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Subject Code 14PHDPHY001 14PHDPHY002 14PHDPHY003 14PHDPHY004 14PHDPHY005 14PHDPHY006 14PHDPHY007 14PHDPHY008 14PHDPHY009 14PHDPHY010 Compulsory

Name of the Subject Mathematical Physics Classical Mechanics and Statistical Mechanics Quantum Mechanics Atomic, Nuclear and Molecular Physics Solid State Physics, Nano Science and Technology Material Science Physics of Liquid Crystals and Polymer Science Flourescence Spectroscopy and X-ray Crystallography Laser Physics and Biophysics Electronics and Instrumentation Research Methodology

UNIT-1: Differential equations Partial differential equations: Classifications, systems of surfaces and characteristics,
examples of hyperbolic, parabola and elliptic equations, method of direct integration, method of separation of variables.
Special differential equations Power series method for ordinary differential equations, Legendre's differential equation: Legendre polynomials and their properties, Generating functions, Recurrence Formulae, orthogonality of Legendre's polynomial. Bessel's differential equation: Bessel's polynomial - generating functions, Recurrence Formulae, orthogonal properties of Bessel's polynomials. Laguerre's equation, its solution and properties. Hermite differential equation: Hermite polynomials, generating functions, recurrence relation.
UNIT-2: Laplace transforms Laplace transforms: Linearity property, first and second translation property of LT,
Derivatives of Laplace transforms, Laplace transform of integrals, Initial and Final value theorems, Transform of Dirac delta function, periodic function and derivatives. Methods for finding LT: direct and series expansion method, Method of differential equation. Inverse Laplace transforms: Linearity property, first and second translation property, Convolution property, Solution of linear differential equations with constant coefficients. Physical applications.
UNIT-3: Fourier series and integrals Fourier series definition and expansion of a function, Fourier's theorem. Cosine and sine
series. Change of interval. Complex form of Fourier series. Fourier integral. Extension to many variables. Fourier transform. Transform of impulse function. Constant unit step function and periodic function. Some physical applications. UNIT-4: Group Theory
Groups, subgroups, classes. Homomorphism and isomorphism. Group representation. Reducible and irreducible representations. Character of a representation, character tables. Construction of representations. Representations of groups and quantum mechanics. Lie groups. The three dimensional rotation group SO(3). The special unitary groups SU(2) and SU(3). The irreducible representations of SU(2). Representations of SO(3) from those of SU(2) Some applications of group theory in physics
UNIT-5: Numerical Techniques Numerical Methods. Solutions of algebraic and transcendental equations: Bisection,
iterative and Newton-Raphson methods. Interpolation: Newton's and Lagrange's methods.

Curve fitting: Method of least squares. Differentiation: Newton's formula. Integration: Trapezoidal rule, Simpson's 1/3 and 3/8 rules. Eigen values and eigen vectors of a matrix. Solutions of ordinary differential equations: Euler's modified method and Runge-Kutta methods.
REFERENCE BOOKS: 1. Mathematical Physics by P K Chattopadhyay, Wiley Eastern Ltd., Mumbai. 2. Mathematical Physics, B D Gupta, 3rd Edition, Vikas Publishing House Pvt. Ltd., 2006. 3. Mathematical Physics by Satya Prakash, S Chand and Sons, New Delhi. 4. Introduction to Mathematical Physics by C Harper, PHI. 5. Mathematical Physics, B S Rajput, 17th Edition, Pragati Prakasam, 2004. 6. Advanced Engineering mathematics, Erwin Kreyszig, 7th Edition, Wiley Eastern Limited Publications, 1993. 7. Mathematical Methods for Physics, G Arfken, 4th edition, 1992. 8. Special Function, W W Bell, 1996. 9. Introductory Methods of Numerical Analysis: S. S. Sastry, PHI, 1995. 10. Numerical Methods: E. Balagruswamy (TMH, 2001).

