# 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 $m = 0$.

## 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.

## Answers and Replies

Homework Helper
Gold Member
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}##?