Physical Chemistry , 1st ed.

Here is an example of degeneracy by accident. The corresponding wavefunc-
tions have no common quantum numbers, but their energy eigenvalues are ex-
actly the same. If we recognize that E 151 and E 115 also have the same energy,
the level of degeneracy in this example becomes fourfold.A diagram of the en-
ergy levels of the 3-D particle-in-a-box is shown in Figure 10.13 and illustrates
the degeneracies of the energy levels.

Example 10.15

Write the four wavefunctions of a cubical box that have energy of 27(h^2 /8ma^2 )

to show that they are indeed different eigenfunctions.

Solution

Using the combinations of quantum numbers (3, 3, 3), (5, 1, 1), (1, 5, 1), and

(1, 1, 5) in the 3-D particle-in-a-box wavefunction:

 333 

a

8

    3 sin^

3

a

x

sin

3

a

y

sin

3

a

z

10.12 Degeneracy 305

(nx  ny  nz) 

30

Energy (in units of

(^2) h
/8
ma
2 )
(3,3,3), (5,1,1), (1,5,1), (1,1,5)
(4,3,1), (4,1,3), (3,4,1), (3,1,4), (1,4,3), (1,3,4)
(4,2,2), (2,4,2), (2,2,4)
(3,3,2), (3,2,3), (2,3,3)
(4,2,1), (4,1,2), (2,4,1), (2,1,4), (1,4,2), (1,2,4)
(3,3,1), (3,1,3), (1,3,3)
(4,1,1), (1,4,1), (1,1,4)
(3,2,2), (2,3,2), (2,2,3)
(3,2,1), (3,1,2), (1,3,2), (1,2,3), (2,3,1), (2,1,3)
(2,2,2)
(3,1,1), (1,3,1), (1,1,3)
(2,2,1), (2,1,2), (1,2,2)
(1,1,2), (1,2,1), (2,1,1)
(1,1,1)
Distinct energy levels [with (nx, ny, nz) labels]
25
20
15
10
5
3
Figure 10.13 The energy levels of the 3-D particle-in-a-(cubical)-box. In this system, differ-
ent wavefunctions can have the same energy. This is an example of degeneracy.