A quantum mechanical rigid roator

[hokhani]

Homework Statement

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

Homework Equations

The eigenfunctions of [itex]H_0[/itex] 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?

The Attempt at a Solution

 

[DrClaude]

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##?

 

[hokhani]

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

 

[DrClaude]

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.

 

[hokhani]

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

 

[DrClaude]

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

 

[hokhani]

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

Ok, it is very easy to calculate it:
[itex]\langle m' | H' | m \rangle=-\frac{\mu E}{2}(\delta(m',m+1)+\delta(m',m-1))[/itex]
Making the [itex]H'[/itex] 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?

 

[DrClaude]

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