CHAPTER 11
THE TWO-SLIT MYSTERY
This chapter is called "The two-slit mystery". What
are the two slits and what is the mystery? The two slits are just that, two
slits cut parallel to each other in an opaque screen. They are used to perform
the two-slit experiment, an experiment devised by Thomas Young, the early
nineteenth century physicist who solved the mystery of the Rosetta stone and
disentangled Egyptian hieroglyphics. But Young left behind him his own mystery,
the two-slit mystery.
In the last two chapters I have been
concerned principally with an explanation of the wave-particle duality of
matter. In this chapter I will
look at the wave-particle duality of electromagnetic radiation. And that is where the two-slit
experiment comes in. It was Niels Bohr who stressed the dual nature of matter
and radiation. He spoke about
complementarity, saying that the wave and particle aspects of matter were like
two sides of a coin. A coin may be
seen as either heads or tails but not both at the same time. According to Bohr,
the reality of particles like electrons, protons, and neutrons must be
described using the complementary models of wave and particle. Perhaps we should refer to them not as
particles but as "wavicles".
This notion of complementarity seemed to Bohr to be very fundamental. As Feuer reports:
When Niels Bohr formulated his principle of
complementarity in 1926, he proposed that physicists renounce the hope of
achieving a system or theory based on one model, either wave or particle...
This duality of complementaries seemed to Bohr "a fundamental
feature" in the nature of all human knowledge. [1]
I suppose you might say that he thought that there is
ambiguity in nature and our knowledge of it must reflect this uncertainty. This gives rise to the big question of
whether this uncertainty is subjective, connected with the state of our
understanding of the real world, or objective, as I have suggested. I have attributed the apparent dual
nature of particles to the fact that all particles suffer random influences
from their environment which are superimposed on whatever influences there are,
due to other nearby particles. If there
are no other particles nearby we would get completely random motion,
much like Brownian motion.
But my explanation would probably be much
too mundane for Bohr. He was greatly influenced by the philosophy of
Kierkegaard as presented by Hoffding.
This is Feuer's account:
From the proposition that theology is psychology,
Kierkegaard (as Hoffding expounded his views) argued that we must simply drop
the idea of truth-in-itself, the objective truth; all that we can have is
psychological truth, hence subjectivity is truth... Thus were sown the first seeds of the notion that
theoretical perspectives on physical experience which seemed contradictory were
but complementary standpoints, depending... on the personal decisions of an experimenter, on the
particular, experimental arrangement he devised to measure and report on his
physical experiences. [2]
This philosophical (or theological) point
of view played a profound role in Bohr's vision of physical truth. Bohr's idea was that
ultimately precise knowledge of physical reality was
not possible. Always there would
be uncertainty. Actually this particular philosophy comes very close to my own
thesis (shades of Kierkegaard perhaps).
But I must try to clarify the difference. Bohr was concerned about
whether our knowledge of the universe was subjective or objective. I maintain
that we should not be able to tell from our knowledge of the universe whether
it arises by chance or by design.
If there were laws this would be evidence for design so I believe that
there are no general laws, only facts about specific things that recur
naturally. The wave-particle
nature of all particles of matter looks dangerously like a general law, whether
it appears as the Heisenberg uncertainty principle or de Broglie's wave nature
of particles and I must explain it.
Heisenberg himself struggles with the subjective-objective nature of
these facts concerning uncertainty in his book Physics and Philosophy:
... such a description [of a measuring device]
contains all the uncertainties concerning the microscopic structure of the
device which we know from thermodynamics, and since the device is connected
with the rest of the world, it contains in fact the uncertainties of the
microscopic structure of the whole world.
These uncertainties may be called objective insofar as they are simply a
consequence of the description in the terms of classical physics and do not
depend on any observer. They may
be called subjective insofar as they refer to our incomplete knowledge of the
world. [3]
In a way, Heisenberg's attitude is not very different
from mine when he mentions "the uncertainties of the microscopic structure
of the world" as being at the root of the uncertainty in our knowledge.
But the fact is that we do not need
to know the details of the influences of the rest of the universe. We
assume that they are random and use probability ideas to compute average
behavior. We stop concerning ourselves with what we can never know in detail
and carry on. And it turns out
that it does not really matter. The behavior of atoms, although subject to
random influences, is probabalistically predictable. The predictions will never be otherwise; but that is good
enough for all our purposes.
The reason that I go into all this
wave-particle discussion is that many scientists believe that it is somehow
connected with the nature of reality.
