Calicut University 2006 M.Sc Physics phy 201 - MATHEMATICAL - II - Question Paper
SECOND SEMESTER M.Sc. DEGREE EXAMINATION, AUGUST 2006
Physics
phy 201 - MATHEMATICAL PHYSICS - II
(2003 admissions)
C 17061 (Pages : 2) Name.....................................
Reg. No.................................SECOND SEMESTER M.Sc. DEGREE EXAMINATION, AUGUST 2006
Physics
Phy 201MATHEMATICAL PHYSICSII (2003 admissions)
Time : Three Hours Maximum : 80 Marks
Section A
Answer any five questions.
Each question carries 4 marks.
1. Classify the singularities of a complex function. Give examples.
2. \Define metric tensor and express it for 3 dimensional Euclidean space in spherical polar
co-ordinates.
3. Define 2nd rank mixed tensor and contract it form a scalar. Prove the contracted quantity transforms
as a scalar.
4. ''Define Cauchy-Riemann conditions satisfied by an analytic function / (z).
5. Explain period doubling with an example.
6. Define Permutation group Sn. Show that any finite group of order n is a subgroup of Sn.
7. State and prove Schurs lemma.
8. Define isomorphism and homomorphism of groups with examples.
(5 x 4 = 20 marks)Section B
Answer two questions.
Each question carries 20 marks.
9. (a) State and prove Cauchy theorem and Cauchys integral formula. Obtain the Laurent series
expansion of a function around an isolated singular point.
(5 + 5 + 10 = 20 marks)
Or
(b) Find the symmetry group of 3 dimensional SchriOdinger equation for a particle in a central potential. Discuss the relation between symmetry and degeneracy.
(15 + 5 = 20 marks)10. (a) For a 3 dimensional Hamiltonian system find the fixed points and classify them. Discuss poincase maps and limit cycles and their classification.
Or
(b) From Varional method with constraints develop Rayleigh*Ritz method for calculating eigenvalues and eigne vector for the Sturm-Lioville equation. Illustrate it with calculation of the ground state eigenvalue and eigen function of a vibrating string.
(15 + 5 = 20 marks) [2 x 20 = 40 marks]
Turn over
11. Evaluate using Residue theorem :
C 17061
Section C
Answer any two questions. Each question carries 10 marks.
oo _i3x
dx.
(x*+2y (e +1)
12. Show that SU (2) and SO (3) groups are homomorphic.
13. Find the Euler equation satisfied by y (x) with the constraint
y2 = a2 f (y, yx;x) = v(y) Jl -
for (a) v (y) = ay + by2 + c.
(b) v{y) = eay-*y't with a, b, c, a, p constants.
14. Given the metric tensor for a 3 dimensional space with co-ordinates r, (p and % as
r cosh <p 0 0
0 r sinh <p cosh % 0
0 Or sinh <p sinh x
and the contravariant vector
A* =
sinh (p tanh % find the components of covariant vector Aj.
(2 x 10 = 20 marks)
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