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Parts (a) and (b) not needed. Thank you.
(b) Consider a rigid rod-shaped molecule which is constrained to lie in the x-y plane, at an angle to the x-axis. The rotational energy of this molecule is 1 Ho=27¹²2 where I is its moment of inertia. Write down normalized wavefunctions (6) for eigenstates of L₂ with quantum number m and state the allowed values of m. Identify two conserved quantities in the dynamics, and state the symmetry giving rise to each. [5 marks] (c) The molecule possesses an electric dipole moment d. A time-dependent electric field E(1), which vanishes as →→∞, is applied in the direction of the x-axis, introducing an additional energy V(1) = -dE(t) cos(0). Before the field is applied (→∞) the rotating molecule has a definite value of the quantum number . Determine, to first order in the field, the possible final values of the angular momentum quantum number m' m, after the field has been applied (→→ +00). Determine also the probabilities of these transitions. [For integer n: (π if n = ±1 S™* cos(8) 10 = [o otherwise ] [4 marks] (d) Consider the case of an electric field which is a constant, E(1) = Eo, over a time interval of duration 7, and zero otherwise. Obtain the transition probabilities for this case. Sketch the dependence of the transition probability on the frequency of the transition, w=(Em - Em)/ti. [4 marks] (e) For sufficiently strong fields, Eo, corrections to the first-order result from higher- order terms become important. Obtain an estimate of the field strength at which this occurs for very short pulses. [4 marks]