Physical Chemistry , 1st ed.

By convention, only the positive square root is used. The 2 in the expres-

sion above is usually converted into the fourth root of 4 (that is,^44 , or 41/4)

so that all of the powers can be combined and the normalization constant can

be rewritten as

N

4







1/4

The complete wavefunction for the n1 level is, after resubstituting in

terms ofx:

 1 

4







1/4

H 1 (1/2x) ex

(^2) /2
The normalization constants for the harmonic oscillator wavefunctions n
follow a certain pattern (largely because the formulas for the integrals involve
Hermite polynomials) and so can be expressed as a formula. The general for-
mula for the harmonic oscillator wavefunctions given below includes an ex-
pression for the normalization constant in terms of the quantum number n:
(n) 






1/4

(^) 
2 n

1

n!



1/2

Hn(1/2x) ex

(^2) /2
(11.19)
where all of the terms have been previously defined.
Determining whether a function is odd or even can sometimes be useful, since
for an odd function ranging from to and centered at x0, the inte-
gral of that function is identically zero. After all, what is an integral but an area
under a curve? For an odd function, the positive area of one half of the curve is
canceled out by the negative area of the other half. Recognizing this eliminates
the need to mathematically evaluate an integral. Determining whether a product
of functions is odd or even depends on the individual functions themselves, since
(odd) (odd) (even), (even) (even) (even), and (even) (odd) 
(odd). This mimics the rules for multiplication of positive and negative num-
bers. The following example illustrates how to take advantage of this.
Example 11.6
Evaluate xfor  3 of a harmonic oscillator by inspection. That is, evaluate
by considering the properties of the functions instead of calculating the av-
erage value mathematically.
Solution
The average value of the position of the harmonic oscillator in the state  3
can be determined using the formula
xN^2 


[H 3 () ex
(^2) /2
]*xˆ[H 3 () ex
(^2) /2
] dx
where Nis the normalization constant and no substitution has been made for
the variable x(and it will not matter). This can be simplified, especially by
remembering that the position operator xˆis multiplication by the coordinate
x, and all other parts of the integrand are being multiplied together:
xN^2 


x [H 3 ()]^2 ex
2
dx
11.4 The Harmonic Oscillator Wavefunctions 327