Syllabus for
Master of Science (Physics)
Academic Year  (2021)

The postgraduate programme in physics helps to provide in depth knowledge of the subject which is supplemented with tutorials, brainstorming ideas and problem-solving efforts pertaining to each theory and practical course. The two-year MSc programme offers 16 theory papers and 7 laboratory modules, in addition to the foundation courses and guided project spreading over four semesters. Foundation courses and seminars are introduced to help the students to achieve holistic development and to prepare themselves to face the world outside in a dignified manner. Study tour to reputed national laboratories, research institutions and industries, under the supervision of the department is part of the curriculum.

Continuous internal assessment (CIA) forms 50% and the end semester examination forms the other 50% of the marks in both theory and practical. For the Holistic and Seminar course, there is no end semester examination and hence the mark is awarded through CIA. CIA marks are awarded based on their performance in assignments (written material to be submitted and valued), mid-semester examination (MSE), and class assignments (Quiz, presentations, problem solving etc.). The mid-semester examination and the end semester examination for each theory paper will be for two- and three-hours duration respectively. The CIA for practical sessions is done on a day to day basis depending on their performance in the pre-lab, the conduct of the experiment, and presentation of lab reports. Only those students who qualify with minimum required attendance and CIA marks will be allowed to appear for the end semester examination.

The course enables students to understand the basic concepts of Newtonian mechanics and introduce other formulations (Lagrange, Hamilton, Poisson) to solve trivial problems.  The course also includes constraints, rotating frames, central force, Kepler problems, canonical transformation and their generating functions, small oscillations and rigid body dynamics.

Mechanics of a particle, mechanics of a system of particles, constraints and their classification, principle of virtual work, D’Alembert’s principle, Generalized co-ordinates, Lagrange’s equations of motion, applications of Lagrangian formulation (simple pendulum, Atwood’s machine, bead sliding in a wire), cyclic co-ordinates, concept of symmetry, homogeneity and isotropy, invariance under Galilean transformations.

Rotating frames, inertial forces in the rotating frame, effects of Coriolis force, Foucault’s pendulum, Central force: definition and examples, Two-body central force problem, classification of orbits, stability of circular orbits, condition for closure of orbits, Kepler’s laws, Virial theorem, applications.                                                       

Canonical transformations, generating functions, conditions of canonical transformation, examples, Legendre’s dual transformation, Hamilton’s function, Hamilton’s equation of motion, properties of Hamiltonian and Hamilton’s equations of motion, Poisson Brackets, properties of Poisson bracket, elementary PB’s, Poisson’s theorem, Jacobi-Poisson theorem on PBs, Invariance of PB under canonical transformations, PBs involving angular momentum, principle of Least action, Hamilton’s principle, derivation of Hamilton’s equations of motion from Hamilton’s principle, Hamilton-Jacobi equation. Solution of simple harmonic oscillator by Hamilton-Jacobi method.

Types of equilibrium and the potential at equilibrium, Lagrange’s equations for small oscillations using generalized coordinates, normal modes, vibrations of carbon dioxide molecule, forced and damped oscillations, resonance, degrees of freedom of a free rigid body, angular momentum, Euler’s equation of motion for rigid body, time variation of rotational kinetic energy, Rotation of a free rigid body, Eulerian angles, Motion of a heavy symmetric top rotating about a fixed point in the body under the action of gravity.

[2].    Goldstein, H. (2001). Classical mechanics (3^rd ed.): Addison Wesley.

[3].    Rana, N. C., & Joag, P. S. (1994).  Classical mechanics. New Delhi: Tata McGraw Hill.

[1].    Greiner, W. (2004). Classical mechanics: System of particles and Hamiltonian dynamics. New York: Springer-Verlag.

[2].    Barger, V., & Olsson, M. (1995). Classical mechanics - A modern perspective (2^nd ed.): Tata McGraw Hill.

