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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
, at a distance
.
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{\pa...
...\frac{\partial^2}{\partial \phi^2} \right]
\psi(r) = E \psi(r)
\end{displaymath}](img309.png) |
(123) |
After a little effort, the eigenfunctions can be shown to be the
spherical harmonics
, defined by
![\begin{displaymath}
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}
\end{displaymath}](img311.png) |
(124) |
where
are the associated Legendre functions.
The eigenvalues are simply
 |
(125) |
Each energy level
is
-fold degenerate in
, since
can
have values
.
Next: The Hydrogen Atom
Up: Some Analytically Soluble Problems
Previous: The Harmonic Oscillator
Contents
David Sherrill
2006-08-15