PH 2808 – QUANTUM MECHANICS

Date : 22/04/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART A (10 x2 m = 20 m)

Prove that the momentum operator is self adjoint.
In what way is the orthonormal property of eigenfunctions belonging to discrete set of eigenvalues different from those of continuous set?
Define parity operator. What is its action on a wave function ψ ( r, θ, φ)?
What is a rigid rotator and what are its energy eigenvalues?
Expand an arbitrary state vector in terms of certain basis vectors. Define projection operator.
Explain the term the ‘wave function’ of a state vector ׀ψ>.
Given that [ J_X,J_Y] = iħ J_Z and its cyclic, verify that [ J₊, J_–] = 2ħ J_Z, where J₊ = J_X + iJ_Y and J_– = J_X– iJ_Y.
Prove that the Pauli spin matrices anticommute.
Write down the Hamiltonian of a hydrogen molecule.
Explain the terms classical turning points and the asymptotic solution in the context of WKB approximation method.

PART B ( 4×7 ½ m = 30 m)

ANSWER ANY FOUR QUESTIONS

(a) Verify the identities [ AB, C) = A [ B, C] + [ A, C] B and [ A, BC] = B[A, C] + [ A, B] C . (b) Determine [ x², p²], given that [ x, p] = iħ.
Evaluate ( u_m, x u_n) where u_n’s are the eigenfunctions of a linear harmonic oscillator.
Prove that “the momentum operator in quantum mechanics is the generator of infinitesimal translations”.
(a) Prove that ( σ.A) (σ.B) = B + i σ. ( A xB) where σ’s are the Pauli spin matrices , if the components of A and B commute with those of σ. (b) Determine the value of (σ_x +i σ_y)².
Estimate the ground state energy of a two-electron system by the variation method.

PART C ( 4x 12 ½ m=50 m)

ANSWER ANY FOUR QUESTIONS

(a) State and prove closure property for a complete set of orthonormal functions. (b) Normalize the wave function ψ(x) = e ^–^׀x^׀
Discuss the simple harmonic oscillator problem by the method of abstract operators and obtain its eigenvalues and eigenfunctions.
(a) The position and momentum operators x_op and p_op have the Schrödinger representations as x and –iħ ∂/∂x”-Verify this statement.(b) Explain the transformation of Schrödinger picture to Heisenberg picture in time evolution of quantum mechanical system.
Determine the eigenvalue spectrum of the angular momentum operators J², J_z ,J₊ and J_–, starting with the postulate [ J_x, J_y] = iħ J_zand its cyclic.
Outline the perturbation theory of degenerate case with specific reference to the two-dimensional harmonic oscillator.