PH 2811 / 2808 – QUANTUM MECHANICS

Date : 19-04-2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions: (10×2=20)

Prove [ [A,B], C]+[ [B,C], A]+[ [C, A], B] = 0
State Heisenberg’s uncertainty principle
What are spherical harmonics? Are they mutually orthogonal?
Prove that the square of the angular momentum commutes with its z-component.
If A and B are two operators, then show that [A^-1[A,B]] = 2B
What are unitary transformations?
Show that commuting operators have simultaneous eigenfunctions.
What are indistinguishable particles?
What is Rayleigh ratio?
Find the separation between any two consecutive energy levels of a rigid rotator.

PART – B

Answer any FOUR questions: (4×7.5 = 30)

Obtain the equation of continuity in Quantum mechanics.
Obtain the normalized wave function for a particle trapped in the potential
V(x) = 0 for 0 < x < a and V(x) =
(a) With an example explain linear operator (b) A and B are two operators defined by AY(x) = Y(x) + x and BY(x) = + 2Y(x) check for their linearity (2.5 +5)
If + μω²x then show that
and ii. x (4+3.5)
Obtain the second order correction for a non-degenerae energy level.

PART – C

Answer any FOUR questions: (4×12.5 = 50)

State and prove Ehernfest’s theorem
Solve the Schrodinger equation for a linear harmonic oscillator. Sketch the first two eigenfunctions of the system.
Determine the eigenvalue spectrum of angular momentum operators J^z and J_z
What are symmetric and antisymmetric wave functions? Show that the symmetry character of a wave function does not change with time. Explain how symmetric and antisymmetric wave functions are constructed from unsymmetrized solution of the schrodinger equation of a system of indistinguishable particles. (3+3+6.5)
Using perturbation theory, explain the effect of an electric field on the energy levels of an atom (Stark effect).