Weblog

No Moving Parts

2013 September 9

Image by Colin M. L. Burnett.

Every physicist knows the atom does not really look like this,
but the archetype has a kind of insidious, mythical power
that is hard to escape. One notes first the roundness of this
"atom", a quality which the ancient Greeks would have admired.
The *p, d,* and *f* states of real atoms, as imagined
in the Twentieth Century quantum-mechanical paradigm, inherit
the complicated forms of the Legendre polynomials, but when one
thinks of an atom, one automatically thinks of spheres, not of
spherical harmonics. It is interesting that, while the simple Bohr
model (with its atom as flat as the ecliptic) is the only part of
Old Quantum Mechanics still routinely taught to students today,
it is Sommerfeld's modification, with ellipses pointing in all directions
for approximate spherical symmetry, that seems to inform the
popular stereotype.

Likewise, although the standard Copenhagen interpretation insists that electrons are not really particles and not really moving, intuition tells us that they are planets in a Rutherfordian solar system, hurtling around the nucleus. However much one insist that only a probability distribution can be measured, and that "momentum" is just an imaginary gradient-operator in Hilbert space, the notion of an orbital path along which the electron moves with some velocity remains unexorcised. "The unusual properties of gold are due to the relativistic speed at which the electron goes ripping by the nucleus ..."

I still recall my first encounter with the Bohm theory, in the form
of an extremely abbreviated synopsis presenting little more than the
main idea of rewriting Schrödinger's equation in terms of
action and quantum potential. I immediately sat down and tried to
solve the hydrogen atom, but stopped in amazement when I saw that an
electron in the ground state must be *at rest*. Surely I had made
some mistake! Unable to see any, I walked to the library (as one did in those
pre-Wikipedian days) to find a fuller account of Bohmian mechanics.
Having confirmed that
my result was indeed standard, I temporarily lost interest in the theory:
an electron that stands still seemed too ludicrous to consider.

Given the strength of these prejudices, it is surprising to realise that they are not, in fact, very old. The sphericity of atoms was (surprisingly) not considered self-evident by the Greeks or their successors; the curious shapes advocated by Dalton at the beginning of modern chemistry are of course well knowm. When chemical atoms ceased to be "atomic" and were resolved into constituent parts, the motion of these parts was not taken for granted: raisins do not hurtle about in a plum pudding!

Physicists traditionally study basic astronomy and celestial mechanics in their student days; probably for this reason, the solar-system atom and its modification by Bohr rapidly gained acceptance in the physics world. Chemists, on the other hand, are more at home with crystals, and the concept of the atom as a kind of "unit cell" persisted among chemists well into the 1920s.

The most famous advocate of this approach was G. N. Lewis, whose
"dot diagrams", still used today, were originally meant to illustrate
the properties of a *cubical* atom with valence electrons sitting
motionless at its vertices. A cube has eight corners -- hence,
according to Lewis, the Octet Rule.

Among the most brilliant advocates of this general approach --- he did not see eye-to-eye with Lewis on every detail --- was the great Irving Langmuir. Here we see him on the cusp of the quantum age, defending the chemical view against the Rutherford-Bohr onslaught.

The Structure of the Static Atom
by Irving Langmuir
[*Science 53*, 290 (1921)]
Langmuir's words are in **bold.**

**
In attempting recently to derive the
conditions which determine the stability of
chemical molecules I was impressed by the
importance of the part played by Coulomb's law
of inverse square forces between charged
particles. In fact, by considering a static
arrangement of electrons according to the
models which I proposed two years ago, and
calculating the total potential energy by
Coulomb's law, I have found it possible not
only to determine the relative stability of
various substances but to calculate with
reasonable accuracy the heats of formation of
compounds even of widely varying types.
**

**
In all such calculations, however, it is
necessary to assume that the electrons are kept
from falling into the nucleus by some
undetermined force, for Coulomb's law alone can
not account for this. According to Bohr's
theory of atomic structure, the requisite
repulsive force is nothing more than centrifugal
force due to rotation of the electrons
about the nucleus. This theory has been so
remarkably successful in accounting for the
spectra of hydrogen and helium that the
fundamental assumption of movement about
the nucleus has seemed justified, notwithstanding
the fact that this violates all our
classical laws regarding the radiation of
energy from accelerated electrons.
**

