The purpose of this article is to foster the development of intuitive
concepts in atomic physics so that knowledge in this field will become more accessible and
understandable to a larger group of people than is currently the case. The term,
"atomic physics" will include quantum mechanics, but will not be limited to just
this one particular field or its particular method of exposition. Conceptual problems in
physics manifest themselves mostly at the extremes of the very small, very large, very
slow, or very fast. They are just as apparent in astronomy and astrophysics, for example,
as they are in quantum mechanics.
Physicist P. C. W. Davies notes that "Quantum
mechanics is one of those subjects that usually comes right in the end, even though it can
seem horribly obscure when only half-learned." Indeed, your first encounter with a
book on quantum mechanics will probably leave you with the impression that it is an
arcane, abstract, almost impenetrable topic. This particular science is filled with
mathematical "maps of hell" written in strange notation seemingly
incomprehensible to all but sorcerers or geniuses. The concepts you will encounter are
also disorienting, namely things like matter waves, probability amplitudes,
non-locality, negative energy, tunneling, various paradoxes, clumsy, inelegant fudge
factors, inexplicable constants, "principles" of indeterminacy and uncertainty,
intrinsic spin, ridiculous models of the atom, quantum jumping, and so forth. And, as
though all these problems were not enough, they raise "fundamental philosophical
questions about the nature of reality."
Instead of quantum mechanics becoming clear "in the
end" as Davies notes, it would be very desirable to make it clear "from the
beginning." I hope that this goal can be achieved by giving a synopsis of the
perplexing factual issues, some suggestions about crucially important intuitive concepts,
and in finding an introductory "middle ground" for the mathematics. I hope
this will make early encounters with this topic a bit less abrasive for everyone
interested in this topic.
I am not a physicist. Nor do I have the time or resources or background
to pursue a full exposition of this important topic. I can only offer some suggestions
about how conceptual problems in quantum mechanics could be resolved. It is my hope that
others will build on these insights, and share their findings just as I am attempting to
do. I regret that I do not have the time to write an actual article on this topic; the
presentation here uses the "under construction format" and is therefore a bit
disjointed. Nevertheless, I hope it will be of some value to the diligent and patient
reader. And I hope readers will share their insights with me. My prime reason for
publishing this "stuff" is so I can learn more myself.
Near the beginning of the twentieth century, scientists were confronted with many
factual mysteries pertaining to microphysical phenomena. The classical, or Newtonian,
mechanics was completely inadequate to deal with these mysteries, and so a new type of
mechanics was invented: quantum mechanics. If you love good mysteries, you will find these
to be among the best that the physical universe has to offer. I rate them as best in
cleverness and best in ultimate importance. Despite their superficial simplicity, they
have not been solved, not even by the most brilliant minds using the best equipment
available to science. Quantum mechanics gives us "recipes" to get useful
numerical answers, but despite 75 years of research, there is STILL no generally accepted
explanation for the mysteries described below. The human mind will not rest until they are
solved. Can you solve them?
What is Quantum Mechanics?
Quantum mechanics is "The
modern theory of matter, of electromagnetic radiation, and of the interaction between
matter and radiation; it differs from classical physics, which it generalizes and
supersedes, mainly in the realm of atomic and subatomic phenomena." (McGraw-Hill
Dictionary of Scientific and Technical Terms, 5th ed.)
__________
""Quantum mechanics" is the
description of the behavior of matter and light in all its details and, in particular, of
the happenings on an atomic scale. Things on a very small scale behave like nothing that
you have any direct experience about. They do not behave like waves, they do not behave
like particles, they do not behave like clouds, or billiard balls, or weights on springs,
or like anything that you have ever seen. . . . Because atomic behavior is so unlike
ordinary experience, it is very difficult to get used to, and it appears peculiar and
mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not
understand it the way they would like to. . . . We know how large objects will act, but
things on a small scale just do not act that way. So we have to learn about them in a sort
of abstract or imaginative fashion and not by connection with our direct experience."
(The Feynman Lectures on Physics, R. P. Feynman, R. B. Leighton, M. Sands, 1965
(Addison-Wesley ), Vol 3, p. 1-1 under "Atomic Mechanics")
__________
"In the short period of 1925 to 1928,
Heisenberg, Schrodinger, Born, Dirac and many others laid the foundations of what is one
of the greatest theories of all time, the theory of quantum mechanics. In generality and
in range of application, it is unsurpassed. It has been so successful that one cannot
discuss atomic and nuclear matters without some understanding of this basic theory.
"Because the predictions of quantum mechanics agree with so many different types of
accurate, careful, repeated experiments,—the last court of appeal for all theories—this theory is almost certain
to become a permanent part of man's equipment for understanding and analyzing a large and
very important part of nature. However its conceptual foundations or philosophy may change
in the future, it has already, in a thousand ways proved its utility and power." (Introduction
to Quantum Mechanics, Chalmers W. Sherwin, 1959, (Holt, Rinehart and Winston ),
pages 6,7)
__________
"Why is it so hard to learn? Students find
quantum mechanics tough going for two reasons, one conceptual, the other technical.
Familiar concepts like speed, size, acceleration, momentum and energy take on weird
features, or even become meaningless. Intuition gained from daily experience is of no
help, or can even be misleading. The student must learn to think about mechanical concepts
in a completely different way. Some of the conceptual issues are still a matter of dispute
among physicists and have raised fundamental philosophical questions about the nature of
reality and the role of the observer in the physical universe. . . . On the technical
side, the mathematical description of quantum processes is rather abstract, and not very
obviously related to the subject of its description. Physical quantities are represented
by mathematical objects with unusual properties. Some of the mathematics is also often new
to the student and learning it can be an additional burden" (Quantum Mechanics,
P. C. W. Davies, 1984, (Chapman and Hall), pages ix, x)
__________
"Quantum mechanics provides a good example
of the new ideas. It requires the states of a dynamical system and the dynamical variables
to be interconnected in quite strange ways that are unintelligible from the classical
standpoint. . . . The justification for the whole scheme depends, apart from internal
consistency, on the agreement of the final result with experiment. (The Principles of
Quantum Mechanics, P.A.M. Dirac, 4th ed., (1958), p. 15)
__________
Newtonian mechanics, also known as classical
mechanics, is the mechanics used to describe the behavior of familiar, ordinary
objects like cars, levers, gears, the forces on ships in the sea, the stresses in bridges
and buildings, the trajectories of cannon balls, etc. Quantum mechanics describes the
behavior of extremely tiny discrete objects like atoms and photons (light). Quantum
electrodynamics is the modern extension of quantum mechanics to include the realms of
electricity, magnetism, and light. In quantum mechanics, the discrete or
"quantized" nature of matter and energy is prominent. In continuum mechanics
(classical field theory) this discrete nature is unimportant.
Quantization is actually a fairly ordinary concept. The real heart of quantum mechanics
is revealed by a phenomenon called "interference". It is, in physicist Feynman's
words "the only mystery". It is "impossible, absolutely
impossible, to explain in any classical way." It also points to one of the most
fascinating things you will learn in the study of quantum mechanics: the Universe
seems strangely "overbuilt." It uses fantastic, seemingly unimaginable
machinery, to create ordinary appearances.
The Basis
for Intuitive Concepts
"Intuitive", as used here, is intended to mean "credibly constructed,
logically and conceptually clear". If you have a well-educated intuition, it might
even mean "self-evident", but most readers won't find the ideas presented here
to be that obvious. I would also like to think that mathematics qualifies as natural and
intuitive. Unfortunately, most people regard math as arcane and abstract. So for
now, I have kept the math to a minimum in this presentation.
Two realizations are necessary to see a path to intuitive concepts in
quantum mechanics:
1. All physical phenomena and physical entities in the physical universe
can be described in terms of pure space/time or time/space ratios. The ratios may have
active (i.e., progressing) components in time, space, or both. They can be active in one,
two, or three dimensions. The ratios may be inherently linear or rotational. Ratios may
enter into relationships with other ratios.
2. The commonly used space-time reference system is incapable of fully
depicting the true, ultimate character of these ratios.
My view here is somewhat like that of the
Copenhagen Interpretation: "Quantum theory is not a representation, much less a
description, of quantum reality, but a representation of the relationship between
our familiar reality and the quon’s utterly inhuman realm." (Quantum Reality,
Nick Herbert, 1985, p 144) In other words, quantum mechanics is the bridge
between our accessible world and a strange inaccessible world. It is simply a pragmatic
way of coping with the limitations of the reference system.
The first step in developing intuitive
concepts in quantum mechanics or physics in general, is to realize how the commonly used
reference system misleads our intuition:
Space and time must always be treated as a ratio. But our
reference system treats space and time as though they are independent and unrelated.
Space is viewed like a container or background,
"in" which events occur. Physical phenomena, however, especially that in quantum
physics, may involve only 1 spatial unit. There is no "inside" to this space.
Nor are there positions or trajectories, at least not in the way we commonly think about
them. All activity at one spatial unit must be temporal.
Time is similarly viewed as a setting by which the
occurrence of events can be ordered. But time is actually three-dimensional, and our
contact with it through the reference system is one-dimensional. Strict notions of causality and determinism ("A comes
before B and causes C") break down. The temporal relationships of the phenomena are
reduced to mere probability amplitudes in the context of the reference system.
For large scale phenomena, a position in
coordinate time maps into the spatial system in a random manner. Phenomena that are
extremely isotropic and homogeneous like cosmic rays, and cosmic background radiation,
represent temporal phenomena (or temporal structures) as seen from a spatial reference
system.
Because a particle has two types of position (one in
coordinate space and another in coordinate time), there are also two types of
"mechanics" to describe its motion. Spatial motion (s/t) is described by
Newtonian mechanics in terms of paths, trajectories, and forces. A particle's temporal
motion component, however, has no path or trajectory as seen from a spatial reference
system. A path in three-dimensional time simply cannot be described directly with spatial
terminology. Instead, its description requires "non-path" mathematical tools
like the expression for total energy and potentials as found in the Hamiltonian. Changes in location, rather than
the locations themselves, can also be described, and this leads to differential equations
like Schrödinger's wave
equation. Temporal locations are "non-local" in the spatial system (the temporal
origin is anywhere/everywhere because it is a "when" description, not a
"where" description. It just "doesn't care" about spatial locations.)
This requires mathematical tools that have "infinite reach" like Schrödinger's wave
equation, Heisenberg's infinite matrices, and Feynman's method of "sum over all
possible paths". These characteristics also lead naturally into the concepts of
superposition of multiple states, probability amplitudes, and of "reality" being
intertwined with the measurement system, rather than existing in an independent way. The
type of mechanics that addresses these temporal aspects is called "Quantum
Mechanics".
The two kinds of locality offer a conceptual framework for
single photon interference in a two-slit appartus. The spatial version of the photon goes
through one or the other slit. But the temporal version is delocalized (is
anywhere/everywhere from the spatial standpoint) and "encounters" both
slits. The effects combine in a way that results in what is called "single photon
interference". (see discussion below)
Intrinsic temporal motion (t/s)
may be attached to a spatial object. The most obvious example is gravitation. Again, such
a motion has no path in space. It simply does not know or care about spatial direction.
Hence, the motion is non-directional in a spatial reference system. It can be described as
a spherically distributed motion, or as an inverse square force that causes such motion.
Such motions are described with "field equations" like Maxwell's instead of
force equations like Newton's.
Phenomena that seem to be connected, but not in or
through space ("non-locality" ), have their connections in coordinate time which
simply cannot be seen in a spatial reference system. The EPR paradox is a good example of
this. Two correlated photons can remain in the same temporal location even though their
spatial locations become widely separated. They are "together" in one sense and
"apart" in the other.
The maximum speed that the spatial reference
system can portray as a change of position in space, is, c, the speed of
light. The space/time (or time/space) phenomena itself is not affected by reference system
limitations and can display enormous energies, enormous redshifts, and other
characteristics that are incomprehensible in the context of such a reference system.
Speeds in space are always less than c, and speeds in time are always greater
than c.
Physicists themselves hope for a theory
that is simpler than the mess we have today, and even believe such a thing is possible:
"We have come to the conclusion that
what are usually called the advanced parts of quantum mechanics are, in fact, quite
simple. The mathematics that is involved is particularly simple, involving simple
algebraic operations and no differential equations or at most only very simple ones. The
only problem is that we must jump the gap of no longer being able to describe the behavior
in detail of particles in space." The Feynman Lectures on Physics,
Feynman, Leighton, Sands, (1965) Vol. 3, p. 3-1
"All physical theories . . . ought to
lend themselves to so simple a description that even a child could understand them."
-Albert Einstein
"However, if we do discover a complete theory, it should in time be
understandable in broad principle by everyone, not just a few scientists. Then we shall
all, philosophers, scientists, and just ordinary people, be able to take part in the
discussion of the question of why it is that we and the universe exist." —A
Brief History of Time, Steven Hawking, 10th ed. (1998) p. 191
"Richard Feynman was able, in his PhD thesis, to reformulate quantum
mechanics into a single, complete system of mechanics that includes all of classical
mechanics as well. Feynman's mechanics, based on the Lagrangian, is all you need to
explain all of mechanics, from the motions of the stars to the motions of electrons. For
obvious reasons, this is often known as the path integral formalism of quantum theory. . .