Reality they believe, is designed that way. Bohr did not say this; he
said that it was "a fundamental feature in the nature of human
knowledge". If I were to make
any statement about "the nature of human knowledge" I would say that
it is strictly limited by the fact that we are trapped inside the universe and
we must recognize that what we know of the universe can not be considered to be
unaffected by this. Have you ever
heard people say that they do not like an "I" novel, where everything
that is written is what is known to one person. They feel trapped, limited in what they can know about other
characters in the novel. As individuals we are all trapped inside our own
bodies and what we know of the universe must come in to us, through our
senses. We are all trapped inside
the universe. We cannot step
outside and have a look. We were
recently able to step outside the earth (vicariously) and look back at it; but
we can never get outside the universe. Nor can our space ships. Our information
is strictly from inside and if you are inside something, it is not a strange
idea that the something (the rest of the universe) is affecting the objects you
are observing. It would be strange
if they were not affected in any way.
Classical physics does not take into account the effect of the
universe. Bohm says:
This error [uncertainty] arises essentially not
because of a lack of knowledge on our part, but rather because of the neglect
of objective factors existing outside the context under investigation. [4]
What, I believe, we have done is to assign to the
particles themselves a nature that really is the result of influences on them
from outside. These influences are
basically classical, that is, describable by Maxwell's laws (or Newton's
laws). Bohm says:
In physics, the influence of any process on its
"background" is even more strikingly brought out by Newton's law that
action and reaction are equal.
From this law, it follows that it is impossible for any one body to
affect another without itself being affected in some measure. Thus, in reality, no perfectly constant
background can exist. [5]
Now it is time to move to the
wave-particle nature of radiation which to me is a more complicated riddle than
the wave-particle nature of matter. Electromagnetic radiation is the name we
give to the effect that is produced by an electric charge that is
accelerating. The radiation field
associated with the acceleration decreases in magnitude inversely as the
distance from the source charge to the field point so, at appreciable distances
from the source charge, it is the dominant component of the field. If the charge is oscillating, the
radiation field oscillates at the same frequency and we say that there are
electromagnetic waves. The speed
with which the wave pattern travels away from the oscillating charge is the
speed with which the electromagnetic interaction travels. The frequency of the
oscillation has nothing whatever to do with the speed of travel. We say that electromagnetic waves of
all frequencies travel at the same speed, which we call c. The
waves travel at speed c because
the electromagnetic interaction travels at speed c. If we
ask what is travelling in an electromagnetic wave we should equally well ask
what is travelling in any electromagnetic interaction. What additionally is
travelling in a wave that is not travelling in any electromagnetic field is a
pattern - the wave pattern.
An analogy that might help here is to
imagine a hose squirting water out of a nozzle. When the nozzle is held still
the water follows in a certain path. The path would be a straight line if it
were not for gravity and wind. If there were no wind it would go in a parabolic
path landing on the ground eventually. If, instead of holding the nozzle still,
you moved it back and forth you would see a pattern move out. Each drop of
water would move as before in a parabolic path but a moving wave-like pattern
would be formed by the stream.
When the idea of energy conservation was
worked out by Poynting for electromagnetic interaction, energy was said to be
stored in the field. In a static
field, the energy just remains in position but, in a dynamic field which is
changing with time, it moves around.
It flows outward from an oscillating charge into space. The charge sends energy out. This energy "comes" from
whatever is accelerating to produce the waves. All very pretty but, since I say
that energy is "all in the mind" and is just an alternative way to
describe some of the information in Maxwell's laws, there is no need to ask
what carries the energy or whether it is "pure energy" as Weinberg
calls it. We can really just
forget all about it. Its
conservation is assured; it was all defined so that, if Maxwell's laws hold,
energy is conserved. But
unfortunately it gets dragged into a lot of the discussion about
electromagnetic radiation. In
particular it gets dragged into the particle picture of radiation.
After Maxwell's work, up until the early
years of the twentieth century, everyone was convinced that there were such
things as electromagnetic waves and that light was simply a range of
frequencies of these waves that happen to stimulate our eyes. Our primary
source of light is the sun and we call its light "white light". As Newton had showed, white light can
be dispersed into colors, the colors of the rainbow, by letting it pass through
a glass prism. This spectrum of
colors from the sun is brightest in the green part of the spectrum and tapers
off in intensity as you go to the extreme ends of red and violet. The brightness of the green is not just
because the eyes are most sensitive to this color but measuring instruments
show that it is really the most intense.
The intensity of light, as measured in say energy units, for sunlight
has a very typical relationship to color.
