Example 4.11: Partition of energy weighting factor of diatomic molecule Consider the energy of a diatomic molecule as the sum of three independent contributions trans, rot and vib arising from translation, rotation and vibration. Consider the relative motion of one of the atoms about the other as if one of them had an infinite mass which acts as a center for the other with reduced mass m0 = m1m2/(m1 +m2). Then, using polar coordinates as in Example 3.6, with pθ = m0r2 ˙θ and pφ = m0r2 ˙φ sin2 θ, the relevant energy is

=

1 2

m0r˙2 +

1 2

m0r2θ˙2 +

1 2

r2 ˙φ2 sin2 θ + vib =

p2r 2m0

+

p2θ

2I

+

p2

φ

2I sin2 θ

+ vib, (4.68)

where I is the moment of inertia of the molecule. Approximating vib by an harmonic potential about an equilibrium position r0, i.e. setting trans + vib =

p2r 2m0

+

1 2a2

(r − r0)2, a= const., (4.69)

and quantizing this in the usual way, we can set trans + vib = hν



n +

1 2



, n= 0, 1, 2, . . . , and ν =

1 2πa√m0

. (4.70)

With rot also quantized we have

=

J(J + 1)2 2I

+ hν



n +

1 2



. (4.71)