Eigenfunctions of Rigid Rotator

[genloz]

Homework Statement

Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the centre (but the centre point itself is fixed).

Homework Equations

(a) Show that the allowed energies of this rigid rotator are
E=(h-bar^2n(n+1))/(ma^2) for n = 0, 1, 2, ...
(b) What are the normalised eigenfunctions for this system? What is the degeneracy of the nth energy level?

The Attempt at a Solution

(a) I realize that this is related to the total angular moment (E=L^2/ma^2)... I'm just a little unsure as to how L^2 becomes (h-bar^2n(n+1)).

(b) I know that the degeneracy is 2n+1 for spherical harmonic based systems but I really am unsure how to work out the eigenfunctions and the steps to getting to that '2n+1' figure.

Thankyou very much!

 

[malawi_glenn]

$$E_{\text{kin}} = \dfrac{I \omega^{2}}{2} = \dfrac{I^2 \omega^{2}}{2I} = \dfrac{L^2}{2I}$$

I is moment of inertia.

Now if L operates on a eigenfunction, the eigenvalue is $$\sqrt{n(n+1)}$$ , where n is the quantum number for that pequliar eigenfunciton (its eigenvalue).

 

[genloz]

ok, thankyou... so I understand part (a), but how do I work out the normalised eignfunctions and the degeneracy?

 

[malawi_glenn]

maybe this can be of help:

http://www.chem.uncc.edu/faculty/sisk/CHEM3141/lecture_17_05.PDF

http://web.ift.uib.no/~lipniack/fys201_v07/sphericallysymmetric.pdf

$$\mu$$ is the reduced mass of the system.

Also, I believe that the most common nomenclature for eigenvalue-numeration for QM-rotators is $$l$$ not n, so you don't get confused.