We know now that the sensation of color that we experience is related to
the frequency of the light waves.
The violet light has the highest frequency of the waves in the visible
spectrum; the red, the lowest. You
can plot a graph of energy density against frequency for sunlight and it is
always the same. It is the
characteristic mix of frequencies produced by our sun. If we use instruments (not the eye)
that are sensitive to all frequencies, we "see" that the sun produces
ultraviolet and infrared radiation as well and the graph of energy density
against frequency always shows the results for the full spectrum, not just the
visible part of it.
The graph of the sun's radiation can be
described by an equation and this same equation can describe the graphs of
radiation from many different hot objects, like furnaces or incandescent light
bulbs. We call these bodies black
bodies, although you might call them hot bodies. The graphs for all blackbody radiation are describable by
the same kind of equation. The
equation shows that the radiation depends on the temperature of the hot
body. Any hot body at the same
temperature as the sun produces white light, with its highest intensity in the
green. Cooler bodies peak in the
red; they look red hot, not white hot.
At lower temperatures, the peak is in the infrared; you can feel the
radiation as heat, but you cannot see it. Think of a wood stove.
I tell you all this, which you probably
know, because it was in trying to explain the graph for blackbody radiation
that Max Planck in 1901 reluctantly offered the suggestion that the oscillators
producing the radiation in a hot body change energy by an amount that is
proportional to the frequency of the radiation. If the change in energy of an oscillator is Ethen
E = h * f
where the constant h, multiplying the frequency f, has come to be called Planck's constant. Energy is packaged or quantized. An oscillator's energy does not change
continuously but in jumps. Abrupt transitions were not part of classical
physics. With Planck, modern
physics was born. This is the
origin of the quantum theory.
Planck said that the energy of an oscillator changed by a quantum, not
that the radiation energy was quantized.
It was left to Einstein to nail down the quantization of energy in
radiation. Feuer notes this:
Einstein was consciously trying to develop a new
foundation for physical science; his intent was revolutionary. Whereas Planck was a reluctant
revolutionist, unwilling but compelled by the sheer weight of experimental
facts to break with the traditional mode of thought... [6]
It is interesting that Einstein was
unhappy with the shape of quantum theory the moment it became connected with
probabilistic prediction as it did
with Heisenberg. He constantly
said "God does not play dice", to which Bohr is said to have replied
"Who are you to tell God what to do". But in 1905 Einstein was young, with revolutionary ideas, questioning
everything. He worked on the
theory of the photoelectric effect.
This effect had been discovered by Hertz when he produced
electromagnetic waves, of frequencies lower than the visible, from oscillating
electric circuits. Hertz found
that light falling on metal surfaces caused electrons to be emitted from the
metal; this became known as the photoelectric effect. The effect did not occur, no matter how brilliant the light,
unless the frequency was high enough.
Einstein said that the amount of energy required to permit the electron
to escape had to arrive all at once and could not be dribbled in
gradually. He said low frequency
light would not produce the effect because the radiation itself was packaged
with the energy $E$ in the package being related to the frequency f by the equation
E = h * f
This is Planck's equation but it now applies to a
quantum of radiation rather than to the energy jump of an oscillator. Just as it requires a certain energy to
get a rocket away from the earth, it requires a certain energy to get an
electron away from the metal. If
the radiation energy quantum, called a photon, was large enough it could free
the electron from the metal; if the frequency of the radiation was lower than
this threshold the electron would not come out. No one low frequency quantum would be big enough and the
effect was not cumulative. Even
though you provide lots of quanta below the threshold size, nothing happens.
Higher energy quanta than threshold size give the electron a speed when it
comes out. Einstein's
photoelectric equation is very
simple.
It is
h * f =
"threshold energy" + m
* v 2/2
It says: the energy of the incoming quantum h*f is equal to the threshold energy plus the energy of
motion of the electron (m*v2/2). This is a statement
of energy conservation on the microscopic level. (Remember: the second law of thermodynamics seems, to me, to
imply that microscopic energy conservation is impossible due to the random influences
of the rest of the universe. But I
leave that for the moment.)
Einstein's explanation of the
photoelectric effect convinced people of the quantum nature of electromagnetic radiation,
and with it the reality of photons.
Planck's work started it but was perhaps too hard to understand. Einstein's 192l Nobel prize was given
for the theory of the photoelectric effect, not for his theory of special
relativity. Relativity was hard to
understand and seemed "unproved" at the time.