. It is actually much easier, in terms of the mathematics, to work with the Lagrangian
than with the alternative Hamiltonian approach (which, through a historical accident, is
the way most people are introduced to mechanics); John Wheeler, who was Feynman's PhD
supervisor, says that his thesis, presented in 1942, marked the moment 'when quantum
theory became simpler than classical theory'. If only teachers of physics in schools had
the sense to teach mechanics from the beginning using the Lagrangian formalism, students
could learn both classical and quantum mechanics at once, using equations that are easier
to manipulate. One of the main reasons why quantum mechanics often seems difficult when
students do encounter it is that they have to unlearn all the old stuff first." (Q
is for Quantum: An Encyclopedia of particle physics, John R. Gribbin, 1998,
p.202-203) See also http://www.eftaylor.com/pub/CallToAction.pdf
, http://www.eftaylor.com/software/ActionApplets/LeastAction.html
, http://www.eftaylor.com/pub/ForceEnergyPredictMotion.pdf
, http://www.eftaylor.com/leastaction.html
, http://www.courses.fas.harvard.edu/~phys16/Textbook/ch5.pdf
In spite of this wish, things have only
gotten worse. The public can now watch the development of String Theory, with its multiple
universes and eleven dimensions, on public television. This is certainly another theory
that is headed off into the weeds. At the other extreme we have the "Saturday evening
theoreticians" who come up with their own theories that have obvious flaws. Ask any
university professor and he will tell you that there are thousands of them (and that he
gets emails from all of them). What I am hoping for is a fresh start, one that begins with
a sound premise, uses sound methodology, and produces clear "ideas" that will
lead into the development of a complete theory.
Now let's consider some of the factual and
conceptual problems physicists have had to wrestle with.
Thomas Young's classical double slit experiment
of 1801 provided unambiguous and convincing evidence that light has a wave nature. He used
an apparatus like the one represented in the diagram below:
If light consisted of particles traveling in
straight lines, one would notexpect the maximum light
intensity to appear directly behind the "shadow" of the central blockage between
the two slits. And if light were particles, there would be no periodic waviness of the
intensity of the pattern on the screen. If light were particles, another slit should
mean more light. And where more light is expected, there should be more light,
not darkness. However, all these problems are resolved if light is behaving as a wave.
This was one of those historical experiments that
every student of physics is practically required to perform. In college physics we would
smoke up a microscope slide and create two slits in the smoke film by placing two razor
blades together and lightly scoring a double line in the smoke film. Then we would hold
the slide up to our eye and look through it at a bare-filament bulb (or even a distant
street lamp) . The (multicolored) diffraction/interference pattern could easily be seen. A
more modern version of this experiment uses an ordinary classroom pointing laser. The
laser is shined on the double slit and the light projects onto a white card or paper
behind it. The diffraction/interference pattern can be clearly seen on the paper (do not
look at the laser beam directly).
The math was simple too. We could calculate the
interference pattern intensity with little more than simple trigonometric relations.
So what is the big mystery? It is simply this:
The light source can be dimmed down to the point where there is, on the average, only one
photon traversing the apparatus at a time. Yet the diffraction/interference pattern still
appears. In fact, in 1909, G. I. Taylor performed a similar experiment by photographing a
diffraction pattern of a needle (instead of a double slit). He used an extremely feeble
light source. A 2000 hour time exposure allowed the diffraction pattern to manifest itself
on a photographic plate. The pattern was every bit as distinct as that obtained from a
short exposure with a bright light source.
Similar experiments have been repeated many times. Short exposures with feeble sources
result in photographs that have a very grainy, seemingly random pattern of exposed spots.
Somewhat longer exposures are also very grainy, but also reveal that a pattern is
beginning to emerge. Much longer exposures with the same dim source finally show a full,
distinct diffraction pattern. Light is acting like a particle (definite position and
energy) when it hits the photographic plate, yet acts like a wave (spread out in space)
when passing through both slits and creating the diffraction pattern. It acts like a
particle when sent one-at-a-time through the apparatus, but the pattern on the
photographic plate shows a pattern characteristic of a wave.
To further confuse matters, similar
experiments were performed with electrons and neutrons. We commonly regard these as particles
(possessing a definite position, trajectory, momentum, etc.). Yet they too produced an
interference pattern. Such patterns are characteristic of waves, not particles.
How can waves act like particles, and particles act like waves? And like the
experiments with light, the interference pattern will appear even if only one electron or
neutron at a time is traversing the apparatus.
Clever experimenters have tried to determine
which slit the photon or electron goes through. Suppose electrons are sent into a similar
apparatus and we have a little light secretly waiting behind one of the two slits (like a
traffic cop with a radar gun). As the electron flies by, a little flash of light will be
reflected, and we will know which of the two slits the electron actually went through.
When we actually try to do this (not exactly in this way), nature seems to get very
devious. The interference pattern simply disappears and goes back to the single slit
pattern (it knows the traffic cop is watching!). All sorts of clever schemes have been
tried, and all end up with the same result. If the detector can somehow distinguish
between the particle paths, even in principle, then there is no interference pattern!
These results are so
counterintuitive no one would believe this actually happens unless the experimental
evidence were as overwhelming as it is.
This diffraction/interference from one-at-a-time photons is commonplace and occurs in
all sorts of optical systems, from complex to the simplest, whether we notice it or not.
Consider the following illustrations:
The modern explanation of this paradox—the wave/particle duality—is that both light and particles
have BOTH a wave nature and a particle nature. The wave amplitude, and
specifically the square of the wave amplitude, represents the probability density
that a photon (or electron) will appear in some position on the photographic plate. It is
as though the photons are being directed by an abstract mathematical wave as they fly
through the apparatus. The wave is like an invisible traffic cop who splits oncoming
traffic, car by car, into a bunch of different directions, according to a definite
pattern. The cars do not interact among themselves and no particular car knows where
the other cars are going.
Physicist Dirac has a discussion of the
interferometer problem in his book The Principles of Quantum Mechanics:
"Suppose we have a beam of light which
is passed through some kind of interferometer, that it gets split up into two components
and the two components are subsequently made to interfere. We may . . . take an incident
beam consisting of only a single photon and inquire what will happen to it as it goes
through the apparatus. This will present to us the difficulty of the conflict between the
wave and corpuscular theories of light in an acute form."
He then describes the difficulties, presents
some aspects of the theory of superposition of wave functions and emphasizes how the
photon situation differs from that in classical mechanics. Then he notes the necessity of
applying the probability principle to one photon at a time:
"Sometime before the discovery of quantum mechanics people realized
that the connexion between light waves and photons must be of a statistical character.
What they did not clearly realize, however, was that the wave function gives information
about the probability of one photon being in a particular place and not the
probable number of photons in that place. The importance of that distinction can be made
clear in the following way. Suppose we have a beam of light consisting of a large number
of photons split up into two components of equal intensity [think of the interferometer
illustration here]. On the assumption that the intensity of a beam is connected with the
probable number of photons in it, we should have half the total number of photons going
into each component. If the two components are now made to interfere, we should require a
photon in one component to be able to interfere with one in the other. Sometimes these two
photons would have to annihilate one another and other times they would have to produce
four photons. This would contradict the conservation of energy. The new theory, which
connects the wave function with probabilities for one photon, gets over the difficulty by
making each photon go partly into each of the two components. Each photon then
interferes only with itself. Interference between two different photons never
occurs." (The Principles of Quantum Mechanics, P.A.M.Dirac, 4th ed., 1958,
pages 7-9)
In other words, this type of
interference is not an en masse phenomena, like the interference of water waves
that can be observed in a ripple tank. The photon in the interferometer ends up in one
optical leg or the other, and is somehow directed to the photographic plate in just
such a special way that will contribute to the build-up of a precise interference pattern,
even though it may be the only photon in the apparatusduring that instant of
time.
Understanding Dirac's concern about
the conservation of energy is also crucial. Two ordinary waves, like water waves, can
"cancel" each other if they are 180 degrees out of phase. In a ripple tank you
can see two sets of waves approaching each other and you can note the spot where they
would have the 180 degree phase difference. At that precise spot, the valley of one wave
is filled in by the peak of the other wave, and the water surface ends up at normal
height. The phenomena is well known and is called "destructive interference." It
finds practical application in the electronic devices that emit "antinoise" to
make work areas quieter and in antireflection coatings on camera lenses.
Likewise, two photon waves can
"cancel" if they are 180 degrees out of phase. But people who think about
this, soon find a problem with the concept. Each photon has a discrete amount of energy.
If two photons were to interfere destructively, what happens to their energy? It
cannot just disappear. Mathematically they cancel, but what happens physically? The
Conservation of Energy principle is inviolate. The energy still has to go some place. Like
two bullets, the two photons cannot just vanish, not even for an instant. So in quantum
mechanics the wave is interpreted to denote the probability that a single photon
will appear in a certain place. In quantum technobabble it is called "the expectation
value of an observable." This "gets over the difficulty", to use Dirac's
own words, of the Conservation of Energy problems that would otherwise be created by both
constructive and destructive interference.
This interpretation has a lot of
factual support and has been used very successfully to predict the outcome of all sorts of
extremely varied phenomena. But while it is a good description of what nature
does, it leaves us utterly mystified about the how. How would a single
photon, or electron, know where to hit the photographic plate or detector? Surely it does
not compute its own wave function. Surely it does not say "Let's see, I am
going towards a photographic plate in an interferometer, and 60% of us are supposed to end
up here, 35% there, and the other 5% next to that." Surely it does not know how many
photons have already arrived in any particular spot. If you were the Designer and Maker,
how could you build something that automatically, and by its very nature, acted this way
so easily and reliably? (Job 38:19-24)
__________
The mystery gets even deeper when one discovers that the quantum type of
interference is also possible temporally, not just spatially. Note what physicist
Mark P. Silverman has to say on this topic:
"The
potential for quantum interference exists whenever a particle can propagate from its
source to the detector by alternative spatial pathways under experimental conditions such
that the exact pathway taken cannot be known. The archetypal example is the Young's
two-slit experiment in which the particle, when probed, passes through one slit or the
other. Unprobed, the resulting particle distribution is explicable only in terms of
probability amplitudes that seemingly propagate through both slits. There is a direct
temporal analogue to the two-slit experiment in which the linearly superposed amplitudes
represent—not alternative spatial pathways—but rather the evolution of
alternative indistinguishable events in time. . . .
The phenomenon of quantum beats . .
. is intrinsic to each atom and not a cooperative interaction between atoms. In other
words, the spontaneous emission from single atoms is not modulated, but registers at the
detector as one quantum of light at a time; the pattern of beats (measured at one location
in real time or, equivalently, at different spatial locations along an accelerated atomic
beam) can nevertheless be built up by the decay of many such single atoms. This is again
the old "mystery" of quantum interference translated to the time domain: How can
independently excited, randomly decaying, noninteracting atoms produce a pattern of photon
arrivals that oscillates in time? Note that the synchronization required for the beats to
survive ensemble averaging does not imply that emitting atoms communicate with or
influence one another. Rather, an apt analogy, if there be any, would be that of a large
number of independent clocks all separately wound and set to the same time by the
clockmaker." (See More Than One Mystery : Explorations in Quantum
Interference, Mark P. Silverman, 1995, p. 100 to 102, ISBN 0-387-94376-5)
In Young's
experiment the interference pattern is most easily seen on a screen placed behind the two
slits. The pattern of fringes is stable and varies with spatial position, but not with
time. Of course, it is also possible to map out the interference pattern by
spatially moving one little photon detector back and forth behind the two slits, and
then plotting the detected intensity versus spatial position on a graph. This might
be done, for example, when a screen or photographic plate cannot be used. The graph shows
the existence of an interference pattern, much like what is seen on the photographic
plate. In the experiment described above by Silverman, the detector remains in one
spatial location, and the intensity of photon arrival varies over time, instead of
over spatial position. When the intensity is plotted versus time, a graph showing similar
interference effects is produced. Roughly speaking, the light seems to be winking on and
off with the passage of time. The specific experimental conditions imposed require this to
be a temporal manifestation of an interference effect. At the single photon level, this is
just as mysterious and counterintuitive as that for Young's double slit experiment.
Space/time ratios do not necessarily progress only as a change of position in
coordinate space or as a change of position in coordinate time. They can also
progress as a change of direction. The ratio is thus "doing something"
even though it is not "going somewhere." This type of activity will
be called "intrinsic spin" and is essentially the same thing as what physicists
mean by the term. It is not the the spin "of" something. It is a space/time
ratio that manifests itself mathematically as a rotation. Spin can be a space/time ratio
or a time/space ratio, and may involve multiple dimensions of either space or time. Such
ratios are inherently in motion (or are motion), yet because this motion is a
change of direction, and not of position, atoms can form stable positional relationships
with other atoms. This allows for the existence of molecules and atomic aggregates in
general (material "things").
In the current vernacular spin is
"the intrinsic angular momentum of an elementary particle or nucleus, which exists
even when the particle is at rest, as distinguished from orbital angular momentum" (McGraw-Hill
Dictionary of Scientific and Technical Terms, 5th ed., "spin") In my
personal concept of the atom, there are no electrons orbiting the nucleus, and the nucleus
is actually the atom itself. Consequently, whatever concepts and mathematics are currently
attached to "orbital angular momentum" must find some alternative
interpretation.
