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Friday, October 22, 2010

SYLLABUS FOR PHYSICAL SCIENCES PAPER I AND PAPER II

The full Syllabus for Part B of Paper I and Part B of Paper II.

The syllabus for Part A of Paper II comprises Sections I-VI.

I. Mathematical Methods of Physics

Dimensional analysis; Vector algebra and vector calculus; Linear algebra, matrices, Cayley Hamilton theorem, eigenvalue problems; Linear differential equations; Special functions (Hermite, Bessel, Laguerre and Legendre); Fourier series, Fourier and Laplace transforms; Elements of complex analysis: Laurent series-poles, residues and evaluation of integrals; Elementary ideas about tensors; Introductory group theory, SU(2), O(3); Elements of computational techniques: roots of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, solution of first order differential equations using Runge-Kutta method; Finite difference methods; Elementary probability theory, random variables, binomial, Poisson and normal distributions.

II. Classical Mechanics

Newton’s laws; Phase space dynamics, stability analysis; Central-force motion; Two-body collisions, scattering in laboratory and centre-of-mass frames; Rigid body dynamics, moment of inertia tensor, non-inertial frames and pseudoforces; Variational principle, Lagrangian and Hamiltonian formalisms and equations of motion; Poisson brackets and canonical transformations; Symmetry, invariance and conservation laws, cyclic coordinates; Periodic motion, small oscillations and normal modes; Special theory of relativity, Lorentz transformations, relativistic kinematics and mass–energy equivalence.

III. Electromagnetic Theory

Electrostatics: Gauss’ 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 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; Dispersion relations in plasma; Lorentz invariance of Maxwell’s equations; Transmission lines and wave guides; Dynamics of charged particles in static and uniform electromagnetic fields; Radiation from moving charges, dipoles and retarded potentials.

IV. Quantum Mechanics

Wave-particle duality; Wave functions in coordinate and momentum representations; Commutators and Heisenberg's uncertainty principle; Matrix representation; Dirac’s bra and ket notation; Schroedinger equation (time-dependent and time-independent); Eigenvalue problems such as particle-in-a-box, harmonic oscillator, etc.; Tunneling through a barrier; Motion in a central potential; Orbital angular momentum, Angular momentum algebra, spin; Addition of angular momenta; Hydrogen atom, spin-orbit coupling, fine structure; Time-independent perturbation theory and applications; Variational method; WKB approximation;

Time dependent perturbation theory and Fermi's Golden Rule; Selection rules; Semi-classical theory of radiation; Elementary theory of scattering, phase shifts, partial waves, Born approximation; Identical particles, Pauli's exclusion principle, spin-statistics connection; Relativistic quantum mechanics: Klein Gordon and Dirac equations.

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; Phase space, micro- and macrostates; Microcanonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First- and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; Principle of detailed balance; Blackbody radiation and Planck's distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to nonequilibrium processes; Diffusion equation.

VI. Electronics

Semiconductor device physics, including diodes, junctions, transistors, field effect devices, homo and heterojunction devices, device structure, device characteristics, frequency dependence and applications; Optoelectronic devices, including solar cells, photodetectors, and LEDs; High-frequency devices, including generators and detectors; 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.

VII. Experimental Techniques and data analysis

Data interpretation and analysis; Precision and accuracy, error analysis, propagation of errors, least squares fitting, linear and nonlinear curve fitting, chi-square test; Transducers (temperature, pressure/vacuum, magnetic field, 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.

Applications of the above experimental and analytical techniques to typical undergraduate and graduate level laboratory experiments.

VIII. Atomic & Molecular Physics

Quantum states of an electron in an atom; Electron spin; Stern-Gerlach experiment; Spectrum of Hydrogen, helium and alkali atoms; Relativistic corrections for energy levels of hydrogen; Hyperfine structure and isotopic shift; width of spectral lines; LS & JJ coupling; Zeeman, Paschen Back & Stark effect; X-ray spectroscopy; Electron spin resonance, Nuclear magnetic resonance, chemical shift; Rotational, vibrational, electronic, and Raman spectra of diatomic molecules; Frank – Condon principle and selection rules; Spontaneous and stimulated emission, Einstein A & B coefficients; Lasers, optical pumping, population inversion, rate equation; Modes of resonators and coherence length.