[3].    Gupta, K. C. (1988). Classical mechanics of particles and rigid bodies: Wiley Eastern Ltd. 

[4].    Takwale, R. G., & Puranik, P. S. (1983).  Introduction to classical mechanics. New Delhi: Tata McGraw Hill.

This module introduces the students to the applications of analog and digital integrated circuits. First part of the module deals with the operational amplifier, linear applications of op-amp., active filters, oscillators, non-linear applications of op-amp, timer and voltage regulators. The second part deals with digital circuits which exposes to the logic gates, encoders and decoders, flip-flops registers and counters.

The ideal op-amp - characteristics of an op-amp., the ideal op-amp., Equivalent circuit of an op-amp., Voltage series feedback amplifier - voltage gain, input resistance and output resistance, Voltage follower. Voltage shunt feedback amplifier - virtual ground, voltage gain, input resistance   and output resistance, Current to voltage converter. Differential amplifier with one op-amp. voltage gain, input resistance.

Linear applications: AC amplifier, AC amplifier with single supply voltage, Summing amplifier, Inverting and non-inverting amplifier, Differential summing amplifier, Instrumentation amplifier using transducer bridge, The integrator, The differentiator.                                                                                                               

Active filters and oscillators: First order low pass filter, Second order low pass filter, First order high pass filter, Second order high pass filter, Phase shift Oscillator, Wien-bridge oscillator, Square wave generator.

Non-linear circuits: Comparator, Schmitt trigger, Digital to analog converter with weighted resistors and R-2R resistors, Positive and negative clippers, Small signal half wave rectifier, Positive and negative clampers.                                                         

Logic gates - basic gates - OR, AND, NOT, NOR gates, NAND gates, Boolean laws and theorems (Review only). Karnaugh map, Simplification of SOP equations, Simplification of POS equations, Exclusive OR gates.

Combinational circuits: Multiplexer, De-multiplexer, 1-16 decoder, BCD to decimal decoder, Seven segment decoder, Encoder, Half adder, Full adder                  

Flip flops: RS flip-flop, Clocked RS flip-flop, Edge triggered RS flip-flop, D flip-flop, JK flip-flop, JK master-slave flip-flop.

Registers: Serial input serial output shift register, Serial input parallel output shift register, Parallel input serial output shift register, Parallel input parallel output shift register, Ring counter.

Counters: Ripple counter, Decoding gates, Synchronous counter, Decade counter, Shift counter - Johnson counter. 

[2].      Leach, D. P., & Malvino, A. P. (2002). Digital principles and applications. New York: Tata McGraw Hill.  

[2].      Morris Mano, M. (2018). Digital logic and computer design: Pearson India.

[3].      Jain, R. P. (1997). Modern digital electronics. New York: Tata McGraw Hill.

This course being an essential component in understanding the behaviour of fundamental constituents of matter is divided into two modules spreading over first and second semesters. The first module is intended to familiarize the students with the basics of quantum mechanics, exactly solvable eigenvalue problems, time-independent perturbation theory and time-dependent perturbation theory.

Review - origin of quantum mechanics (particle aspects, wave aspects and wave-particle duality), uncertainty principle, Schrodinger equation, time evolution of a wave packet, probability density, probability current density, continuity equation, orthogonality and normalization of the wave function, box normalization, admissibility conditions on the wave function, Operators, Hermitian operators, Poisson brackets and commutators, Eigen values, Eigen functions, postulates of quantum mechanics, expectation values, Ehrenfest theorems.

Bound and unbound systems. Application of time independent Schrodinger wave equation - Potential step, rectangular potential barriers - reflection and transmission coefficient, barrier penetration; particle in a one-dimensional box and in a cubical box, density of states; one dimensional linear harmonic oscillator - evaluation of expectation values of x² and p_x²; Orbital angular momentum operators - expressions in cartesian and polar coordinates, eigenvalue and eigenfunctions, spherical harmonics, Rigid rotator, Hydrogen atom - solution of the radial equation.

