Page 158 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.4 The rigid rotor 149
L i ˆ m i r i v i sin (ð=2) ˆ m i r i v i ˆ ùm i r 2 (5:63)
i
where equation (5.61) has been introduced.
We next apply these classical relationships to the rigid diatomic molecule.
Since the molecule is rotating freely about its center of mass, the potential
energy is zero and the classical-mechanical Hamiltonian function H is just the
kinetic energy of the two particles,
p 2 1 p 2 2 2 2
1
1
H ˆ ‡ ˆ m 1 v ‡ m 2 v 2 (5:64)
1
2
2
2m 1 2m 2
If we substitute equation (5.61) for each particle into (5.64) while noting that
the angular velocity ù must be the same for both particles, we obtain
2
2
2
1
1
H ˆ ù (m 1 r ‡ m 2 r ) ˆ Iù 2 (5:65)
2 1 2 2
where we have de®ned the moment of inertia I by
2
I ˆ m 1 r ‡ m 2 r 2 2 (5:66)
1
In general, moments of inertia are determined relative to an axis of rotation.
In this case the axis is perpendicular to the interparticle distance R and passes
through the center of mass. Thus, we have
r 1 ‡ r 2 ˆ R
and
m 1 r 1 ˆ m 2 r 2
or, upon inversion
m 2
r 1 ˆ R
m 1 ‡ m 2
(5:67)
m 1
r 2 ˆ R
m 1 ‡ m 2
Substitution of equations (5.67) into (5.66) gives
I ˆ ìR 2 (5:68)

where the reduced mass ì is de®ned by
m 1 m 2
ì (5:69)
m 1 ‡ m 2
The total angular momentum L for the two-particle system is given by
2
2
L ˆ L 1 ‡ L 2 ˆ ù(m 1 r ‡ m 2 r ) ˆ Iù (5:70)
1 2
where equations (5.63) and (5.66) are used. A comparison of equations (5.65)
and (5.70) shows that
L 2
H ˆ (5:71)
2I