If you are having some trouble
picturing "intrinsic spin," rest assured you are not alone. It is a problem for
physicists too:
"Although one can try to picture the spin of an electron as
analogous to the diurnal rotation of the Earth about its axis, this is not really
satisfactory. High-energy scattering experiments probing the internal structure of the
electron indicate (in contrast to the proton and neutron) that the electron is a
‘point’ particle composed of no more fundamental subunits to within an
experimental limit of about 10-16 cm. [10-8
Angstrom] If one models the electron as a spinning charged sphere of radius equal to
the so-called classical electron radius . . . the resulting linear velocity of a point on
the ‘equator’ of the electron surface . . . exceeds the velocity of
light by a factor of over 170. If a smaller radius is adopted, then the violation of
relativity is even greater. No classical model of electron structure, in fact, has proved
adequate. It seems therefore, that spin must simply be accepted, and not structurally
interpreted." (And Yet It Moves: Strange Systems and Subtle Questions in Physics,
Mark P Silverman, 1993, p. 216-217)
"Evidently it is necessary to rotate a particle
with intrinsic spin twice, i.e. through 4p, before its original physical state is restored. This weird double-valued aspect
of intrinsic spin sets it apart from ordinary angular momentum (and intuition). It
cautions us not to attach too literal a meaning to the word spin. The classical image of a
body rotating about an axis is totally inadequate to describe the peculiar geometrical
properties of intrinsic spin. The nature of the spin of a particle such as an electron has
no direct counterpart in the macroscopic world of our experience." (Quantum
Mechanics, P. C. W. Davies, 1984, p. 83)
". . . the rotation of a spin-½ particle
through 360° . . . changes the sign of its spin state . . . . There appears to be a
general requirement that actions which for spin-1 particles restore original states
completely (such as 360° rotations and particle interchanges), must be applied twice
to recover the initial state of a spin-½ system." (Quanta:
a handbook of concepts, P.W. Atkins, 2nd ed., p. 268)
"In an introductory treatment of quantum mechanics it is very
difficult to give a really adequate account of electron spin . . . . the idea of spin is
intimately associated with the concept of rotation, and yet we do not succeed in
demonstrating the connection between spin and ordinary angular momentum. . . . [certain
spin phenomena] are clearly demonstrated in one-dimensional systems where ordinary angular
momentum cannot even be defined. Since some of the important features due to spin appear
without any reference to ordinary angular momentum, the "intrinsic angular
momentum" asociated with spin must be regarded as only one of the several
aspects demonstrated by the matter waves of the Dirac theory." (Introduction to Quantum Mechanics, Chalmers W. Sherwin,
1959, p. 279)
There are apparently two types of intrinsic spin. I refer
to them as one-dimensional and two-dimensional spins:
A one-dimensional (2p) spin can be visualized
as a disk. The angular displacement required for one rotation can be computed by the
common formula: S=rq. Solving for q gives q = S/r and so one rotation is 2pr/r
or simply 2p radians.
A two-dimensional (4p) spin can be visualized as a sphere.
"The unit used for measuring a solid angle is the steradian. Since the area of a
sphere of unit radius equals 4p , the total solid angle about a
point is 4p steradians" (Van Nostrand's
Scientific Encyclopedia, 3rd ed., under "angle") Hence,
one complete revolution in the two-dimensional sense represents the surface area of a
sphere instead of twice the arc length of a circle. The solid angle for one revolution is:
q = 4pr2/r2 which reduces to 4p steradians.
The 4p spin is the counter intuitive one that seems to have
"to go around twice to get back where it started." But this would be the case if
you visualized the 4p spin like the stacked disks in a
combination lock where you have to turn the knob backwards twice, to get the
disks back into their original position. An alternative interpretation is that this is
ONE two-dimensional spin which can be decomposed into TWO
one-dimensional spins that are orthogonal. Perhaps one
of my illustrations from Cartoomb™ #
7 can make this easier to visualize:
The photon and the (uncharged) electron appear to have the 2p
spin. Trying to measure their diameter may be an
exercise in futility. Other simple particles like the neutron appear to use the 4p spin system. The periodicity of the Periodic Table suggests that
atoms in their simplest forms (no isomers, isotopes, ionization, etc) are composed of
three orthogonal spin systems: two 4p spins and one 2p spin. If you insist on trying to
picture this, it would probably be something like a football (American type) with an
end-over-end spin imposed on it ( a "bounced football"). All this applies to
what we currently call the "nucleus" because, as I have said before, my
version of the atom does not have any electron shells.
I can just hear some of my readers saying, "So you
think the atom is made up of these perpendicular spinning sphere things. How can spheres
be perpendicular?" Perhaps some clarifications are in order lest the concepts soon
become mangled beyond all recognition.
Firstly, I mean "orthogonal" in the general
sense of "independent", not in its more restricted geometric sense of
"perpendicular". The atom can be represented by a set of numbers, say {a,b,c},
where each of these numbers represents a rotational magnitude (a frequency) and where two
of the numbers, say a and b, each represent a separate rotation of the 4p type and the remaining one represents a rotation of the 2p type. Each make their own special contribution. If one were
missing, the characteristics it contributes cannot be made up by those that remain (in
other words, entire rows (or columns) of the Periodic Table would disappear).
Secondly, it is best to think of the 4p rotation as ONE
rotation, not two. It is ONE rotation of the two-dimensional type. This may
seem like a strange concept, but actually most of my readers have encountered a very
similar concept in highschool algebra: complex numbers. A complex number is just ONE
number that is comprised of an ordered pair of two real numbers. The two components are
kept separate by the device of multiplying one of them by Ö-1
(usually denoted by the letter i in mathematical literature and j in electrical
engineering literature, where i is conventionally used to denote an element of electric
current). A complex number is often written out as z = x + iy. We say it consists
of a real component (x) and an imaginary component (iy). We visualize the real numbers as members of a number line.
Complex numbers are visualized as members of a number plane (called the
"z-plane" or "complex plane" or "Gaussian plane").
It is important to understand that the x and y
components of a complex number are independent (orthogonal). For this reason, they are
plotted on axes that are perpendicular and only cross at zero. Sometimes readers confuse this
use of x and y with the other x and y that were used in highschool algebra to map
functions. In that case x was the independent variable, and y was the dependent
variable. My readers probably remember doing many of those y = f(x) graphs. The
function, f(x) maps a value, x, from one real number line onto another real number line
which contains the value y. If you were to do a similar thing with complex functions, say
w = f(z), you would need a complex plane to represent z, and (usually)
another complex plane to represent w (because if w is a function of a complex
variable it could also be a complex number itself).
The uses and properties of complex numbers are
fascinating topics, especially in calculus. And you will make use of complex numbers if
you study quantum mechanics (see Some Complex Fun below
for some teasers about complex numbers). But for now the only essential point is to be
able to think of a complex number as ONE number. If you "look under the hood," you will
find two components; you could say that it is ONE two-dimensional number. But conceptually, it is easier to
visualize it as one (special) number.
Similarly the 4p spin is just ONE spin
of the two-dimensional type.
The dimensional effects of spin are probably most readily
manifested in polarization experiments. Light, which has the 2p spin system (one-dimensional), cannot pass through two crossed
polarizers that are in series, but does go through if the polarizers have the same
orientation:
A beams of atoms, which have the 4p spin system (two-dimensional), can
also be polarized. Atoms, however, behave quite differently than light when sent through
crossed polarizers in series:
See also: The Feynman Lectures on Physics, Feymen, Leighton, and Sands, 1965, Vol 3,
lecture 5 and The Structure and Interpretation of Quantum Mechanics, R.I.G. Hughes,
1989, "The Stern-Gerlach Experiment", p. 1-8.
The idea that "two-dimensional motion" can be
just ONE motion is a lot easier to visualize if the motion is linear,
instead of rotational. And again you already have seen examples, although you probably
never thought of them in this way. Next time you watch TV, try to visualize what is
happening to the points of light on the display when the camera zooms in on a scene. The
points of the scene all start moving outward and away from each other, as though they were
on the surface of a (huge) expanding balloon. This happens simultaneously in the vertical
and horizontal dimensions. Note that this motion is just ONE motion of the
two-dimensional linear type (I personally have a lot of difficulty trying to think of it
as two separate vertical and horizontal motions). This could be called "scalar
motion"—motion that has no direction. It is simply an
"outward" or "away" type of motion, and can be described with one
number like +1. The expansion of the Universe is a similar scalar motion, except that it
is ONE motion of the three-dimensional linear type. It is also regarded as
centerless.
Using Microsoft Windows on a computer suggests another analogy. To enlarge a viewing
window you can use the familiar click-and-drag operation on an edge. This will enlarge one
dimension of the Window at a time. You can also click-and-drag on a corner and enlarge two
dimensions (height and width) all at once. In other words you can enlarge a Window by
using two, one-dimensional motions (drags), or one, two-dimensional motion to produce the
same effect. Note that there are four corners and therefore (more generally) four ways to
apply the motion. This reasoning can be extended to enlarging a cube by using three,
one-dimensional motions or one, three-dimensional motion. But note that the cube has eight
corners and eight ways of applying the motion to produce the same effect. (The numeric
magnitudes are important when figuring out how the effects of intrinsic spin present
themselves to a spatial reference system.)
The intrinsic spin of the atom opposes the general
expansion of the Universe and results in a phenomena that we call gravitation. It is
likewise a scalar motion, but in this case it is a motion that is "towards"
(i.e., decreasing spatial separation) rather than "away". Unfortunately, in the
case of two objects, like iron balls, we tend to view gravitation as two separate motions,
the motion of A towards B, and the motion of B towards A. But in reality, it is ONE
motion. An "antigravity" device acting on A will decrease the motion of A
towards B, but it will also instantaneously decrease the motion of B towards A. There
would be no propagation effect because there is only ONE motion that is
affected.
The inverse square forces, commonly known as electric, magnetic, and gravitational
forces, all have different underlying scalar dimensions when treated as motions. Electric
appears to be one-dimensional, magnetic is two-dimensional, and gravitational is
three-dimensional, or at least has a three-dimensional distribution. A factor of c (the
speed of light) would be expected to appear when relating one type to another. The
relationship between electric and magnetic field intensity, for example, would
be expected to be E = c B, which is in fact the case. There should be another
relation like, E= c2m which relates the strength of an electric field to that
of a gravitational field. However, the only thing that appears on the scene is E= mc2
which expresses a relationship between energy and mass. It is not clear
whether this dilemma results from a poorly framed question, or is a result of the mess of
problems and inconsistencies in gravitational and electromagnetic theory. (see table for another instance of the c
factor)
Effects of
Spin
Spin is at the very foundation of certain features of
quantum mechanics. It offers us a conceptual basis for the de Broglie wavelength,
the all-important interference effects, the Heisenberg uncertainty principle, and a poorly
understood effect I will call "localization". Crucial to the understanding
of these properties is the idea that spin does not exist independently, but is intimately
interrelated to translational motion. The (charged) electron has been the most
studied in this respect:
"The fact that "spin effects" appear in
plane waves implies that one should not regard "spin" as an independent
kinematical property of the electron, but rather as an essential aspect of its
translational motion." —Introduction to Quantum
Mechanics, Chalmers W. Sherwin, 1959, p. 300
More recent studies of the Dirac electron theory by
physicist David Hestenes have led to similar conclusions:
"The Dirac equation has a hidden geometric structure that is
made manifest by reformulating it in terms of a real spacetime algebra. This reveals an
essential connection between spin and complex numbers with profound implications for the
interpretation of quantum mechanics. Among other things, it suggests that to achieve a
complete interpretation of quantum mechanics, spin should be identified with an intrinsic
zitterbewegung."
. . .
"A related mystery that has long puzzled me is why
Dirac theory is almost universally ignored in studies on the interpretation of quantum
mechanics, despite the fact that the Dirac equation is widely recognized as the most
fundamental equation in quantum mechanics. . . . I hope to convince you that Dirac theory
provides us with insights, or hints at least, that are crucial to understanding quantum
mechanics and perhaps to modifying and extending it. Specifically, I claim that an
analysis of Dirac theory supports the following propositions:
(P1) Complex numbers are
inseparably related to spin in Dirac theory. Hence spin is essential to the
interpretation of quantum mechanics even in Schroedinger theory.
(P2) Bilinear observables are geometric
consequences of rotational kinematics, so they are as natural in classical mechanics as in
quantum mechanics
(P3) Electron spin and phase are
inseparable kinematic properties of electron motion (zitterbewegung)."
Later, in a section about spin and Zitterbewegung ("jittering motion"), he offers this
comment:
"At
last we are ready to grapple with the most profound insight and the deepest mystery in the
real Dirac theory: The inseparable connection
between quantum mechanical phase and spin! This
flies in the face of conventional wisdom that phase is an essential feature of quantum
mechanics, while spin is a mere detail that can often be ignored. We have seen that it is
a rigorous feature of real Dirac theory, though it remains hidden in the matrix
formulation." (Annales de la Fondation Louis de Broglie, Volume 28 no 3-4,
2003, "Mysteries and Insights of Dirac Theory", David Hestenes, 2003; all
italics are his; see also http://www.ensmp.fr/aflb/AFLB-283/aflb283p367.pdf
, http://modelingnts.la.asu.edu/pdf/MysteriesofDirac.pdf
, http://modelingnts.la.asu.edu/pdf/ZBW_mod.pdf
, http://modelingnts.la.asu.edu/pdf/Kinematic.pdf
)
If you want to know more
about David Hestenes' insights, please consult the references. I am simply trying to point
out that physicists recognize an intimate connection between spin, translational motion,
the de Broglie wavelength, and quantum mechanical phase.
