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: [itex]\hat{H} = \frac{\hat{L^2}}{2I} - dE \cos{\theta}[/itex], where the polar angle [itex]\theta[/itex] 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 [itex]\hat{V} = −dE \cos{\theta}[/itex] on the ground-state wave function [itex]Y_{00}[/itex]. How does the function [itex]Y_{00}[/itex] depend on angles? Use this to show that there is only one state which is coupled to the ground state by [itex]\hat{V}[/itex], the state with [itex]\ell = 1[/itex] and [itex]m = 0[/itex].

Homework Equations

The Attempt at a Solution

I have stated that [itex]Y_{00}[/itex] does not depend on angles, but have no idea how to show that [itex]Y_{10}[/itex] 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}##?