Loyola College M.Sc. Physics April 2012 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2012

PH 2812 – MATHEMATICAL PHYSICS

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions:                                                                                                              (10×2=20)

  • Evaluate the complex line integral around the closed loop C: |z| = 1.
  • Determine the residue at Z= 0 and at Z = I of the complex function f (z) = .
  • Define Dirac delta function . What is its Laplace transform?
  • Express the function f (t) = 2 if 0<t<π, f(t) = 0 if π<t<2π and f(t) = sin t if t >2π in terms of the unit step function.
  • What are the two possible initial conditions in the vibration of a rectangular membrane? Explain the symbols used
  • Solve .
  • Use the Rodrigue’s formula to evaluate the 3rd degree Legendre polynomial .
  • State the orthonormality property of the Hermite polynomials.
  • List the four properties that are required by a group multiplication.
  • What is irreducible representation of a group?

 

PART – B

Answer any FOUR questions:                                                                                                 (4×7.5 = 30)

 

  • Verify the Cauchy’s integral theorem for the contour integral for the contour C: the triangle with vertices at 0, 1, and 1+i.
  • A capacitor of capacitance C is charged so that its potential is V0. At t = 0 the switch in figure is closed and the capacitor starts to discharge through the resistor of resistance R. using the Laplace transformation, find the charge q(t) on the capacitor.
  • Use the method of separation of variables to solve the partial differential equation , where u( x,0) = 6 e-3x.
  • (a) Prove that J-n(x) = (-1)n Jn(x) if n is a positive integer where Jn(x) is the Bessel function of first kind.

(b) Determine the value of J -1/2(x).                      ( 4 ½ +3)

  • Work out the multiplication table of the symmetry group of the proper covering operations of a square. Write down all the subgroups and divide the group elements into classes. What are the allowed dimensionality of the representation matrices of the group?

 

PART – C

Answer any FOUR questions:                                                                                                  (4×12.5 =50)

  • (a) Using the contour integration, evaluate the real integral,

(b) Evaluate the following integral using Cauchy’s integral formula dz, where C is the circle |Z |= 3/2.                   ( 6 ½ + 6)

  • Find the Fourier transform of (i) f(x) = exp( -x2) and (ii) f(x) =1 – |x| if |x| <1 and f(x) = 0 for |x| >1

( 6 ½ + 6)

  • Solve the one- dimensional wave equation by the separation of variable technique and the use of Fourier series. The boundary conditions are u(0,t) =0 and u(L,t) = 0 for all t and the initial conditions are u ( x,0) = f(x) and ∂u/∂t = g(x) at t =0. ( Assume that u (x,t) to represent the deflection of stretched string and the string is fixed at the ends x = 0 and x = L)
  • (a) Solve the Legendre differential equation (1 – x2) – 2x  + n (n+1)y = 0 by the power series method.

(b) Establish the orthonormality relation where  is the  Legendre polynomial of order n.                 ( 6 ½ + 6)

  • (a) Prove that any representation by matrices with non-vanishing determinants is equivalent to

a representation by unitary matrices.

(b)  Enumerate and explain the symmetry elements of CO2, H2O and NH3 molecules. ( 6 ½ +6)

 

Go To Main page

 

 

© Copyright Entrance India - Engineering and Medical Entrance Exams in India | Website Maintained by Firewall Firm - IT Monteur