Biomedical Engineering Reference
In-Depth Information
and 5896 Å, is due to transitions from two closely spaced energy levels, degener-
ate in nonrelativistic theory. In spite of its successes in some areas, semiclassical
atomic theory did not work for many-electron atoms and for such simple systems
as some diatomic molecules, for which it gave unambiguous but incorrect spectra.
As a guide for discovering quantum laws, Bohr in 1923 introduced his corre-
spondence principle. This principle states that the predictions of quantum physics
must be the same as those of classical physics in the limit of very large quantum
number
n
. In addition, any relationships between states that are needed to obtain
the classical results for large
n
also hold for all
n
. The diagram of energy levels in
Fig. 2.3 illustrates the approach of a quantum system to a classical one when the
quantum numbers become very large. Classically, the electron in a bound state has
continuous, rather than discrete, values of the energy. As
n
→∞
, the bound-state
energies of the H atom get arbitrarily close together.
Advances toward the discovery of quantum mechanics were also being made
along other lines. The classical Maxwellian wave theory of electromagnetic radi-
ation seemed to be at odds with the existence of Einstein's corpuscular photons
of light. How could light act like waves in some experiments and like particles in
others? The diffraction and interference of X rays was demonstrated in 1912 by
von Laue, thus establishing their wave nature. The Braggs used X-ray diffraction
from crystal layers of known separation to measure the wavelength of X rays. In
1922, discovery of the Compton effect (Section 8.4)—the scattering of X-ray pho-
tons from atoms with a decrease in photon energy—demonstrated their nonwave,
or corpuscular, nature in still another way. The experimental results were explained
by assuming that a photon of energy
E
has a momentum
p
=
E
/
c
=
h
ν/
c
, where
ν
is the photon frequency and
c
is the speed of light. In 1924, de Broglie proposed
that the wave/particle dualism recognized for photons was a characteristic of all
fundamental particles of nature. An electron, for example, hitherto regarded as a
particle, also might have wave properties associated with it. The universal formula
that links the property of wavelength,
λ
, with the particle property of momentum,
p
, is that which applies to photons:
p
=
h
ν
λ
. Therefore, de Broglie proposed
that the wavelength associated with a particle be given by the relation
/
c
=
h
/
h
p
=
h
γ
mv
,
(2.22)
λ =
where
m
and
v
are rest mass and speed of the particle and γ is the relativistic factor
defined in Appendix C.
Davisson and Germer in 1927 published the results of their experiments, which
demonstrated that a beam of electrons incident on a single crystal of nickel is dif-
fracted by the regularly spaced crystal layers of atoms. Just as the Braggs mea-
sured the wavelength of X rays from crystal diffraction, Davisson and Germer
measured the wavelength for electrons. They found excellent agreement with Eq.
(2.22). The year before this experimental confirmation of the existence of electron
waves, Schroedinger had extended de Broglie's ideas and developed his wave equa-
tion for the new quantum mechanics, as described in the next section.