Such connections would be expected. In the
space/time ratio interpretation, intrinsic spin is a continuous change of direction
instead of position, but it qualifies as a "motion" nevertheless. The motion
causes it to not participate in the general expansion "nothing datum" of the
Universe that I have described in Advanced Stellar
Propulsion Systems. The result is a "not-nothing" entity that moves
with multidimensional linear motion just like that of our familiar gravitational reference
system, and such an entity can therefore come to rest in the context of that system. The
(charged) electron (4p spin system) is an example.
If it also moves with an additional ordinary motion,
it acquires a de Broglie wavelength (l = h/p). Note that the
wavelength is inversely proportional to momentum which has the space/time dimensions of t2/s2
and not velocity, which has the dimensions of s/t. Momentum is regarded as the product of
mass and velocity. In space/time dimensions, that is p = mv or t2/s2
[=] (t3/s3)(s/t). This is important because temporal motions
are non-directional in the spatial reference system.
When an electron moves towards a so-called
"scattering center", its temporal description is "re-emitted" or
"updated". (For our purposes a scattering center is some kind of spatial
discontinuity; it might be a dust particle in air or a tiny hole in a metal film.) The
re-emission is necessarily non-directional and is conceptualized as a spherical wave
spreading out in all (spatial) directions. Physicists use the term "Huygens
wavelets" to describe this situation. You can see drawings of them in almost any
introductory college physics textbook.
If the electron is directed towards a hole in a thin
metal foil, the points around the edge of the hole (the entire circumference) serve as
multiple scattering centers (we will say millions of them). This results in multiple
re-emissions and multiple spherical waves. (Remember that the temporal description is
anywhere/everywhere in the spatial system; three o'clock in the kitchen is also three
o'clock everywhere else in the house. Hence, the single temporal description can be
"reused" multiple times on multiple scattering centers.) If the hole is large
compared to the de Broglie wavelength, the envelope of all the wavelets looks
like one hemispherical wave on the other side of the hole. The wavelets can also be
thought of as producing an interference pattern, but if the hole is large, the pattern is
so fine and lacking in contrast that it is normally not visible.
If the hole size is smaller and more comparable to
the wavelength, the interference pattern becomes coarse and is readily seen. A stream of
electrons will build up an exposure pattern on a phosphor screen downstream from the hole
that looks like a bullseye archery target (the so-called "Airy disc" is
the bright central spot; for a vivid
visual simulation of this effect see http://www.kw.igs.net/~jackord/df/d1.html
. Run the Circular Aperture simulation by selecting a d value and then clicking
Run. For example, select d=0.4 and then click the Run button )
If the hole is rectangular (a slit), instead of circular, the same
arguments apply, but the pattern appearing on the screen with be that of a slit, not a
hole. The image will be surrounded by rectangular "diffraction fringes"
instead of the archery target pattern. (All ordinary images are actually formed by
diffraction; see Optical Physics, Lipson, Lipson, and Tannhauser, 3rd ed.,
Chapter 12)
If the round hole is made still smaller, it becomes essentially one
scattering center, and the illumination beyond the hole becomes (guess what) very uniform
and very dim. This is because each electron will be scattered in a completely random
direction.
So when an electron is
diffracted from a hole, its momentum acquires a spread in values. Each electron tends to
move radially away (perpendicular) from the original line of motion to various extents.
The amount of uncertainty in this direction is directly related to the size of the hole
and the de Broglie wavelength. The math reduces to the following:
Dx Dp > h/4p
which is the well-known uncertainty relation.
As the hole closes down (Dx
decreasing) the radial spread of the momentum increases ( Dp increasing) and the stream of
electrons illuminate a larger and larger area on the screen beyond the hole. Note that the formula refers specifically to momentum, not
velocity; the latter has a "path" and is not particularly important in
quantum mechanics; also, because of spin, momentum and velocity are not necessarily
collinear; momentum or "quantity of motion" is a better fit to the problem.
As pointed out above, the diffraction occurs
even when electrons (or photons) are sent into the apparatus one-at-a-time. There is no
way to predict where any individual electron will go; but the overall diffraction pattern
which builds up after a long exposure is very predictable and very definite. The
overall pattern is determined by quantum mechanical phase, and that, as Hestenes has
pointed out, is linked to spin.
Interference arises from the phase of the
wavelets combining constructively or destructively. In quantum mechanics the various
phases are described mathematically and then are summed to allow them to interfere. The
square of the resulting amplitude is taken to get the probability density that an electron
will appear in a particular place. These are the "matter waves",
"pilot waves", or "probability waves" that you read about in the
literature of quantum mechanics. They are usually diagrammed as transverse waves, but keep
in mind that "There is no evidence that matter waves actually consist of transverse
vibrations." (Sherwin, p. 297)
These non-physical, abstract,
mathematical matter waves give us a description where the electron is most likely to be
found, or most likely to be absent. But the math alone does not give us a clue about the
underlying reason or mechanism. In a transverse wave like a water wave, the most
water will be found where the wave height (amplitude) is highest. In a longitudinal wave
like a sound wave in air, the most particles will likewise be found where the amplitude is
highest. But matter waves aren't anything "physical" as far as we know. Why is
the "whereness and thereness" or "localization" of an electron
described so effectively by a heap of abstract mathematics?
We suspect that it has something to do with
spin, but unfortunately the exact mechanism has so far eluded description. When we talk
about electrons getting together or avoiding each other, the word "spin"
(instead of charge) enters the conversation. Two electrons can have only two spin
orientations with respect to each other: parallel or antiparallel. There are no
intermediate values or orientations. These orientations somehow affect what I call
"localization". Here is an an example that has to do with Fermi holes and Fermi
heaps:
"Because of the quantum mechanical
effect of spin correlation, two electrons with the same spin cannot be found at the same
point. Thus, a plot of the probability of finding a second electron relative to the
location of the first falls to zero at zero separation. . . . There is a corresponding
decrease in amplitude of the wave function for the location of the second electron, and
that wavefunction is close to zero in a small region surrounding the location of the first
electron. This region of almost zero amplitude is the Fermi hole in the
wavefunction of the second electron.
If the two electrons have opposite spins,
there is an enhanced probability of finding the second electron close to the first. That
is, instead of a Fermi hole, there is a corresponding Fermi heap . . .
which is an enhanced amplitude in the wavefunction of the second electron wherever the
first electron happens to be at any instant." (Quanta,: A Handbook of
Concepts, P.W. Atkins, 2nd ed. (1994), p. 122-123; see also http://quantum.bu.edu/notes/GeneralChemistry/FermniHolesAndHeaps.html
)
Finally, this is probably a good place to remind
readers that our concepts of space and time come from motion. Motion is what we actually
observe and measure. Our concepts of space and time are derived from the type of motion we
observe. Motion is the primary concept, whereas space and time are secondary concepts (or
abstractions). This can lead to some counterintuitive notions. A misunderstanding of
translational temporal motion leads to the paradoxes in Special and General Relativity,
for example. The concept of rotational spatial motion leads us to a better understanding
of the electron as a unit of rotational space, and to the concepts in electromagnetics
that I have already mentioned in Advanced
Stellar Propulsion Systems. In Quantum Mechanics the spatial component of motion is
fixed at one unit, and so quantum mechanical motion can be translational temporal motion
or rotational temporal motion . The latter leads to the concept of, literally,
"rotational time". If you were in a world of rotational time, what would your
world look like? Besides expecting it to be non-local, you would also expect it to be
repetitive or periodic. So it is not surprising that the math of quantum mechanics
involves many periodic functions like sine and cosine, complex exponentials, and various
"wave" representations. These too are indeed the "effects of spin".
The concept of spin, along with the concept of
temporal motion, could probably explain all the major mysteries of quantum mechanics in a
satisfying way if we could just shake ourselves loose from our current blindspots and
misconceptions.
__________
On a side note, I should say something about the
terms "uncertainty" and "indeterminancy." These have acquired various
meanings over the years (see summary). I prefer
to use the term "indeterminate" in reference to how a temporal motion maps into
a spatial reference system. Conceptually, it either has no position, direction,
state, etc. or an infinite number of simultaneously existing potential positions,
directions, states, etc., that can be "superposed". One of these
potential states will materialize during an interaction with the spatial system. An act of
measurement, for instance, forces the temporal phenomenon to participate in the
spatial system.
"Uncertainty", on the other hand, seems to
have more to do with the product of two quantities equating to a constant. The uncertainty
in their values is inversely related, and consequently I sometimes use the term
"inverseness" to emphasize this aspect. See article.
Zitterbewegung
: "An oscillatory motion of an electron suggested in some interpretations of
the Dirac electron theory, having a frequency greater than 4pmc2/h . . . or approximately 1.5 x 1021
hertz" (McGraw-Hill Dictionary of Scientific and Technical Terms, 5th
edition, 1994)
"In
general, the momentum p is not collinear with the local velocity v = v(x), because it includes a contribution from the spin." ("Mysteries and Insights of Dirac Theory", David Hestenes,
p.9 http://modelingnts.la.asu.edu/pdf/MysteriesofDirac.pdf
See also: O. Costa de Beauregard, “Noncollinearity of Velocity
and Momentum of Spinning Particles,” Found. Physics 2: 111–126 (1972)).
html 1/05
The Origin of Intrinsic Spin
The article Advanced Stellar Propulsion Systems proposed
that the "nothing datum" for the physical universe consists of ratios of
space and time expanding in three linear dimensions. These innumerable units of nothing
are quantized, are all independent of each other, and move at the speed of light (the
ratio). Collectively they are the source of the properties of space and time which are
familiar to all of us, and define the datum for an empty but physical
universe. Now, how can this universe become populated with "not nothings" or
"things"?
To simplify matters, let's imagine a scaled down universe of one
dimension. Lets create a "flow field" of "nothing datums" that moves
right-to-left as shown in the illustration below (left). These units are all independent
of each other. We could also imagine another flow field that goes top-to-bottom (not
shown). These units are likewise independent of each other and are also
independent of the right-to-left units. Through an Act of Creation let's borrow one of the
top-to-bottom units and attach it to a right-to-left unit. Furthermore, let's assign the
top-to-bottom unit the property that no matter which way the right-to-left unit goes, the
unit that was originally a top-to-bottom unit always remains perpendicular (and in the
same plane) to the unit that was originally right-to-left. The result of this is shown in
the diagram below (right).
The original right-to-left unit now moves in a circle. The speed
of the (original) right-to-left unit remains constant, but its direction is
changing at a constant rate. This altered unit no longer progresses with the linear
flow of the general "nothing datum" because it is constantly and uniformly
changing direction. The orthogonal motion has resulted in rotation, and this is a
drastically different behavior.
Let's pause now and note the presence of some concepts that seem to be
quietly imbedded in this picture that are especially relevant to Quantum Mechanics. A
physicist would probably see the following almost immediately:
a. The picture embeds the concept of Intrinsic Spin. Shrink
the circle down to a point and you have "pure spin" with no translational
motion. Physicists use the term "intrinsic spin" to refer to what seems to be a
particle's rotational motion that is inherent in its very existence, not because some part
of it is actually spinning.
b. The picture embeds the concept of Uncertainty. A
small circle has less uncertainty in location than a large one. If you dropped a coin and
knew it was within 50 feet of you, its position is known with more certainty than if
you knew it was within one mile of you. In the picture, more orthogonal motion
shrinks the circle. The development below will use additional orthogonal motion in
additional dimensions. Massive particles like atoms and molecules have multiple rotating
systems, which implies even more orthogonal motion. The positions of atoms can be
precisely localized.
c. The picture embeds the concept of simple harmonic motion and
wavelength. Two linear, but inherently orthogonal, motions result in
circular motion. A projection of circular motion results in a harmonic
oscillation. A harmonic oscillation with yet another perpendicular linear motion
imposed, results in a sinusoidal wave trace. Such "transverse waves" have been
used to describe electromagnetic waves, de Broglie waves, and the abstract "matter
waves" of Quantum Mechanics.
A projection of circular motion results in a
back-and-forth
harmonic motion, which, when projected in an
additional dimension, results in a sinusoidal wave trace
d. The picture intrinsically connects
rotation with linear motion. As mentioned in the preceding section, quantum mechanical spin is intimately
connected with translational motion. The orthogonal motion can apply only to
an already existing motion. But in the reference system we commonly use, objects can be
stationary. The wavelength effects (diffraction, interference) become apparent
only when the object is moving. The combination of translational motion
and inherently orthogonal motion is the origin of intrinsic spin and the de Broglie
wavelength.
e. The picture embeds the concept of
a "location" or a "locality." In physics we need the concept
of "position." This is how we are are going to get it.
f. The picture embeds the essence of
quantum mechanics: xp - px = ih/2p. In this well known commutation rule, the h is
Planck's constant and implies quantized momentum. The i is a mathematical notation
representing orthogonality. The x and p are position and momentum respectively. Because
they are orthogonal, the combination of x and p implies angular momentum. (Remember
that the uncertainty relation of the photon and electron diffraction experiments, Dx is perpendicular to the line of
travel (momentum) of the photon or electron. It is not a Dx along the path but represents a deviation from
the path.) See Quanta: A handbook of Concepts, 2nd ed., 1991,
"Operators", p. 252. See also Commutation
and Angular Momentum below.
