In a diatomic molecule the atoms can rotate about each other. This rotation can be shown to be equivalent to a reduced mass μ=m1m2/(m1+m2) rotating in three dimensions about a fixed point. According to classical physics, the energy of a three-dimensional rigid rotator is given by E=L2/2I, where I is the moment of inertia and L2=Lx2+Ly2+Lz2 is the magnitude squared of the orbital angular momentum.
(a) What is the energy operator for this three-dimensional rotator? What are the energy eigenstates and corresponding energy eigenvalues? Take the moment of inertia I to be a constant. (A complication is that molecules ``stretch'' as they rotate faster, so the moment of inertia is not a constant.) Show in the limit of large l (l≫1) that
ElEl−El−1→0
which means that the discrete nature of the energy levels becomes less apparent as l increases.
(b) Determine the frequency of the photon that would be emitted if the rotator makes a transition from one energy level labeled by the angular momentum quantum number l to one labeled by the quantum number l−1.
(c) Show in the limit in which the orbital angular momentum quantum number l is large (l≫1) that the frequency of the photon from part (b) coincides with the classical frequency of rotation of the rotator (recall L=Iω). This result illustrates the correspondence principle, namely how the results of classical physics and the results of quantum mechanics can coincide in the appropriate limit. It also shows how we might use the correspondence principle to deduce the existence of a selection rule, in this case Δl=±1. Note: Here Δl means the change in l, not the uncertainty in l.