Physical Chemistry , 1st ed.

numbers themselves exchange values, the energies would be exactly the same

even though the wavefunctions are different. This condition is called degener-

acy.Different, linearly independent wavefunctions that have the same energy

are called degenerate.A specific level of degeneracy is indicated by the number

of different wavefunctions that have the exact same energy. If there are two, the

energy level is called twofold(or doubly) degenerate; if there are three wave-

functions, it is threefold(or triply) degenerate; and so on.

From equation 10.25, the specific energy is determined by what values the

quantum numbers have. We can label each energy as Exyzwhere the x,y, and z

labels indicate what the appropriate quantum numbers are. Thus,

E 111 

8 m

h^2

a^2

(1  1 1)  3

8 m

h^2

a^2

E 112 

8 m

h^2

a^2

(1  1 4)  6

8 m

h^2

a^2

E 113 

8 m

h^2

a^2

(1  1 9)  11

8 m

h^2

a^2

and so forth. (It is easier to illustrate this point by leaving the energies in terms

ofh,m, and ainstead of evaluating their exact values in terms of joules.) E 112

is the eigenvalue of the wavefunction that has nx1,ny1, and nz2. We

also have the following two wavefunctions:

 121 

a

8

    3 sin^

1

a

x

sin

2

a

y

sin

1

a

z

 211 

a

8

    3 sin^

2

a

x

sin

1

a

y

sin

1

a

z

where we are now starting to label the wavefunctions as xyz, like the energies.

These are differentwavefunctions. You should satisfy yourself that they are dif-

ferent. (One has the quantum number 2 in the xdimension and the other has

the quantum number 2 in the ydimension.) Their energies are

E 121 

8 m

h^2

a^2

(1  4 1)  6

8 m

h^2

a^2

E 211 

8 m

h^2

a^2

(4  1 1)  6

8 m

h^2

a^2

E 121 and E 211 are the same as E 112 , even though each energy observable corre-

sponds to a different wavefunction. This value of energy is threefold degenerate.

There are three differentwavefunctions that have the same energy. (Degenerate

wavefunctions may have different eigenvalues of other observables.)

This example of degeneracy is a consequence of a wavefunction in three-

dimensional space where each dimension is independent but equivalent. This

might be considered degeneracy by symmetry.One can also find examples of

accidentaldegeneracy. For example, a cubical box has wavefunctions with the

sets of quantum numbers (3, 3, 3) and (5, 1, 1), and the energies are

E 333 

8 m

h^2

a^2

(9  9 9)  27

8 m

h^2

a^2

E 511 

8 m

h^2

a^2

(25  1 1)  27

8 m

h^2

a^2

304 CHAPTER 10 Introduction to Quantum Mechanics