Biomedical Engineering Reference
In-Depth Information
Fig. 2.4
Examples of standing waves in string of length
L
stretched between two fixed points
P
1
and
P
2
. Such waves exist
only with discrete wavelengths given by
λ =
2
L
/
n
,where
n
=
1, 2, 3,
...
.
nonrelativistically
(γ = 1)
that
2
πr
=
nh
/
mv
,or
mvr
=
n
. One thus arrives at Bohr's
original quantization law, Eq. (2.3).
Schroedinger's wave equation is nonrelativistic, and he proposed a modification
of it in 1926 to meet the relativistic requirement for symmetry between space and
time. As mentioned earlier, the Schroedinger differential equation is second order
in space and first order in time variables. His relativistic equation, which contained
the second derivative with respect to time, led to a fine structure in the hydrogen
spectrum, but the detailed results were wrong. Taking a novel approach, Dirac pro-
posed a wave equation that was first order in both the space and time variables. In
1928 Dirac showed that the new equation automatically contained the property of
intrinsic angular momentum for the electron, rotating about its own axis. The pre-
dicted value of the electron's spin angular momentum was
/2, the value ascribed
experimentally in 1925 by Uhlenbeck and Goudsmit to account for the structure of
the spectra of the alkali metals. Furthermore, the fine structure of the hydrogen-
atom spectrum came out correctly from the Dirac equation. Dirac's equation also
implied the existence of a positive electron, found later by Anderson, who discov-
ered the positron in cosmic radiation in 1932. In 1927 Dirac also laid the foundation
for quantum electrodynamics—the modern theory of the emission and absorption
of electromagnetic radiation by atoms. The reader is referred to the historical out-
line in Section 1.3 for a chronology of events that occurred with the discovery of
quantum mechanics.