Now it is time to get down to the
two-slit mystery. Before Maxwell there was considerable controversy about the
nature of light. Newton believed
that light was a stream of particles; Christian Huygens, Newton's contemporary,
believed that light was a wave.
Huygens was Dutch and often
observed the ripples in water produced by stones dropping in the canals. If the path of the ripples was
obstructed by a barrier the waves were reflected from the barrier just as light
is reflected by a mirror. If the
barrier had a hole in it the waves were obstructed (and reflected) except where
the hole was. As the unobstructed part of the wavefront went through the
barrier, instead of proceeding like a sort of slice of the original wave
pattern, it fanned out, just as if the hole were a new source of waves. This phenomenon is called diffraction,
which means spreading out, and is thought to be characteristic of waves.
Huygens said that if you could isolate any point on any wavefront, as you do
when you obstruct all but a small part, you would find that it acts like a
source of wavelets moving out in circles from it. These wavelets, called Huygens' wavelets, do not show in an
ordinary wave spreading out because they interfere with one another to produce
the overall effect of circular wavefronts moving out from the real source of
the waves. If there were two holes
in a barrier the waves emerging from the two holes would interfere with each
other and produce an interference pattern.
That brings us to Young and the two
slits. Young believed that he
could demonstrate that light was a wave by letting it pass through two
slits. If it were a wave, each
slit would act like a new source of waves just like a hole in the barrier for
water waves. The waves from the two slits would interfere with each other, just
like the waves from two holes in a barrier for water waves. In certain directions the waves would
reinforce and there would be light; in others they would cancel each other and
there would be darkness. If you
put a second screen up on the side of the opaque slit-screen away from the
source, you would see strips of light and dark called interference fringes,
parallel to the slits themselves.
Young did the experiment, saw the fringes, and settled the argument
about the nature of light. It was a wave.
But, if you think of light as a stream
of photons as Einstein suggested, what is happening in the two-slit experiment?
This is when the trouble starts. Here is an account by Dicke and Wittke in
their book on Quantum Mechanics.
They describe the two-slit experiment of Young where the slits are called A and
B and the screen where the interference fringes are observed is replaced by an
array of photoelectric detectors.
They say:
The result is paradoxical in several ways... The photoelectric effect [at the
detectors] can be understood only on the basis of the photon picture of
light. However, a photon
sufficiently small to affect only one electron could presumably not go through
both slits A and B. In fact, a
photon detector placed at either A or B catches only whole photons or none,
never a part of a photon. This
raises the question of how a photon which passes through A can be influenced by
the presence of B. One obvious possibility is that some photons pass through A
and some through B, and that the separate photons act on one another in such a
way as to arrive only at the bright fringes on the screen P. This explanation must be incorrect, as
can be seen by reducing the intensity of the light to the point where on the
average only one photon per minute passes through the system. Even in this case the photons continue
to arrive at only the bright fringes!
One striking thing about this experiment is that the
behavior of any given photon is largely unpredictable... The intensity
distribution over a fringe merely serves to give a probability distribution for
the arrival of any given photon; it does not allow an exact prediction of where
the photoelectron will appear... If either slit A or B is closed, photons begin
to arrive at locations where there were previously dark fringes: a decrease in
the number of paths by which a photon can get from S [the source] has resulted
in an increased probability [of arrival]. [7]
You can see that it certainly is not simple. Here is Heisenberg himself on exactly
the same theme, this time with a photographic plate as a detector:
If one describes this experiment in terms of the wave
picture, one says that the primary wave penetrates through the two holes; there
will be secondary spherical waves starting from the holes that interfere with
one another, and the interference will produce a pattern of varying intensity
on the photographic plate.
The blackening of the photographic plate is a quantum
process, a chemical reaction produced by single light quanta. Therefore, it must also be possible to
describe the experiment in terms of light quanta. If it would be permissible to
say what happens to the single light quantum between its emission from the
light source and its absorption in
the photographic plate, one could argue as follows: The single light
quantum can come through the first hole or the second one. [8]
One way often used to get around this
perplexing problem is to say that the photon picture of radiation applies only
to the emission process, as Bohr claimed, and to the absorption process, as
Einstein claimed, and has no validity in between. What happens during the transmission remains a mystery. Not very satisfactory as far as I am
concerned. I can remember when I
first studied interference, wrestling with light's dual wave-particle nature
\(me I could not reconcile the duality no matter how I tried. It all seemed like double-talk. Many scientists welcome the mysterious;
it makes science more fascinating. It makes me uneasy. I always feel that I have somehow just
been stupid in not getting the point.
