Physical Chemistry , 1st ed.

b.The vibrational frequency expected for a hydrogen atom having a mass of

1.674 

10 ^27 kg and a vibrational force constant of 967 kg/s^2 is given by



2

1





m

k





2

1





1.67

9

4

66



.7

1

k

0

g



/



s

2

2

 (^7) kg

1.209
1014 Hz
This is a somewhat lower frequency, about 2^12 % lower, than is found experi-
mentally. This illustrates that using the reduced mass does have an effect on
the calculation. The effect is even more obvious when the two particles have
similar masses. Repeat this example using H 2 (see example 11.9) and HD,
where D ^2 H.
In all cases where multiple particles are moving relativeto each other in our
system, the reduced mass must be considered in place of the actual mass. In
the harmonic oscillator, two particles are moving relative to each other, and so
the reduced mass is used. In a purely translational motion, two masses are
moving through space but remain in the same positions relative to each other.
Therefore, the sum of the masses, the total mass, is the correct mass needed to
describe the translational motion.

11.6 Two-Dimensional Rotations

Another model system consists of a mass traveling in a circle. A simplistic di-

agram of such a system is shown in Figure 11.7. The particle having mass mis

moving in a circle having a fixedradius r. There may or may not be another

mass at the center, but the only motion under consideration is that of the par-

ticle at radius r. For this system the potential energy Vis fixed and can be ar-

bitrarily set to 0. Since the particle is moving in two dimensions, chosen as the

xand ydimensions, the Schrödinger equation for this system is



2



m

2







x

2

 2 



y

2

 (^2) E (11.29)
This is actually not the best form for the Schrödinger equation. Since the
particle is moving at a fixed radius and changing only its angle as it moves in
a circle, it makes sense to try and describe the motion of the particle in terms
of its angular motion, not its Cartesian motion. Otherwise, we would have to
be able to solve the above Schrödinger equation in two dimensions simultane-
ously. Unlike the 3-D particle-in-a-box, we cannot separate the xmotion from
the ymotion in this case, since our particle is moving in both xand ydimen-
sions simultaneously.
To find eigenfunctions for the Schrödinger equation, it will be easier if we
express the total kinetic energy in terms of angular motion. Classical mechan-
ics states that a particle moving in a circle has angular momentum,which was
defined in Chapter 9 as Lmvr. However, we can also define angular mo-
mentum in terms of linear momenta,pi, in each dimension. If a particle is con-
fined to the xyplane, then it has angular momentum along the zaxis whose
magnitude is given by the classical mechanics expression
Lzxpyypx (11.30)
11.6 Two-Dimensional Rotations 333
m r
Figure 11.7 Two-dimensional rotational mo-
tion can be defined as a mass moving about a
point in a circle with fixed radius r.