Course Catalog

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Undergraduate thesis is a research project, spread over two consecutive semesters, in which students will work extensively on a research problem of current interest under the guidance of a faculty member.

This course (Topics in Quantum Many-Body Theory) aims to introduce the student with ample knowledge of Quantum Mechanics (qualified all of IQM, QM-I and QM-II) to the complexity of the many-body problem, mainly in the field of Condensed Matter Physics, though some of the methods go well beyond the scope of Condensed Matter Physics. Many-body Physics is the study of systems with a very large number of coupled degrees of freedom, typically involving a system of many (often ~ Avogadro’s Number ~ 1023) interacting particles. An exact solution of this would ideally involve the solution of ~ coupled Schrödinger Equations, which is essentially and unsolved problem! So the methods of Many-body Physics involve making useful and valid approximations to extract useful information about the system, without having to do a full exact solution of the many-body problem. This could involve, for example, the reduction of the fully interacting problem to a non-interacting or weakly interacting problem via certain “canonical transformations”. The prototypical example in this case is the resolution of the complex motion of crystal lattices into independent and non-interacting oscillator modes called “Phonons”, in the harmonic approximation. The interactions between these Phonons when anharmonic effects are included is weak, compared to that between the original lattice atoms. Similarly “elementary excitations” of the strongly coupled Heisenberg Spins on a lattice, are non-interacting spin-waves or “magnons”, to a first approximation, and magnon-magnon interactions are relatively weak. This course will familiarize students with the concepts of “Elementary Excitations” in many-body systems, like “quasi-particles” and “collective excitations” etc. Also the concepts of “Broken Symmetry” and the idea of “Emergent Complexity” will be emphasized, following the prophetic article “More is Different” by P.W. Anderson. It also inspects Spin Systems and related complexities in some detail. Other approximations and methods of calculations like “Mean-field Theories”, “Green’s Functions and the Renormalization method” etc. will be dealt with. “Linear Response Theory” and “Kubo Formulae” that connects theory to experiments will also be covered. We will also try to cover “Many-body Perturbation Theory” and some aspects of “Strong Correlations”.

We begin with an overview of special theory of relativity and proceed to give the definitions of tensor, connection, parallel transport and covariant differential with the aim of providing the description of gravity as arising from a curve space. From Riemann geometry and the Christoffel symbols we move to geodesic equations and to Riemann tensor outlining its various properties. We explain how one can formulate Einstein equations from fundamental principles. We also derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action. We define the energy-momentum tensor for matter and show that it obeys a conservation law. We take up the study of the black hole type solution and derive the one for Schwarzschild black hole. We touch upon the Birkhoff theorem and explain the important differences between energy-momentum conservation laws in the absence and in the presence of the dynamical gravity. We discuss gravitational waves and give an introduction to cosmology including cosmic microwave background radiation, dark matter and dark energy.

Review of Klein-Gordon and Dirac equations Solutions of Dirac Equation, Properties of Dirac matrices Free Klein Gordon Field Theory   Self-Interacting Scalar Field Theory            Complex Scalar Field Theory                      Dirac Field Theory                     Feynman diagrams

PHY 308 is a lab course offering an opportunity for hands-on learning through physics experiments based on various physics concepts covering Condensed matter physics and interaction of matter and energy.

1. Advanced Angular Momentum Theory  2. Advanced Topics in Perturbation Theory  3. Identical Particles and Second Quantization  4. Central Potentials and Potential Scattering Theory  5. Interaction of Radiation with Matter  6. Symmetries in Quantum Mechanics

1. Invitation to Condensed Matter Physics  2. Geometrical Description of Crystals and Scattering  3. The Sommerfeld Free Electron Theory of Metals  4. One Electron Theory and Energy Bands  5. Lattice Dynamics of Crystals : Phonons

