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

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Total No. of Questions : 7] [Total No. of Pages : 3 P883 [3722]-104 M. Sc. PHYSICS PHY UTN - 504 : Quantum Mechanics - I (New Course) (Sem. - I) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Question No. 1 is compulsory. 2) Attempt any four from the remaining. 3) Figures to the right indicate full marks. 4) Draw neat diagrams wherever necessary. 5) Use of Mathematical tables and pocket calculator is allowed. Q1) Attempt any four of the following : a) [16] The spectral density of a matter wave-packet is in the form of a symmetrical exponential as : (k) = Ae | k | Find the form of wave function (x). b) The x-component of a linear momentum Px is classically expressed as Px = m dx . By using Schroedinger equation, show that dt + Px = c) * i x (x, t ) ! (x, t ) dx . Define : i) adjoint of an operator A ii) Hermitian operator and () + Show that A B = B+ A + d) Explain unitary operator. Show that the norm of a state functions do not changes under unitary transformation. P.T.O. e) In three dimension, the ground state wave-function of hydrogen atom is 1 2 1 e r / a0 where a0 is Bhor radius. Then show that the 3 a0 (x ) = 3 2 1 2a probability in momentum space is : C ( p) = 0 ! f) 2 )] 2 a0 / ! 2 + 1 2 The operators for angular momentum are : J+ = Jx + iJy and J = Jx iJy Show that : (i) Q2) a) [(p 1 [J + , J ] = 2! J z and (ii) [J z , J ] = ! J A particle of mass m is moving in potential well : V(x) = V0 for x < a = O for a < x < a = V0 for x > a When energy of a particle E < V0; then show that there exists at least one bound state. [8] b) Obtain Clebsch-Gordan coefficients for a system of two non-interacting particles with angular momenta : j1 = Q3) a) b) 1 1 and j2 = . 2 2 [8] Using ladder operator method, obtain the energy eigen values and eigen functions of a one dimensional harmonic oscillator. [10] What are observables? Using expansion postulate show that [6] i) ii) Q4) a) b) [3722]-104 Eigen functions belonging to discrete eigen-values are normalizable. Eigen functions belonging to contineous eigen values are of infinite norm. 2 Obtain the eigen value spectrum of L and Lz operators. [8] Describe Schroedinger and Heigenberg pictures regarding the evolution of a system with time. [8] -2- Q5) a) What is spin angular momentum and spin space of a particle having spin S. For spin = i) 1 particles; explain Pauli spin matrices and show that 2 2 2 x + y + z2 = 3 and ii) x y = i z [8] b) Apply the evolution equation of Heigenberg s picture, to construct the position operator x (t ) and momentum operator p(t ) for harmonic oscillator. [8] Q6) a) When the unitary transformation is induced by rotation of co-ordinate system then show that [8] i) ii) b) Q7) a) Lz plays the role of the generator of infinitesimal rotation ! x = x ei n .L / ! . Explain completeness property and prove the closure relation. Define projection operator. Show that the sum of all the projection operators leaves any state vector unchanged. [4] 1 . 2 b) Obtain matrices for Jx and Jy when j = c) In Dirac formulation of Quantum Mechanics; explain the terms : i) [4] Hilbert space Use Dirac notation to prove that the Eigen values of Hermitian operator are real. [4] rrrr [3722]-104 [4] State vectors ii) d) [8] -3-

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Additional Info : M.Sc. PHYSICS, PHY UTN - 504 : Quantum Mechanics - I (New Course) (Semister - I), Pune University
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