Physical Chemistry , 1st ed.

This product of three integrals is relatively easy to evaluate, despite its length.

The xand zintegrals are exactly the same as the one-dimensional particle-in-

a-box wavefunctions being evaluated from one end of the box to the other,

and they are normalized. Therefore the first and the third integrals are each 1.

The expression becomes

py  1 

b

y 0

2

b

sin^

2

b

y



i





y

2

a

sin^

2

b

y

dy 

1

For the ypart, evaluation of the derivative part of the operator is straight-

forward, and rewriting the integral, bringing all constants outside the integral

sign, yields

py 

2

b

2

b

i 

b

y 0

sin

2

b

y

cos

2

b

y

dy

Using the integral table in Appendix 1, we find that this integral is exactly

zero. Therefore,

py  0

This should not be too much of a surprise. Although the particle certainly has

momentum at any given moment, it will have one of two opposite momen-

tum vectors exactly half the time. Because the opposing momentum vectors

cancel each other out, the average valueof the momentum is zero.

The above example illustrates that although the triple integral may look dif-
ficult, it separates into more manageable parts. This separability of the integral
is directly related to our assumption that the wavefunction itself is separable.
Without separability of, we would have to solve a triple integral in three
variables simultaneously—a formidable task! We will see other examples of
how separability ofmakes things easier for us. Ultimately, the issue of sep-
arability is paramount in the application of the Schrödinger equation to real
systems.

10.12 Degeneracy

For the one-dimensional particle-in-a-box, all of the energies of the eigen-
functions are different. For the general 3-D particle-in-a-box, because the to-
tal energy depends on not only the quantum numbers nx,ny, and nzbut also
the individual dimensions of the box a,b, and c, one can imagine that in some
cases the quantum numbers and the lengths might be such that different sets
of quantum numbers {nx,ny,nz} would yield the sameenergy for the two dif-
ferentwavefunctions.
This situation is very possible in systems that are symmetric. Consider a cu-
bical box:abc. Using the variable ato stand for any side of the cubical
box, the wavefunctions and energies now become

(x,y,z) 

a

8

3    sin^

nx

a

x

sin

ny

a

y

sin

nz

a

z

(10.24)

E

8

n

m

x^2 h

a

2

2  (^8)

n
m
y^2 h
a
2
2  (^8)
n
m
z^2 h
a
2
2  (^8) m
h^2
a^2
(nx^2 ny^2 nz^2 ) (10.25)
The energy depends on a set of constants and the sum of the squares of the
quantum numbers. If a set of three quantum numbers adds up to the same
total as another set of three different quantum numbers, or if the quantum
10.12 Degeneracy 303