MSc Physics (Electronics I) SEM II 2018 19 May 2018-19 PHYSICS PAPER III QUANTUM MECHANICS II Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one:--- 8 marks
   • (i) Explain time independent perturbation theory for non-degenerate states Obtain the first-order perturbation corrections to the energy eigenvalues and
   • (ii) Consider an isotropic harmonic. oscillator dimensions. The Hamiltonian is given by
   • A. What are the energies of the two lowest -lying states? Is there any
   • B. A perturbation, = exy, 1) is applied on the system. Find the first-order correction to the ground and first excited states
   • (b) Attempt any one :--- 4
   • (ii) eigenfunctions for the infinite square well of width a is: Find the first order correction to the energies and eigenfunctions for the
2. Q2 (a) any one :--- 8 marks
   • (i) Use the variational method to estimate the ground state energy of a particle of mass min the potential given by Take W(x) = as the trial wave function where a is the variational parameter and A is the normalization constant
   • (ii) Use approximation to find the tunneling probability through a
   • (b) Attempt any one :--- 4
   • (i) Show that variational method gives the upper bound of the ground state
   • (ii) Obtain energy eigenvalues of harmonic oscillator using WKB
3. Q3 (a) Attempt any one :--- 8 marks
   • (i) A particle with mass m, is scattered elastically by a particle of mass m, at rest in the Lab frame
   • A) Find the relation between the scattering angles of m, in Lab frame and the Centre of mass frame
   • B) Find the relation between differential scattering cross-section in Lab and centre of mass frame
   • (ii) Calculate the differential cross-section in the Born approximation for the potential V(r) = 2) Also calculate the total cross-section
   • (b) Attempt any one :--- 4
   • (i) the total cross-section for low energy (S-wave) scattering of a particle of mass m from the following potential
   • (ii) Discuss the validity conditions of Born approximation
4. Q4 (a) Attempt any one :--- 8 marks
   • (i) Consider a system of three noninteracting particles confined in a one dimensional infinite potential well of length a. Determine the energy and wavefunction of the ground state and first excited state when the particles
   • A) distinguishable with masses m, < mz <
   • B) Identical Bosons
   • (ii) Obtain the plane wave solution for the spin half particle in the relativistic formalism. Write the wavefunctions corresponding to positive and negative energies and two spin states
   • (b) Attempt any one:--- 4
   • (i) What are negative energy states? What is a hole? Obtain the equation of continuity from the Klein- Gordon equation
5. Q5 Attempt any four.--- 12 marks
   • (a) A particle is initially in its ground state in a one-dimensional harmonic potential A perturbation, H’ = is turned on at t = 0. Calculate the probability that the particle will be found in its first excited state after a sufficiently long time
   • (b) A hydrogen atom is in a constant uniform electric field ‘E’ that points in the z direction. Calculate the first order correction to the ground state energy of the Given: Unperturbed ground state of hydrogen atom is = ao)
   • (c) Use WKB approximation to estimate the transmission coefficient of a particle of mass m and energy (E < Vg) moving through the following potential barrier
   • (d) Discuss the validity condition of WKB approximation
   • (e) What is scattering amplitude? How is it related to scattering cross section?
   • (f) Explain optical theorem
   • (g) Obtain Klein-Gordon equation from relativistic energy relation
   • (h) Show that: = io’, where are Dirac matrices

Read from the scan above, so a character or two may differ. The scan is the original.