E-field Perturbation of 2D rotor. Show Y_10 couples ground state?

[fandango92]

Homework Statement

Consider a molecule with an electric dipole moment d. The Hamiltonian of a molecule in the external electric field E is: \hat{H} = \frac{\hat{L^2}}{2I} - dE \cos{\theta}, where the polar angle \theta characterises the orientation of the molecule. (We have chosen the field direction as the angular momentum quantisation axis.) Let us find field-induced contribution to the ground-state energy.

Consider the action of the perturbation operator \hat{V} = −dE \cos{\theta} on the ground-state wave function Y_{00}. How does the function Y_{00} depend on angles? Use this to show that there is only one state which is coupled to the ground state by \hat{V}, the state with \ell = 1 and m = 0.

Homework Equations

The Attempt at a Solution

I have stated that Y_{00} does not depend on angles, but have no idea how to show that Y_{10} is the only state that couples with the ground state.

 

[TSny]

What does it mean mathematically to say that a state $Y_{l,m}$ is coupled to the ground state $Y_{0,0}$ by $\hat{V}$?