Page 293 - Essentials of physical chemistry

The Quantized Harmonic Oscillator: Vibrational Spectroscopy 255

It can be shown by matching the second derivative of the Morse potential to the second
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
k=2D e [3,4], so we can plot both the harmonic
derivative of the parabolic potential that a ¼
and Morse potential on the same graph. With D e in electron volts and r 0 in Å along with the
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
k=2D e we obtain data points to plot in Figure 12.1.
calculated value of a ¼
The graph shows that the harmonic and Morse potential wells superimpose fairly well at low
energy and probably the first few energy levels correspond for both models. That may be all we
need for ground states most of the time on planet Earth. The exact potential well can also be
constructed from spectroscopic data but it is adequate to introduce the Morse potential to provide a
useful model. Both the parabolic and Morse potentials are incorrect in that the short range part
crosses the vertical axis which is meaningless so the potentials are only shown for x > 0 but the fit
for low energy levels is still useful.
Just as with the PIB model, the particle (effective mass of the vibrating parts of the molecule) will
have kinetic energy and rattle back and forth in the parabolic well. The commutator idea in the
previous chapter could be used to completely solve this problem. However, our goal here is to show
how the ‘‘essential’’ quantized energy condition occurs. While the calculus here is usually found in a
further course in differential equations (recommended), we think it is better to use functions and
derivatives instead of formal operators. If we follow a derivation based on operator commutators, it
would add another unfamiliar aspect to the problem so we will present the polynomial method in the
assumption that a teacher can put the steps on the board or that a student can follow it on scratch
paper. There is just no way around using some form of detailed mathematics to show how this
problem was solved! However, here we take the time-honored path used first by Schrödinger [1] and
beautifully detailed by Pauling and Wilson in 1935 [7] to solve this problem with a polynomial
expansion technique, which uses standard calculus methods. We do this partly so that we can allude
to the polynomial expansion method when we solve the H-atom using Schrödinger’s method. Note
the Schrödinger H-atom treatment is a problem in three dimensions (compared to the flat Bohr
model) while the harmonic oscillator is our best chance to give a complete solution in only one
dimension. At this point, a student easily frustrated by mathematical details is advised to go to the
section marked Harmonic Oscillator Results to obtain the key results and then return to check out
the rigorous details of the mathematical basis of the facts related to the quantized harmonic
oscillator.

HARMONIC OSCILLATOR DETAILS

As is now usual, we start by writing the Hamiltonian operator and attempt to solve the differential
equation. The basic strategy is that we recall the idea of a Taylor power series expansion, which can
represent any function as a (potentially infinite) power series. Thus we hope we can write
P
n
1 a n x and find some way to evaluate the values of a n . However, we have to first suffer
c(x) ¼ n¼0
through a few changes in variable to achieve a ‘‘simple’’ equation! We give more details than most
texts at this point so that you can follow the derivation with pencil and paper or the teacher can put
2 2 2
d kx
h
where x (r r 0 ) and we
2m dx 2
these steps on the board for slow appreciation. H ¼ 2 þ
2
2
d c kx 2
h
will define the meaning of the mass later. Thus þ c ¼ Ec. Now let x ¼ bj and absorb
2m dx 2 2
2 d c k
2
h
2 2
all the physical units into b so that j is without units. Then þ b j c ¼ Ec and this is
2
2mb dj 2 2
h 2 k h
2 2 ffiffiffiffi . What follows is
useful for unit analysis related to energy. In units E 2 b ) b ¼ p
2mb 2 mk
a mind-bending sequence of variable changes so we seek as much meaning as possible from the