**
From the chemical point of view it is a
matter of comparative indifference what the
cause of the repulsive force is, so long as it
exists. I therefore endeavored to find what
law of repulsive force between electrons and
positive nuclei would produce an effect
equivalent to the centrifugal force of Bohr's
theory.
**

**
According to Bohr the average kinetic
energy in any atom or molecule is half as
great as the average potential energy, but
opposite in sign. I therefore now assume that
this energy, which Bohr called kinetic, is
another form of potential energy dependent
upon certain quantum changes in the electron.
From this potential energy it is then easy to
determine the law of repulsive force.
**

**
The result of this analysis is that we may
regard the force between any nucleus of
charge Ze and an electron of charge e as
consisting of two parts which act independently.
The first is the Coulomb attractive force F_{e}
given by
**

**
**

**
The second force, which we may call the
quantum force, is a repulsive force F_{q}
given by
**

**
**

**
In these equations r is the distance between
the electron and the nucleus, m is the mass of
the electron, h is Planck's quantum, and n is
an integer denoting the quantum state of the
electron. This repulsive force, varying
inversely as the cube of the distance, is
remarkable in that it is independent of the
charge on the nucleus.
**

**
It is to be noted
especially that an electron which has not
undergone any quantum change and for which
therefore n = 0, follows Coulomb's law
accurately. Thus presumably β-rays in passing
through an atom will be acted on only by
the usual law.
**

**
It can be readily shown that under the
influence of these two forces an electron will
be in stable equilibrium when it is at a
distance from the nucleus equal to
**

**
**

**
where a_{0} is given by
**

**
**

**
This result is identical with that for the
radius of the orbit in Bohr's theory, but of
course the law of force was chosen to give
just this result.
**

**
If W is the total energy of the system with
its sign reversed we obtain
**

**
**

**
where
**

**
**

**
Equation (5) has no equivalent in Bohr's
theory for it applies to the transitions between
stationary states. The first term in the
second member represents the Coulomb potential
while the second corresponds to the quantum potential.
**

**
When an electron has settled down into its
position of equilibrium, the value of W
becomes
**

**
**

**
This also is identical with the result obtained
by Bohr for the total energy in any stationary
state. It follows from this that the Rydberg
constant, the Balmer series and all other
series calculated by Bohr can be obtained by
this method without assuming electrons
moving about the nucleus.
**

**
If the electron is disturbed from its position
of equilibrium it oscillates about this position.
From equation 5 the frequency of this oscillation
is found to be
**

**
**

**
This is identical with the frequency of
revolution of the electron in Bohr atom.
From this we may draw a definite physical
picture of the mechanism of the transition
between two states, at least when Z is large.
Bohr has shown that under these conditions
the frequency radiated when an electron
passes from one circular orbit to the next
inner one is the same as the frequency of
revolution. According to the present theory,
if the quantum number of an electron in a
stable position decreases by one unit, the
electron is no longer stable but falls towards
its new position of equilibrium, and oscillates
about this position. It then radiates its
energy of oscillation according to the usual
laws of electro-dynamics.
**

**
One of the greatest successes of the Bohr
theory is that it accounts for certain slight
differences between hydrogen and helium lines
by the nuclear mass correction. This correction
is taken care of in the present theory
with the same accuracy if we assume a slight
modification to our law of quantum repulsion,
viz. replace equation (2) by
**

**
**

**
where M is the mass of the nucleus. This
seems to indicate that the quantum force is
due to an interaction between the electron
and the nucleus in which both masses play
a similar rôle. For example, it may be
imagined that both are set into rotation in
opposite directions about the axis connecting
them.
**

