The difference in the energies is
E4.96
10 ^19 J
Using Eh , it can be shown that this energy difference corresponds to the
absorption of a photon having frequency of
7.49
1014 s^1
Using c
, this frequency corresponds to a wavelength of
4.00
10 ^7 m
which is 400. nm. This compares very well to the experimentally measured
absorption appearing at 404 nm.The above example shows that the 3-D rotational model is applicable to a real
system, just as the particle-in-a-box and 2-D rotational motion can be applied
to real systems. Other transitions of C 60 can also be fit to the 3-D rotation equa-
tions, but they are left for exercises. These examples show that even though these
are model systems, they do have application to the real world. The situation is
similar to that of the ideal gas: we have equations to express the behavior of an
ideal gas. And although there is no such thing as an ideal gas, real gases can ap-
proximate ideal gas behavior, so ideal gas equations have a useful purpose in real
life. These equations from quantum mechanics have the same applicability as the
equations for an ideal gas. These model systems do not exist in reality, but some
atomic or molecular systems do exist that can be approximated with these sys-
tems. The quantum mechanical equations work reasonably well. They work
much better than anything that classical mechanics could ever have provided.
11.8 Other Observables in Rotating Systems
There are other observables to consider, starting with the total angular mo-
mentum. The operator is ˆL^2 , so the eigenvalue will be the square of the total
angular momentum. Since the total energy can be written in terms of the
square of the total angular momentum, it should be no surprise that the spher-
ical harmonics are also eigenfunctions of (total angular momentum)^2. Like the
energy eigenvalues, the analytic demonstration of the eigenvalue equation is
complex. Here, only the ultimate result is presented:
Lˆ^2 ,m(1)^2 ,m (11.52)
The square of the total angular momentum has the value (1)^2 .The
total angular momentum is the square root of this expression, so the total,
three-dimensional angular momentum of any state described by the quantum
numbers and mis
L (1) (11.53)
The total angular momentum is not dependent on the mquantum number.
Nor is it dependent on the mass of the particle, or the dimension of the sphere.
These ideas are again counter to the concepts of classical mechanics.
Example 11.18
What are the total angular momenta of an electron in the 4 and 5
states of C 60 (see Example 11.17 above)?11.8 Other Observables in Rotating Systems 347