UNIT-1: Newtonian mechanics and Lagrangian formulation
Single and many particle systems - Conservation laws of linear momentum, angular momentum and energy. Application of Newtonian mechanics: Two-body central force field motion, Kepler's laws of planetary motion. Scattering in a central force field; Scattering cross-section; The Rutherford scattering problem.
Constraints in motion. Generalised co-ordinates, Virtual work and D'Alembert's principle. Lagrangian equations of motion. Symmetry and cyclic co-ordinates. Hamilton variational principle, Lagrangian equations of motion from variational principle. Simple applications.
UNIT-2: Hamiltonian formalism , Relativistic mechanics, Continuum mechanics
Hamilton's equations of motion - from Legendre transformations and the variational principle. Simple applications. Canonical transformations. Poisson brackets - Canonical equations of motion in Poisson bracket notation. Hamilton-Jacobi equations.
a. Relativistic mechanics: Four-dimensional formulation-four-vectors, four-velocity, fourmomentum, and four-acceleration. Lorentz co-variant form of equation of motion
b. Continuum mechanics: Basic concepts, Equations of continuity and motion; Simple applications
UNIT-3: Microcanonical, Canonical and Grandcanonical ensembles Microcanonical distribution function, Two level system in microcanonical ensemble,
Gibbs paradox and correct formula for entropy, The canonical distribution function. Contact with thermodynamics - Two level system in canonical ensemble, Partition function and free energy of an ideal gas, Distribution of molecular velocities.
Equipartition and Virial theorems, The grand partition function, Relation between grand canonical and canonical partition functions.
Fluctuations in canonical, grand canonical and microcanonical ensembles. The Brownian motion and Langevin equation. Random walk, diffusion and the Einstein relation for mobility. Fockker-Plank equation. Johnson noise and shot noise.
UNIT-4: Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann Distributions Bose-Einstein and Fermi-Dirac distributions, Thermodynamic quantities, Fluctuations in
different ensembles, Bose and Fermi distributions in microcanonical ensemble - MaxwellBoltzmann distribution law for microstates in a classical gas - Physical interpretation of the classical limit, Derivation of Boltzmann equation for change of states without and with collisions, Boltzmann equation for quantum statistics, Equilibrium distribution in Boltzmann equation.

UNIT-5: Thermodynamics, Microstates and Macrostates Basic postulates of thermodynamics, Fundamental relations and definition of intensive
variables, Intensive variables in the entropic formulation, Intensive variables in the entropic formulation - Equations of state, Euler relation, densities - Gibbs-Duhem relation for entropy Thermodynamic potentials and extensivity properties, Maxwell relations, Energy differential and thermodynamic potentials of systems in external magnetic field - Thermodynamic relations, Microstates and macrostates, Ideal gas, Microstate and macrostate in classical systems, Microstate and macrostate in quantum systems, Density of states.
REFERENCE BOOKS: 1. Classical Mechanics: H Goldstein, (Addison-Wesley, 1950). 2. Introduction to Classical Mechanics: R G Takawale and P S Puranik (TMH, 1979). 3. Classical Mechanics: N C Rana and P S Joag (Tata McGraw, 1991). 4. Mechanics: Landau L D and Lifshitz E M (Addition-Wesley, 1960). 5. An Introductory Course of Statistical Mechanics, Palash B. Pal, Narosa Publishing House, New Delhi, 2008. 6. Elements of Statistical Mechanics, Kamal Singh & S. P. Singh, S. Chand & Company, New Delhi, 1992. 7. Statistical Mechanics an Elementary Outline, Avijit Lahiri, University Press, Hyderabad, 2002.

UNIT-1 a. Physical basis of quantum mechanics: Experimental background, inadequacy of classical physics, summary of principal experiments and inferences, Uncertainty and complementarity. Wave packets in space and time, and their physical significance. b. Schrodinger wave equation: Development of wave equation: One-dimensional and extension to three dimensions inclusive of forces. Interpretation of wave function: Statistical interpretation, normalisation, expectation value and Ehrenfest's theorem. Energy eigen functions: separation of wave equation, boundary and continuity conditions. One dimensional: Square well and rectangular step potentials, Rectangular barrier, Harmonic oscillator. Three dimensional: Particle in a box, Particle in spherically symmetric potential, Rigid rotator, Hydrogen atom.
UNIT-2: General formalism of quantum mechanics Hilbert space. Operators-definition and properties, eigen values and eigen vectors of an
operator; Hermitian, unitary and projection operators, commuting operators, complete set of commuting operators. Bra and Ket notation for vectors. Representation theory: matrix representation of an operator, change of basis. Co-ordinate and momentum representations.
The basic formalism: The fundamental postulates, expectation values and probabilities; quantum mechanical operators, explicit representation of operators, uncertainty principle. Matrix method solution of linear harmonic oscillator.
Quantum dynamics: Equations of motion, Schrodinger, Heisenberg and Interaction pictures. Poisson brackets and commutator brackets.
UNIT-3 a. Approximation methods for stationery states: Time-independent perturbation theory; non-degenerate and degenerate cases, perturbed harmonic oscillator. The variation method: Application to ground state of Helium. WKB method: Application to barrier penetration. Bohr-Sommerfield quantum condition. b. Theory of scattering: Scattering cross-section, wave mechanical picture of scattering, scattering amplitude. Born approximation. Partial wave analysis: phase shifts, scattering amplitude in terms of phase shifts, optical theorem; exactly soluble problem- scattering by square well potential. c. Time-dependent phenomena Perturbation theory for time evolution, first and second order transition amplitudes and
their physical significance. Applications of first order theory: constant perturbation, wide and closely spaced levels-Fermi's golden rule, scattering by a potential. Harmonic perturbation: interactions of an atom with electromagnetic radiation, dipole transitions and selection rules; spontaneous and induced emission, Einstein A and B coefficients. Sudden approximation.

