MPH-1 EM 2026 SOLVED ASSIGNMENT

Description

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS

M.Sc. (Physics) Programme
(MSCPH)
MATHEMATICAL METHODS IN PHYSICS
Valid from 1st January, 2026 to 31st December, 2026

Course Code: MPH-001

Assignment Code: MPH-001/TMA/2026

Max. Marks: 100 Note: Attempt all questions.

The marks for each question are indicated against it.

PART A

1. a) Reduce the following PDE into three ODEs:

0 ) , , ( ) , , ( 2 2 2 2 2 2 2 = + ú ú û ù ê ê ë é ¶ ¶ + ¶ ¶ + ¶ ¶ zyx f k zyx f z y x (5)

1. b) Derive an integral equation corresponding to the ODE: 0 2 = -¢¢ y y subject to the conditions: 2 )0( ;4 )0( -= ¢ = y y (5) c) Use the method of separation of variables to reduce the Laplace’s equation 0 2 = Ñ f into three ODEs. (5)
2. d) Using the generating function for Bessel functions of the first kind and integral order å ¥ -¥ = =ú û ù ê ë é ÷ ø ö ç è æ – = n n n t x J t t x txg ) ( 1 2 exp ) , ( Obtain the recurrence relation ) ( 2 ) ( ) ( 1 1 x Jx n x J x J n n n = + + – Also using the generating function show that 1 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 6 4 2 0 = + + + + + + L L x J x J x J x J x J k (10)
3. a) Obtain orthogonality relation for Hermite polynomials using the generating function: å ¥ = – = 0 2 ! ) ( 2 n n n t xt n t x H e (10)
4. b) i) Show that the following vectors

ú ú ú û ù ê ê ê ë é ú ú ú û ù ê ê ê ë é ú ú ú û ù ê ê ê ë é 1 1 0 , 0 1 0 , 0 0 1 (2) are linearly independent. ii) The first Pauli matrix is ú û ù ê ë é = s 0 1 1 0 1 calculate L- s q -= sq + = sq = q 2 1 2 1 1 1 2 1 ) ( exp ) ( i i U For real q, show that 1U is unitary and has determinant 1. (3) 4

1. c) Obtain the eigenvalues and eigenvectors of matrix A: ú ú ú û ù ê ê ê ë é – – = 2 0 1 0 2 0 1 0 2 A (5)
2. d) Define covariant and cotravariant tensors of rank
3. Prove that j ij i vg a = transform covariantly, where ijg are the components of the matrix tensor of rank 2 and iv the components of a contravariant vector. (5)

PART B

3. a) i) Obtain the analytic function whose real part is y e yxu xcos ) , ( = (3) ii) Locate and name the singularity of the function: 2 2 2 2 2 + + – z z z z (2)
4. b) Calculate the value of the integral ò C dz z z cos when C is the circle .2=z (5)
5. c) Show that the Series å ¥ = – 1 ) 1( n n z z converges for 1<z and find its sum. (5)
6. d) Obtain the Laurent series expansion of 2)1 ( -z ez about z = 1. Determine the type of singularity and the region of convergence.
7. e) Evaluate the value of the contour integral ( ) ò + – C z z dz z )9 )1 ( 6 2 where C is a circle defined by .4=z (5)
8. a) Evaluate the integral òp q + q 2 0 cos 1 p d by the method of residues when 1 1 < < – p . (10)
9. b) Consider a triangle T in the z-plane with vertices at i, 1–i, 1+i. Determine the triangle 0T into which T is mapped under the transformations i iz w – + = 2 (5)
10. c) Obtain the Fourier cosine transformation of the function: 0 , 0 , ) ( > ¥< < = – p x e x f px (5)
11. d) Define homomorphisms. When do the homomorphisms become endomorphism and isomorphism?