Advanced Mathematical Physics – III — DU B.Sc. (Hons) Physics Previous Year Question Papers

B.Sc. urons.) Physics NEP: UGCF-2o22 Advanced Mathematical Physics (DSE) (b) Is the permutation (:2334::::;) even or odd? (c) What are the generators of the cyclic group G = < a > of order 10? (e) Show that the set of orthogonal matrices forms a SO(2) group under matrix multiplication. (I) Construct Green's function for the inhomogeneous equation y"(x) - f (x) with the boundary conditions y(0} = 0 and y'(1) = 0 . (g) Show that Green's function is symmetric with respect to ri and r2 i.e. (a) Solve the following differential equation using Laplace transform (10) y-y-F2J2J with initial conditions y(0) = 1 and y'(0) = 0 . (b) Find Laplace transform of the following function. .(a) Plot the function f (t) - sin t , 0 = t = 7t - 0, 7t = t = 27t extended periodically with period 27t.. Find L[f(t)I. (b) Using convolution theorem of Laplace transforms, evaluate. ) Consider a group G = {e,a,b,C} under a binary operation '`*''having Cayley group table: Show that following map is isomorphic where F :G + S4 (b) Show that the general form of the elements of 5If (2) group with respect to matrix multiplication is given by Also find all the generators of 5U(2)` group. I(8) (a) Prove that the symmetry operations of an equilateral triangle form a dihedral group D!. Write the (2X 2} matrix representation of all the elements of DE. Show that the symmetry operations` and the group of representation matrices are isomolphic to each other. (10) (b) Consider the cyclic permutations generated by (1234} in Permutation/Symmetric group, f4. Check whether these cyclic permutations form a subgroup of s4. (8) (a) A pendulum in the earth gravitational field is driven by an external time dependent force F(t). If the driven y(t) from the equilibrium position remains small, we can model this system by a harmonic oscillator whose differential equation is given by y(t)+co2y(t)-F(t),t>o with. homogeneous initial conditions y(0) = y(0) = 0. By using Green's function approach, solve the differential equation. (10) (b)Obta::;h(ey;)r:ei';fu(n;t;n)fo_r¥i)n::ry;perator for x(1) = 0 and finite at x(0). in be the constant. (8)