IX. 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; Diamagnetism, paramagnetism, and ferromagnetism; Electron motion in a periodic potential, band theory of metals, insulators and semiconductors; Superconductivity, type – I and type - II superconductors, Josephson junctions; Defects and dislocations; Ordered phases of matter, translational and orientational order, kinds of liquid crystalline order; Conducting polymers; Quasicrystals.

X. Nuclear and Particle Physics

Basic nuclear properties: size, shape, charge distribution, spin and parity; Binding energy, semi-empirical mass formula; Liquid drop model; Fission and fusion; Nature of the nuclear force, form of nucleon-nucleon potential; Charge-independence and charge-symmetry of nuclear forces; Isospin; 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; Nuclear reactions, reaction mechanisms, compound nuclei and direct reactions; Classification of fundamental forces; Elementary particles (quarks, baryons, mesons, leptons); Spin and parity assignments, isospin, strangeness; Gell-Mann-Nishijima formula; C, P, and T invariance and applications of symmetry arguments to particle reactions, parity non-conservation in weak interaction; Relativistic kinematics.

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Monday, October 18, 2010

Waves in continous medium

  • There are essentially two ways of transporting energy from one place to another (a) Actual transport of matter for example a fired bullet and (b) Waves carry energy but there is no transport of matter for example sound waves carry energy so thay can move diphagram of the ear.
  • Here we will consider the oscillations of open or unbounded systems i.e., systems having no outer boundaries.
  • If such system is disturbed , waves travel in the system with a speed determined by the properties of the system.
  • Waves are not reflected back in such a system.
  • The waves generated by driving force are called travelling waves ; these waves travel from the point where the driving force produces the disturbance.
  • If the driving force produces a harmonic disturbance the travelling wave it produces are called harmonic travelling waves.
  • In the steady state, all moving parts of the system oscillates with simple harmonic motion at the driving frequency.
  • Waves where the displacements or oscillations are transverse (i.e., perpandicular) to the direction of wave propagation is called transverse wave.
  • The wavelength (denoted by λ) of the wave is defined as the distance, measured along the direction of the propagation of the wave, between two nearest points which are in the same state of viberation.
  • Wavelength λ is just the distance travelled by the wave during one time period T of particle oscillation. Thus wave velocity
    v=λ/T=λν
    where ν=1/T - is the frequency of the wave.
  • This relation between wave velocity, frequency and wavelength also holds for longitudinal waves in which the displacements or oscillation in the medium are parallel to the direction of wave propagation.
  • Waves in spring and sound waves are longitudinal waves.
  • Wavelength for longitudinal waves is the distance between two successive compressions or rarefactions.
  • Sound waves are also compressional.
  • Assumptions that are made while obtaining wave equation are:-
    1. Amplitude A of particle oscillations does not change in course of the propagation of wave.
    2. The medium is isotopic and homogeneous so that velocity of wave does not chance from place to  place
  • Displacement of particle at x at any time t is
    Ψ(x,t) = A sin{2π(t-x/v)/T)}
  • The function Ψ(x,t) repeats itself in a distance λ . Wavelength of a wave is also known as spatial periodicity of the wave.
  • The wave is thus doubly periodic. It has temporal periodicity T and spatial periodicity λ.
  • Let us define quantities
    k=2π/λ and
    ω=2π/T
    then wave function can be written as
    Ψ(x,t) = A sin{ωt-kx}
    where quantity k is known as wave number of the wave and ω is called angular frequency of particle oscillations in wave.
  • Harmonic wave travelling in
    + x direction    :     Ψ(x,t) = A sin{ωt-kx}
    - x direction    :     Ψ(x,t) = A sin{ωt+kx}
    above equations can also be equally well described by cosine function.
  • Classicsl wave equation is



  • Important inferences from above wave equation
    1. Whenever second order time derivative of any physical quantity is related to second order space derivative as in above equation , a wave of some sort must travell in the medium.
    2. Velocity of that wave is given by the square root of the coefficent of second order space derivative.
  • Individual derivatives which makes up the medium do not propagate through the medium with the wave; they merely oscillates ( transversly or longitudinally) about there equilibrium positions.
  • It is their phase relationship which we observe as wave.
  • Wave velocity is also called phase velocity with which crest or troughs in case of transverse wave and compressions or rarefactions in case of longitudinal waves travell through the medium.
  • The phase velocity is given by
    v=λ/T=λν
    or,
    v=ω/k
  • Ψ(x,t)=f(vt-x) is the solution of the above given wave equation.
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