l. Planar rotator is a one-degree of freedom physical system that can execute rotational motion in a plane. Its classical Hamiltonian function is M? '2! where M; is the projection of the angular momentum i'ector onto the z-axis that is perpendicular to the plane of rotation and I is the moment of inertia of the system about the Eaxis. The Hamiltonian operator of the planar rotator has the form H: {13. .. 1 l M: s33: , . a H = = f 2 _ [ 2! at up where qt: is the rotation angle about the z-axis. {a} Gite at least 3 examples of planar rotators. {b} 1d'hat physical quantities are conserved for the planar rotator? {c} 1d'hat are the normalized eigenfunctions and eigenvalues of the operator :2? {d} Determine the stationary state wave functions and energy levels of the planar rotator. Hon is this question related to your answers in part [c]? {e} 1d'hat is the deg enerac'; of the energy levels? 2. The planar rotator of previous problem is in a state with the wave function us (p) = Ccos p. (1) where C is the normalization constant. (a) Determine the probability of different values of the projection of the angular momentum, its average value and the dispersion. A quick way to expand the wave function (1) in eigenfunctions of l determined in previous problem is to apply Euler's formula to Eq.(1). Also remember to normalize the wave function (1). (b) Determine the probability of different values of the energy, its average value and the dispersion. (c) Write down an expression for the time-dependent wave function of the planar rotator, show that it is periodic in time and find its period