# Direct PhD

**Core Courses**

· PHL601 | Classical and Mathematical Physics (3-0-0)

· PHL602 | Quantum and Statistical Physics (3-0-0)

· PHL603 | Physics of Electromagnetic Waves (3-0-0)

· PHL604 | Physics of Atoms, Molecules, and Solids (3-0-0)

· PHL605 | Numerical Methods (3-0-0)

**Pass Course**

· PHL606 | Pre-thesis literature survey and seminar (pass course)

**Open Elective**

· PHL6xx | One Elective (3-0-0) | Click **here** for the choice of PHYSICS elective courses

**Details of the PhD Courses**

**PHL601 | Classical and Mathematical Physics (3-0-0)**

Mathematical Physics: Fourier series; Fourier and Laplace Transforms: Sine and Cosine Transforms, Convolution and Correlations, Application-oriented problems; Linear vector space: properties, Gram-Schmidt orthogonalization; Matrices: inverse, rank, eigenvalues and eigen functions, diagonalization; Solution of linear equations: linear transformation, change of basis; Tensors: rank, products, contraction, tensors with special forms, examples in physics; Complex calculus: Integration in complex plane, Cauchy’s integral formula, singularities, residue theorem, definite integration; Group Theory: Isomorphism and homomorphism, cyclic and permutation group, reducible and irreducible representation, character tables, finite and infinite groups, crystallographic and molecular symmetries

Classical Mechanics: Lagrange’s and Hamilton’s equations of motion: scope of the application, introductory problems, Poisson’s bracket, Liouville equation; Damped and forced harmonic oscillation, Q-factor; Small oscillations of coupled systems, Normal modes; Classical theory of harmonic crystal: monoatomic and diatomic one-dimensional chain, dispersion relations; Special theory of relativity: Length contraction and time dilation, four-vector notations, Lorentz transformations

**PHL602 | quantum and statistical physics (3-0-0)**

Quantum Mechanics: Schrodinger equation in time-independent potential: Particle in a box, Barrier and well (Outline of calculations, emphasis on essential physics), tunnelling and bound state; resonant tunnelling in two quantum wells; Harmonic oscillator: Outline of calculations in wave function and operator approach, Comparison with barrier and well; vibrational modes of a linear chain of coupled harmonic oscillators: phonons; Electron energy levels in periodic potentials: Bloch’s theorem; band structures; density of states; concept of holes; Particle in a central potential: Hydrogen atom, Outline of calculations; angular momentum algebra, symmetry; Case studies: rotation of diatomic molecules, charged particle in a magnetic field (Landau levels); Scattering of a particle by a fixed center of force: cross sections, Fermi’s Golden Rule, partial waves, phase shift, optical theorem; Born approximation.

Statistical Mechanics: microcanonical, canonical, and grand canonical ensembles; examples, partition functions; paramagnetism, negative temperatures; Ideal Bose systems: thermodynamic properties, examples: black-body radiation, liquid Helium II; Ideal Fermi systems: Thermodynamic properties, examples: Pauli paramagnetism, Landau diamagnetism.

**PHL603 | Physics of Electromagnetic Waves (3-0-0)**

Classical Electrodynamics: Maxwell’s equations: Energy and momentum of electromagnetic field, Radiation pressure, Boundary conditions of electromagnetic field at interfaces, reflection, refraction, and transmission, Brewster’s angle; Solution in free space, concept of polarization, Stokes’ parameters, Jone’s matrix; Solution in a dielectric media: theory of local field and polarization, Clausius-Mossotti relation, atomic polarizability, Kramers-Kroenig relation, Lorentz-Lorentz formula for dispersion, normal and anomalous dispersion, electrical conductivity in a metal, plasma frequency, negative refractive index; Radiation from electric dipole, multipole radiation, Radiation of a uniformly moving charged particle; Relativistic electrodynamics: electromagnetic field tensor, Lorentz force in vacuum, energy-momentum tensor in material media, radiation reaction; mechanical property of electromagnetic field of a charge.

Optics: Light propagation inside a metal: reflection and refraction, optical constant of metal; Light propagation inside a crystal: Fresnel’s formula, optical properties of uniaxial and biaxial crystals, double refraction, interference; Nonlinear susceptibility, phase matching and second harmonic generation; Intermolecular Forces: Forces between atoms and molecules, Forces between particles and surfaces.

**PHL604 | Physics of Atoms, Molecules, and Solids (3-0-0)**

Quantum Mechanics: Review of quantum mechanics: Hydrogen atom, scattering theory; Quantization of electromagnetic field, emission and absorption of photons by atoms, Einstein’s A and B coefficients, Concepts of lasers; Rayleigh, Thomson, and Raman scattering, Resonance Fluorescence, Dispersion relation; Relativistic quantum mechanics: Klein-Gordon equation, Dirac equation, simple solutions, relativistic covariance; negative energy solutions, hole theory; revisit of Hydrogen atom problem; Perturbation theory: time-independent theory up to second order, Zeeman and Stark shifts; time-dependent theory with sinusoidal fields, Rabi oscillation; Addition of angular momentum, electron spin, L-S coupling in atoms, Zeeman effect revisited, electric dipole transition and selection rules.

Applications: Molecular spectra: selection rules, rotational and vibrational spectra, Raman and IR spectra; Crystal structure: reciprocal lattice, X-ray diffraction and Bragg’s law, Bravais lattice and Brillouin zone; Electron gas in metals: Sommerfeld’s theory, Fermi surface, thermodynamic properties; Case of weak periodic potentials: perturbation approach, energy levels near Bragg plane, bands and Brillouin zones, effective mass of electron and holes.

**PHL605 | Numerical Methods (3-0-0)**

Basic concepts and ideas in numerical analysis: Introduction to error, accuracy and stability. Data Reliability assessment: parameter sensitivity, experimental perturbations, Interpolation (Lagrange’s method, divided difference formula, splines), integration (Simpson’s method, Romberg’s method, quadrature formula), extrapolation, discretization, convergence, regression, quadrature, Random numbers, Solution of linear algebraic equations (elimination methods, LU decomposition), Eigenvalue problems, numerical solution of ordinary differential equations: explicit and implicit methods, multistep methods, Runge-Kutta and predictor-corrector methods. Introduction to numerical solutions of partial differential equations; Von Neumann stability analysis; alternating direction implicit methods and nonlinear equations.

Computer organization and programming: Fundamental computer concepts in hardware, software, Programming in C++ for scientific computations (hands-on with basic programming), Use of graphical software: Basics of curve fitting. Introduction to grid/parallel computing. Fast Fourier Transforms and Convolution, Monte Carlo techniques.

**PHL606 | Pre-thesis literature survey and seminar **(pass course)

Supervised study of a special topic in the area of interest (choice of material and allotment of time according to individual needs) –preparation & presentation of scientific review of studied reports/publications. The topic would be decided by the Supervisor. Students are expected to do thorough literature survey and will submit a report to the Coordinator of this Course. Students will also present a seminar on his/her topic, based on the literature survey.