Perturbation Theory and WKB Method

Abstract

A plane rigid rotator has the following Hamiltonian

$${\hat H_0} = \frac{{\hat L_z^2}}{{2I}}$$

where I is the momentum of inertia and ${\hat L_z} = - i\hbar \tfrac{d}{{d\phi }}$ is the component of the angular momentum in the z direction (ϕ is the azimuthal angle). A small perturbation

$${\hat H_1} = \lambda \cos (2\hat \phi ) \lambda \ll 1$$

is applied to the rotator. Determine the average value of ${\hat H_1}$ on each unperturbed eigenstate. Then, determine the off-diagonal elements of the perturbation matrix between the degenerate states. Finally, find the first order correction to the energy of the ground state and the splitting induced on the enery of the first excited state.