Advanced Quantum Mechanics – II — DU B.Sc. (Hons) Physics Previous Year Question Papers

Serial Number of Question Paper Adv. Quantum Mechanics-II B.Sc. Hons (Physics) NEP: UGCF-2022 VIII- Semester a)SE Paper) All questions cany equal marks. a) For a linear harmonic oscillator, the force constant changes from k to k + A, where A << k. Treat the change in force constant as a perturbation and write the perturbation Hamiltonian. b) A variational trial wave function gives expectation energy values 6ev, 5ev,, and 4.5ev for three different choices of the variational parameters. Which value gives the best variational estimate of ground-state energy? Justify your answer. c) A perturbation acts on a quantum system for time interval 6t. Determine whether the evolution is sudden or adiabatic if 6£ >> fi/6E where 6E is a characteristic energy difference of the system. Justify your answer. d) A particle in a box of length I suddenly expands to length 2L. If initially, the particle was in the ground state, what is the probability that after the expansion, it would still be in the new ground state? e) In the first-Born approximation, the scattering amplitude is f (G) = # where A = 2 x 10-11 in. Calculate the differential cross-section at G = 30°. I) Explain the meaning of the scattering matrix (S-matrix) in quantum scattering theory. What physical information is obtained from its diagonal elements? g) For a scattering process, the forward scattering amplitude is f (0) = (2 + Z5) X 10-11 in. The wave number of the incident particle is k = 2 X 10£°m-1. Using the optical theorem, calculate the total scattering cross-section. a) Consider a quantum system with Hamiltonian H = Ho + AV, where Ho has a complete set of non-degenerate eigenstates. Using time-independent perturbation theory, derive the expressions for the first-order correction to the energy and the first-order correction to the eigenfunction. [9] b) Consider the one-dimensional quantum anharmonic oscillator described by the Hamiltonian H = H(0) + H' with H(0) = g; + :mco2£2 and H' = 1£4 a small perturbation. Using a characteristic length scale cZ = and the ladder operator representation of the position operator, £ = i(a+at) , express the perturbation in terms of the creation and annihilation operators. Using first-order time-independent perturbation theory, compute the correction to the ground state energy due to the perturbation. [9] a) Combine the relativistic and spin--orbit corrections for the hydrogen atom to obtain the flne-structure energy correction. Show that the corrected energy levels depend only on the principal quantum number ;7 and the total angular momentum quantum number/.. What is the physical significance of the remaining degeneracy of states having the same values of7€ and/.? [9] b) Starting from the Lippmann-Schwinger equation, derive the first-Born approximation for the scattering amplitude f(O, ¢). Show that, for a central potential V(r), the scattering amplitude depends only on the scattering angle a and can be written as