LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – PHYSICSFIRST SEMESTER – NOVEMBER 2012PH 1820 – MATHEMATICAL PHYSICS – I
Date : 09/11/2012 Dept. No. Max. : 100 Marks Time : 1:00 – 4:00 PART – AAnswer ALL questions: (10×2=20) Write down the formula for the Euler modified method for solution of ordinary differential equation. Explain the underlying difference between Newton-Raphson method and the Regula Falsi method of solving nonlinear equations Integrate Re z from 1+i to 3+2i along the straight line path. Find the singular points and the corresponding residues of the complex function f (Z) = . List the properties of scalar or inner product of two vectors in a linear vector space . Write down the matrix representations of the orthonormal basis vectors in 3-dimensional real space and the associated projection operators . Write down the transformation equations and its inverse in the Cartesian coordinates x,y,z and the spherical polar coordinates r,θ, φ. What is meant by contraction of a tensor? Use Rodrigue’s formula for the Legendre polynomial to evaluate the 3rd order polynomial. Write down the generating function for the Bessel functions.PART – BAnswer any FOUR questions (4×7.5=30) Solve the system of linear equations x + 2y + z = 3; 2 x + y + 3 z = 8; and 3 x + y + 2 z= 7 by Gauss elimination method with pivoting . (a) Integrate f(z) = (( z+1))/(( z^2+1)) counterclockwise around the circle C :|z -i| = 3/2. (3.5)(b) Find the Maclaurin series of f(z) = tan–1z, given that (df(z))/dz = 1/(( 1+z^2)). (4) (a) Prove that the eigenvalues of a Hermitian matrix are real and any two eigenvectors belonging to distinct eigenvalues are orthogonal to each other. (3.5)(b) Prove that the eigenvalues of an anti-Hermitain matrix are either zero or pure imaginary and any two eigenvectors belonging to distinct eigenvalues are orthogonal to each other. (4) (a) If a contravariant tensor of rank two is symmetric in one coordinate system, show that it is symmetric in any coordinate system. (3.5)(b) Show that in a cartesian coordinate system, the contravariant and the covariant components of a vector are identical. (4) Show that the beta function B(x,y) is related to the gamma functions by B(x,y) = (Γ(x)Γ(y))/(Γ(x+y)) and establish that B( x+1,y) = x/(x+y) B( x,y) . PART – CAnswer any FOUR questions (4×12.5=50) Apply Newton- Raphson method to find an approximate solution of the equation ex – 3x = 0 correct upto three significsnt figures( Assume x = 0.4 as an approximate root of the equation). Verify your result by Regula Falsi method . (a) Using the contour integration, evaluate the following real integral: (6.5) (b) Evaluate the contour integral, ∮▒dz/(z^2+1) with the contour C being (i) |z+i | = 1, and (ii) |z -i | = 1, counterclockwise. (3+3) (a) Explain the Schmidt orthogonalisation procedure. (8)(b) Construct an orthonormal set of three vectors from the given set of vectors a = (■(1@1@0)); b = (■(2@1@0)) ; c = (■(1@1@1)). (4.5) (a) Obtain an expression for the fractional increase in volume (dilation) associated with a deformation in terms of strain tensors. (6.5)(b) Obtain the relation between the angular momentum and the angular velocity of a system of particles in terms of the moment of inertia tensor. (6) Establish the orthonormality relation ∫_(-1)^(+1)▒P_n (x) P_m (x)dx={█(0 if n≠m@2/(2n+1 ) if n=m)┤ where Pn(x) is the Legendre polynomial of order n.