Chem 120A
The Harmonic Oscillator, part I
03/01/06
Spring 2006
Lecture 18
READING:
Engel (QCS): Chapter 7, 8.3
Introduction to the harmonic oscillator
Along with the hydrogen atom, the harmonic oscillator is one of the most important systems in elementary
quantum mechanics. Its significance far exceeds the physical chemistry textbook example of a diatomic
molecule. Every quantum mechanics textbook treats this system, as does every physical chemistry textbook.
Harmonic
potential
Interatomic potential
V(R)
R
Figure 1:
Interatomic potential for a diatomic
molecule compared to that of a
harmonic oscillator centered at r
0
.
0
R
0
E
0
We begin with a classical discussion in order to point out a few points. Consider a mass
m
attached to a
spring. The potential energy as a function of
x
is
V
(
x
) =
1
/
2
kx
2
. The force is
F
=
−
kx
. Another situation
is a diatomic molecule, for example HCl. It turns out that the potential energy between the two atoms as
a function of relative distance looks like that shown in Figure 1. (This interaction potential is actually a
derived result and is one of the great triumphs of quantum mechanics.) Now,
V
(
R
)
can be expanded about
its minimum at
R
0
:
V
(
R
) =
V
0
+
1
2
d
2
dR
2
V
(
R
)
R
=
R
0
(
R
−
R
0
)
2
+
...,
(1)
where
d
2
dR
2
V
(
R
)
R
=
R
0
is the curvature of the potential at the minimum, which we will call
k
. We will also
define
R
−
R
0
=
x
. Thus, near the minimum of the interaction potential, the relative motion undergoes small
oscillations.
This is what we call a ’harmonic oscillator.’
The deeper the well the more ’harmonic’ the
motion is. In other words, to second order within the Taylor expansion of
V
(
R
)
, we can approximate the
Chem 120A, Spring 2006, Lecture 18
1
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potential shown in Fig. 1 as a parabola, and the deeper the minimum , the more closely the parabola will
match the potential near the minimum. Since
V
0
is just a constant, it does not effect the relative motion of
the diatomic, so we will ignore it.
Now, as we’ve said, the initial positions and momenta determine all future motion.
The energy is also
determined initially, but it is a conserved quantity and it does not change with time. Suppose that the total
energy of the system is
E
. This bounds the possible position,
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 Spring '07
 Whaley
 Physical chemistry, Atom, pH, Energy, Even Parity, CHEM 120A, Schr¨ dinger equation

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