M.Sc. DEGREE EXAMINATION, DECEMBER First Year. Physics. Paper I MATHEMATICAL PHYSICS. Answer any FIVE questions.

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1 (DPHY 01) M.Sc. DEGREE EXAMINATION, DECEMBER 009. Paper I MATHEMATICAL PHYSICS (5 0 = 100) 1. (a) Obtain the series solution for Legendre s differential equation. (b) Evaluate : J ( ) 1.. (a) Starting from the differential equation, prove the orthonormal property of Laguerre polynomial. 3 (b) Epress 3 interms of Hermite s polynomial. 3. (a) State and prove Cauchy s residue theorem. (b) Prove that 0 d a bsin a b if a b. 4. (a) Define the conditions for a function to be analytic in a region R. Given 3 u 3 y y y, verify whether Harmonic, if so, find v such that f ( z) u iv is analytic. e (b) Evaluate : ( z ) C z dz where C is a circle z (a) Components of a first rank tensor in rectangular Cartesian co-ordinate system are given as coordinates. y, y z, z. Obtain the covariant components in spherical (b) Define inner product and outer product of two tensors show that every contraction of mied tensor reduces its rank by.

2 6. (a) Prove that g ij is a second rank covariant symmetric tensor. (b) Find the metric of a Euclidean space referred to cylindrical coordinates. 7. (a) Epress the function f ( ) sin in Fourier series in the interval. Hence deduce / 4. (b) Find the Fourier sine and cosine transform of f ( ), (a) Find the inverse Laplace transform of (i) (ii) (s ( s ( s 3 5s 4) s s) ( s 1). 6s 5) e (b) Evaluate : 0 t e t 3t dt using the knowledge of Laplace transform. 9. Answer any TWO of the following : (a) State and prove the Rodrigue s formula for Legendre polynomial and hence evaluate P ( ). (b) Epand the function 1 f ( z) in a Laurrent series valid for ( z 1)( z 3) (i) 1 z 3 (ii) z 3 (iii) z 1. (c) Show that the covariant derivative of a contravariant vector is a mied tensor of rank two. (d) Find the Fourier transform of, f ( ) 0, a a Comment on for f ( ) 1, 0 a a.

3 (DPHY 0) M.Sc. DEGREE EXAMINATION, DECEMBER 009. Paper II CLASSICAL MECHANICS AND STATISTICAL MECHANICS All questions carry equal marks. 1. (a) Define D'Alembert's principle. (b) Derive Lagrange's equation from the Hamiton's principle.. (a) With a neat diagram, eplain Euleri angles. (b) Derive Euler's equation for the motion of a rigid body. 3. (a) What do you understand from canonical transformations? (b) Discuss simple eamples of canonical transformations. 4. (a) What is Hamilton's characteristic function? (b) Discuss Hamilton-Jacobi equation for Hamilton's characteristics function. 5. (a) Define the three types of ensembles and eplain them. (b) Compare and contrast the properties of ensembles. 6. (a) State and prove equi-partition theorem. (b) State and eplain Gibb's parado. 7. (a) Eplain the postulates of quantum statistical mechanics. (b) Eplain the classical limit of the variational principle. 8. (a) Discuss the theory of white dwarf stars. (b) Eplain the essential features of Bose-Einstein condensation. 9. Write notes on any TWO of the following: (a) Coriolis force (b) Action-angle variables (c) Grand canonical ensemble (d) The partition function.

4 (DPHY 03) M.Sc. DEGREE EXAMINATION, DECEMBER 009. Paper III QUANTUM MECHANICS All questions carry equal marks. 1. (a) Give the physical interpretation of a wave function. Eplain the concept of probability current density and derive equation of continuity. (b) Define a Hermitian operator. Prove that two eigen functions of a Hermitian operator belonging to different eigen values are orthogonal.. (a) Obtain the energy eigen values of linear Harmonic oscillator. (b) State and eplain the uncertainity principle with two eamples. 3. (a) State and eplain Rayleigh-Ritz variational method for the estimation of ground state energy. (b) Apply the above method to the ground state of Helium atom. 4. (a) Discuss the time dependent perturbation theory. (b) State and eplain sudden and adiabatic approimations with the conditions for their validity. 5. (a) State and prove the properties of Pauli spin matrices. (b) Obtain commutation relations for the components of angular momentum and show that L commutes with any of the three components. 6. (a) Eplain the concept of general angular momentum and obtain matrices for the operators J and J for j 1. z (b) What are Calebsch-Gordon coefficients? Obtain C-G coefficients for 1 1 j 1, j. 7. (a) Obtain Dirac s relativistic equation for a free particle. (b) Show that Dirac s equation logically endows the electron with spin. 8. Write short notes on any TWO of the following :

5 (a) Wigner-Eckert theorem. (b) Dirac s hole theory. (c) WKB method. (d) Klein-Gordon equation.

6 (DPHY 04) M.Sc. DEGREE EXAMINATION, DECEMBER 009. Paper IV ELECTRONICS All questions carry equal marks. 1. (a) What is an op-amplifier? Eplain how it can be used as an integrator and differentiator. (b) Eplain the principle and operation of voltage follower.. (a) Draw the circuit diagram and operation of a Wien s bridge oscillator. (b) Eplain 555 timer and how it generates square wave and triangular wave. 3. (a) Give an account on microwave resonators. (b) What is magic T and eplain its working? 4. (a) Discuss the generation of AM waves with the help of waveforms. (b) Eplain the working of Foster-Seelpy discriminator. 5. (a) Eplain the truth-tables of OR, AND, NOT and NAND gates with their logic symbols. (b) Eplain the working of multipleer encoder. 6. (a) Eplain the working of JK master slave Flip-Flop. (b) Eplain the working of a 4-bit Shift Register. 7. (a) With suitable eamples, eplain the various addressing modes of 8085 CPU. (b) Write an assembly language program to create a delay subroutine. 8. (a) Eplain the instruction and addressing modes of 8086 CPU. (b) Write a program to add two 8 bit numbers. 9. Write notes on any TWO of the following : (a) Common Mode Rejection Ratio (b) Klystron (c) Demorgan theorems (d) Sky wave propagation.

(DPHY01) ASSIGNMENT - 1 M.Sc. (Previous) DEGREE EXAMINATION, MAY 2019 PHYSICS First Year Mathematical Physics MAXIMUM : 30 MARKS ANSWER ALL QUESTIONS

(DPHY01) ASSIGNMENT - 1 M.Sc. (Previous) DEGREE EXAMINATION, MAY 2019 PHYSICS First Year Mathematical Physics MAXIMUM : 30 MARKS ANSWER ALL QUESTIONS (DPHY01) Mathematical Physics Q1) Obtain the series solution of Legendre differential equation. Q2) a) Using Hermite polynomial prove that 1 H n 1( x) = ( x 1)H n 2( x) + H n( x) 2 b) Obtain the generating