Physical Chemistry Third Edition

678 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

23

0

1

Energy/

hn

or wave function (arbitrary units)

2

3

4

22 2 101 23

ŒWa z

Figure 15.6 Harmonic Oscillator Wave Functions.

A comparison of these graphs with those for the particle in a one-dimensional box

in Figure 15.4b shows that in both cases a wave function with a larger number of

nodes corresponds to a higher energy. In addition to the nodes at infinite|x|for the

harmonic oscillator and at the ends of the box for the particle in a box, the lowest-energy

wave function has no nodes, the next-lowest-energy wave function has one node, and

so on.

The frequency of oscillation of the wave function has some surprising properties.

EXAMPLE15.6

Calculate the frequency of oscillation of the wave function corresponding to thev0,v1,

andv2 states of a harmonic oscillator. Find the difference between the frequencies for two

adjacent values ofv.

Solution

The time-dependent factor of the wave function is:

η(t)e−iEt/ ̄he−^2 πiEt/hcos(2πEt/h)−isin(2πEt/h)

So that the frequency isvE/h. Forv0,Ehνclass/2 andννclass/2. Forv1,

E 3 hνclass/2 andν 3 νclass/2. Forv2,E 5 hνclass/2 andν 5 νclass/2. The differ-

ence between any two frequencies is equal toνclass. A potential energy can have an arbitrary

constant added to its value, which adds the same constant to all energy eigenvalues of the

system. The frequency of the wave function thus can have a constant added to it, indicating

that the actual frequency of oscillation of a de Broglie wave is not meaningful. Only differ-

ences in frequency are meaningful.

In Section 14.2 we described how two masses connected by a spring oscillate har-

monically, and this version of a harmonic oscillator is a good model for the vibration

of a diatomic molecule. The only change required in our formulas is the replacement

of the mass of the oscillator with the reduced mass of the two nuclei.