Huygens' principle is used to explain the
behavior of light waves at a barrier. But where did Huygens' principle come
from? From the observation of water waves? It should be possible to derive it from Maxwell's laws of electromagnetism. Maxwell's laws are consistent with an
alternative formulation which focuses on the electromagnetic interaction
between charges. In this
alternative formulation it is clear that all electric effects travel at speed
$c$ and in straight lines. How can light be diffracted if it comes to an opaque
screen with a slit in it? It seems
to go through the slit and then spread out; some light seems to go straight
through; some changes direction. This is completely inconsistent with Maxwell's
laws. The same things could be
said about the refraction of light as it passes through a prism. Its direction is changed. How can this be consistent with the
idea that light travels in straight lines? Most scientists will say that light travels more slowly in a
medium like glass and this is why it changes direction. Different frequencies
of light travel at different speeds in a medium. This is why light is spread
out in a spectrum by a prism; different frequencies are refracted different
amounts. We call it the dispersion of light into a spectrum. What about
diffraction? The explanation you often get is that light changes direction at a
narrow opening because it is a wave, and waves bend around corners.
For many years I had always happily used
Huygens' principle to "understand" the behavior of light waves in a
medium and passing through small openings and never questioned what was
necessary to justify such a principle. The apparent slowing down of light in a
medium, other than vacuum (or air which is not very far from a vacuum compared
to solids like glass, or liquids like water) can be explained by considering
that the atoms of the medium are caused to produce secondary electromagnetic
waves by the incoming waves. The
total electromagnetic field in the medium is the result of the superposition of
the incoming field and the secondary waves produced by the atoms of the medium. These secondary waves have the same
frequencies as the incoming waves but are not exactly in phase with them. The net result is the illusion that the
wave is travelling more slowly in the medium. It is an illusion however since each wave that interferes to
produce the resultant wave is actually travelling at the speed c The incoming wave continues through the medium at
speed c and superimposed on it
are waves from the atoms of the medium, all of which travel at speed c. They all add up, by superposition, to a resultant
wave that seems to be traveling
more slowly than speed c. Very
tricky! What is more, the resultant wave can be in a different direction from
the incoming wave, depending on how the incoming wave is oriented to the
interface between the air and the medium.
This is the refraction or bending that we observe. No wave is actually bent. The incoming wave keeps right on
going. The secondary waves from
the atoms of the medium move straight out in all directions from their source
atoms. You will recall that one of our basic facts about electromagnetic
interaction is that different fields superimpose but do not interfere with each
other. What we call the
interference of waves really should be called the superposition of waves.
What about diffraction; do waves bend
then? Of course not. The wave that comes up to the screen
goes right through the screen. As
it does, it sets the atoms in the screen oscillating so that each is a source
of secondary waves. There are no
secondary waves along where a slit is.
The resultant of all these waves superimposed is exactly the same as if
there were no incoming wave or secondary sources in the screen but instead a
series of secondary sources where the slits are. It is an illusion but a
completely convincing one.
The simple mathematical argument for this
is not very difficult to understand. Imagine an opaque screen with two slits
cut out, but instead of throwing away the parts that you cut out you leave them
in the openings. So the sreen really has no holes in it. I will call the hole
fillers "plugs". On the shadow side of the screen the total electric
field E is zero; there is no
light. Remember: it is an opaque screen. But this field is the sum of the
fields of the light source E
(source) and the field due to the atoms in the screen E (screen). This latter field is the sum of two parts,
the field of the screen with slits E (slit-screen) and the field of the plugs E (plugs). So we can write the equation
E (source) + E (slit-screen) + E
(plugs) = 0
The electric field on the shadow side with the source
and plugged screen in place is zero. The fields of the source and screen are
superimposed and cancel exactly.
Now what is the field when we remove the plugs from the holes? It is
E
(source) + E (slit-screen)
But we know by looking at the other equation that
this must be
ÐE (plugs)
The field on the shadow side is the same size as if
there were just oscillating charges in the plugs. The intensity of light
depends on the square of the field so the minus sign does not matter.
So Huygens' principle, which says that
the wavefront across the slit opening acts like a series of secondary sources,
is right because it is exactly the opposite of what is there,that is, secondary
sources all along the screen everywhere but at the opening.
It makes quite a difference to your view
of what is happening when you realize that a principle works because it is
completely wrong, as far as light is concerned. Light waves do not ever really change speed or
direction. If they appear to, it
is always an illusion produced by a cooperative effect of many atoms. I found
this interpretation quite shattering when I first read it. It is like hearing
that black is white. I am sure that it opened my eyes to the possibility that
other ideas I accepted might be exactly opposite to what I had been told.
