Physical Chemistry , 1st ed.

Example 11.11

An electron is traveling in a circle having radius 1.00 Å. Calculate the energy

eigenvalues of the first five 2-D rotational wavefunctions; that is, where

m0,1, and 2.

Solution

First, we calculate the moment of inertia of the electron. Using me

9.109 

10 ^31 kg and the given radius of 1.00 Å 1.00 

10 ^10 m, the mo-

ment of inertia is

Imr^2 (9.109 

10 ^31 kg)(1.00 

10 ^10 m)^2 9.11 

10 ^51 kg m^2

11.6 Two-Dimensional Rotations 337

m  3

m  2

m  1

m  0

Circular representation Linear representation

0

1

 2 

 2 

0

1

 2 

 2 

0

1

 2 

 2 

0

1

 2 

 2 

Figure 11.9 The first four 2-D rotational wavefunctions. The circular representations mimic

the true geometry of the system, and the linear representations clarify what the wavefunctions

look like. Each linear representation represents one circuit (2radians) of the rigid rotor.