Chapter 4 - UCSB HEP

WORK AND ENERGY
attracted tothesurface, leaves. Theenergydetivered to the sur-
Tace is so large that the platinum glows brightly. A third atom
can also carry off the excess energy, but for this to happen the
two atoms must collide when a third atom is nearby. This is a rare
event at low pressures, but it becomes increasingly important at
higher pressures. Another possibility is for the two atoms to lose
energy by the emission of light. However, this occurs so rarely
that it is usually not important,
4.10 Small Osdllatio~s in a Bound System
The interatomic potential we discussed in the last section illus-
'0 r trates an important feature of all bound systems; at equilibrium
the potential energy has a minimum. As a result, nearly every
bound system oscillates like a harmonic oscillator if it is slightly
perturbed from its equilibrium position. This is suggested by the
appearance of the energy diagram near the minimum-U has
the parabolic shape of a harmonic oscillator potential. If the total
energy is low enough so that the motion is restricted to the region
where the curve is nearly parabolic, as illustrated in the sketch,
the system must behave like a harmonic oscillator. It is not difficult
to prove this.
As we have discussed in Note 1.1, any "well behaved" function
f(x) can be expanded in a Taylor's series a bout a point xo. Thus
Suppose that we expand U(r)about ro, the position of the potential
minimum. Then
= 0. Furthermore,
for sufficiently small displacements, we can neglect the
terms beyond the third in the power series. In this case,
However, since U is a minimum at TO, (dU/dr) I,
This is the potential energy of a harmonic oscillator,
kx2
U(x) = constant + -. 2