Physical Chemistry , 1st ed.

the space a particle inhabits. What equation 10.8 usually means is that wave-

functions must be multiplied by some constant so that the area under the

curve of*is equal to 1. According to the Born interpretation of,nor-

malization also guarantees that the probability of a particle existing in all space

is 100%.

Example 10.7

Assume that a wavefunction for a system exists and is (x) sin ( x/2),

where xis the only variable. If the region of interest is from x0 to x1,

normalize the function.

Solution

By equation 10.8, the function must be multiplied by some constant so that

^10 *dx1. Note that the limits are 0 to 1, not to , and that d

is simply dxfor this one-dimensional example. Let us assume that is mul-

tiplied by some constant N:

→N

Substituting for into the integral, we get



1

0

(N)*(N) dx

1

0

N*Nsin

2

x

(^) sin
2
x
(^) dx
Since Nis a constant, it can be pulled out of the integral, and since this sine
function is a real function, the has no effect on the function (recall it
changes every ito i, but there is no imaginary part of the function in this
example). Therefore, we get

1
0
NNsin
2
x
(^) sin
2
x
(^) dxN^2 
1
0
sin^2
2
x
dx
Normalization requires that this expression equal 1:
N^2 
1
0
sin^2
2
x
dx 1
The integral in this expression has a known form and it can be solved, and
the definite integral from the limits 0 to 1 can be evaluated. Referring to the
table of integrals in Appendix 1, we find that
sin^2 bx dx
2
x

4

1

b

sin 2bx

In our case,b

/2. Evaluating the integral between the limits, we find that

the normalization requirement simplifies to

N^2 

1

2

(^)  1
Solving for N:
N 2 
where the positive square root is assumed. The correctly normalized wave-
function is therefore (x)  2 [sin ( x/2)].
284 CHAPTER 10 Introduction to Quantum Mechanics