Let's continue with the original line of
reasoning. At this juncture we have a circle, and it has become stationary in one
dimension of the flow field. Let's add more dimensions, both for flow, and for orthogonal
motion.
If we create a second flow field that moves
INTO the paper, the circle will be swept away into the paper. The circle has a definite
diameter and we are using it to represent a rotation with a certain speed. If we give the
circle a phase mark, we can see the rotation. The mark will trace out a helical path (like
threads on a screw) as the whole combination moves in this new dimension going into the
paper. (this may remind us of the Zitterbewegund
of electron motion.)
If this circle is now given another
rotation in the new dimension, it will become stationary in TWO linear dimensions of flow.
The circle now has the form of a sphere, and the sphere becomes stationary in the two
dimensional flow field.
We could compare the first (one-dimensional)
case to a bunch of transparent race cars on a big field. The cars represent the
"nothing datums" and are moving from right-to-left. The individually
applied orthogonal motion makes some of them turn and go in a circular path. These
"new locations" become manifest in an otherwise featureless flow field.
For the second case, imagine a little piece
of paper (or maybe a little steel ball) on the surface of an expanding balloon. The
expanding surface represents the two-dimensional flow field (again, remember that
there are two dimensions of flow, not just two dimensions of position or
two dimensions of extension). The piece of paper is the thing with the two
dimensions of orthogonal motion. It is not carried along with the surface expansion.
If there were two such pieces of paper, they would not participate in the expansion but
would maintain the same distance from each other even as the surface of the balloon is
expanding. (This analogy is somewhat misleading, however. The expanding balloon surface is
ONE two-dimensional motion. Likewise, the bits of paper each have ONE opposite
two-dimensional motion, instead of two one-dimensional motions, to oppose the expansion.)
As explained in a preceding section, nature
has something akin to these two types of rotation. They are the 2p and 4p rotations in Quantum Mechanics. The 2p rotation can be visualized as a disk. It represents one dimension
of rotation. The 4p
rotation can be visualized as a sphere. The latter represents ONE (single) two-dimensional
rotation. (Because this is hard to visualize, I illustrate it as though it were a
combination of two rotations, but the concepts are not identical.) These rotations can be
used to infer the structure of photons, massless particles, and atoms, as will be done
below. But for now just think of them as part of a conceptual exercise.
Let's add a third dimension of flow. The flow
field is now like a room full of compressed air that has leaky, porous walls. As the
air thins out, the air molecules become increasingly distant from each other. The
action is like a centerless expansion. Our "spherical location" is being swept
along in the third dimension. We impose yet a third rotation on it. If it follows the
previous pattern, we would expect this to be an 8p rotation. But no such thing has ever been found in nature.
Apparently, what is possible is only a rotational distribution. This does not have the
same fundamental character of the 2p and 4p rotations. It is not ONE three-dimensional rotation.
Instead, it is apparently a two-dimensional rotation with an additional rotation added on
to it. The extra rotation brings the particle to a full stop (spatially) in the expanding
flow. (The lack of an 8p
rotation is apparently the "loophole" that permits the development of
antigravity technology)
If the rotations are in space, the time flow
continues to progress, and carries the particles (atoms, we shall say at this juncture)
with it. Although the atoms are stopped in space, they continue to move apart in time.How will this look to an observer in a spatial reference system? In a
conceptual system based on space/time ratios, more time separation equates to
less spatial separation. This means that the particles will be coming together in space. Note
that the original spatial motion was actually stopped by a rotation; this
"extra" spatial motion is caused by the progression of time. We shall call it
temporal motion.
How do motions
in three-dimensional time manifest themselves to an observer in a three
dimensional spatial reference system? If we gave a marble or a coin spatial
motion, we know exactly what we would see. But what would we see if we gave it temporal
motion? We can surmise the following characteristics:
1. Temporal motion is motion in
three-dimensional time, not space. Hence this kind of motion cannot have a spatial path or
a spatial trajectory. We are looking for some sort of "motionless motion."
2. Temporal motions can have no directional
preference in a spatial reference system. Temporal motions are in a world of
"when" instead of "where." Additionally, in a spatial reference
system, time has only a magnitude (no direction or location; but it does seem to have a
"polarity": forwards or backwards).
3. Temporal motions are "non-local"
(a consequence of #2). Hence, temporal motion must be infinite in (spatial) extent.
4. Temporal motion is actual motion and must
express the theme of "motion" somehow, even though it is still "motionless
motion." In other words, we would expect that temporal motion would still manifest
traits of momentum, energy, work, power, etc., although these manifestations could
be very different from the more familiar spatial traits.
What we seem to have is a description of a force
field, specifically a gravitational field in this case. If we have two
iron balls with this kind of motion:
1. The motion clearly has no particular
spatial path or trajectory. In fact, we usually think of gravitation as a force that
causes motion, not motion itself.
2. The motion (or force) clearly has no
directional preference. The balls can have any relative orientation and the motion is
still simply "towards." It is indifferent to north, or south, or east, or west,
or above, or below (as is time). In this sense it is "scalar motion" —motion that has no inherent direction,
only a magnitude.
3. Gravitational motion (or force) is
normally understood as extending to infinity (or out to a quantization limit).
4. There is clearly a connection with the
theme of motion here. The usual technical term is a potential for motion. There
is also a theme of "perpendicularity" inherent in this kind of motion. See Motion Cancellers and also Motion Couplers for some illustrations.
Due to its inherent lack of direction,
temporal motion has an inverse square distribution (the intensity of it falls off rapidly
with distance). See Advanced Stellar Propulsion Systems for additional details, and the slide below
about a characteristic of gravitational motion.
Besides gravitation, there are also electric
and magnetic force fields. These are also directionless and follow the inverse square law.
In passing we might note the following:
1. Electrical charges have a
non-directional motional distribution and are thought of as potentials instead of
actual spatial motion. There is a "scalar potential" from which an electric
field is derived, and a "vector potential" from which a magnetic field is
derived. These characteristics imply temporal motion.
2. Temporal motion can be "towards"
or "away". (This may be analogous to our intuition that time flow has
"polarity" (forwards or backwards) but no inherent direction.).
3. Charges appear to be created in pairs.
Symmetry and conservation arguments imply the simultaneous creation/presence of both a
"towards" motion and an "away" motion.
4. The two kinds of motion imply
"poles" of some sort with opposite polarity. Electric poles can be on separate
objects (monopolar). Magnetic poles are always on the same object (dipolar).
5. A temporal motion may consist of one, two,
or three dimensions of time progression. All "three flavors" are non-directional
in a spatial reference system, but the intensity of the motions, as seen from the spatial
system, will correlate with the dimensions of the time progression. A one-dimensional
time progression will map into a spatial system with full intensity, whereas a
two-dimensional time progression will be reduced by a factor of the speed of light.
6 Charges do not have a de Broglie
wavelength. Particle wavelength is dependent only on Planck's constant and particle
momentum. Lack of wavelength implies either a lack of momentum or a lack of
"sufficient rotation". Charges must therefore have a simpler space/time
structure than photons, massless particles, or massive
particles like atoms. It could be argued that charges are more akin to a
"situation" than they are to "things."
7. Charges are always attached to something.
They do not exist independently. This implies that they are a modification of an already
existing space/time relationship.
8. Charge modifies the
gravitational motion of a particle. Each particle in a collection of gravitating
particles, is pursuing its own independent course, namely, "anti" to the
expansion of the general progression of space/time. None of the particles
"knows" what the other particles are doing, nor are they attracting each other.
Charge modifies this type of motion, but the motion still retains the aspect of
independence.
This independence of motion
is very non-intuitive upon first encounter. So picture this illustration. An advertiser
sets up a big flashing sign in the middle of town that says "win something free
here". May people like "winning" and many like getting something
"free". The advertisement "pulls in" a lot of people of like mind, but
it may also "repel" others who do not like this kind of thing. In any case, the
people are responding separately and independently to the advertisement, not to
each other.
Similarly, particles will
have a like-kind of motion if they have a like-kind of spin modification relative to one
reference system. Two electrically charged pith balls seem to recognize each other and
respond to each other. But if we put a powerful, stationary magnet near the charged pith
balls, they seem not to respond to the magnet at all. The magnet, however, responds to the
presence of another stationary magnet. Our perception is that these objects
really seem to sense each other's presence, provided they have a nature of the same kind.
The reference system here is
not the invisible flow field described above, but is now a collection of visible
objects. We think they are stationary, but this gravitational reference system of our
ordinary everyday experience is not only moving, it is accelerating (which is why
your feet, chair, and desk remain pressed against the earth). In other words we
are immersed in a non-intuitive reference system which we mistakenly think is simple and
intuitive at a fundamental level. If you don't believe this, you can get into my Mustang
and I'll accelerate it at 9.8 m/sec2 while you conduct physics experiments in
the back seat. Taking the seat as "stationary" and ignoring the
acceleration, you will get some very non-intuitive and puzzling results during your
experiments. You will probably also wonder about the claims of "zero point
energy" as you see the scenery outside flying by you in the "stationary"
Mustang.
9. Gravitational motion at
the quantum mechanical level has a lot to do with interatomic bonding and the physical
"state" (solid, liquid, gas) of matter. A "spin
flipper" type of motion canceller could
conceivably change the state of matter. That is, solid matter would have one dimension of
its "towards" motion momentarily cancelled, and would therefore act like a
liquid for an instant. This implies that antigravity technology could have some
troublesome side effects, and that the "disintegrator beams" of science fiction
might be not so farfetched after all. However, the ability to induce a "liquid
state" at ordinary temperatures would be a metallurgist's dream come true. And it
could be handy in mining and tunneling. (See also Ray guns, Nuclear Isomers, Rydberg Atoms
and Melted atoms
or a melted aggregate? )
10. As is suggested in The
Atomic Spin System (below) atoms are made of "shells" of these basic spins.
The spins may be 2p or 4p, and may
be rotations in space or rotations in time (which must be built upon a net spatial
rotation). The Periodic Table indicates that each atomic number associates with two 4p spins and one 2p spin as a set that is
built upon a previous set. The sets ("subshells) can interact with one another and
give rise to quite a variety of physical properties, including spin-spin and spin-orbit
coupling. They may be considered individually (j + l) or as a whole (J+S). The outer set
of spins is probably the source of chemical valence and stereochemistry. Inequalities in
the energies of the 4p spin systems apparently result in what
are called "shape isomers", "hyperdeformed nuclei" and "nuclear
isomers". There is also a possibility that "instantaneous chance alignment"
of (rotating) subshells in the 4p systems might somehow be a
factor in radioactive decay time intervals; if so, it should be possible to both increase
or even decrease radioactive decay rates by the use of external magnetic and electrical
fields. (See Extending radioactive half-livesUpdate 2-27-04
,Update 11-11-06)
11. In the illustration above, if the original
rotation was in time, with space still progressing, the ultimately resulting
"atom" would be localized in coordinate time (instead of coordinate space). Such
atoms would gravitate temporally and form temporal stars and galaxies. But these could not
be seen as discrete objects by observers in a spatial reference system because time is
non-local in such a reference system. Instead, with space still progressing, they
would be seen from our standpoint as a sea of "particles" having an extremely
isotropic and homogeneous spatial distribution, and moving at the speed of light and with
high energies. They would also possess some sort of "inverse mass" and
"inverse chemistry". (Cosmic rays seem to have these characteristics. )
12. The original combination of orthogonal motions
that results in rotation does not occur spontaneously in the picture presented above. If
it did, the Universe would ultimately fill up with "stuff" that it self-creates
automatically (matter, antimatter, and photons). This implies that we live in a finite
Universe that has been created, and which has a fixed amount of matter-energy. We would
not expect to see true, actual violations of the principle of conservation of
matter-energy (even occasionally) in such a Universe.
Hopefully this section will
clarify the origin of what physicists call "intrinsic spin", as well as the more
fundamental concept that I call "intrinsic rotation". An accurate and
detailed understanding of spin is crucial to understanding and feeling
comfortable with practically everything in Quantum Mechanics and Quantum
Electrodynamics. It could also lead to some startling practical applications.
A Fun Fact
about the Coulomb Constant
". . . The
constant 2.306 x 10-28 newton-m2/(elem. ch.)2 tells us
directly the force between two elementary charges one meter apart. They exert just 2.306 x
10-28 newton on each other. No doubt that force looks absolutely negligible.
