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Monday, 9 March 2015

Quantum Mechanics- Model question paper for MSc/PhD course work



1(a) Assume that a particle is represented by a wavepacket and show that the uncertainty principle follows naturally from the de Broglie hypothesis. (6 Marks)
(b) What is the significance of commutability of operators? Discuss Heisenberg’s commutation relations. (6 Marks)
(c) What is momentum wavefunction? Explain its significance. (4 Marks)
(d) Use the uncertainty principle to calculate the ground state energy of an electron in hydrogen atom. (4 Marks)
2(a) Generalize the expression for transmission coefficient to a potential hill of any shape. (6 Marks)
(b) Discuss the solution of the Schrödinger equation for a three-dimensional potential box and show that the degeneracy depends on the symmetry of the box. (10 Marks)
(c) What are creation and annihilation operators? (4 Marks)
3(a) Setup and solve the Schrödinger equation for a rigid rotator with free axis for its eigenvalues and eigenfunctions. (10 Marks)
(b) Find the average value of the distance of the electron from the nucleus in the ground state of a hydrogen atom. (6 Marks)
(c) Write Laguerre differential equation. Comment on its solutions. (4 Marks)
4(a) What is Hilbert space? (5 Marks)
(b) What is Hermitian matrix? Show that the eigenvalues of a Hermitian matrix are real and that the diagonalising matrix of a Hermitian matrix is unitary. (10 Marks)
(c) Write a note on secular equation. (5 Marks)
5(a) Use the time independent perturbation method to obtain the ground state energy of a helium atom. Compare the result with the experimental value. (8 Marks)
(b) Explain the Heitler-London theory of hydrogen molecule and discuss the results obtained. (6 Marks)
(c) Use the WKB approximation to calculate the decay constant of nuclei emitting alpha particles. (6 Marks)
6(a) What is sudden approximation? What is the nature of eigenfunctions in this process? (6 Marks)
(b) Explain the Hamiltonian of a charged particle in an electromagnetic field and use it to find the transition rate in dipole approximation. (6 Marks)
(c) What is Born approximation? (4 Marks)
(d) Write a note on Eikonal approximation. (4 Marks)
7(a) Show that three Pauli matrices along with a 2 x 2 unit matrix form a complete basis of algebra. (6 Marks)
(b) What is the central field approximation? Discuss the Thomas-Fermi statistical model used for the purpose. (6 Marks)
(c) What are pure quantum state and mixed quantum state? (4 Marks)
(d) Write a note on symmetrisation of wavefunctions. (4 Marks)
8(a) Derive Klein-Gordon equation. (6 Marks)
(b) Show that the spin-orbit interaction is a consequence of the Dirac relativistic equation. (6 Marks)
(c) How does the quantization of an electromagnetic field in vacuum lead to the concept of photon? (4 Marks)
(d) Find the commutation relations for the electric and magnetic vectors of the electromagnetic field. (4 Marks)