Physical Chemistry , 1st ed.

To normalize, the wavefunction  0 must be multiplied by some constant N

such that

N^2 





(c 0 ex

(^2) /2
)*(c 0 ex
(^2) /2
) dx 1 (11.18)
Since Nand c 0 are both constants, it is customary to combine them into a
single constant N. The complex conjugate of the exponential does not change
the form of the exponential, since it does not contain the imaginary root i.The
integral becomes
N^2 


ex
2
dx 1
The final change to this integral begins with the understanding that because
the xin the exponential is squared, the negative values ofxyield the same val-
ues ofex
2
as do the positive values ofx. This is one way of defining an even
mathematical function. [Formally,f(x) is even if, for all x,f(x) f(x). For
an oddfunction,f(x) f(x). Examples of simple odd and even functions
are shown in Figure 11.3.] The fact that the above exponential has the same
values for negative values ofxas for positive values ofxmeans that the inte-
gral from x0 to is equal to the integral from x0 to . So instead
of our interval being xto , let us take it as x0 to and take
twicethe value of that integral. The normalization expression becomes
2 N^2 

0
ex
2
dx 1
The integral  0 ex
2
dxhas a known value,^12 (a)1/2. In this case,a .
Substituting for this and solving for N, one finds
N






1/4

The complete wavefunction  0 is therefore

 0 







1/4

ex

(^2) /2
It turns out that the set of harmonic oscillator wavefunctions were already
known. This is because differential equations like those of equation 11.6, the
rewritten Schrödinger equation, had been studied and solved mathematically
11.4 The Harmonic Oscillator Wavefunctions 325
f(x)
f(x)  f(x)
x
(a)
f(x)
f(x)  f(x)
x
(b)
Figure 11.3 Examples of odd and even functions. (a) This function is even, so that changing
the sign on x(from xto x) yields the same value as for f(x), as the arrow shows. (b) This func-
tion is odd, where changing the sign on xyields f(x), as the arrow shows.