1. The Fundamentals of Statistical Mechanics  1. Introduction  2. The Microcanonical Ensemble.  3. Entropy and Temperature  4. The Canonical Ensemble  5. The Partition Function , Energy and Fluctuations, Entropy, Free Energy  6. The Chemical Potential  7. Grand Canonical Ensemble, Grand Canonical Potential, Extensive and Intensive Quantities  2. Classical Gases.  1. Ideal Gas, Equipartition of Energy, Boltzmann's Constant, Gibbs's Paradox  2. Maxwell Distribution, Kinetic Theory  3. Diatomic Gas, Interacting Gas, Mayer f Function, Virial Coecient  van der Waals Equation of State, The Cluster Expansion.  3. Quntum Statistical Mechanics  1. The Postulate of Quantum Statistical Mechanics  2. Density Matrix  3. Ensembles in Quantum Statistical Mechanics  4. The Third Law of Thermodynamics  5. Fermi Systems, Bose Systems.  4. Phase Transitions  1. Liquid-Gas Transition, Phase Equilibrium, The Clausius-Clapeyron Equation,The Critical Point  2. The Ising Model, Mean Field Theory, Critical Exponents, Validity of Mean Field Theory.  3. Some Exact Results for the Ising Model, The Ising Model in d= 1 Dimensions 2d Ising Model.  4. Landau Theory, Second Order Phase Transitions, First Order Phase Transitions,  5. Landau-Ginzburg Theory, Correlations, Fluctuations.

1. Introduction: Definition of biophysics, why to study, examples.    2. Thermodynamics: Entropy, Enthalpy, The free energy of a system, Chemical potential, Redox potential, Bioenergetics   3. Biophysical properties: Brownian motion, Osmosis, Dialysis, Colloids  4. Membrane biophysics: Structure of bio-membrane. Structure-function relation.  5. Application of Radiation to Biological system: Introduction, particles and radiations of significance, physical and biological half-lives, macroscopic absorption of radiation, activity and measurements, units of dose, relative biological effectiveness and action of radiation at molecular level.  6. Experimental methods in biophysics:  (a) Microscope: Light characteristics, microscopes- compound, phase contrast, polarization, fluorescent and electron microscopes – Transmission Electron Microscope, Scanning Electron Microscope, and Scanning tunneling electron microscope, Atomic Force Microscopy  (b) Spectroscopy: Electronic structure of atoms, Bond formation, hybridization of orbitals, Molecular orbitals, Bond energy, Ultraviolet & Visible spectroscopy-Beer Lamberts law- spectrophotometer. Infrared spectroscopy, Raman spectra, Circular Dichroism, Fluorescence spectroscopy, NMR spectroscopy.

PHY207 is a bridge course specially designed for students who have already taken PHY101 and PHY102 instead of PHY103 and PHY104. The course supplements and develops their understanding of Newtonian physics and classical electromagnetism.  The content is as follows:  Review of Newtonian mechanics  Solving planetary motion using a personal computer A brief introduction to Lagrangian formulation of mechanics Review of simple harmonic motion Introduction to coupled oscillators and normal modes Introduction to special theory of relativity, space-time diagrams and four vectors Review of electrostatics and magnetostatics Review of Maxwell’s equation Wave equation from Maxwell’s equation, plane wave solution, polarization Light as an electromagnetic wave

1. Oscillations of Systems with Many Degrees of Freedom (a) Review of the Harmonic Oscillator (b) Systems with More than One Degree of Freedom (c) Linearity, Normal Modes and the Matrix Equation of Motion (d) Forced Oscillations and Resonance is Systems with More than One Degree of Freedom (e) The Infinite System and Translational Invariance (f) Forced Oscillations and Boundary Conditions 2. Traveling Waves (a) The Continuum Limit of a Discrete System (b) Longitudinal Oscillations and Sound (c) Harmonic Traveling Waves in One Dimension Phase Velocity (d) Index of Refraction and Dispersion (e) Impedance and Energy Flux 3. Modulations, Pulses, and Wave Packets (a) Group Velocity (b) Pulses (c) Fourier Analysis of Pulses (d) Fourier Analysis of Traveling Wave Packet 4. Waves in Two and Three Dimensions (a) Harmonic Plane Waves and the Propagation Vector (b) Water Waves (c) Electromagnetic Waves (d) Radiation from a Point Charge   5. Polarization (a) Description of Polarized States (b) Production of Polarized Transverse Waves (c) Double Refraction (d) Bandwidth, Coherence Time, and Polarization 6. Interference and Diffraction (a) Interference between Two Coherent Point Sources (b) Interference between Two Independent Sources (c) How Large Can a “Point” Light Source Be? (d) Angular Width of a “Beam” of Traveling Waves (e) Diffraction and Huygen’s Principle (f) Geometrical Optics

