Physical Chemistry , 1st ed.

The Hermite polynomial H 3 () contains only odd powers ofx, but upon

squaring it becomes a polynomial having only even powers ofx. Therefore,

it is an even function. The exponential has x^2 in it, so it is an even function.

The term xitself is an oddfunction. (The dxis not considered, since it is part

of the integration operation, not a function.) Therefore the overall function

is odd, and the integral itself, centered at zero and going from to ,is

identically zero. Therefore x0.

This property of odd functions is extremely useful. For even functions, the

integral mustbe evaluated. Probably the best method of doing so at this point

is to substitute for the form of the Hermite polynomial, multiply out the terms,

and evaluate each term according to its form. Several integrals from Appendix

1 may be useful. However, odd functions integrated over the proper interval

are exactly zero, and such a determination can be made by an inspection of the

function rather than evaluation of the integral—a timesaving routine, when

possible.

Plots of the first few harmonic-oscillator wavefunctions are shown in Figure

11.4. Superimposed with them is the potential energy curve of a harmonic os-

cillator. Although the exact dimensions of Figure 11.4 depend on what mand

kare, the general conclusions do not. Recall that in a classical harmonic oscil-

lator, a mass goes back and forth about a center. When passing the x0 cen-

ter, the mass has minimum potential energy (which can be set to zero) and

maximum kinetic energy. It is moving at its fastest speed. As the mass extends

farther away from the center, the potential energy grows until all of the energy

is potential, none is kinetic, and the mass momentarily stops. Then it begins

motion in the other direction. The point at which the mass turns around is

called the classical turning point.A classical harmonic oscillator never extends

beyond its turning point, since that would mean that it has more potential en-

ergy than total energy.

As seen in Figure 11.4, wavefunctions for quantum-mechanical harmonic os-

cillators exist in regions beyond the point where classically all energy would be

328 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

n  0

n  1

n  2

n  3

n  4

x

x  0

V ^12 kx^2

Figure 11.4 Plots of the first five wavefunctions of the harmonic oscillator. They are super-

imposed against the potential energy for the system. The positions where the wavefunctions go

outside the potential energy are called the classical turning points. Classically, a harmonic oscil-

lator will never go beyond its turning point, since it does not have enough energy. Quantum me-

chanically, there is a nonzero probability that a particle acting as a harmonic oscillator will exist

beyond this point.