Time independent perturbation theory- First and second order perturbation theory applied to non-degenerate case; first order perturbation theory for degenerate case, application to normal Zeeman effect and Stark effect in hydrogen atom.

Time-dependent perturbation theory - First order perturbation, Harmonic perturbation, Fermi’s golden rule, Adiabatic approximation method, Sudden approximation method. 

Scattering cross-section, Differential and total cross-section, Born approximation for the scattering amplitude, scattering by spherically symmetric potentials, screened coulomb potential, Partial wave analysis for scattering amplitude, expansion of a plane wave into partial waves, phase shift, cross-section expansion, s-wave scattering by a square well, optical theorem.                                                      

[1].    Zettli, N. (2017). Quantum mechanics. New Delhi: Wiley India Pvt Ltd.

[6].    Crasemann, B., & Powell, J. H. (1998). Quantum mechanics: Narosa Publishing House.

[8].    Griffiths, D. J. (1995). Introduction to quantum mechanics: Prentice Hall Inc.

[10].Landau, L. D., & Lifshitz, E. M. (1965). Quantum mechanics: Pergamon Press.

A sound mathematical background is essential to understand and appreciate the principles of physics. This module is intended to make the students familiar with the applications of tensors and matrices, special functions, partial differential equations and integral transformations, Green’s functions and integral equations. 

Vectors and matrices: Review (vector algebra and vector calculus, gradient, divergence & curl), transformation of vectors, rotation of the coordinate axes, invariance of the scalar and vector products under rotations, Vector integration, Line, surface and volume integrals - Stoke’s, Gauss’s and Green’s theorems (Problems), Vector analysis in curved coordinate, special coordinate system - circular, cylindrical and spherical polar coordinates, linear algebra matrices, Cayley-Hamilton theorem, eigenvalues and eigenvectors.

Tensors: Definition of tensors, Kronecker delta, contravariant and covariant tensors, direct product, contraction, inner product, quotient rule, symmetric and antisymmetric tensors, metric tensor, Levi Cevita symbol, simple applications of tensors in non-relativistic physics.

Beta and Gamma functions, different forms of beta and gamma functions. Dirac delta function. Kronecker delta, Power series method for ordinary differential equations, Series solution for Legendre equation, Legendre polynomials and their properties, Series solution for Bessel equation, Bessel and Neumann functions and their properties, Series solution for Laguerre equation, it's solutions and properties (generating function, recurrence relations and orthogonality properties for all functions).

Method of separation of variables, the wave equation, Laplace equation in cartesian, cylindrical and spherical polar coordinates, heat conduction equations and their solutions in one, two and three dimensions.  

Review of Fourier series, Fourier integrals, Fourier transform, Properties of Fourier sine and cosine transforms, applications. Laplace transformations, properties, convolution theorem, inverse Laplace transform, Evaluation of Laplace transforms and applications.

Dirac delta function, properties of Dirac delta function, three dimensional delta functions, boundary value problems, Sturm-Liouville differential operator, Green’s function of one dimensional problems, discontinuity in the derivative of Green’s functions, properties of Green’s functions, Construction of Green’s functions in special cases and solutions of inhomogeneous differential equations, Green’s function- symmetry of Green’s function, eigenfunction expansion of Green’s functions, Green’s function for Poisson equation.

Linear integral equations of first and second kind, Relationship between integral and differential equations, Solution of Fredholm and Volterra equations by Neumann series method.

[1]. S. Prakash: Mathematical Physics, S. Chand and Sons, 2004.

[2]. H. K. Dass: Mathematical Physics, S. Chand and Sons, 2008.

[3].G. B. Arfken, H. J. Weber and F. E. Harris: Mathematical methods for physicists, 7th Edn., Academic press, 2013.

[1]. Murray R. Spiegel, Theory and problems of vector analysis, (Schaum’s outline series)

[2]. M. L. Boas: Mathematical Methods in the Physical Sciences, 2nd Edn, Wiley 1983.

