CHAPTER
10
THE STOCHASTIC ATOM
At the end of the last chapter I
suggested that we might think of particles of matter like electrons and protons
as undergoing a random, jittery, zigzag, Brownian motion just as a spore or smoke particle can be observed to execute
when you look at it under a microscope. This seems reasonable because a
relationship between the uncertainties of position and momentum exists in both
cases. In this chapter I will describe a model of the atom called the
stochastic atom in which the electron moves about the proton in a Brownian-like
way.
Brownian motion has played a very
interesting role in physics. The
theory of Brownian motion was developed by Einstein. He showed that the average energy of the Brownian particles
was the same as the average energy of the atoms of the gas that was causing the
motion. Mach was convinced finally by the theory of Brownian motion that atoms
did exist. For many years he had
thought that atomic theory was nonsense and said so many times:
However well fitted atomic theories may be to
reproduce certain groups of facts, the physical inquirer who has laid to heart
Newton's rules will only admit those theories as provisional helps, and will strive to attain, in some more
natural way, a satisfactory substitute. [1]
It seems rather strange that Brownian motion finally
won Mach over to the idea of the
reality of atoms and has led me to doubt wave-particle duality. But I have been
unsatisfied for many years with
quantum theory and have longed for some "more natural"
substitute. I do not seem to be alone in
wanting to explain quantum theory in a less mystical way than invoking a dual wave-particle nature to
matter and radiation. Bridgman
says this:
There is a sense in which all the revolutionary
aspects of quantum theory can be subsumed under the single point of view that
the operation of isolation always fails eventually. [2]
Bridgman finds quantum mechanics
"revolutionary" and suggests that environmental influences might
explain everything much more simply.
Others find that the wave-particle dual nature of matter and radiation
point of view is very natural and satisfying. Feynman says:
One of the consequences [of quantum mechanics] is
that things which we used to consider as waves also behave like particles, and
particles behave like waves; in fact everything behaves the same way. [3]
Feynman
rejoices in this unity of nature. Wave-particle duality to him accords with the
idea of design in the universe. But I am set against this point of view and
must explain the facts in a way that does not imply design.
Before the wave-particle duality of
matter and the uncertainty principle were introduced in the 1920's, Niels Bohr
had devised a model of the hydrogen atom in an attempt to explain the spectrum
of light which hydrogen gas produces when it is excited. The spectrum of
hydrogen consists of a series of lines of different colors rather than the
continuous spectrum you get from the sun's light. This means that the hydrogen atom is producing light of a
number of discrete frequencies. In Bohr's model, an electron moved in orbit
around a proton which was the nucleus of the hydrogen atom in much the same way as the earth moves
in orbit around the sun. The big
difference is that it is an electric attraction between the electron and proton
rather than a gravitational attraction as it is with the earth and sun. As I
said, the spectrum of hydrogen
consists of a number of particular frequencies of light. Bohr said that these
were produced by the electron first being given energy, and thus excited to
move in an orbit farther away from the proton, then eventually jumping back to
the original orbit where it normally moved (the ground state). The different spectral frequencies
corresponded to jumps between different possible excited orbits and the ground
state
Bohr had a formula for prescribing exactly where the excited
orbits might be. It was called a
quantum rule. The permitted orbits
were called "stationary" states because they were believed to be
semi-stable. The electron would
stay in an excited orbit for some little time before it jumped to some other
"stationary" state or to the ground state. The ground state was truly stable. The frequency f
of radiation emitted when the electron jumped from an orbit of energy E2 to an orbit of lower energy E1 he gave by what became called Bohr's frequency
condition
f = (E2ÐE1)/h
Here the constant h is Planck's constant. Planck's constant had been introduced into physics a decade
before Bohr's work. Remember: Planck and Einstein said that a quantum of
electromagnetic radiation of frequency f has an energy equal to h
times f. If we rewrite Bohr's frequency condition as
h*f = E2ÐE1
we can interpret it as a statement that energy is
conserved in what Bohr called a quantum jump. The equation says the energy of the quantum of radiation (h*f) given out is equal to the change in energy of the
electron as it moves from orbit 2 to orbit 1. Here are Bohr's own words:
1. That energy radiation is not emitted (or absorbed)
in the continuous way assumed in the ordinary electrodynamics, but only during
the passing of the systems between different 'stationary' states.
