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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 onedimensional
system by considering the rigid rotor to have one mass fixed at the
origin, which is orbited by the reduced mass ,
at a distance r.
The Schrödinger equation is (cf. McQuarrie [1],
section 6.4 for a clear explanation)

(123) 
After a little effort, the eigenfunctions can be shown to be the
spherical harmonics
,
defined by

(124) 
where
P_{J}^{M}(x) are the associated Legendre functions.
The eigenvalues are simply

(125) 
Each energy level E_{J} is 2J+1fold degenerate in M, since M can
have values
.