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Pune University - Sem - II : Quantum Mechanics - II, April 2010

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Total No. of Questions : 7] P887 [Total No. of Pages : 3 [3722]-204 M. Sc. PHYSICS PHY2 UTN - 604 : Quantum Mechanics - II (New Course) (2008 Pattern) (Sem. - II) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Question No. 1 is compulsory. Solve any four questions from remaining. 2) Draw neat diagrams wherever necessary. 3) Figures to the right indicate full marks. 4) Use of Logarithmic tables and calculator is allowed. Q1) Attempt any four of the following : a) Explain the conditions of the validity of W. K. B. approximation. [4] b) What is perturbation? Develope the first and second order perturbation equations for stationary states. [4] c) Find the energy levels and eigen functions of Hamiltonian 1+ H= where << 1, corrected upto first order in by 1 + using perturbation theory. d) [4] The scattering amplitude by partial wave analysis is : 1 f ( )= k (2l + 1) exp (i l) sin l Pl (cos ) where symbols have their l=0 usual meaning. Hence obtain optical theorem. e) [4] Show that the angle of scattering in Laboratory frame ( L) and in the centre of mass frame ( C) are related by : tan L sin C = m1 + cos m2 C [4] P.T.O. f) Explain the terms : i) Identical particles and ii) Symmetric and Antisymmetric wave functions. [4] Q2) a) Apply non-degenerate time independent perturbation theory to find i) first order change in energy and ii) second order change in energy. [8] b) Evaluate the scattering amplitude and differential cross-section for Yukawa potential V(r ) = Born approximation. Q3) a) V0 exp ( r ) where V0 and are constant use r [8] In time dependent perturbation theory, the first order transition amplitude is a f = (i! ) (1) 1 t iw t 0 H fi (t ) e fi dt . Hence obtain Fermi-Golden rule for constant perturbation. [8] b) Obtain the Slater determinant for N-identical particles and explain the Pauli exclusive principle. [8] Q4) a) By using Green s function technique, obtain the expression for the [8] scattering amplitude. b) What is stark effect in hydrogen atom? Show that i) in the ground state, it is absent ii) in first order excited level of hydrogen atom : 3eFa 0 3eFa 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0, 0 a 0,0 a 1, 0 = W (1) a 1, 0 a 1,1 a 1,1 a 1, 1 a 1, 1 where F = external electric field. 1 (Given : 100 = and 210 = [3722]-204 2 a0 e r / a0 ; 200 = r 2 e r / 2 a0 3 a0 4 2 a0 r r / 2 a0 e cos ) 3 4 . 2 a0 a0 1 -2- 1 [8] Q5) a) Describe the variation method used to estimate ground state energy and apply it to find ground state energy of a one-dimensional harmonic 2 [10] oscillator when trial wave-function is = Ae x . b) Using partial wave analysis, obtain the expression for phase shift of s-wave scattering from hard sphere (V(r) = for o r a and V(r) = 0 for r > a). Show that in low energy limit, the total scattering cross-section is 4 a2. [6] Q6) a) Explain the First Born approximation. Show that for spherically symmetric scattering centre, the Born approximation amplitude is f B( b) Q7) a) 1 ) = r sin Kr U(r) dr . Hence for the Coulomb screen potential K0 Ze 2 r V(r ) = e ; find the Born approximation amplitude fB( ). [8] r Write down the connecting formulae in W. K. B. approximation, hence obtain Bohr-Sommerfeld quantum rule. [8] p2 1 The anharmonic Hamiltonian of oscillator is H = + m 2 x 2 + bx 3 . 2m 2 Show that the second order correction to ground state energy is 11 b 2! 2 8 m 3 4 . (Given : a un = n un 1 and a+ un 1 = n un) [4] b) Define exchange operator P12 for identical particles and show that it commutes with Hamiltonian H(1, 2). [4] c) With necessary diagram, explain scattering event and define differential as well as total scattering cross-section. [4] d) Explain the dipole approximation. Which of the following transitions are allowed : [4] i) 1s 2s ii) 2p 3d and iii) 3s 5d. rrrr [3722]-204 -3-

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Additional Info : M.Sc. PHYSICS, PHY UTN - 604 : Quantum Mechanics - II (New Course) (2008 Pattern) (Semister - II), Pune University
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