2. That
the dynamical equilibrium of the system in the stationary state is governed by
the ordinary laws of mechanics while these laws do not hold for the passing of
the systems between the different stationary states. [4]
Although the Bohr theory has now been
discarded completely as being wrong, the Bohr frequency condition with its
energy interpretation is maintained. I call this Bohr's legacy.
I believe that the second law of
thermodynamics says that microscopic conservation of energy is not possible so
I must reject the interpretation of Bohr's frequency condition as an expression
of microscopic energy conservation. How can I do it? I must say that I believe
that radiation quanta are artifacts and have absolutely no correspondence with
reality. I do not believe that it
is appropriate to say that h*f is
the energy of a quantum of radiation.
There has been, from the beginning, something very wrong about Bohr's
original idea that radiation only emerges when the electron jumps from one
allowed orbit to another allowed orbit.
According to electromagnetic theory an electron in an orbit should
radiate at a frequency equal to the frequency of the motion in orbit. L.I. Schiff in his book Quantum
Mechanics notes the contradiction in
Bohr's theory:
It was difficult to understand why the electrostatic
interaction between a hydrogen nucleus [in a Bohr atom] and an electron should
be effective when the ability of the accelerated electron [in orbit] to emit
electromagnetic radiation disappeared in a stationary state. [5]
Bohr wanted electrostatic interaction to hold the
electron in orbit but, he did not want the radiation field due to its
acceleration in orbit. He just
said baldly that an electron in a stationary orbit did not radiate. The argument went: If it did radiate it
would lose energy and gradually spiral in to the nucleus and collapse. So scientists became convinced that
atoms would all collapse if they did radiate all the time.
Well, the Bohr picture of the atom was
discarded when the new wave-particle uncertainty view appeared. The idea of
orbital motion keeping the electron from moving right in to the proton was
abandoned. The present picture of the atom is much vaguer. Here is Feynman
speaking of the current view:
What keeps the electrons [in an atom] from simply
falling in [to the nucleus]? This
principle [the uncertainty principle]... We cannot know where they [the
electrons] are and how fast they are moving, so they must be continually
wiggling in there! [6]
This explanation says that atoms do not collapse
because of a principle (really!).
Even Feynman is not convinced and ends by saying "they must be
continually wiggling in there".
Why do they wiggle? Two answers: They are born wigglers, or, they are
being buffeted about. Take your
pick; I pick the latter. As you
might expect I am not alone in this choice. David Bohm says this:
Indeed, a rather similar behaviour [to an electron in
an atom] is obtained in classical Brownian motion of a particle [like pollen in
a container of water] in a gravitational field, where the random motion which
tends to carry the particle into all parts of the containers is opposed by the
gravitational field, which tends to pull it towards the bottom. In this case, the net effect is to
produce a probability distribution [formula], which describes a tendency for
the particles to concentrate at the bottom and yet occasionally in their random
motions to be thrown up to great heights. [7]
I call the model of the atom which treats the motion
of the electron as due to a superposition of the electric attraction of the
proton as described classically, and random influences from the rest of the
universe, the stochastic atom. It
might equally well be called the Brownian motion atom. "Stochastic" applies to
things in which there is a random input, an element of chance. For such an atom, probabilistic or
statistical predictions are all that would be possible.
I became interested in this stochastic
model of the hydrogen atom a long time ago and thought that I might be able to
simulate its behavior on a computer.
