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)

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

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

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

where $P_J^{\vert M\vert}(x)$ are the associated Legendre functions. The eigenvalues are simply

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

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