Physical Chemistry Third Edition

15.4 The Quantum Harmonic Oscillator 677

Solution

a.

∆EE 2 −E 1 

3

2

hν−

1

2

hνhν

h

2 π

√

k

m

ν

1

2 π

√

k

m

b.The frequency of the photon is identical with the classical frequency of the oscillator.

In the classical picture, if an oscillator is charged, it will emit or absorb electromagnetic

radiation with the same frequency as its frequency of oscillation, so that this result seems

to correspond with the classical picture.

The first few energy eigenfunctions of the harmonic oscillator are

ψ 0 S 0 e−ax

(^2) / 2
c 0 e−ax
(^2) / 2

(a
π

) 1 / 4

e−ax

(^2) / 2
(15.4-10)
ψ 1 

(

4 a^3

π

) 1 / 4

xe−ax

(^2) / 2
(15.4-11)
ψ 2 
(a
4 π

) 1 / 4

(2ax^2 −1)e−ax

(^2) / 2
(15.4-12)
ψ 3 

(

9 a^3

π

) 1 / 4 (

2 ax^3

3

−x

)

e−ax

(^2) / 2
(15.4-13)
The value of the constant factor in each formula corresponds to normalization, which
will be discussed in the next chapter. Other energy eigenfunctions can be generated
from formulas for the Hermite polynomials in Appendix F, which also contains some
useful identities involving Hermite polynomials.
Exercise 15.10
Using information in Appendix F, verify the formula forψ 3 for the harmonic oscillator. Omit
evaluation of the constant factor.
Figure 15.6 shows graphs of the energy eigenfunctions forv0,v1,v2, and
v3. Each wave function is plotted on a separate axis that is placed at a height
representing the energy eigenvalue. The potential energy as a function ofxis also
plotted with the same energy scale. The classical turning point for any given energy is
the point at which the total energy is equal to the potential energy (the point at which
a given axis and the potential energy curve cross). Classical mechanics asserts that
the particle cannot oscillate past the turning point. However, each wave function is
nonzero in the regions past its turning points. This correspond to penetration of the
oscillating particle into the classically forbidden regions, which is called “tunneling.”
We will discuss this phenomenon in the next chapter.