Page 291 - Essentials of physical chemistry

12 The Quantized Harmonic

Oscillator: Vibrational

Spectroscopy

INTRODUCTION
Now that we have learned some of the principles which apply in quantum mechanics, we move on
to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a
rigorous application of the polynomial method of solving differential equations on a relatively
simple case and to provide some insight into how the Schrödinger equation was first solved [1].
Then we proceed to application in the form of worked examples. We have tried to give sufficient
details of this solution to allow a student to follow the derivation with pencil and paper but do not
forget to ponder over the spectroscopic applications! This author would agree that it is more
important to absorb the main conclusions of this material than to master the derivations. In fact,
the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the
conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.
The previous particle-in-a-box (PIB) and particle-on-a-ring (POR) problems both had V ¼ 0 and
only dealt with the kinetic energy operator. The essence of the harmonic oscillator is a parabolic
kx 2
where k is the ‘‘spring constant’’ or ‘‘restoring force constant.’’ According
potential energy V ¼
2
to the idea that a force is the negative derivative of a potential, we have
dV 2kx
¼ kx so k is the proportionality factor of the force of a spring stretched
f (x) ¼ ¼
dx 2
or compressed away from x ¼ 0. The classical case can be solved easily by equating two
" ! #
r ffiffiffiffi
2
d x k
forces f ¼ ma ¼ m ¼ kx. The solution from sophomore physics is x ¼ A sin t
dt 2 m
" ! #
r ffiffiffiffi r ffiffiffiffi
dx k k
and this can be shown by direct substitution. ¼ A cos t and
dt m m
" ! #
r ffiffiffiffi
2
2

d x k k k d x
¼ A sin t ¼ x.So m ¼ kx, Q.E.D.
dt 2 m m m dt 2
This could have been solved to obtain the same result in more complicated ways but we want to
build on your experience from problems in physics and we note that ‘‘A ¼ amplitude.’’
Here we are faced with the quantum mechanical treatment of the harmonic oscillator. This
potential does not take into account the fact that a chemical bond will eventually break when
stretched more than about 10 Å but it does provide a very good approximation to the small
vibrations of a bond in low energy states. An interesting fact plays a role here in that on planet
Earth at average temperatures around 258C, almost all molecules are in very low vibrational states.
This is fortuitous for the use of a potential which is most accurate at low energies. We see in
Figure 12.1 a sketch of a parabolic potential superimposed on a model of a dissociative bond
potential. Here the value of the potential is a minimum at r ¼ r 0 and the energy is negative for a
‘‘bound’’ state but we will measure the energy as positive above the minimum of the parabolic
potential. We see that the parabolic potential makes a good fit to a real potential for a considerable
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