Question

A quantum mechanical rigid rotor constrained to rotate in one plane has moment of inertia I about its axis of rotation and electric dipole moment µ (in the plane).

This rotor is placed in a weak uniform electric field \(\varepsilon\), which is in the plane of rotation. Treating the electric field as a perturbation, find the first non-vanishing corrections to the energy levels of the rotor.

Answer 1

Take the plane of rotation of the rotor as the xy plane with the z-axis parallel to \(\varepsilon\) as shown in Fig. In the absence of external electric field the Hamiltonian of rotor is

When the external electric field acts on the system and may be treated as perturbation, the Hamiltonian of perturbation is

H' = -µ.\(\varepsilon\) = -µ\(\varepsilon\) cos\(\theta\)

The first order energy correction is

The second order energy correction is