UNIT-4 a. Identical particles and spin: Indistinguishability of identical particles. Symmetry of wave function and spin. Bosons and Fermions. Pauli exclusion principle. Singlet and triplet states of He atom and exchange integral Spin angular momentum, Pauli matrices. b. Angular momentum: Definition, eigenvalues and eigenvectors, matrix representation, orbital angular momentum. Addition of angular momenta, Clebsch-Gordon coefficients for simple cases: j1 = ½, j2 = ½ and j1 = 1, j2 = ½ .
UNIT-5 a. Symmetry principles: Symmetry and conservation laws, symmetry and degeneracy. Space-time symmetries, Displacement in space- conservation of linear momentum, Displacement in time, conservation of energy, Rotation in space, conservation of angular momentum, Space inversion,parity. Time reversal invariance. b. Relativistic wave equations: Schrodinger's relativistic equation: free particle, electromagnetic potentials, separation of equations, energy level in a coulomb field. Dirac's relativistic equation: free particle equation, Dirac matrices, free particle solutions, charge and current densities. Electromagnetic potentials. Dirac's equation for central field: spin angular momentum, approximate reduction, spin orbit energy. Separation of the equation. The Hydrogen atom, classification of energy levels and negative energy states.
REFERENCE BOOKS: 1. Quantum Mechanics: L. I. Schiff (McGraw-Hill, 1968). 2. Quantum Mechanics: F. Schwabl (Narosa, 1995). 3. Text book of Quantum Mechanics: P. M. Mathews and K. Venkateshan (TMH, 1994). 4. Quantum Mechanics: V. K. Thankappan (Wiley Eastern, 1980). 5. Quantum Mechanics: B. K. Agarwal and Hari Prakash (Prentice-Hall, 1997).

UNIT-1 a. One electron System: Quantum states of one electron atoms, atomic orbitals, hydrogen spectrum. Spectra of alkali elements, spin-orbit interaction and fine structure in alkali spectra. (Ref: 1, 6, 7) b. Two electron Systems: LS-coupling, equivalent and non-equivalent electrons, spectral terms, Pauli exclusion principle, coupling schemes for two electrons, interaction energies for LS coupling, fine structure splitting for sp electron configuration, Lande interval rule. jj-coupling- spectral terms, interaction energies for jj-coupling, fine structure splitting for sp electron configuration. Qualitative consideration of selection and intensity rules for LS and jj-coupling. Hyperfine structure for one and two electrons and Lande interval rule. (Ref: 1, 6, 7)
UNIT-2 a. Weak magnetic field effects: Normal and anamolous Zeeman effect, magnetic moment of a bound electron and Lande g-factor, magnetic interaction energy, selection rules, Zeeman pattern for principal series doublet, intensity rules. Zeeman effect for two
electrons-magnetic moment of the atom and g-factors, expression for magnetic interaction energy, selection rules, Zeeman pattern transitions for diffuse-series singlet, intensity rules. (Ref: 1, 6, 7)
b. Strong magnetic field and Electric field effects: Paschen-Back effect, expression for total energy shift, transitions for principal series doublet. Qualitative treatment of PaschenBack effect and complete Paschen- Back effect for two electrons. Isotope structure. Stark effect-first and second order Stark effects in hydrogen. Width of spectral lines (qualitative). (Ref: 1,6,7)
UNIT-3: Microwave, Infra-red spectra ,
UV-Visible spectra Types of molecules- linear, symmetric top, asymmetric top and spherical top molecules.
Theory of rotational spectra for rigid and non-rigid rotator diatomic molecules, energy levels, intensity of rotational lines. Microwave spectrometer and applications.
Vibrational energy of diatomic molecule as simple harmonic and anharmonic oscillators, Morse potential energy curve, energy levels and vibrational spectra. Diatomic molecule as a vibrating-rotator, vibration-rotation spectra-P,Q,R branches. IR- spectrometer and applications. (Ref: 2-7)