(a) Linear transformations of the plane     i. Affine planes and vector spaces     ii. Vector spaces and their affine spaces     iii. Euclidean and affine transformations    iv. Representing linear transformations by matrices     v. Areas and determinants   (b) Eigenvectors and eigenvalues     i. Conformal linear transformations     ii. Eigenvectors and eigenvalues    iii. Markov processes   (c) Linear differential equations in the plane     i. Functions of matrices     ii. Computing the exponential of a matrix     iii. Differential equation and phase portraits     iv. Applications of differential equations                     (d) Scalar products     i. The Euclidean scalar product     ii. Quadratic forms and symmetric matrices     iii. Normal modes  iv. Normal modes in higher dimensions     v. Special relativity: The Poincare’ group and the Galilean group    (e) Calculus in the plane i. The differential calculus and the examples of the chain rule: the Born approximation and Kepler motion   ii. Partial derivatives and differential forms.   iii. The pullback notation   iv. Taylor’s formula   v. Lagrange multiplier   (f) Double integrals   i. Exterior derivative   ii. Two-forms iii. Pullback and integration for two-forms   iv. Two-forms in three space   v. Green’s theorem in the plane

 1. The Kinetic Theory of Gases Macroscopic and  microscopic description of matter, thermodynamic variables of a system, State function, exact and inexact differentials, Basic assumptions of the kinetic theory, Ideal gas approximation, deduction of perfect gas laws, Maxwell’s distribution law, root mean square and most probable speeds. Collision probability, Mean free path from Maxwell’s distribution. Degrees of freedom, equipartition of energy. Nature of intermolecular interaction : isotherms of real gases. van der-Waals equation of state.    2. Transport Phenomena  Viscosity, thermal conduction and diffusion in gases. Brownian Motion: Einstein’s theory, Perrin’s work, determination of Avogardo number.    3. Thermodynamics of Photon Gas  Spectral emissive and absorptive powers, Kirchoff’s law of blackbody radiation, energy density, radiation pressure. Stefan-Boltzmann law, Planck’s law    4. First Law of Thermodynamics Zeroth law and the concept of temperature. Thermodynamic equilibrium, internal energy, external work, quasistatic process, first law of thermodynamics and applications including magnetic systems, specific heats and their ratio, isothermal and adiabatic changes in perfect and real gases.   5. The Second Law of Thermodynamics and its Statistical Interpretation (a) Second law of thermodynamics: different formulations and their equivalence (b) Entropy: The statistical postulate. (c) Equilibrium of an isolated system: Temperature (d) Illustration: The Schottky defects. (e) Equilibrium of a system in a heat bath: Boltzmann distribution; Kinetic interpretation of the Boltzmann distribution.   6. Thermodynamic Functions  Enthalpy, Helmholtz and Gibbs’ free energies; Chemical potential, Maxwell’s relations; thermodynamic equilibrium and free energies.   7. Change of State  Equilibrium between phases, triple point, Gibbs’ phase rule and simple applications. First and higher order phase transitions, The phase equilibrium and the ClausiusClapeyron equation,. JouleThomson effect, third law of thermodynamics   8. Applications of Thermodynamics. (a) Heat engines and Refrigerators: Derivation of limits on efficiency from the laws of thermodynamics; Carnot cycle; realistic cycles for internal combustion engines, steam engines, and refrigeration (b) Thermodynamics of rubber bands: Gibbs free energy, Entropy (c). Paramagnetism: A paramagnetic solid in a heat bath. The heat capacity and the entropy. An isolated paramagnetic solid.  Negative temperature.

1.Systems of Linear Algebraic Equations: Gauss Elimination Method, LU               decomposition, Choleski’s Decomposition Method, Symmetric and Banded Coefficient Matrices, Pivoting, Matrix Inversion, Iterative Methods    2.Symmetric Matrix Eigenvalue Problems: Jacobi Method, Power and Inverse Power Methods, Eigenvalues of Symmetric Tridiagonal Matrices, Computation of Eigenvectors     3. Two-Point Boundary Value Problems: Shooting Method  4. Solution of Partial Differential Equations: Separation of variables, Finite             Difference Method, The Relaxation Method, The matrix method for difference Equations.