But in fact it is huge. The gravitational attraction between two hydrogen atoms at this
distance is about 2 x 10-64 newton. The electric force between two elementary
changes is 1036 times as big. . . . The force is so big that two moles of
elementary charge (two collections of 6 x 1023 charges) placed one on each side
of the earth would push on each other with a force of half a million newtons —50 tons. Two moles of the heaviest
atoms placed at opposite ends of a diameter of the earth would have no appreciable
gravitational effect on each other." Physics, Physical Science Study
Committee, 2nd ed., 1965, D. C. Heath and Company, part IV Electricity and Atomic
Structure, p. 500)
A Fun Fact
about de Broglie Waves
"These properties of
relativistic invariance led de Broglie, before the discovery of quantum mechanics, to
postulate the existence of waves . . . associated with the motion of any particle. They
are therefore known as de Broglie waves." ( The Principles of
Quantum Mechanics, P.A.M. Dirac, 4th ed, 1958, p. 120)
When I was a kid I kept looking for
illustrations of the photon (also known as light and electromagnetic waves) in physics
textbooks. I wanted to know what the structure of the photon was, and how it managed to
move through a vacuum. The usual textbook model of the photon is that of an
electromagnetic wave which has time-varying electric and magnetic fields that are at right
angles to each other, and both of which are perpendicular to the direction of propagation
of the wave. For a while I accepted this model because it seemed to make sense in the
context of my experience. In those days one of my hobbies was amateur radio. Inside
a radio transmitter there is a thing known as a "tank circuit". It consists of
an inductor and a capacitor connected in series or parallel. The capacitor stores
electricity in the form of an electric field, and the inductor stores it as a magnetic
field. Because they are connected together, they discharge electricity back and forth into
each other in an oscillatory form. When the capacitor is losing its electric charge, the
magnetic field in the inductor is gaining strength, and vice versa. A tank circuit is
needed because the "carrier" in a radio wave is a smoothly oscillating
electromagnetic wave, but the circuits which generate this wave are most efficient when
the power is delivered in pulses. The tank circuit is used to smooth out these pulses. The
effect is much like that of a parent pushing a child on a swing. The parent delivers the
power in pulses, but the overall effect is smooth oscillatory motion.
And so I thought that light was a kind of energy like that
in a tank circuit. It tossed energy back and forth in the form of electric and magnetic
fields and it moved because these fields were leapfrogging over each other, pulling
each other along . But I became dissatisfied with this model for two reasons. The first
was that most analytic descriptions of the photon were presented in terms of the electric
field; the magnetic field was practically ignored. I began to get the impression that this
"electromagnetic" thing was basically an "electric" thing. And
secondly, I noticed that in the diagrams, the magnetic field was always exactly in phase
with the electric field. That destroyed my concept of leapfrogging fields. And when I
finally came across the E=cB equation, I realized that the magnetic field was a result
or an effect of the motion of the electric field through space at the speed of
light.
My concept of the photon then became something like that
of the "electric vector" presented in the textbooks. This can be visualized as a
little arrow that starts out with zero length, grows to some definite length and points
upward, shrinks back to zero, and then grows in the downward direction to the same length
and then shrinks back to zero again. This cycle repeats indefinitely. Its motion, which is
entirely a separate issue, is perpendicular to the arrow. The tip of the arrow therefore
traces out a sine wave. This model seemed to explain phenomena like interference,
diffraction, and linear polarization.
Later, I learned that photons carry momentum and can exert
an impact force (albeit a very small one) when they strike an object. They can also carry angular
momentum and exert a torque when striking an object. The latter property caused me a lot
of conceptual confusion. It is easy to picture an electric vector rotating as it moves,
and its tip tracing out a helix like the threads on a screw, and that this twisting motion
could exert a torque on the object it hits. But this seems to require two sine
waves oriented perpendicularly and with a 90 degree phase difference. This could easily be
accomplished with a stream of photons which combine in such a way as to produce a
resultant electric field that rotates as it progresses. But individual photons
possess angular momentum. Furthermore, photons have a quantum mechanical property called
"spin". The electric vector concept thus became inadequate, even
confusing. Photons seemed to involve some sort of "twoness" and also the
property of intrinsic rotation.
Moreover, transverse waves, like waves on the surface of
the ocean or waves on a rope attached to a doorknob, exert forces that are in line with
the up-and-down motion of the wave. A ship on an ocean wave goes up and down, and the
reaction on the doorknob with a transverse wave on the string, is also up and down, not in
or out along the line of the string. A transverse electromagnetic wave (light), however,
exerts its momentum along the direction of its travel, not in the direction
of the waveform itself. This produces phenomena like the photoelectric effect, in which
electrons are kicked out of the surface of metals when struck by light, and radiation
pressure which is used to compress the hydrogen in the hydrogen bomb. The electric vector
model was not very good at explaining this.
And I realized I had not a clue about how
the photon moved. The equation E=mc2 suggested that the photon was a definite,
discrete physical entity, just as definite as mass, and was therefore NOT a
"disturbance" or a "wave" propagating through space like waves on a
string or rope. Photons clearly did move, but this did not seem to be a property of the
photon itself. (I later concluded that photons are actually
stationary!)
I realized that intrinsic rotation cannot be portrayed in
a linear extensional reference system, except by projecting the rotational magnitude along
the line of the reference system. This is done mathematically by using what is variously
called a dot product, inner product, or scalar product (usually taught in a first
year college calculus or vector spaces class). The inner product concept is well utilized
in quantum mechanics. After playing around with the rotations, I came up with the
following model:
It is simply TWO one-dimensional rotations
that are orthogonal. They are structurally coupled (for lack of a better term) but are
otherwise independent. Note that this is not the same thing as the ONE two-dimensional rotation discussed above in the previous section (the 4p rotation). The
two rotational magnitudes of the photon have to be independent because phase shifts and
changes in rotational magnitudes are required (phase is to the world of rotation what
position is to our everyday world of extension). I doubt if this is physically possible
with the 4p
system, even though the math is similar.
This model accounts for linear, circular,
and elliptical polarizations just as per the text books ( see for instance Optics,
Eugene Hecht, 2nd ed. Chapter 8, "Polarization") It is consistent
with the following properties of photons:
Plane polarization: In this case, the two sinusoidal components are in phase and of equal magnitude.
The electric vector therefore has a fixed orientation but varies sinusoidally in length.
Plane polarization is also known as linear polarization. In the conventional wave
representation, linear polarization can be regarded as the superposition of two
circularly polarized waves of opposite sense.
Circular polarization: In this case, the two sinusoidal components have a 90
degree phase difference. The electric vector therefore rotates but has a fixed length. Its
tip traces out a circle. As the photon advances, the tip traces out a helix. In the
conventional wave representation, circular polarization can be regarded as the
superposition of two linearly polarized waves at right angles and 90 degrees out of phase.
Elliptical polarization: In this case, the sinusoidal components have a phase
difference other than 0 or 90 degrees and may also have different magnitudes (represented
by the diameter of the circles). The electric vector rotates and varies in length. Its tip
traces out an ellipse.
Optical anisotropy: In an optically anisotropic
medium the speed of one spin will be retarded more than the other. The retardation is
caused by the intrinsic spins of the transparent medium (such as calcite) through which
the photon passes. This will cause the phase of the two spins to shift. If the
photon entered as plane polarized, it could emerge as circularly polarized if the phase
shift is 90 degrees (such a shift can be deliberately caused by a 1/4
wave plate).
Optical activity: In this case the intrinsic spins of asymmetric molecules
(such as common sugar) can cause axial rotation of the entire photon. This is equivalent
to a momentary rotation of the electric vector. When the photon emerges from the
transparent medium, its polarization plane will have been rotated through some
angle.
Linear momentum: The one dimensional rotations have no mass effect
because gravitational motion requires a three-dimensional distribution of motion. However,
the photon motion does have a distribution in the dimension of its travel, and therefore
will have what could be called a "one-dimensional mass effect" . This should
account for the linear momentum and zero rest mass conventionally ascribed to the photon.
Angular momentum: Circularly polarized light can exert a torque on an
object or an atom. The explanation for this is probably the same as that presented
in the textbooks. See Feynman, op. cit., vol 3, p. 17-10, and others.
Atomic absorption/emission: If the entire photon structure depicted above is
given a third rotation perpendicular to its direction of travel, it presumably would
acquire a gravitational-like motion. This means that photons could be absorbed by atoms or
molecules, and that the atoms would acquire an "excited state" as a result. This
type of rotation however, is still "foreign" to the atom and the photon may be
re-emitted. If the photon is energetic enough (approximately 931 MeV) its rotational
system could convert to the atomic system of rotation and materialize as actual mass.
Spin: Circularly polarized photons can trace out a left or right
handed helix as they travel. This corresponds to a spin state of ±1.
Other properties ascribed to photons actually have more to
do with the concept of locality than with the photon itself. A couple of these were
discussed in the Advanced Stellar Propulsion article. A
summary is offered here for completeness:
Constancy
of the speed of light: Photons have
a location both in coordinate space and coordinate time. These locations progress
outward and away from each other (increasing space and increasing time separation). The
photon has no independent motion relative to this progression and in this sense they are
stationary and not propagated as a wave motion. Atoms of matter move "anti" to
the progression and so we see the photon moving at the speed of light. The photon will
appear to traverse both space and time and this will result in a constant speed.
Even when treated as a wave motion, the speed of the photon is still unique. See Exploration
of the Universe, Abell, Morrison & Wolff, 5th ed., 1987, p 217-223 for
a good discussion of this unique property.
EPR Paradox: If two photons originate in the same event, they will
remain in the same temporal location, even though their spatial
locations become widely separated. An experimental action imposed on one photon will
result in a complementary action being imposed on the other photon, even though it might
be miles away spatially from the first photon. Scientists call this problem
"non-locality", "action-at-a-distance", "over-connected
Universe", "entangled states", etc. ("Hyper-Entangled Photon
Pairs" http://www.aip.org/pnu/2005/split/754-1.html
)
Superposability: If space and time are quantized, two photons can never really
be in the same spot. If they appear to be in the same place spatially, they are not in the
same place temporally. Paradoxically, this allows them to appear to be superposed and
bunched in the same spatial location. Physicists use the term "boson" to
describe this behavior.
I hope this photon model will prove to be a substantial
improvement over the ones I have used in the past.
__________
Afterword: The photon structure
described above is one that has been derived from the Periodic table. I have wondered if
this "atomic photon" structure is truly irreducible, or whether an even simpler
version of the photon exists. For example, a radio frequency version of this photon could
be created with an antenna system that has two physically crossed dipole antennas
fed 90 degrees apart in electrical phase (temporal phase). The dipole antennas would be at
the back of the illustration, with their axes both perpendicular to the direction of
photon propagation. Either dipole will emit radio waves (photons), although both are
required for circularly polarized radiation. This raises an obvious question: what
kind of photon do we get with only one dipole? This implies that a photon can consist of a single
rotation. Do photons come in two versions or just one? Is there both a "single
dipole" version and a slightly more complicated "crossed dipole" version?
Previously I had believed that the photon has no
independent motion relative to the expansion of space/time. Yet the Origin of Intrinsic Spin discussion seems to imply that
the photon may "gravitate" in at least one dimension. That would be
consistent with the fact that photons possess momentum and with the concept of
equivalent "rest mass" used by physicists. In this view, a photon is
simply an even "less massless" version of massless particles (which
"gravitate" in two dimensions instead of the three used by atoms; either sort of
massless particle will still be moving through a gravitational reference system at the
speed of light). Answers to these questions may also give some insights about the
origin of Bose-Einstein statistics.
The above example might need some clarification.
Each photon has an angular momentum of h/2p, and its angular momentum is independent
of frequency. Photon energy, however, is proportional to frequency (E = hf). A
one megajoule pulse of radio frequency photons must therefore have many more photons than
a one megajoule pulse of visible light photons. The radio frequency pulse will therefore
exert much greater angular momentum than the visible light pulse. (For yet another
twist on light see "Light With Orbital Angular Momentum" at http://www.aip.org/png/2005/229.htm , http://www.aip.org/pnu/2005/split/721-3.html
)
If the atomic rotational system is based
on combinations of 4p and 2p intrinsic spins then the Periodic Table suggests how these spin
systems might be structured. The number of elements on each row of the Periodic Table are
commonly displayed as 2, 8, 8, 18, 18, 32, 32. This can be expressed as the following
pattern:
The proposed interpretation of this pattern is as follows:
The exponent of 2 implies a two-dimensional rotation (the 4p intrinsic spin system). Allowable
magnitudes appear to be 1, 2, 3, and 4.
The fact that there are generally two rows with the same
squared term (e.g.: two rows use 3
2 ) implies that there are two such 4p spin systems in the atom, and that
they are orthogonal (independent).
The periodicity implies a "missing row" based on
1
2 .This "row" would contain two "subatoms" or
particles that are "less than atomic". They are based on the 4p intrinsic spin system, and they are
part of the periodicity, but simply do not have the properties of mass and chemistry.
The 2n
2 terms can be viewed as
lump sum quantities —entire rows as a whole.
The n
2 terms imply the
existence an n type of term. This would be the 2p intrinsic spin system. Whereas the 2n2
terms correspond to rows, the "plain n" type of term would correspond to
individual elements within a row. (This, incidentally, also implies the existence of two
more subatomic particles based on the 2p spin system.)
