Rotator in Field

Consider a rigid body with moment of inertia I, which is constrained to rotate in the xy-plane, and whose motion is given by the Schrodinger equation

(1)

a) Find the eigenfunctions and eigenvalues.

b) Assume the rotator has a fixed dipole moment p in the plane. An electric field E is applied to the plane. Find the changes in the energy levels to first and second order in the field.

SOLUTION

a) The eigenfunctions and eigenvalues are

(1)

(2)

b) The electric field interacts with the dipole moment to give an interaction

(3)

This problem is almost identical to the previous one. The quantity mr² of the previous problem is changed to the moment I in the present problem. The perturbation results are similar. The first-order perturbation vanishes since ⟨n|V|n⟩ = 0. The second-order perturbation  after changing mr² to I and er to p:

(4)