Physical Chemistry , 1st ed.

^2 (12 sin^3 ) 

8

7

2

0



e^3 i

 12 ^2 3, 3

So, invoking the postulate that the value of an observable is equal to the

eigenvalue from the corresponding eigenvalue equation:

Lˆ^2 3, 3  12 ^2 3, 3

or

L^2  12 ^2

Since the eigenvalue of the square of the total angular momentum is 12^2 ,

the value of the total angular momentum must be the square root of this, or

12 . This value, numerically, is 3.653 

10 ^34 J s, or 3.653 

10 ^34 kg m^2 /s.

The value for the energy can be determined from

E

L

2 I

2



which is

E

12

2



I

2



The exact numerical value of the total energy will depend on the moment of

inertia of the system (which is not given, so we cannot calculate the energy

numerically). The zcomponent of the angular momentum,Lz, is determined

by the eigenvalue equation

Lˆzi







3, 3 i









8

7

2

0



e^3 isin^3 

i(3i)

8

7

2

0



e^3 isin^3  3 

8

7

2

0



e^3 isin^3 

 3 3, 3

So the value of the zcomponent of the angular momentum is given by the

eigenvalue 3, which equals 3.164 

10 ^34 J s.

In all three cases, the predicted observables are the same as those deter-

mined by the analytic formulas of each observable.

It would have been easier (and shorter!) to use the formulas for the energy
and momenta to determine the values of these three quantized observables. But
it is important to understand that these differential equations actually do work
when the wavefunctions are operated on by them. The above example shows
that all of the operators do yield the appropriate values of the observables.
There are a few other analytically solvable systems, but most are variations
on the themes presented here and in the last chapter. For now, we will halt our
treatment of model systems and move on to a system that is more obviously
relevant chemically. But before we do, it is important to re-emphasize a few
conclusions about the systems we have treated so far. (1) In all of our model
systems, the total energy (kinetic potential) is quantized. This is a result of
the postulates of quantum mechanics. (2) In some of the systems, other ob-
servables are also quantized and have analytic expressions for their quantized
values (like momentum). Whether other observables have analytic expressions

11.8 Other Observables in Rotating Systems 351