Physical Chemistry , 1st ed.

Since the values ofLxand Lyare indeterminate for specified and m,
the graphical representations of Figure 11.14 are better represented three-
dimensionally as cones instead of vectors. Such a representation is shown in
Figure 11.15. This figure shows the angular momentum vectors superimposed
on a sphere, representing our 3-D surface. Again, the “length” of each cone is
constant. The orientation of each cone with respect to the z-axis is different,
and is determined by the value of the mquantum number.
Since the energy of 3-D rotational motion depends only on the quantum
number and has 21 possible values ofm, each energy level has a de-
generacy of 21.

Example 11.19

What are the degeneracies of the 4 and 5 levels for C 60 if the elec-

trons are assumed to behave like particles confined to the surface of a

sphere?

Solution

The 4 energy level has 2(4)  1 9 possible values ofm, so this energy

level has a degeneracy of 9. Similarly, the 5 energy level has a degener-

acy of 11.

Example 11.20

Construct the complete spherical harmonic for 3, 3 and use the operators

for E,L^2 , and Lzto explicitly determine the energy, the total angular mo-

mentum, and the z-component angular momentum. Show that the values of

these observables are equal to those predicted by the analytic expressions for

E,L^2 , and Lz. (The objective of this example is to illustrate that the operators

do in fact operate on the wavefunction to produce the appropriate eigenvalue

equation.)

11.8 Other Observables in Rotating Systems 349

Lz  0   Ltot   6 

Ltot   6  

Ltot   6  

Lz  1  

Ltot   6  

Lz  2  

Lz   2     Ltot   6 

z

Lz   1    

Figure 11.15 Because quantum mechanics does not address the angular momentum compo-
nents in the xor ydimension, the proper diagram relating Land Lzis a cone, where the total
angular momentum and the zcomponent of the total angular momentum are quantized, but the
xand ycomponents are indeterminate and can have any value.