Biomedical Engineering Reference
In-Depth Information
Rydberg constant for a stationary (infinitely heavy) nucleus, with
m
r
=
m
. Then the
Rydberg constant for ions with different nuclear masses
M
is given by
R
∞
1+
m
/
M
.
(2.21)
R
M
=
Problem 29 shows an example in which the reduced mass plays a significant role.
Problem 31 indicates how Eq. (2.20) can be derived for motion in one dimension.
2.4
Semiclassical Mechanics, 1913-1925
The success of Bohr's theory for hydrogen and single-electron ions showed that
atoms are “quantized” systems. They radiate photons with the properties described
earlier by Planck and by Einstein. At the same time, the failure of the Bohr theory
to give correct predictions for other systems led investigators to search for a more
fundamental expression of the quantum nature of atoms and radiation.
Between Bohr's 1913 theory and Heisenberg's 1925 discovery of quantum me-
chanics, methods of semiclassical mechanics were explored in physics. A general
quantization procedure was sought that would incorporate Bohr's rules for single-
electron systems and would also be applicable to many-electron atoms and to mole-
cules. Basically, as we did above with Eq. (2.5), one used classical equations of mo-
tion to describe an atomic system and then superimposed a quantum condition,
such as Eq. (2.3).
A principle of “adiabatic invariance” was used to determine which variables of
a system should be quantized. It was recognized that quantum transitions occur
as a result of
sudden
perturbations on an atomic system, not as a result of gradual
changes. For example, the rapidly varying electric field of a passing photon can re-
sult in an electronic transition with photon absorption by a hydrogen atom. On the
other hand, the electron is unlikely to make a transition if the atom is simply placed
in an external electric field that is slowly increased in strength. The principle thus
asserted that those variables in a system that were invariant under slow, “adiabatic”
changes were the ones that should be quantized.
A generalization of Bohr's original quantum rule (2.3) was also worked out (by
Wilson and Sommerfeld, independently) that could be applied to pairs of variables,
such as momentum and position. So-called phase integrals were used to quantize
systems after the classical laws of motion were applied.
These semiclassical procedures had some successes. For example, elliptical or-
bits were introduced into Bohr's picture and relativistic equations were used in
place of the nonrelativistic Eq. (2.5). The relativistic theory predicted a split in some
atomic energy levels with the same quantum number, the magnitude of the energy
difference depending on the fine-structure constant. The existence of the split gives
rise to a fine structure in the spectrum of most elements in which some “lines”
are observed under high resolution to be two closely separated lines. The well-
known doublet in the sodium spectrum, consisting of two yellow lines at 5890 Å