I remembered how difficult it was to model a planet going around the sun
by computer; because the errors
(or uncertainties) in the precision with which the calculations were being
carried out were not negligible, the orbit
precessed. I imagined that if
uncertainties or random influences were increased (rather than decreased) the
motion would not be like an orbit at all but at least the atom would not
collapse.
When I ran the program for the model on
the computer I had hoped to find that it would take up stable (or semi-stable)
states with various average distances of the electron from the proton just as
Bohr's atom did. I would then be
able to predict the frequencies of the radiation spectrum. But no such thing happened. There was no sign of equilibrium; in
fact, my atom, instead of collapsing, just got larger and larger. The longer I ran the simulation model the
farther the electron got in its drunken motion from the proton. Failure.
I came again at the whole project after
several years in an entirely
different way. But let me digress. The quantum mechanical model for the
atom which incorporates the uncertainty principle was developed by both Erwin
Schrodinger and by Werner Heisenberg. I will concentrate on describing
Schrodinger's approach, which is often called wave mechanics. The two formulations have been shown to
be equivalent even though they look very different. In Schrodinger's model all calculations about the atom are
probabilistic and are made by computing a function called the probability
amplitude function (sometimes called wave function). Using the probability amplitude function, average values can
be calculated for the distance the electron is away from the proton, the energy
the electron has, and so on. The
probability amplitude functions are found by solving Schrodinger's equation for
them. Since the solutions of the equation were to be interpreted as giving the
probability of finding the electron in a particular region around the proton,
only certain mathematically possible solutions would make acceptable solutions as
far as the probability interpretation was concerned. For one thing the function had to predict that the electron
was somewhere. The total
probability, added up over the whole region surrounding the proton, had to be
equal to one. That meant it was somewhere. This restriction, placed on all possible solutions, yielded
a set of functions which were called proper functions or, in German,
eigenfunctions. For each one of
these acceptable solutions, the average energy could be calculated and this was
found to give a precise value, not a value with any uncertainty about it.
Using these average energy values, which
were put into a one to one relationship with Bohr's orbits, and the Bohr
frequency condition, the observed spectrum of hydrogen was predicted (or I
should say fitted). Bridgman
comments on the naturalness of using probabilistic methods:
It is perhaps not too surprising that particles are
so easily treated by probability methods, for having come to the end of the
effects of structure we have also come to the end of the possibility of explanation
in causal terms. [8]
I do not object to probabilistic laws any more than
causal laws as long as they can be shown to record the facts about specific
things like the fundamental particles or the universe as a whole. I cannot
tolerate general laws that cannot be explained. I had attempted to explain the uncertainty principle, from
which Schrodinger's equation can be derived, by seeing a particle as having, as
well as its usual classical electromagnetic interaction with other local
particles, a random influence from the rest of the universe that buffets it about
in a Brownian-type motion.
There were two problems: Why had my
Brownian computer model not shown these special privileged states predicted
from Schrodinger's equation and how could I keep my atom from exploding (rather
than collapsing)? The solution to my problems lay in rejecting Bohr's legacy to
quantum mechanics, his quantum jumps.
But how? Let me again go back a bit.
Niels Bohr was very interested in philosophy
and was influenced by the philosopher Hoffding. Feuer indicates Hoffding's position:
[Hoffding] was affirming that a personal choice was
as operative in the ultimate methodological commitments of scientists and in
their analogies, as it was among religious philosophers and metaphysicians...
Hoffding's unique contribution to scientific thought was his insistence on the
heuristic potentialities of the notion of discontinuities in existence. [9]
You can see how quantum jumps would appeal to Bohr if
he shared Hoffding's enthusiasm for "discontinuities in
existence". Bohr changed
scientists' thinking radically.
