Page 157 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

148 Angular momentum

as the procedure using ladder operators. However, the Frobenius method may
not be used to obtain the eigenvalues and eigenfunctions of the generalized
^
^ 2
angular momentum operators J and J z because their eigenfunctions do not
have a spatial representation.

5.4 The rigid rotor

The motion of a rigid diatomic molecule serves as an application of the
quantum-mechanical treatment of angular momentum to a chemical system. A
rigid diatomic molecule consists of two particles of masses m 1 and m 2 which
rotate about their center of mass while keeping the distance between them ®xed
at a value R. Although a diatomic molecule also undergoes vibrational motion
in which the interparticle distance oscillates about some equilibrium value, that
type of motion is neglected in the model being considered here; the interparti-
cle distance is frozen at its equilibrium value R. Such a rotating system is
called a rigid rotor.
We begin with a consideration of a classical particle i with mass m i rotating
in a plane at a constant distance r i from a ®xed center as shown in Figure 5.2.
The time ô for the particle to make a complete revolution on its circular path is
equal to the distance traveled divided by its linear velocity v i
2ðr i
ô ˆ (5:60)
v i
The reciprocal of ô gives the number of cycles per unit time, which is the
frequency í of the rotation. The velocity v i may then be expressed as
2ðr i
v i ˆ ˆ 2ðír i ˆ ùr i (5:61)
ô
where ù ˆ 2ðí is the angular velocity. According to equation (5.1), the
angular momentum L i of particle i is
L i ˆ r i 3 p i ˆ m i (r i 3 v i ) (5:62)
Since the linear velocity vector v i is perpendicular to the radius vector r i , the
magnitude L i of the angular momentum is

v i
m i

r i

Figure 5.2 Motion of a rotating particle.