If we go back now to the photon picture,
where does that leave us? L.I. Schiff in his book on Quantum Mechanics addresses the problem:
From the point of view of the particle picture [of
light], we may then ask how it is that a stream of independent photons, each of
which presumably can go through only one of the slits, can produce a
diffraction pattern that appears only when both slits are open... In this
question is implicit the assumption that the photon actually does go through a
particular one of the two slits. [9]
The waves that produce the effect go right through
the whole screen including the slits and secondary waves are produced by the
atoms in the screen. It is
difficult to see how a single incoming photon can produce the effect. It really must spread out over the
whole screen and interact with all the atoms. The two-slit mystery gets much worse. But after all this trickery and
illusion, perhaps the existence of photons is an illusion. Perhaps it is useful because it is
exactly wrong. There are
scientists who believe in a semiclassical quantum theory. They believe that light is a wave only and that particles of matter give rise to the
peculiarity we label uncertainty.
These semiclassical theory (SCT) people say that the reason photons can
be successfully used to describe the emission and absorption of energy is
because the matter does not behave classically. This is basically my position
(I would say because microscopic energy conservation is impossible).
Semiclassical theory has actually been been successful in explaining the
photoelectric effect although I cannot give the argument here. It is rather a
shock again to learn that the photoelectric effect which is supposed to have proved that photons exist can be explained without any
reference to them whatsoever. In SCT it is assumed that quantum mechanics
describes the behavior of atoms and that light is adequately described as a
wave. Most of the arguments about
the need to have radiation quantized involve the assumption that, in
microscopic processes, energy is conserved. This is, from my point of view, a weakness since I believe
that microscopic energy conservation cannot be a fact because of the
impossibility of isolating the system from the random influences of the rest of
the universe.
It is very easy to confuse the statements
"that Einstein's theory of the photoelectric effect proved the existence
of photons" and "that scientists became convinced about the reality
of photons by Einstein's arguments concerning the photoelectric
effect". Semiclassical theory
explains the photoelectric effect without any reference to the idea of photons
or to the idea of microscopic energy conservation. Does this mean that we can
dispense with photons? Not quite so fast! What about Bohr's relation for the
frequency spectrum of hydrogen? His relation is
f = (E2ÐE1/h
If it is rewritten as
h*f = E2ÐE1
it becomes a statement of microscopic energy
conservation. The energy
of the photon emitted by the atom h*f is equal to the difference between the initial
energy state of the atom E2 and the final energy state of the atom E1. Again
we find the two ideas coupled: photons and microscopic energy
conservation. But Bohr's relation
too can be derived without any reference to either photons or energy
conservation, using semiclassical theory. What you use is Schrodinger's
equation for the atom and Maxwell's equations for the electromagnetic waves.
My idea that microscopic energy
conservation is not true can actually be argued from Schrodinger's
equation. Rather than speaking
about the hydrogen atom I will refer to the linear harmonic oscillator. In a classical oscillator, like a
pendulum bob swinging back and forth, the energy of the oscillator depends on
the amplitude of the swing. If the
swing is bigger, the energy is bigger.
A classical oscillator can be stationary and have zero energy. A quantum oscillator cannot be
stationary. A quantum oscillator
of frequency $f$ has, as its lowest energy, the energy
h * f / 2
This is called the zero-point energy of the
oscillator because if a group of quantum oscillators could be cooled to a
temperature of absolute zero, where all motion is supposed to cease, they would
still each have this energy. In
the Brownian motion picture of quantum mechanics the oscillators would still be
in motion; Brownian motion is a perpetual motion. (Of course it is not a
macroscopic perpetual motion.)
If you examined the probability distribution
function for the quantum oscillator in its ground state you would find that the
probability of the particle in the oscillator being at the equilibrium
position, at the center of its oscillation, would be the highest and then it
would taper off on both sides of the equilibrium position. If a classical oscillator had an energy
of h*f/2 there would be a limit
to how far away from equilibrium the particle would get. The probability of the quantum particle
being outside this classical limit,
as determined by solving Schrodinger's equation for the harmonic
oscillator, is not zero, even though it tapers off rapidly outside the
classical limit. But how can energy of position be larger than the total
energy, which it must be if the particle is outside the classical limits of
oscillation? Certainly the energy of motion cannot be negative. The energy of
the oscillator must in fact be greater than the average energy h*f/2 whenever it is outside the classical limit. Many say
that there is an uncertainty relation between energy and time that is similar
to the one between position and velocity.
