A quantum pendulum: Consider the Hamiltonian of quantum pendulum that is swinging in a xed vertical plane under the inuence of gravity. Denoting the operator corresponding to the angle made by the pendulum with a vertical line in the plane of oscillation as 6': and the operator corresponding to the angular momentum as L: the Hamiltonian is given by '2 H = E? + ngu mm? (1) where I is the moment of inertia: R is the radius of the pendulum: m is the mass: 9 is the acceleration due to gravity. Angular momentum L and angle 6 are canonically conjugate: i.e.= [85 L] = i. (a) First= consider the approximation that the oscillations of the pendulum are small. In this approximation: one may Taylor expand the cosine term in H to the quadratic order. Let us denote this approximate Hamiltonian as H0. Find the eigenvalues of H0. Also nd the ground state wavefunction of H0 in the angular representation= i.e.: (SLED): where |Eg) is the ground state of H0. (b) Let us denote HHO as H1. When the oscillations are small: it is reasonable to treat H1 as a perturbation. Find the leading nonzero correction to the ground state energy and the ground state wavefunction due to this perturbation