dependent perturbation theory: Interaction picture, Fermi's golden rule, sudden and adiabatic approximations.

Scattering theory: Transition rates and cross sections, Lippmann-Schwinger equation, scattering amplitude, Green's functions; Born approximation; phase shifts and partial waves.

Second quantization: Fock-space representation for bosons and fermions, representation of many-body operators.

Electron-electron interaction: Hartree and Hartree-Fock approximations; Quasiparticles, Landau-Fermi liquid theory for interaction between quasiparticles, equilibrium properties of normal Fermi liquid, 3He: ideal Fermi liquid, transport of quasiparticles, current density.