They argue that the particle can have an energy larger than permitted by
energy conservation, but only for a limited time. This sounds to me suspiciously like saying that, on the
average, energy is conserved, but energy is not conserved microscopically. The
microscopic system, the quantum oscillator, has an everage energy of h*f/2 but from time to time has more or less energy.
Energy is not constant in a microsystem.
If we accept as a basic premise that, in
reality, microscopic energy is not conserved then the picture, begun by Bohr
and taken directly into quantum mechanics, that his frequency relation
describes an electron jump is a lot of nonsense. The Bohr condition is a clear
statement of energy conservation in a microsystem. You can't have it both ways.
So I reject Bohr's interpretation. I reject quantum jumps and I reject photons.
Thi
means I reject the particle picture of electromagnetic
radiation. Let me summarize all this.
Although I reject a particle picture of radiation, I affirm a particle
picture of matter. If an electron
is really a particle, the probability amplitude for it in the stable excited
states in an atom or an oscillator cannot be right. Recall that I said that each of these has nodes, places
where the probability becomes zero.
The probability amplitude is non-zero on either side of each node. So the particle, if it is a particle
and cannot disappear and reappear like a Cheshire cat, cannot be in such a
state. It can be in a ground state
because that has no nodes. But we know that the frequencies in the hydrogen
spectrum are related to the average energies associated with the stable excited
states as described by the Bohr frequency condition. How can this be reconciled
with a view that the stable excited states are exactly those states where the
electron cannot be?
Mathematically the wave function for any
dynamic state of the quantum system can be described as a combination of the
wave functions for the stable excited states. But I would say that no pure
stable excited state is possible.
The electromagnetic radiation produced by an atom then will be related
to the energies associated with the stable excited states exactly according to Bohr's relation
but the system cannot ever be in any of these states. It is a situation just
like Huygens' principle: Bohr's relation is right because it is exactly
wrong. Bohr said the particle can
only be in one or other of the set of stable excited states that are proper
solutions of Schrodinger's equation.
I say these stable excited states are all improper in that they contain
nodes which means that, unless the particle is an escape artist, it cannot
exist in such a state.
I tried to simulate the Brownian-motion
atom on the computer, you may remember, and found no excited stable
states. In fact my atom kept
expanding. How do you prevent
this? In terms of energy, you need
to radiate some energy all the time.
In equilibrium you should radiate just as much as is coming in from the
rest of the universe to keep the Brownian motion going. And that must be what happens if there
is such Brownian motion of the electron in the atom. It must be radiating all
the time because it is being accelerated all the time. In the ground state it
radiates a continuous spectrum. In
any disturbed condition which you get if you give it extra energy, it radiates
discrete frequencies, as given by the Bohr relation, as well as a continuous
spectrum of frequencies due to the continuing Brownian motion.
The radiation from atoms even in the
ground state could be exactly what is producing the Brownian motion in other
atoms. And that brings me to the
other cornerstone of the dual picture of radiation Ð Planck's blackbody
radiation equation. Many believe that Planck's blackbody radiation proves the need for quantized energy states in atoms. It
has been shown by Boyer and Theimer that Planck's equation could be derived
instead by assuming a zero-point energy of h*f/2. It is clear from what I say where this would come
from. Theimer says:
Some fascinating new ideas concerning the physical
meaning of the quantum theory have been developed in a series of papers by
Boyer and a related paper by Nelson. In Boyer's work the main new concept is
the existence, at the absolute zero of temperature, of a classical fluctuating,
electromagnetic background radiation which is, in some unknown fashion,
equivalent to the ground state of the radiation field in quantum
electrodynamics. Boyer demonstrates that incorporating this radiation
background into classical statistical physics makes possible a classical
derivation of Planck's blackbody spectrum. He also suggests that the universal
background radiation might be the source of the random perturbations,
postulated by Nelson, which transform continuous classical particle motion into
an equivalent random-walk [Brownian] process.
He continues:
What is the origin of the zero-point radiation? ...
The zero-point radiation is a self-consistent radiation field in dynamical
equilibrium with all the electrically charged particles in the universe. These
particles perform a complicated Brownian motion, in the spirit of Nelson's
work, which is caused by random absorption and emission of the self-consistent
zero-point radiation. And this radiation has such an energy density that there
is no net time-averaged energy exchange between matter and radiation at the
absolute zero of temperature. [10]
So the continuous radiation spectrum from atoms that
I believe exists even in the ground state is not just hiding in the blackbody
radiation; it is necessary in order to give it the distribution that it has.