The 2 in front of the n
2 may imply some kind of plus and minus effect
(or opposite space/time sense). In other words, a distance of 2n2 would
be interpreted as going from -n2 to 0 to +n2
The last point, and its problems, require some
elaboration. If an intuitive
J picture
of atomic structure is to be developed, a zero would be a nice number to place at the
inert elements. A series of 4p spins and zero 2p spins would get us to the end of a row on the Periodic Table. This
in fact corresponds to the inert gases. The atomic number, element name, and mathematical
pattern can be given as follows:
2 Helium 2(1)2
10 Neon 2(1)2 + 2(2)2
18 Argon 2(1)2 + 2(2)2 + 2(2)2
36 Krypton 2(1)2 + 2(2)2 + 2(2)2 + 2(3)2
54 Xenon 2(1)2 + 2(2)2 + 2(2)2 +
2(3)2 + 2(3)2
86 Radon 2(1)2 + 2(2)2 + 2(2)2 +
2(3)2 + 2(3)2 + 2(4)2
Addition of another
2(4)2 takes
us to element 118, the last member of the Periodic Table.
Note that the above table consists only
of 4p spins. The effect of the 2p spin will thus be to "walk
across the rows" and fill in the other elements between the inert elements. But a
difficulty becomes apparent. The inclusion of a zero term creates 2n2+1 terms for a row, instead of the 2n2 actually shown in the Periodic Table. Although the
zero is nice to have, we now have to get rid of an "extra" element somehow.
Let's back up a bit and play with some possibilities
—the kind you would
scribble out on a napkin while waiting for your order at a restaurant. Hmmmm . . . Maybe
the pattern goes like this:
Li Be B C N O F Ne
1 2 3 4 3 2 1 0
This pattern produces the required number of 8 elements.
It also produces a zero at the inert gas Neon. But the pattern also raises some obvious
questions. Why count up to carbon and then back down? And what (numerically) would
distinguish B from N, or Be from O, or Li from F?
A scheme that uses both positive and negative numbers
would be more natural, and would allow the numbers to be different. In this example, they
would sweep from -4 to +4. Our second (or third, or fourth) guess might look like this:
Li Be B C C N
O F Ne
1 2 3 4 -4 -3 -2 -1 0
This seems more elegant and symmetrical, but introduces
new questions. Carbon is "something plus 4" as well as "something minus
4". This seems intuitively clumsy, and it gets worse with the other elements.
Ytterbium would be "something plus 16" and "something minus 16". It is
hard to believe that this kind of math would point to the same element.
However, an alternative to this situation would be to
claim the rows in the Periodic Table actually split in the middle, not at the ends. If the
magnitude of one of the 4
p rotations goes up a notch in the middle of the row, then maybe we
can subtract from it and end up with the same magnitude. This would mean that carbon is
both "something plus 4" and "something bigger minus 4".
The splits would occur at Carbon, Silicon, Cobalt,
Rhodium, Ytterbium and Nobelium. Mathematically these elements will have two forms.
Whether this implies a subtle physical difference is not clear. Chemically, they would
have to be indistinguishable in order to get rid of the "extra element" caused
by the use of +/- numbers for the 2p term. But physically they could be
space/time allotropes or space/time tautomers.
In this picture, the Periodic Table and the
"ordinary" subatomic particles can be specified by a set of three numbers, {n1,
n2, n3}. Both n1 and n2 specify the magnitudes of the 4p spin systems and can take on values of 1,2,3,4 (but not in
an arbitrary order). The third term, n3, specifies the magnitude of the 2p spin system and can take on values
ranging from -n2 to + -n2 , including zero.
A word of caution: The scheme presented above is very
foundational and corresponds to the Periodic Table at absolute zero and with no isotopes,
isomers,
charges, ionizations, radioactivity, etc. The pattern is based entirely on
atomic numbers, not atomic masses.
The subatomic particles introduced as the "missing
row" at the top of the Periodic Table, present additional challenges. None of these
particles would have mass, (i.e., the "towards motion" in all three dimensions;
without this motion, the particles would be moving at the speed of light in one or more
dimensions). Presumably, these subatomic particles would all be stable (there is nothing
in the picture to suggest they would be unstable).
Available candidates seem to be the electron and positron
for the 2
p spin system and the neutron and neutrino for the 4p spin system.
The electron and positron could be viewed as massless if
the charge is actually an
additional
rotation (or some kind of motion) which would distribute the 2p rotation in the other
dimensions. This would cause a small mass effect to become manifest. We normally think of electrons and charge as inseparable, but
electric wires can carry considerable current and yet do not bristle with static
electricity. A buss bar carrying 10,000 amps at 5 volts seems, electrostatically, like one
that has no current at all. The uncharged form of the electron would therefore be the one
that occurs within material aggregates and the charged one would be the
one commonly seen in free space or on the surface of such aggregates.
Similar arguments would apply to the positron, but the positron tends to disappear from
the environment. Unlike the electron, atoms apparently have an unlimited capacity to
absorb them.
The neutrino seems to qualify "as is". It is
massless, stable, and subatomic (no chemical properties). The commonly observed neutron,
in contrast, is apparently not part of this picture. It is unstable and has mass. This
leaves a place for an undiscovered particle. (If you are studying "cold fusion"
and are missing some neutrons, I suggest you look here.)
__________
We could also play with the notational
representation and see what we come up with. The first combination we attempt would be
{0,0,0}. This denotes something
composed of no 4p spins and
no 2p spins. In other
words, it is a non-entity, and is probably nothing more than a meaningless mathematical
artifact.
The next set would be {0,0,-1} and {0,0,0} and {0,0,+1}.
This too is probably meaningless. It includes the meaningless {0,0,0}and also has a
+/- 1, which according to the rules implied by periodicity, is illegal. There has to be at
least one full 4p spin
before there can be a 2p
spin. Said differently, there has to be an n2 term before there can be a range
around it of -n2 to + -n2 (which of course includes a zero).
Hence, the first legitimate set seems to be:
{0,1,-1}
{0,1, 0}
{0,1,+1}
The {0,1,0} entity is probably the "inert gas edition
of the photon". According to the discussion on the photon spin system (above), the
photon is composed of two one-dimensional (2p)
spins.
The {0,1,0} entity is composed of one two-dimensional (4p) spin. Reread that carefully. We could
say that this is sort of a Periodic Table version of the photon, or a non-photon photon,
ghost photon, stationary photon, etc. It is not an actual observable photon.
We have to guess what the {0,1,-1} and {0,1,+1} entities
are. These would be ghost photons with an additional 2p rotation, and the two rotations are opposite in sense. It is very
tempting to say that they are probably the electron and positron (or vice versa). There
are also suggestions here of the annihilation/pair production reactions. But if this is
the case, the
{0,1,0}entity has to be
a "nothing datum" of some sort (like the speed of light was, above). Otherwise,
combining the {0,1,-1} and {0,1,+1} would result in {0,2,0}—another illegal combination according to the periodicity
rules. But this might just be a notation problem. Maybe we should have used {1,0,0}
for the ghost photon and then {1,0,-1} and {1,0,-1}for the other two particles. Combining
them gets us back to {1,0,0} (the
"nothing datum" in altered notation) which is at least a legitimate entity
according to the periodicity rules. This would again be the ghost photon (one 4p spin).
Using the altered notation, the next set would be:
{1,1,-1}
{1,1, 0}
{1,1,+1}
The
{1,1,0}
is probably the neutron. It is the subatomic particle counterpart of the inert gases. As
mentioned before, it has to be massless and stable and cannot therefore be identified with the experimentally
observed neutron. One of the other combinations is probably the neutrino, and the
remaining one is apparently an undiscovered particle (another mystery for the "cold
fusion" people to ponder).
The next set, again using the altered notation, would at
first seem to be
{2,1,-1}
{2,1, 0}
{2,1,+1}
but because n is now 2, the third number can range from -4
to +4. So the pattern is more likely:
{2,1,-4} ???
{2,1,-3} ???
{2,1,-2} ???
{2,1,-1} Hydrogen ?
{2,1, 0} Helium ?
{2,1,+1} Lithium ?
{2,1,+2} Beryllium ?
{2,1,+3} Boron ?
{2,1,+4} Carbon ?
{2,2,-4} Carbon ?
{2,2,-3} Nitrogen ?
{2,2,-2} Oxygen ?
{2,2,-1} Fluorine ?
{2,2, 0} Neon ?
{2,2,+1} Sodium ?
{2,2,+2} Magnesium ?
{2,2,+3} Aluminum ?
{2,2,+4} Silicon ?
We now seem to be getting into the actual
Periodic Table. The {2,1,0} combination is probably Helium and the {2,2,0} is probably
Neon. Note that the inert gases are in the middle of these two sets, whereas in the
Periodic Table as commonly displayed, they are in the rightmost column. Allowing for this,
the {2,1,+1}would be Lithium. The {2,1,+4} and {2,2,-4} would be Carbon. Note that Carbon
occurs at the hoped-for "split in the middle" mentioned above. (The
"something" is the 2,1 and the "something bigger" is the 2,2). Silicon
will follow a similar pattern with {3,2,-4} being the other form.
But another problem becomes apparent. If
{2,1,-1}is Hydrogen, what do the {2,1,-2}, {2,1,-3}, {2,1,-4} represent? More undiscovered
subatomic particles? Some undiscovered elements? Another mathematical artifact? The
{2,1,-4} is particularly disturbing. It seems to equate to {0,1,0}which takes us all the
way back to the ghost photon. Apparently, the third number in this situation can only
range from -1 to +4, instead of -4 to +4. An explanation for this will have to be
found.
A picture of the atom, as described above, might be somewhat as shown.
It consists of:
two 4p spins (depicted as two spheres superimposed)
one 2p spin (depicted as a disk)
In conventional terms, "nuclear properties" are identified with the dual 4p spin systems and "atomic properties" (like chemistry,
ionization) are identified primarily with the single 2p spin.
There are no electron shells. The model consists entirely of intrinsic spin systems
(space/time ratios that are changing direction, but not position). The trapping of gamma
rays and massless particles will affect the shapes of the 4p
spin systems. Changes in ordinary energies will affect the shape of the 2p spin system.
See also Ray guns, Nuclear Isomers, Rydberg Atoms for more about atomic shapes.
Clearly, there is a lot more that must be worked out and
explained. I can only offer a starting point and hope that atomic theory will some day
return to common sense and offer models that are far more intuitive than those presently
in use. (A "Summary of Experimental Evidence Which Should Be Represented by the
Model" is given in The Atomic Nucleus by Robley D. Evans, 1955, under
"Models of Nuclei", p. 357-358)
If space
and time exist in discrete units, then how big, or how small, are these units?
Quantum phenomena become most apparent at very small spatial dimensions. So this is
undoubtedly a pertinent question. As two atoms approach each other in space, they
eventually reach a point where they are separated by one unit of space. This space
has no "inside" and so the atoms must stop moving together in space. If
everything in the universe is composed of space/time ratios, then it follows that this
ratio cannot become zero, because a zero would mean that it is not in the physical
universe (I think mathematicians would say "not a member of the set"). This unit
space is therefore the minimum separation possible in space. But this does not
lead to a dead end. The time component of the ratio can still change. In this
case it would represent a change of position of coordinate time instead of a change of
position in coordinate space. Also, instead of a change of position in space or
time, a change of direction appears to be another possible behavior atoms and
other entities can express. We call the latter "intrinsic spin" and it is also
regarded as a quantum characteristic.
The above arguments would apply to space/time
ratios in general. Hence, there must be such things as a unit of space, a unit of time, a
unit of speed, a unit of frequency, a unit of mass, etc. Nature only gives some clues as
to what these might be. We have to make some educated guesses, try them out, and see if
the results make sense.
Let's attempt to use the the Rydberg formula
for spectra to derive some possible unit magnitudes for space, time, and frequency:
This approach yields 912 Angstroms
as the limit of the Lyman series for the hydrogen atom (Quantum Chemistry,
McQuarrie, 1983, p.19). But now there is a problem of interpretation. The number is
derived from a spectral formula and photons are rotational entities. Should we apply a
factor of 2p somehow to
this number? Or because we are looking at a rotational entity from a linear reference
system, maybe we should regard the 912 Angstroms as a projection of a diameter? That would
make the basic unit (the radius) 456 Angstroms instead of 912. Likewise, is the unit of
time 3.05 x 10 -16 seconds or half that? (See also Rydberg
Atom )
We will leave these problems for later
resolution. For now we will say, tentatively, that the Natural Quantity of space is either
912 or 456 Angstroms. The important point is that space is not infinitely divisible. Atoms
can approach each other in space only as close as one spatial unit. After that, the
spatial separation becomes fixed at one unit and all further variation must be in time.
The time magnitude representable in the reference system will make the space unit involved
in the separation appear as 1/t. The space/time ratio thus appears as 1/t/1/t or
1/t2. This suggests how the reciprocal square integer terms in the Rydberg
equation originate. They are a direct result of the quantization of space and time. . . . (11-3-03 note: For more on the reciprocal squares see The
Feynman Lectures on Physics, Vol. 3, p. 19-5, equation 19.24. What I see in the math
is a hint of how the quantum world "connects" with our ordinary world through
the 1/n2 relation.)
"But wait a minute!" you say. "The
separation between atoms in a solid is on the order of a few Angstroms. That is much
smaller than the 912 or 456 Angstroms that are supposed to be the spatial minimum. How do
you explain that?" Well, that is exactly the point. The separation is not
spatial. It is temporal. The question now becomes one of how temporal
"distances" appear to an observer in a spatial reference system. How does an
inherently "when" type of distance manifest itself in an inherently
"where" type of reference system? Can what we think of as duration manifest
itself as a spatial length? How does a location in time relate to a location in space?
