Physical Chemistry , 1st ed.

These are the correct units for the moment of inertia. Now we can consider

the energies of each state. Since m0 for the first state, it is easy to see that

E(m0)  0

For the other states, we recall that the energy is dependent on the square of

the quantum number. Therefore, the energy when m1 is the same as the

energy when m1.

E(m1) 6.10 

10 ^19 J

E(m2) 2.44 

10 ^18 J

The (2)^2 terms in the denominators are on account of. The units come

out to joules, which the following unit analysis illustrates:



k

(

g

J

s

m

)^2

 2 k

J

g

2

m

s^2

 2 kg

J

s

m

2

 (^2) 
kg
s
2
m^2
J
where in the next-to-last step, one of the joule units is broken down into its
basic units.
A diagram of the energy levels of two-dimensional rotational motion is
given in Figure 11.10. As for the particle-in-a-box, the energy depends on the
square of the quantum number, instead of changing linearly with the quantum
number. The energy levels get spaced farther apart as the quantum number m
gets larger.
Because of the square dependence of the energy on the quantum number
m, negative values ofmyield the same value of energy as do the positive val-
ues of the same magnitude (as noted in Example 11.11). Therefore, all energy
levels (except for the m0 state) are doubly degenerate:two wavefunctions
have the same energy.
This system has one more observable to consider: the angular momentum,
in terms of which the total energy was defined. If the wavefunctions are eigen-
functions of the angular momentum operator, the eigenvalue produced would
correspond to the observable of the angular momentum. Using the polar-
coordinate form of the angular momentum operator:
Lˆz i








1

2 

eimi(im)

1

2 

eim

Lˆz m (11.39)

The wavefunctions that are eigenfunctions of the Schrödinger equation are

also eigenfunctions of the angular momentum operator. Consider the eigen-

values themselves: a product of, a constant, and the quantum number m.The

angular momentum of the particle is quantized.It can have only certain values,

and those values are dictated by the quantum number m.

Example 11.12

What are the angular momenta of the five states of the rotating electron from

Example 11.11?

22 (6.626 

10 ^34 J s)^2



2(9.11 

10 ^51 kg m^2 )(2)^2

12 (6.626 

10 ^34 J s)^2



2(9.11 

10 ^51 kg m^2 )(2)^2

338 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

m  0 E  0 2 / 2 I  0

m  1 E  1 2 / 2 I

m  2 E  4 2 / 2 I

m  3 E  9 2 / 2 I

m  4 E  16    2 / 2 I

m  5 E  25    2 / 2 I

m  6 E  36    2 / 2 I

Figure 11.10 The quantized energy levels of
2-D rotation. They increase in energy according
to the squareof the quantum number m.