Hanson describes the difference in his Patterns of Discovery:
Pianistic thinking cannot allow violinistic glissandi: pianos allow a C sharp, or a D, but nothing in
between. Classical physics
regarded nature as a complicated violin: that is, differential equations were
always in place; but we cannot think of the atom thus. [10]
Bohr had introduced discontinuous stationary states
of the atom and the idea of jumps from one state to another. When his orbital
model of the atom was discarded, his quantum jumps survived. Bohr was quick to
embrace the new quantum mechanics of Schrodinger and Heisenberg which
incorporated de Broglie's wave nature of particles and Heisenberg's uncertainty
principle along with his quantum jumps. He led a group in Copenhagen to give
quantum mechanics an interpretation which
emphasized the duality of nature.
This "Copenhagen interpretation" dominated and is generally
accepted today. There are a few
scientists beside Einstein who
question this interpretation of quantum mechanics, among them de Broglie and
Bohm.
I myself for many years delighted in the
many exciting and mysterious ideas in physics. When Donald Ivey and I were preparing television programs we
marvelled at the great material we had at our disposal. Duality and uncertainty have a
strong dramatic appeal. So much so that I began to suspect that
we, as physicists, were indulging ourselves in a pseudo-religious experience,
finding the universe so mysteriously contrived. That is when my real doubts began. I fell out of step with
the orthodox view of things.
Heisenberg tells of how he became
interested in finding a theory for the atom. He was put off by a textbook illustrator's view of
molecules:
In order to explain further just why one atom of
carbon and two atoms of oxygen form one carbonic acid molecule, the artist had
given the atom hooks and eyes... To me this seemed wholly senseless, because
hooks and eyes are Ð as I saw it Ð quite arbitrary forms, which can be chosen
in different ways according to their technical usefulness. But atoms are supposed to be the result
of natural laws, and guided by them in forming molecules. [11]
The natural laws (e.g. the uncertainty principle
devised by Heisenberg), to me, need explanation, and that is why I am attracted
to a stochastic model of the atom. Nelson, in an article in the Physical
Review in 1966, describes such a
model:
If we have, for example, a hydrogen atom in the
ground state, the electron is in dynamical equilibrium between the random force
causing the Brownian motion and the attractive Coulomb [electrostatic] force of
the nucleus. Its trajectory is
very irregular. Most of the time
the electron is near the nucleus, Sometimes it goes farther away, but it always
shows a general tendency to move toward the nucleus, and this is true no matter
what direction we take for time... the electron has states of dynamical
equilibrium at the usual discrete energy levels of the atom. [12]
In the article Nelson shows that a stochastic model
leads to the Schrodinger equation but he runs into a little trouble trying to
explain the character of the probability function for excited states of the
atom. The problem is that the
function that describes the probability for position of the electron in any
excited state has certain places where it becomes zero. We call these positions nodes in the
function. There is a definite
probability that the electron will be found on either side of the node but no
probability that it will be at the node.
How does the electron get from one place to another without passing in
between? Here is Nelson struggling
with the problem:
For real solutions of the time-independent
Schrodinger equation, other than the ground-state solution, the probability
functions have nodal surfaces... However, it can be shown that the associated
Markoff [stochastic] process is well defined in each such region, and that a
particle performing the Markoff process never reaches a nodal surface. [13]
Nelson seems to believe that the particle stays in
one or other of the compartments between nodes. The answer the Bohr disciples
give is that the electron is not really a particle, it has a dual nature and as
such can perform this difficult feat.
It is like going from one room to another without passing through the
connecting door. But that is nature for you \(me mysteriously contrived. After all, the electron is not to be
considered as a particle. But look
at Schrodinger's equation and what do we see: the expression for the energy of
a particle (not a
wave-particle). We use the law of
Coulomb to compute the electrostatic energy of position of the electron
relative to the proton. Now it is, now it is not a particle!