[3]. K.F. Riley, M.P Hobson, S. J. Bence, Mathematical methods for Physics and Engineering,  Cambridge University Press (Chapter 24)

[4]. P. K. Chattopadhyaya: Mathematical Physics, Wiley Eastern, 1990.

[5]. E. Kryszig: Advanced Engineering Mathematics, John Wiley, 2005.

[6]. Sadri Hassani: Mathematical Methods for students of Physics and related fields, Springer 2000.

[7]. J. Mathews and R. Walker: Mathematical Physics, Benjamin, Pearson Education, 2006.

[8]. A W. Joshi: Tensor analysis, New Age, 1995.

[9]. L. A. Piper: Applied Mathematics for Engineers and Physicists, McGraw-Hill 1958.

CIA 1: Assignment /quiz/ group task / presentations before MST - 10 marks.

CIA 2: Mid-Sem Test (Centralized), 2 hours - 50 marks to be converted to 25 marks.

CIA 3: Assignment /quiz/ group task / presentations after MST - 10 marks.

CIA 4: Attendance (76-79 = 1, 80-84 = 2, 85-89 = 3, 90-94 = 4, 95-100 = 5) - maximum of 5 marks.

 The research methodology module is intended to assist students in planning and carrying out research projects. The students are exposed to the principles, procedures and techniques of implementing a research project. In this module, the students are exposed to elementary scientific methods, design and execution of experiments, and analysis and reporting of experimental data.

 Introduction - meaning of research - objectives of research - motivation in research, types of research - research approaches - significance of research -research methods versus methodology - research and scientific method, importance of knowing how research is done - research processes - criteria of good research - defining research problem - selecting the problem, necessity of defining the problem - techniques involved in defining a problem - research design - meaning of research design - need for research design - features of good design, different research designs - basic principles of experimental design. Resources for research - research skills - time management, role of supervisor and scholar - interaction with subject experts. Thesis Writing: The preliminary pages and the introduction - the literature review, methodology - the data analysis - the conclusions, the references (IEEE format)

 Literature Review: Significance of review of literature - source for literature: books -journals – proceedings - thesis and dissertations - unpublished items.

On-line Searching: Database – SciFinder – Scopus - Science Direct - Searching research articles - Citation Index - Impact Factor - H-index etc.

Document preparation system: Latex, beamer, Overleaf-Writing scientific report - structure and components of research report - revision and refining’ - writing project proposal - paper writing for international journals, submitting to editors - conference presentation - preparation of effective slides, graphs - citation styles.

1. C. R. Kothari, Research Methodology Methods and Techniques, 2nd. ed. New Delhi: New Age International Publishers, 2009.

2. R. Panneerselvam, Research Methodology, New Delhi: PHI, 2005.

3. P. Oliver, Writing Your Thesis, New Delhi:Vistaar Publications, 2004.

Experiments are selected to improve the understanding of students about mechanical, magnetic, optical and basic electronic properties of materials.

1.      Elastic constants of glass plate by Cornu's interference method. (Online/Offline)

2.      Study of thermo-emf and verification of thermoelectric laws (Onlilne/Offline)

3.      Wavelength of iron arc spectral lines using constant deviation spectrometer. (Offline)

4.      Energy gap of the semi-conducting material used in a PN junction. (Offline)

5.      Characteristics of a solar cell. (Online/Offline) 

6.      Stefan’s constant of radiation. (Offline)

7.    Study of hydrogen spectra and determination of Rydberg constant (Offline)

1.      Relaxation time constant of a serial bulb. (Offline)

2.      e/m by Millikan’s oil drop method. (Online)

3.      Study of elliptically polarized light by using photovoltaic cell. (Offline)                                     

4.      Study of absorption of light in different liquid media using photovoltaic cell. (Offline/Online)

5.      Determination of Curie temperature of a given ferro magnetic material. (Offline)

6.      Determination of energy loss during magnetization and demagnetization by means of BH loop. (Online/Offline)

2. Sears, F. W., Zemansky, M. W.,& Young, H. D. (1998). University physics(6^thed.): Narosa Publishing House.

4. Collings, P. J. (1980). Simple measurement of the band gap in silicon and germanium, Am. J. Phys., Vol. 48, Page. 197.

