Physical Chemistry , 1st ed.

to indicate this dependence. The acceptable wavefunctions for a one-dimensional

particle-in-a-box are written as

n(x)          2

a

sin^

n

a

x

, n1,2,3,4,... (10.11)

The quantized energies of the particles in this box are

En

8

n

m

(^2) h
a
2
2 (10.12)
What do these wavefunctions look like? Figure 10.6 shows plots of the first
few wavefunctions. All of them go to zero at the sides of the box, as required
by the boundary conditions. All of them look like simple sine functions (which
is what they are) with positive and negative values.
Example 10.11
Determine the wavefunctions and energies of the first four levels of an elec-
tron in a box having a width of 10.0 Å; that is,a10.0 Å 1.00 10 ^9 m.
Solution
Using equation 10.11, the expressions of the wavefunctions are straightforward:
 1 (x) 

2

a

sin^

a

x

 2 (x) 

2

a

sin^

2

a

x

 3 (x) 

2

a

sin^

3

a

x

 4 (x) 

2

a

sin^

4

a

x

Using equation 10.12, the energies are

E 1 

8

1

m

(^2) h
ea
2
2 6.02 10
 (^20) J
E 2 
8

2

m

(^2) h
ea
2
2 24.1 10
 (^20) J
E 3 
8

3

m

(^2) h
ea
2
2 54.2 10
 (^20) J
E 4 
8

4

m

(^2) h
ea
2
2 96.4 10
 (^20) J
The exponents on the magnitudes of the energies have been intentionally
kept the same, 10^20 , to illustrate how the energy changes with quantum
number n. Note that whereas the wavefunctions depend on n, the energies
depend on n^2. You should verify that the units in the above expression do
yield units of joules as the unit of energy.
42 (6.626 10 ^34 J s)^2
8(9.109 10 ^31 kg)(1.00 10 ^9 m)^2
32 (6.626 10 ^34 J s)^2
8(9.109 10 ^31 kg)(1.00 10 ^9 m)^2
22 (6.626 10 ^34 J s)^2
8(9.109 10 ^31 kg)(1.00 10 ^9 m)^2
12 (6.626 10 ^34 J s)^2
8(9.109 10 ^31 kg)(1.00 10 ^9 m)^2
292 CHAPTER 10 Introduction to Quantum Mechanics
 n  5
 n^ ^4
 n  3
 n  2
 n^ ^1
Figure 10.6 Plots of the first few quantum-
mechanically acceptable particle-in-a-box wave-
functions.