Vector Analysis  Electrostatics: Electric Field, Divergence and Curl of Electrostatic Fields, Electric Potential, Work and Energy in Electrostatics, Conductors  Potentials: Laplace's Equation, Method of Images, Multipole Expansion  Electric Fields in Matter: Polarization, Field of a Polarized Object, Electric Displacement, Linear Dielectrics  Magnetostatics: Lorentz Force Law, Biot-Savart law, Divergence and Curl of Magenetic Field, Magnetic Vector Potential  Magnetic Fields in Matter: Magnetization, Field of a Magnetized Object, Auxiliary Field, Linear and Nonlinear Media  Electrodynamics: Electromotive Force, Electromagnet Induction, Maxwell's Equations  Conservation Laws: Charge and Energy, Momentum, Work  Electromagnetic Waves: Waves in One Dimension, Electromagnetic Waves in Vacuum, Electromagnetic Waves in matter

This is a continuation of PHY 101 and is meant for engineers and non-physics majors. The course will introduce students to Electricity and Magnetism, Maxwell’s equations, Light as an electromagnetic wave, and Wave optics.  Electrodynamics:  Vector calculus: Gradient, Divergence, Curl and fundamental theorems of vector calculus. Basic laws in electricity and magnetism, Classical image problem, displacement current and continuity equation, Maxwell’s Equations, electromagnetic wave equation and its propagation in free space, conducting media and dielectric medium, Poynting theorem, Electromagnetic spectrum.  Wave Optics:  Interference of light waves: Young’s double slit experiment, displacement of fringes, Interference in thin films  Diffraction: Fresnel’s and Fraunhofer’s class of diffraction, diffraction from single, double & N- Slits, Gratings.  Polarization: Concept of Polarization in electromagnetic waves, types of polarized waves.

Nonlinear dynamics will deal with fundamental properties of nonlinear systems and the question of non-integrability. This course provides a broad introduction and familiarity to the field of nonlinear dynamics and chaos. It takes an intuitive approach and focuses on both the analytical and the computational tools that are important in the study of nonlinear dynamical systems.

This course introduces the student to detectors, data analysis and other experimental techniques used in experimental particle physics.

This is an advanced course in condensed matter emphasizing the special properties of solids: magnetism, super fluidity and superconductivity, dielectrics and ferroelectrics.

This course has two parts. The first part reformulates classical electrodynamics as a field theory. The second part introduces general theory of relativity.

Explorations in Research

This course covers the basic interaction of matter with electrons, neutrons, ions, energetic particles, elastic and non–elastic scatterings, and characterization techniques: Optical microscopy, Transmission electron microscopy (TEM), Scanning electron microscopy (SEM), Scanning probe microscopy (SPM), Atomic force microscopy (AFM), X-ray diffraction, Energy dispersive X-ray analysis. X- Ray photoelectron spectroscopy (XPS), Secondary ion mass spectrometry (SIMS).

Characterization Of Materials

Complex Fluids

This course develops the basic programming skills to perform numerical analysis of various physical phenomenon by emphasizing on the algorithms and their implementation in the FORTRAN program language.

This is an interdisciplinary advanced level Ph.D. course in which various nanomaterials processing techniques, including chemical and physical vapor deposition, lithography, self-assembly, and ion implantation will be introduced. Tools commonly used to characterize nanomaterials will be introduced. The structural, mechanical, optical and electronic properties which arise due to nano-scale structure will be discussed from the point of view of nano-scale devices and application.

Advanced topics in non-linear dynamics

Review of Quantum Mechanics

Review of Classical Mechanics

Quantum Mechanics

Classical Mechanics

The aim of this course is to introduce the basic concepts of string theory by applying quantum mechanics to a relativistic string. In this manner the student will deepen his or her understanding of quantum mechanics and will also be able to appreciate the diverse areas of physics in which the mathematical description of a string like object is useful.

This course is intended to give an in-depth study of detector, data analysis and other experimental techniques used in particle physics. Modern particle detectors such as micro-pattern gaseous detectors, drift chambers, silicon detectors, calorimeters, Cherenkov detectors and others are discussed along with advanced statistical methods and data analysis techniques to extract results.

This course introduces the techniques of quantum field theory and its application to condensed matter physics and particle physics.

Ion accelerator, instrumentations, basic interaction of matter with ions, energy loss process, elastic and non–elastic scatterings, characterization techniques: Rutherford backscattering spectrometry (RBS), Ion channeling, Resonance channeling, Proton induced X-ray emission (PIXE), Elastic recoil detection analysis (ERDA), Nuclear reaction analysis (NRA), pitfalls in ion beam analysis, and radiation safety.