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The Rigid Rotor

The rigid rotor is a simple model of a rotating diatomic molecule. We consider the diatomic to consist of two point masses at a fixed internuclear distance. We then reduce the model to a one-dimensional system by considering the rigid rotor to have one mass fixed at the origin, which is orbited by the reduced mass $\mu$, at a distance r. The Schrödinger equation is (cf. McQuarrie [1], section 6.4 for a clear explanation)

\begin{displaymath}- \frac{\hbar^2}{2I} \left[ \frac{1}{sin \theta}
...\frac{\partial^2}{\partial \phi^2} \right]
\psi(r) = E \psi(r)
\end{displaymath} (123)

After a little effort, the eigenfunctions can be shown to be the spherical harmonics $Y_J^M(\theta, \phi)$, defined by

\begin{displaymath}Y_J^M(\theta, \phi) = \left[ \frac{(2J + 1)}{4 \pi}
...)!} \right]^{1/2} P_J^{\vert M\vert}(cos \theta)
e^{iM \phi}
\end{displaymath} (124)

where PJ|M|(x) are the associated Legendre functions. The eigenvalues are simply

\begin{displaymath}E_J = \frac{\hbar^2}{2I} J(J+1)
\end{displaymath} (125)

Each energy level EJ is 2J+1-fold degenerate in M, since M can have values $-J, -J+1, \ldots, J-1, J$.