5. Fischer, C. W. (1982). Elementary technique to measure the energy band gap and diffusion potential of pn junctions: Am. J. Phys., Vol. 50, Page. 1103.

Electronics being an integral part of Physics, Laboratory 2, Electronics is dedicated to experiments related to Electronic components and circuits. The experiments are selected to make the students familiar with the commonly used electronic components and their application in electronic circuits. During the course, the students will get to know the use of various electronic measuring instruments for the measurement of various parameters.

1. Transistor multivibrator.
2. Half wave and full wave rectifier using op-amp.
3. Op-amp. voltage regulator.
4. Op-amp. inverting and non-inverting amplifier.
5. Timer 555, square wave generator and timer.
a) RS flip-flop using NAND gates, b) Decade counter using JK flip-flops.

6. Half adder and full adder using NAND gates.
7. Construction of adder, subtractor, differentiator and integrator circuits using the given Op-amp.
8. Construction of a D/A converter circuit and study its performance-R-2R and Weighted resistor network.
9. JK Flip-Flop and up-down counter
10. Differential Amplifier with Op-Amp

The objective of the course, MPH 231- statistical physics is to enable the students to explore the basic concepts and description of various topics such as phase space, ensembles, partition functions, Bose-Einstein and Fermi-Dirac gases, non-equilibrium states, and fluctuations.

 Introduction, phase space, ensembles (microcanonical, canonical and grand canonical ensembles), ensemble average, Liouville theorem, conservation of extension in phase space, condition for statistical equilibrium, microcanonical ensemble, ideal gas. Quantum picture: Microcanonical ensemble, quantization of phase space, basic postulates, classical limit, symmetry of wave functions, effect of symmetry on counting, distribution laws.      

Gibb’s paradox and its resolution, Canonical ensemble, entropy of a system in contact with a heat reservoir, ideal gas in canonical ensemble, Maxwell velocity distribution, equipartition theorem of energy, Grand canonical ensemble, ideal gas in grand canonical ensemble, comparison of various ensembles. Canonical partition function, molecular partition function, translational partition function, rotational partition function, application of rotational partition function, application of vibrational partition function to solids

Bose-Einstein distribution, Applications, Bose-Einstein condensation, thermodynamic properties of an ideal Bose-Einstein gas, liquid helium, two fluid model of liquid helium-II, Fermi-Dirac (FD) distribution, degeneracy, electrons in metals, thermionic emission, magnetic susceptibility of free electrons. Application to white dwarfs , High temperature limits of BE and FD statistics

Boltzmann transport equation, particle diffusion, electrical conductivity, thermal conductivity, isothermal Hall effect, Quantum Hall effect. Introduction to fluctuations, mean square deviation, fluctuations in ensembles, concentration fluctuations in quantum statistics, one dimensional random walk, electrical noise (Nyquist theorem). Fluctuations in FD and BE gases, Winer Khintchine theorem.     

[1].   F. Reif: Statistical and Thermal Physics, McGraw Hill International, 1985.

[2].   K. Huang: Statistical Mechanics, Wiley Eastern Limited, 1991.

[3].   J. K. Bhattacharjee: Statistical Physics: Equilibrium and Non Equilibrium Aspects, Allied Publishers Limited, 1997.

[4].   R. A. Salinas: Introduction to Statistical Physics, Springer, 2^nd Edn,2006. 

[5].   E. S. R. Gopal: Statistical Mechanics and properties of matter, Macmillan, India 1976.