**
Sommerfeld has accounted for the fine-line
structure of spectral lines by considering a
relativity correction due to the rapid motion
of the electron. This would seem to be
excellent proof that the electrons do move.
However, it is possible that the motion resides
within the electron and nucleus. It is probably
significant that the relativity correction
can be taken into account in the present
theory if we substitute in equation 2 in place
of n^{2} the expression
**

**
**

**
where α is a constant calculated by
Sommerfeld. A consideration of this equation may
lead to more definite conceptions of the
structure of the electron and nucleus. The
quantities n_{a}
and n_{r} refer to what Sommerfeld
calls angular and radial quanta. It is not
yet clear just what interpretation is to be
placed upon these in the present theory but
they are evidently only of secondary importance
in determining the forces between the
electrons and the nucleus.
**

**
When we consider other atoms such as
helium it seems as if the new theory may lead
us much further than the usual theory, for
the determination of equilibrium positions
under static forces is extremely simple
compared to the corresponding dynamical problem.
Furthermore we are not troubled by
mysterious quantum conditions which are
theoretically applicable only to periodic orbits
while the calculated orbits in atoms are not
periodic.
**

**
At present I am studying the spectra of
helium and lithium from this viewpoint. The
following tentative conclusions may be stated.
**

**
The quantum force between quantized
electrons is not as simple as between electrons
and nuclei. The quantum force between
electrons on opposite sides of a nucleus is one of
repulsion whose magnitude is approximately
given by equation (2) if the quanta are all
of the "angular" type, but is considerably
less when the quanta are of the "radial " type.
But if the electrons are on the same side of
the nucleus, the quantum force between
electrons is one of attraction, also given
approximately by equation (2). Thus if one of the
electrons in the helium is uniquantic, and the
other one is diquantic, the latter can take
equilibrium positions either on the opposite
side of the nucleus from the uniquantic
electron or on the same side. This perhaps
explains the fact that helium (as well as other
elements with two outer electrons such as
calcium, etc.) has two separate complete sets of
spectra (helium and parhelium). It is also
in accord with the remarkable facts in regard
to the helium spectrum which were recently
pointed out by Franck and Reiche.
**

**
These properties of the electron are in full
accord with those which are needed to
account for chemical relationships. The new
theory fulfills the predictions of G. N. Lewis
who in 1916
[ Jour. Amer. Chem. Soc., 38, 773] wrote
in reference to Bohr's
theory :
**

**
" Now this is not only inconsistent with the
accepted laws of electromagnetics but, I may add, is
logically objectionable, for that state of motion
which produces no physical effect whatsoever may
better be called a state of rest."
**

**
It is also in accord with the conclusion
which I gave in a paper entitled "The properties
of the electron as derived from the chemical
properties of the elements,"
[ Phys. Rev., 8, 300 (1919)]
viz.:
**

**
" How can these results be reconciled with Bohr's
theory and with our usual conception of the
electron? It is too early to answer. Bohr's stationary
states and the cellular structure postulated
above have many points of similarity. It seems
that the electron must be regarded as a complex
structure which undergoes a series of discontinuous
changes while it is being bound by the
nucleus or kernel of an atom. There seems to be
strong evidence that an electron can exert
magnetic attractions on other electrons in the atom
even when not revolving about the nucleus of the
atom."
**

**
Irving Langmuir
Research Laboratory,
General Electric Co.,
Schenectady, N. Y.,
March 8, 1921
**

**
**
Those were confusing times, and although Langmuir's
discussion of electron-electron interactions
seems a bit like special pleading,
one could find instances of similar language used by the
school of Bohr.

A modern reader (or at least *this*
modern reader) cannot read this essay without
being struck by its (admittedly
dim) prefiguration of the Bohmian
attitude -- right down to the name "quantum potential"!
The Lewisite position, in its essence, was that
quantum phenomena are the result of a new force acting on
microscopic objects; this is as close to Bohm's interpretation
of quantum mechanics as the old Bohr atom is to Copenhagen.
One could easily imagine an alternate history in which
Lewisites developed wave mechanics first, and (what we would
call) the Bohmian approach became the standard
one throughout the 1900s.

Lewis's notes, 1902. [California State University Los Angeles]