This makes Planck's notion of quantized energy transitions for oscillators as
an explanation of blackbody radiation unnecessary.
In this chapter I have been looking at
the great difficulty that I have encountered over the years with the dual,
wave-particle picture of electromagnetic radiation. In my undergraduate
education a wave picture was always used to explain the reflection, refraction,
and diffraction of light. Young's double-slit experiment on the diffraction of
light by a screen with two slits was offered as conclusive proof that light was
a wave. You have seen from my quotations what a tangle various physicists get
into when they try to understand the two-slit experiment in terms of the
particle, or photon, picture of light. Most of these explanations refer to the
photon passing from source to screen by way of one or other of the two slits.
But the present wave view of the two-slit
experiment (which many physicists are unaware of) is that the source waves pass
right through both the screen and the slits. These source waves have
superimposed on them waves from the atoms in the screen that are "forced"
to vibrate in sympathy with the source waves passing through. The net result of
all these waves, as they superimpose (or "interfere") is a pattern of
light and dark fringes. As it happens, there are just as "many" waves
of light passing through a dark fringe as a light fringe, only at a dark fringe
they are out of phase and cancel, at a bright fringe they are in phase and add.
It is important to notice that light does
not appear to be diffracted unless there are a lot of sources present Ð the
primary source and the secondary sources in the screen. Light always travels in
straight lines from its source to the point of observation. It is an illusion
that light goes from the source to a slit and then changes direction as it
proceeds to land on a bright spot on the screen.
To me it is a contradiction to say that a
single photon interacts with many atoms in the slit screen; especially when you
say that a single photon, at least, is necessary to trigger a detector in the
second screen. (This latter is held to be true whether the detector is an atom
of a photographic plate or a photoelectric detector.)
You probably have heard it said that
light travels in straight lines unless it passes through a narrow opening in
which case it is bent or changes direction. This bending is an illusion
provided by the presence of matter in the electromagnetic field of the primary
source of the waves.
Exactly the same situation holds for
refraction. You probably have heard that light travels at speed $c$ in a vacuum
but, in a medium, it travels at a lower speed. This "explains" how it
changes direction (is bent or refracted) as it enters a glass prism. But this
is also an illusion. The superposition of the primary source waves and the
waves from the atoms in the glass give the illusion of a wave slowing down as
it enters the glass and changing direction. When a prism produces a spectrum of
light, it is foolish to imagine a red photon being emitted from the source,
travelling at speed $c$ until it reaches the surface of the prism then slowing
down and changing direction in the glass and finally emerging, speeding up,
changing direction again and landing on a spot where the red of the spectrum
is. The photon picture is absolutely inconsistent with the wave picture.
When I was young I often asked my Mother
how a magician had done a certain trick. My Mother's stock answer was
"It's all done by mirrors". When you have an object, say a candle, in
front of a mirror there is an illusion created that there is a second candle,
the image, behind the mirror. If the object candle is hidden from your view,
you might believe that the image candle was a real object. No doubt many magic
illusions do involve the use of mirrors. But the explanation you usually get
about the reflection of light by a mirror is really an illusion. Light from the
candle we say goes up to the mirror and is bounced off (reflected) to your eye.
It then seems (if we assume light always travels in straight lines) to be
coming from behind the mirror, from the image. But the bouncing is an illusion.
Light from the object candle passes right through the mirror. In the mirror,
secondary sources, excited by the incoming light, radiate what comes out in
front of the mirror, superimposing to seem like waves from the image. The light
from the secondary sources that goes behind the mirror cancels the waves from
the primary source and there is the illusion of darkness. But remember: there
are many light waves behind the mirror.
In this book I have been looking for
explanations of physical laws. I can not be satisfied with asnwers that tell me
"It's all done by mirrors". Wave-particle duality of radiation is
such an answer.
SUMMARY
1. Semiclassical
theory treats electromagnetic radiation as a wave, and it treats particles of
matter as described by the uncertainty principle (meaning to most that
particles have a wave nature). The photoelectric effect and Bohr's frequency
relation can be explained by semiclassical theory. I accept a wave theory of
radiation and a particle theory of matter. For me the wave nature of a particle
is due to the random fluctuations in its motion.
2. The energy distribution in blackbody radiation can be explained on the assumption that atoms radiate even in the ground state, rather than that there are quantized excited states. Quantization of energy in radiation is, I believe, not necessary to the explanation of any phenomenon.
Copyright © 1983 J.N.P. Hume All rights in this book reserved