A quantum mechanical rigid roator

1. Jan 18, 2014

hokhani

1. The problem statement, all variables and given/known data

Consider the Hamiltonian for a rigid rotator, constrained to rotate in xy plane and with moment of inertia I and electric dipole moment$\mu$ in the plane, as $H_0=\frac{L_z^2}{2I}$. Then suppose that a constant and weak external electric field,$E$, in the direction of x is applied so the perturbation is $H'=-\mu E cos (\phi)$.

2. Relevant equations

The eigenfunctions of $H_0$ are double degenerate and the degeneracy is not removed even in high orders of perturbation. How should I solve it? could anyone please help me?

3. The attempt at a solution

Last edited: Jan 18, 2014
2. Jan 18, 2014

Staff: Mentor

The problem statement doesn't include a question.

Have you calculated $\langle m' | H' | m \rangle$? And found it to be always zero for $m'\neq m$?

3. Jan 18, 2014

hokhani

The degenerate eigenfunctions are $|m\rangle=\frac{1}{\sqrt {2\pi}} e^(im\phi)$ and $|-m\rangle=\frac{1}{\sqrt {2\pi}} e^(-im\phi)$ and $\langle -m|cos(\phi)|m\rangle =0$ (ms are integer). Hence all the elements of perturbation matrix between degenerate kets would be zero.

Last edited: Jan 18, 2014
4. Jan 18, 2014

Staff: Mentor

I made a quick calculation, and it seems indeed that the degeneracy is not lifted. How is that a problem? The field still shifts the levels.

5. Jan 18, 2014

hokhani

Could you please give me your solution? Or at least, how much does the electric field move the levels?

6. Jan 18, 2014

Staff: Mentor

You should calculate $\langle m' | H' | m \rangle$ yourself. Rewriting $\cos \phi$ in terms of exponentials helps in evaluating the integrals.

7. Jan 18, 2014

hokhani

Ok, it is very easy to calculate it:
$\langle m' | H' | m \rangle=-\frac{\mu E}{2}(\delta(m',m+1)+\delta(m',m-1))$
Making the $H'$ matrix on the two degenerate kets, m and -m, we observe that all the four matrix elements are zero. Could you please tell me what is my mistake?

8. Jan 19, 2014

Staff: Mentor

I'm sorry, but the I don't understand what question you are trying to answer.