Physical Chemistry , 1st ed.

Now the two parts of the solution for equation 11.46 can be combined to

get the entire solution for 3-D rotational motion. The acceptable wavefunc-

tions are



1

2 

eim,m (11.49)

where the following conditions apply:

0,1,2,3,...

m

These wavefunctions are functions that were well known to the people who

developed quantum mechanics. They are called spherical harmonicsand are la-

beled Ym(or Y,m). Once again, classical mathematics anticipated quantum

mechanics in the solution of differential equations. Although the Legendre

polynomials do not distinguish between positive and negative values of the

quantum number m, the exponential part of the complete wavefunction does.

Each set of quantum numbers (,m) therefore indicates a unique wavefunc-

tion, denoted ,m, that can describe the possible state of a particle confined

to the surface of a sphere. The wavefunction itself does notdepend on either

the mass of the particle or the radius of the sphere that defines the system.

Example 11.16

Show that the wavefunction 1,1is normalized over all space. Use the asso-

ciated Legendre polynomial listed in Table 11.3.

Solution

The complete wavefunction consists of the appropriate associated Legendre

polynomial as well as the appropriate (1/ 2 )eimpart. The complete 1,1

wavefunction is

1,1

1

2 

 ei^1  

1

2

 3 sin 

which simplifies to

1,1

2

2

3



 eisin 

The normalization requirement is that the integral of the square of the wave-

function over all space equals 1. So, we set up the wavefunction and integrate

it over and :

1,1 

2 

 0





 0 



2

2

3



 eisin * 

2

2

3



 eisin dsin d

where the final sin term comes from the definition ofd in this two-

dimensional system.

1,1

4

3

2 

 

2 

 0





 0

eisin eisin dsin d

The two exponentials cancel each other. Separating the remaining and 

parts into their respective integrals:

1,1

8

3



 

2 

 0

d 



 0

sin^3 d

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