Concise Physical Chemistry

c18 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

THE HOOKE’S LAW POTENTIAL WELL 289

state to the next higher quantum state. One of the most common laboratory instruments

is the spectrophotometer, which is calibrated to record selected absorbed frequencies

calledresonance frequencies. These are often reported aswave numbersν ̄, which are

frequencies divided by the speed of radiation. This is just the experimental measure-

ment we want in order to calculate force constants. In units of kg m−^2 ,wehave

ν ̄=

ν

c

=

1

λ

=

1

2 π

√

kf

m

kf= 4 π^2 mν ̄^2

The force constant is also expressed in newtons per meter (N m−^1 ) because a

newton is a kg m−^1.

18.2 THE HOOKE’S LAW POTENTIAL WELL

In the harmonic oscillator, potential energy is increased by the work done on the mass

by moving itdzagainst the opposing force–fof the constraining spring. This work,

dw=−fdz, is not lost. It is stored in the spring aspotential energy V:

dV=dw=−fdz=kfzdz

Integrating, we obtain

∫V

V 0

dV=kf

∫z

z 0

zdz=

kfz^2

2

It is convenient to define the potential energy as zero,V 0 =0, at the equilibrium

position of the oscillating mass,z 0 =0 (Fig. 18.1). The oscillating mass passes

throughz 0 =0 on each oscillation but continues along the z axis to nonzero values

in the opposite direction, whereupon

∫V

V 0 dV=V−V^0 =Vand

V=

kfz^2

2

The potential energy defined in this way is always positive. It is symmetrical about

the equilibrium position of a perfect Hooke’s law spring and it is steep for a stiff

spring (largerkf) but open for a weak spring (relatively smallerkf). The excursionsz

away fromz 0 are positive and negative, but the energy always goes up withzbecause

the square ofzis always positive (Fig. 18.2).

Upon solving the Schrodinger equation for this system, one finds that the simple ̈

harmonic oscillator has evenly spaced energy levels and that there is a half quantum

of energy at the bottom of the well (called thezero point energy). Because the energy

of transition from one level to an adjacent neighbor level is the same no matter where