Question

2. A rigid rotor is constrained to move about a fixed axis, with respect to which its moment of inertia is I. The time-independent Schrödinger equation for this system is: 21 do where W is the energy, ф is the angular displacement of the rotor from a fixed direction (0 ps 2 ) , and 111(pf dф gives the probability of finding the rotor at an angular position between ф and ф+49 State the general solution of this equation and apply an appropriate boundary condition to show that the energy levels of the rotor are given by W,nh (2I); n 0,1 Explain your reasoning clearly. Show also that the corresponding normalised eigenfunctions are given by us (ф)-exp( ing)W2r. (a) 5 marks) (b) At the instant t-0 the rotor is in a state described by the un-normalised wave function up)3-2e2ig (i) Write down an expression for the wave function at a later time, (2 marks) (ii) What are the possible outcomes of a measurement of the energy of the rotor at time t, and what is the probability of observing each outcome? (3 marks)

Answer 1

Given the time-independent Schrodinger equation for rigid rotor as: -h^2/2I d^2 u(Phi)/d Phi^2 = w u(Phi) rightarrow d^2/dPhi^2 u(Phi) = -2 I w/h^2 u(Phi) Let, m^2 = 2 I w/h^2: then d^2 u(Phi)/d Phi^2 = -m^2 u(phi) where general solution is u(Phi) = A e^-i m Phi + B e^i m Phi or u(Phi) = C e^plusminus i m Phi Putting boundary condition on u(Phi) (which is essentially the wave fn of the rotor along the circular trajectory and the wave fn must complete the circular trajectory in multiple of it's wavelength, to fit into it) rightarrow u(Phi) = u(Phi + 2 pi) to maintain the continuity of u(Phi). rightarrow C e^i m Phi = c e^i m(Phi + 2 pi) rightarrow e^i m 2n = 1 rightarrow cos (2 pi m) + i sin (2 m pi_/ = 1 rightarrow cos(2 m n) = 1 rightarrow m = 0, plusminus 1, plusminus 2, Ellipsis, infinity