Physical Chemistry , 1st ed.

but also their new directions of motion. If waves have particle properties, per-

haps it is not too far-fetched to consider that matter can have wave properties.

Consider two examples that illustrate the importance of the de Broglie

equation. First, if a baseball having a mass of 150 grams (which is 0.150 kg)

were traveling at a speed of 150 kilometers per hour (which is 41.6 meters per

second), its de Broglie wavelength would be

 1.06 

10 ^34 m

A wavelength of a millionth of a billionth of a billionth of an angstrom is

undetectable even under modern conditions. The wavelength of the baseball

would never be noticed, not by scientists of the late nineteenth century (or

even a baseball player).

The second example is an electron, which is much smaller than a baseball.

Since the de Broglie wavelength is inversely proportional to mass, we would ex-

pect that the de Broglie wavelength of a particle gets larger as the particle gets

smaller. For an electron moving at the same speed as the baseball, its de Broglie

wavelength is

 1.75 

10 ^5 m

which is 17.5 microns. This “wavelength’’ corresponds to the infrared region of

light! Even in the late nineteenth century, this wavelength could have been detected.

Electrons typically move at higher speeds than this, and their de Broglie

wavelengths are typically shorter, in the range equivalent to X rays. Since X rays

were known by then to be diffracted by crystals, why not diffract electrons? In

1925, Clinton Joseph Davisson did just that. After accidentally breaking a vac-

uum tube with a nickel sample in it, Davisson reconditioned the nickel sam-

ple by heating it and formed large nickel crystals. Aware of de Broglie’s ideas,

Davisson (with coworker Lester H. Germer) exposed a nickel crystal to elec-

trons and found a diffraction pattern exactly as one would expect if electrons

were indeed waves. This diffraction of particles showed that the particles did

have wave properties, as predicted by de Broglie. Additional work confirming

the wave nature of electrons was performed later that year by G. P. Thomson,

the son of J. J. Thomson, who in 1897 had discovered the electron as a parti-

cle. The wave-particle dual nature of particles (as well as photons) has been a

cornerstone of modern science ever since.

Example 9.7

Calculate the de Broglie wavelength of a 1000-kg automobile traveling at 100

kilometers per hour and of an electron traveling at 1% of the speed of light

(0.01c3.00 

106 m/s).

Solution

For the automobile:



2.39 

10 ^38 m

6.626^10 ^34 Js

(1000 kg)(100 km/hr)(1 hr/3600 s)(1000 m/km)

6.626^10 ^34 Js

(9.109 

10 ^31 kg)(41.6 m/s)

6.626 

10 ^34 Js



(0.150 kg)(41.6 m/s)

268 CHAPTER 9 Pre-Quantum Mechanics