UV-Visible spectra Electronic spectra of diatomic molecules, Born-Oppenheimer approximation, vibrational
coarse structure- band progressions and sequences, Frank-Condon principleintensity of vibrational-electronic spectra, dissociation energy and dissociation products. Rotational fine structure of electronic-vibration transitions, determination of vibrational and rotational constants. Molecular orbital. Classification of electronic states and multiplet structure, selection rules for electronic transitions and simple electronic transitions. UVVisible absorption and fluorescence spectrophotometers and applications. (Ref: 2-7)
UNIT-4 a. Properties of Nucleus: Nuclear constitution. The notion of nuclear radius and its
estimation from Rutherford's scattering experiment; the coulomb potential inside the nucleus and the mirror nuclei. The nomenclature of nuclei, and nucleon quantum numbers. Nuclear spin and magnetic dipole moment. Nuclear electric moments and shape of the nucleus. b. Nuclear Forces: General features of nuclear forces. Bound state of deuteron with square well potential, binding energy and size of deuteron. Deuteron electric and magnetic moments - evidence for non-central nature of nuclear forces. Yukawa's meson theory of nuclear forces. c. Nuclear Reactions: Reaction scheme, types of reactions and conservation laws. Reaction kinematics, threshold energy and Q-value of nuclear reaction. Energetics of exoergic and endoergic reactions. Reaction probability and cross section. Bohr's compound nucleus theory of nuclear reactions. d. Nuclear Models: The shell model; Evidence for magic numbers, energy level, scheme for nuclei with Infinite Square well potential and the ground state spins. The extreme single particle prediction of nuclear spin and magnetic dipole moments -Schmidt limits. The liquid drop model: Nuclear binding energy, Bethe-Weizsacker's semi empirical mass formula; stability limits against spontaneous fission and nuclear decay.
UNIT-5 a. Nuclear Decays: Alpha decay: Quantum mechanical barrier penetration, Gammow's theory of alpha decay and alpha half-life systematics. Beta decay: Continuous beta spectrum, neutrino hypothesis, and Fermi's theory of beta decay, beta comparative halflife systematics. Gamma decay: Qualitative consideration of multipole character of gamma radiation and systematics of mean lives for gamma multipole transitions. b. Interaction of Radiation with Matter: Interactions of charged particles with matter, ionisation energy loss, stopping power and range energy relations for charged particles. Interaction of gamma rays; photoelectric, Compton and pair production processes. Nuclear radiation detectors-G M counter and Scintillation detector.

c. Nuclear Energy: Fission process, fission chain reaction, four factor formula and controlled fission chain reactions, energetics of fission reactions, fission reactor. Fusion process, energetics of fusion reactions; Controlled thermonuclear reactions; Fusion reactor. Stellar nucleo synthesis.
d. Fundamental Interactions and Elementary Particles: Basic interactions and their characteristic features. Elementary particles, classification; Conservation laws in elementary particle decays. Quark model of elementary particles.
REFERENCE BOOKS: 1. Introduction to Atomic Spectra : H E White, McGraw Hill, 2. Fundamentals of Molecular Spectroscopy: C N Banwell and E M McCash, Tata McGraw Hill, 1999, 4th Edition. 3. Molecular Spectra and Molecular Structure Vol. 1: Spectra of Diatomic Molecules: G. Herzberg, Von Nostrand. 4. Spectroscopy, Vols. 1, 2 and 3: B P Straughan and S Walker ,Chapman and Hall 5. Introduction to Molecular Spectroscopy: G M Barrow, McGraw Hill. 6. Physics of Atoms and Molecules: B H Bransden and C J Joachain, Longman, 1983. 7. Spectra of Atoms and Molecules: P F Bernath, Oxford University Press 1995. 8. The Atomic Nucleus: R D Evans (TMH). 9. Nuclear and Particle Physics : W.E. Burcham and M. Jobes (Addison Wesley, 1998). 3. Nuclear Physics: R R Roy and B P Nigam (Wiley Eastern). 10. Physics of Nuclei and Particles: P Mermier and E Sheldon (Academic Press). 11. Atomic and Nuclear Physics: S N Ghoshal (S. Chand). 6. Nuclei and Particles: E Segre (Benjamin). 12. Nuclear Physics: D C Tayal (Himalaya). 13. Introduction to Nuclear Physics: S B Patel (Wiley Eastern).