But that is the beauty of the Copenhagen
interpretation Ð we can have it both ways. Bridgman does not mind:
The celebrated remark of Wm. Bragg that we seem
forced to use classical [particle] mechanics on Monday, Wednesday and Friday
and wave mechanics on Tuesday, Thursday and Saturday may prove not to be a
reductio ad absurdum as it is usually taken to be but an ultimate and necessary
procedure. [14]
Although he also says this:
The ultimately important thing about any theory is
what it actually does, not what it says it does or what the author thinks it
does, for these are often very different things indeed. [15]
Schrodinger's equation gives the right frequencies of
the spectrum of hydrogen but that does not mean that what he or Bohr or anybody
says about its interpretation need be accepted. After all, Bohr's theory of orbits gave the right frequencies
for the spectrum too.
I was beginning to suspect that the
problem might be that there were no quantized excited states of the atom at
all. First my own computer model
did not show any. The Schrodinger
probability functions for excited states had nodes and would not square with a
real particle (as contrasted with a wave-particle) undergoing a Brownian
motion, superimposed on motion under the attraction of the proton. But how would you get a discrete
frequency spectrum without quantized excited states?
I now had the answer. Bohr's assumption about there being no
radiation in the ground state or the atom would collapse was unnecessary. My atom did not collapse; in fact, it
expanded. I needed some way to
keep it from expanding. The atom could radiate. If I went by
classical electromagnetic theory, the radiation from the ground state would
have all frequencies in it, from the random motion of the electron.
That would mean a continuous spectrum of frequencies even from the
ground state, presumably for any other state as well. But no one has ever reported observing such a continuous
spectrum. Well not exactly
true. We know that all macroscopic
bodies radiate a continuous spectrum all the time called blackbody radiation. That is what Planck was studying when
he stumbled on the idea for the quantization of energy of oscillators. It would be like looking for a needle
in a haystack to find the continuous radiation from the ground state of
hydrogen. So I did not worry about
supposing that one existed. But what about the discrete spectrum? Even though I thought that the
probability functions for excited stable states did not correspond to anything
real and rejected them as not proper functions because they had nodes, I knew
that any non-equilibrium state of the system could be expressed mathematically
as a combination of these functions.
Wait a minute; this is getting rather
complicated. I am trying to explain how a spectrum at discrete frequencies can
be produced even though the atom never stays in any one of the excited
stationary states. First of all let me be clear about what I expect the atom to
do. In the ground state it will radiate a continuous spectrum Ð all frequencies
more or less all the time. The motion of the electron is a complex zigzag
motion which mathematically is analyzable as the sum of oscillations of every
frequency. When the electron in the atom is given additional energy, by being
bumped for example, the atom is excited. It has more energy than it has in the
ground state. Its motion is similar to what it is in the ground state and it
gradually loses energy and returns to the ground state. In the process it
produces a continuous spectrum as it does in the ground state but in addition
it produces the discrete frequency spectrum. The electron does not necessarily
oscillate at these discrete frequencies but its motion is analyzable as the sum
of oscillations at these frequencies.
Perhaps an analogy here would be helpful.
I quoted Hanson earlier in the chapter as saying that "Classical physics
regarded nature as a complicated violin ... but we cannot think of the atom
thus..." . There is a sense in which we can relate a violin to an atom. If
we keep our finger off a violin string and bow it carefully it will produce a
particular discrete frequency just the way one of the strings of a piano does
when it is struck. This frequency is produced by the vibration of the string.
The two ends of the string are held fixed and the center part moves back and
forth. We say there is a node at each end and a loop at the center. The
wavelength associated with this frequency is twice the length of the string.
This is one mode of vibration for the string. It is the fundamental mode. But
the mathematical equation which describes the vibrating string has other
solutions Ð other modes of vibration. One such mode has a node in the center as
well as at each end. This mode of vibration has a frequency which is exactly
twice the frequency of the fundamental mode. We call it the first harmonic of
the fundamental. All the modes of vibration that are solutions of the
mathematical equation have frequencies that are integral multiples of the
fundamental. We call them the normal modes of vibration of the string. But that
does not mean that they are the only ways that the string can vibrate. Most of
the time it vibrates in a more complicated way, but the more complicated
vibration can always be analyzed as a sum of the normal modes of vibration. We
get a mixture of the fundamental and its harmonics. Just as a prism separates
light vibrations which are a mixture of different frequencies into the spectrum
showing the different lines, the ear of a listener can separate the complex
sound vibrations into the different frequencies. We can hear the fundamental
and the harmonics. The ground state of the violin string is no vibration at
all; in the atom there is perpetual motion.
