Wave Mechanics 191
reinforce itself. In the same manner, the trough or minimum of the wave
would also match after orbiting the circle one time. The condition that
the wavelength fits an integer number of times into the circumference, c,
of the orbit of radius R, is given by the formula: c = 2πR = nλ.
The wavelength, λ, is related to the momentum by λ = h/p. Hence, the
condition for stability becomes 2πR = nh/p. Rearranging the terms of this
equation we have L = pR = nh/2π, since the angular momentum for a
circular orbit is just p times R. This condition is exactly Bohr’s angular
momentum condition that we encountered in the last chapter.
De Broglie’s application of his wave hypothesis to the electron
enabled him to explain a key feature of Bohr’s theory of the atom. His
hypothesis regarding the wave nature of the electron applies to all
particles. We must, therefore, ask why the wave nature of particles
had not been observed before de Broglie made his hypothesis. Let us
first consider the wavelength of a macroscopic object, for example, a
baseball that has just been thrown by a player. The mass of the ball is
approximately 0.2 kg and its velocity is approximately 300 m/sec. The
wavelength of the ball is therefore only 10-21 cm.
With wavelengths as small as 10-21 cm one would never expect to
detect any wave behaviour from a macroscopic object. It is only when
we get to atomic size particles that we could possibly observe any wave
behaviour because it is only for these small size objects that the
wavelengths become large enough to detect wave behaviour. De Broglie,
therefore, predicted that only subatomic particles such as electrons would
display wave behaviour.
The only way to test de Broglie’s hypothesis was to observe the
wave behaviour of the electron. This did not happen immediately.
Although de Broglie was able to explain Bohr’s frequency condition, his
contemporaries were quite skeptical about his result. They thought his
work was a wild theoretical scheme unrelated to reality. There was even
some doubt as to whether or not his work would be accepted for his
doctoral thesis. His thesis was finally accepted and ironically turned out
to be the only doctoral thesis, which won its author both a doctor’s
degree and a Nobel Prize. Luckily, de Broglie’s work was brought to the
attention of Einstein who was very favorably impressed by it. Einstein, in
turn, passed the thesis on to others. To Max Born he remarked, “Read it
even though it might look crazy, it is absolutely solid.” With Einstein’s
blessings the experimentalists began a systematic search to detect the
wave nature of the electron suggested by de Broglie.