I hope this analogy will help you
understand how a discrete spectrum at frequencies predicted by Schrodinger's
equation might exist for hydrogen. In any non-stable state, radiation would be
expected at all the different possible discrete frequencies as well as a
continuous spectrum. The system if excited would produce the whole observed
spectrum. The more the atom is
excited, the more prominent the radiation at higher frequencies would be.
As the atom is left alone, it returns to
the ground state where the discrete frequencies die out. So you see I reject
the idea of quantized excited states in an atom with an electron jumping from
one to another thereby producing a photon of a discrete frequency on each
quantum jump from higher energy state to lower energy state. Even Schrodinger
hoped to do away with quantum jumping. At the beginning of quantum mechanics he
said:
If all this damned quantum jumping were really here
to stay, I should be sorry I ever got involved with quantum theory. [16]
Bohr was quick to disagree with him:
... remember the Einstein derivation of Planck's
[blackbody] radiation law. This derivation demands that the energy of the atom should assume discrete
values and change discontinuously from time to time... You can't seriously
be trying to cast doubt on the whole basis of quantum theory! [17]
I will be looking at an explanation of blackbody
radiation that does not require quantized energy states in the chapter called
"The Two-slit Mystery".
The stochastic model of the atom answers
another kind of question: how is it that atoms are so durable. A stochastic
atom would be durable because it has adapted to the environment. In fact it
exists because of, not in spite of, the environment with all its random
influences. Heisenberg wondered at the great durability of atoms:
... chemical elements displayed in their
behavior a degree of stability completely lacking in [classical] mechanical
systems. [18]
On large scale systems like planets the random
influences are not significant but as we go to smaller objects they are
dominant. The atom works, I believe,
by a combination of deterministic influence, of the proton on the
electron, and by random influences, of the rest of the universe on it. C.S. Smith says this:
Significant structure is a mixture of perfection and
imperfection. [19]
SUMMARY
1. The
model of the atom that is consistent with my world view is one called the
stochastic atom. (Nelson et al)
2. If
microscopic energy conservation is impossible, which I believe is evidenced by
the second law of thermodynamics, then the Bohr frequency condition cannot
properly be given an energy interpretation.
3. The
idea of a quantum of radiation is thus not supported by the line spectrum of
atoms.
4. The
natural motion of an electron in an atom is perpetual motion due to the
fluctuating effect of the rest of the universe. Collapse of the atom is not
likely even if the electron is producing electromagnetic radiation in the
ground state.
5. Radiation
produced by an electron moving in a Brownian-like fashion under the attraction
of a proton, as in the stochastic model of the atom, would be a continuous
spectrum of radiation in the ground (unexcited) state.
6. This
continuous spectrum might not be observed because all macroscopic bodies
radiate a continuous spectrum, known as blackbody radiation.
7. If
the stochastic atom is a valid model and is described by Schrodinger's
equation, then solutions of the equation which have nodes cannot be proper
solutions. A real particle cannot have a probability density function of
position which becomes zero between places where it is non-zero.
8. The
only stationary state for a stochastic atom is the ground state. This is the
only state whose probability density function has no nodes. Although no excited
state is stable, every excited state may be described as a combination of the
solutions of the Schrodinger equation that possess nodes.
9. The
discrete line spectrum of hydrogen can be explained by the fact that excited
states are describable as a combination of the solutions of the Schrodinger
equation normally considered proper solutions.
10. An excited atom produces radiation which, when analysed by a device such as a prism, should contain both a continuous spectrum and radiation at certain discrete frequencies simultaneously.
Copyright © 1983 J.N.P. Hume All rights in this book reserved