Understanding what is meant by terms like locality, position, separability, and metric
spaces is central to acquiring an intuitive understanding of quantum phenomena.
Large Scale
Space/Time and Time/Space Phenomena
Consider space and time in a more general
situation. Roll a marble across your desk. You will find that your mind makes some
abstractions in trying to apprehend this phenomena. "It starts out here, and goes to
there, and takes a certain amount of clock time to do this." The motion is the
primary phenomena, but concepts like position, locality, time, trajectory, are
abstractions that are used to describe it. We call the movement of the marble motion in
space.
The same idea applies if you have motion in coordinate time. If you were entirely
within the temporal system it would look exactly the same as the view from entirely within
the spatial. "It starts out here, and goes to there, and takes a certain amount of
time to do this." You of course have no way of knowing that space and time have been
swapped. To you, coordinate time will look like ordinary space, and the clock space (the
"time" you think you are using) will look like ordinary time. Nor would you
guess that the desk and the marble are made of atoms with space/time relationships
that are the inverse of the spatial ones.
All this is very ordinary. But things get really interesting when you are in one system
looking at phenomena that are at least partially in the other. This situation
applies in quantum mechanics, where there is temporal activity or phenomena at unit
spatial distance, but this unit distance is still localized in the spatial system
(e.g., you can take the phenomena and put it under a microscope).
It also applies to high speed phenomena. Speed, as a ratio of quantized space and time,
will itself be quantized also. As the speed of a particle approaches that of light, speed as
a change of position in space, approaches a limiting value, the speed of light.
Speeds above light are possible, but they are in coordinate time, and a change of position
in coordinate time cannot be readily depicted in a spatial reference system. Hence,
quantum phenomena and high speed phenomena have something in common: temporal motion. We
are trying to figure out how this motion appears to an observer in a spatial reference
system. Knowing what happens to s/t and t/s ratios near the speed of light may give us
some additional clues.
The reasoning goes like this:
Note that the s/t = 0 in the illustration is used in the everyday sense. It means that
if an unlimited amount of time becomes associated with a particular spatial position, then
the speed of that position is zero. It is not intended to mean that the speed does not
exist or that it is not part of the physical universe.
The transform implied by the above example can be applied
as shown below:
In case you feel uncomfortable with the idea that
there is another half of the Universe that is an inverse counterpart to the spatial one,
you might find some comfort in the fact that scientists have been playing around with
ideas vaguely similar to this at least since the 1950s:
"[Some] scientists argue that nothing less than an
entire universe of shadow matter, made of particles nearly identical to neutrons, protons,
and electrons, shares our space. We just can't tell it's there.
Welcome to the mirror world, in which every particle in
the known universe could have a counterpart. This cosmos would hold mirror planets, mirror
stars, and even mirror life.
The concept may sound as fantastic as the world that Lewis
Carroll's Alice encountered through the looking glass. But proponents of the mirror world,
a notion that dates back to the 1950s, say its existence would solve a number of puzzles
in physics and cosmology."
See "Through the Looking Glass: Reflections on a
mirror universe" by Ron Cowen in Science News, Vol 158, No 11, p. 173 (Sept.
9, 2000). The world they are proposing is not the same sort as the one I am
attempting to describe, and so there is no point in elaborating. I am just pointing out
that "main stream science" has been playing around with similar ideas.
Unfortunately, they have not proved fruitful because they are based on "facts"
that turn out to be illusions, if not just a lot of outright loony ideas. But we will see
how correct answers come easily, even trivially, when the real situation is understood.
Let's consider, as an example, the existence of what is known to
astronomers as "diffuse background radiation". The temporal half of the Universe
has the same statistical properties as our spatial half. It has the same type of stars
emitting the same type of light (infrared, visible, ultraviolet, X-ray, gamma ray, etc) as
viewed from within that system. And the stellar populations have the same statistical
distributions as stars in our spatial system. The stars that are visible in our sky are
mostly blue. It just happens that blue stars are also the brightest and that is why we see
them. But cooler stars of lesser brightness are actually more plentiful. These would be
infrared or near infrared stars. The same would be true of the temporal system when
viewed from within that system.
Let's apply the Master Transform and see how how this plentiful radiation from infrared
temporal stars would appear in our spatial system. First, we will calculate frequency of
radiation. Let's take infrared as 1013 Hertz, and approximate the Rydberg
frequency as 1015 Hertz (so we can do this mentally). Infrared is thus
1/100 of a Natural Quantity of frequency. The inverse of that is 100, and 100 times the
Rydberg frequency is 1017 Hertz. This radiation is in the low energy X-ray
range. As seen from our sky then, the temporal infrared radiation should appear as
X-radiation. It should also be plentiful, but unlike normal starlight, it will
distributed in a completely uniform and diffuse manner (isotropic and homogeneous). Is
this the case?
Actually, the existence of the X-ray background has been known for
decades:
"Even the most contentious people usually agree that the night sky is dark. Don't
try arguing the point with an astronomer, however. In 1962 researchers discovered that
when seen through instruments sensitive to X-rays, the sky glows with a bright and oddly
uniform intensity. This pervasive radiation, rather unpoetically known as the diffuse
X-ray background, has eluded easy explanation. Roughly 25 to 30 percent of the background
has been attributed to quasars. . . . The origin of the rest has been a persistent
mystery. . . . The spectrum of the X-ray background closely resembles that of a thin, hot
gas. (Scientific American, March, 1991, p.26, "X-ray Riddle: Cosmic background
is still unexplained." See also Astronomy, April 1991, p.22, "X-rays
Light Up Philadelphia")
The same reasoning can be applied to other portions of the electromagnetic spectrum.
There is also a gamma ray background, for example. And you have probably heard of the
cosmic microwave background when the COBE observations were done some years ago. The
microwave background was found to be uniform to 1 part in 100,000 to 1 part in 1,000,000.
It does not vary with time of year or direction in space. This is an extraordinary
uniformity in what we see as an otherwise lumpy universe. Conventional science has not yet
devised a credible explanation for the entire spectrum of background radiation.
Furthermore, temporal stars are made of inverse atoms just like our spatial stars are
made of normal atoms. This implies that there should be cosmic background particles
just like there is cosmic background radiation. The omnipresent gravity of stars
and galaxies tends to sweep our space free of particles. This makes our space transparent,
and also puts most of the particles in a high energy environment inside stars. The
counterpart of this should be happening in the temporal system. According to the
diagram, the temporal zero speed, as seen from the spatial zero, is 2c distant (note that
the diagram is considering only one dimension). The spatial reference system can only
depict speed as a change of position in space up to 1c; speeds in excess of this
will appear as energy. Low speed temporal particles, such as those in interstellar
temporal gas or atoms in a temporal planet, will appear to us spatial observers as a
diffuse background of particles moving through our reference system at the speed of light
and possessing extremely high energies. Atoms in a temporal star will appear likewise,
except that they will have lower energies, and be much more plentiful.
It should be pretty clear that these particles are what the scientific community
has been calling "cosmic rays". They are in fact extremely isotropic and
homogeneous just as we would expect if they have a temporal origin. As for their energies
consider this:
"We find that there is a flux of about 1 particle/cm2 sec at 10 GeV.
Above 1020 eV, we can expect to see only about 5 particles per century
per square kilometer! . . . If we add up all the energy carried by all of the CR [Cosmic
Ray] particles, we find that the rate of arrival of CR energy on the Earth amounts to
about 100,000 kilowatts (105 kw) —about one billion times less than the
energy arriving in sunlight, but comparable to the total energy that we receive in
starlight." (Cosmic Rays, Michael W. Friedlander, 1989, p. 84, 86)
A particle with an energy of 1020 eV has roughly the energy of a golf ball
or baseball in flight.( http://www.sciam.com/0197issue/0197swordy.html
, http://www.sciencedaily.com/releases/1998/12/981216081217.htm)
They are extremely rare, and do not deposit all their energy in one collision.You don't
have to worry about getting hit by one. A natural process that could accelerate
particles to 1020 eV and spray them uniformly all over the Universe is simply
inconceivable. Even earthbound particle accelerators cannot produce particle energies
that are even close to 1020 eV. It is much more reasonable to view these
particles as originating in a temporal system with near zero speed; this will explain both
the extremely high energy and the diffuse nature of these particles when seen from a
spatial system.
Cosmic ray particles also seem to be within the required
mass range. These inverse atoms with inverse masses must be members of an "Inverse
Periodic Table." If we choose 1 a.m.u. as the likely natural unit of mass, then the
mass range of the Table can be worked out. However, cosmic ray particle mass are usally
stated in terms of electron masses. If we equate one amu with 1835 electron masses,
then the mass range of the "Inverse Periodic table" extends from (2/1)(1835) to
(2/118)(1835) electron masses. This range, 3670 to 31 electron masses, apparently does
encompass the range of cosmic ray particle masses. (note that the particles with the
highest atomic number are actually the least massive particles)
Stars we can see in space apparently generate power in pulses. The pulses, in the case
of stars like our Sun, are apparently several years apart, and last only a few to several
seconds. Stars can also blow up in a supernova explosion if the power generation process
is not self-limiting, as appears to be the case sometimes with massive blue supergiants.
The same thing should be true of a temporal star. Under either of these conditions
the interior atoms in such a star will briefly have such an extremely high amount of
energy that their speeds may reach out across the speed of light boundary and approach the
spatial zero. This will "stop" the motion in the context of the spatial
reference system. Such an energetic event would only last a few to several seconds, and it
would localize in space for approximately that amount of time. It is not clear that any
phenomena corresponding to this have been observed. The mysterious gamma ray bursts
("cosmic flashbulbs") observed by astronomers seem to be the only possibility so
far.
Quantum Mechanics and the
Hamiltonian
Back in the 1830s Sir
William Rowan Hamilton devised a system of classical mechanics that would turn out to be
ideally suited, with some reformulation, for use in quantum mechanics nearly 100 years
later. In classical mechanics the Hamiltonian, H, of a system is the sum of the kinetic
energy, T, and potential energy, V, of all the particles present. The relation is
expressed as:
In quantum mechanics the expression must be
converted into an operator:
The Hamiltonian operator appears in Schrödinger's wave
equation in the following form:
One cannot help but wonder why Hamilton's description of motion is
so useful in quantum mechanics. Newton's F=ma and its various derivatives were much more
popular than Hamilton's. Yet in quantum mechanics, we have to use Hamilton's.
In classical form the Hamiltonian is expressed as:
This states that for a conservative system, the total energy is the sum of the kinetic
and potential energy. Why does energy seem to be the critically important quantity in
quantum mechanics?
Where else is energy used as a measure of motion? As you might recall from the
discussion in the section above, energy was used as a measure of motion at very high
speeds. When a particle is accelerated to speeds near that of light, the acceleration
begins to decrease from the classically expected values. A presumably constant
accelerating force does not produce a constant acceleration. The applicable equation is
F=ma and can be rewritten as a = F/m. It is apparent that the
decrease in acceleration could be due to a decrease in Force or an increase in mass.
Scientists chose the latter as an explanation. However, mass, at least in the Periodic
table, seems to come in discrete units and the idea that it could vary continuously is an
uncomfortable premise.
On the other hand, we have readily available examples of force decreasing with
speed. In a common electric motor a magnetic field rotating at constant speed drags the
rotor around until it reaches near-synchronous speed. At synchronous speed the currents
induced in the rotor approach zero, and the accelerating torque does likewise. The rotor
thus approaches a speed limit. It can never go faster than the rotating magnetic field.
Or, if you don't understand motors, try my tennis ball and roller skate illustration. A
roller skate is affixed to a linear track and a line of people on the ground
shoot constant velocity tennis balls at it as it flies by. A little thought will show that
the roller skate can never travel faster than the tennis balls (because if it goes faster,
the tennis balls will never hit it). Again this places a limit on the accelerating force.
But . . . that explanation won't work either. In the case of a particle being
accelerated to speeds near that of light, the energy keeps on increasing even
though the speed does not. As you may have guessed from reading the above
section, we are talking about speed in coordinate space. As one of the diagrams
shows, speeds in coordinate time appear to us spatial observers as a t/s ratio
instead of a s/t ratio. A t/s ratio undoubtedly corresponds to some sort of physical
property. But what is it? It would be nice if it turned out to be energy.
We can determine the space/time dimensions for energy from an equation like E=mc2.
But to do this, we will have to make a choice for the space/time dimensions for mass. As
explained in Advanced Stellar Propulsion Systems, mass
moves opposite to the three-dimensional progression of space and time. The progression
moves everything that has no independent motion (like photons) to increasing spatial and
temporal separation. At the speed of light these motions are equivalent (as explained
above) and so the dimensions are s3/t3 or t3/s3.
(The expansion in time, incidentally, exactly balances out the expansion in space, both
physically and numerically; the "fabric" of the Universe as a complete whole
(spatial + temporal) is "stationary" or "doing nothing" at the speed
of light, neither expanding nor contracting. That is why I view the speed of light as the
nothing datum for the physical universe.) Mass moves opposite to the expansion of
space. That suggests its dimensions are somehow "opposite" to s3/t3.
We seem to have two choices for "opposite": use a minus sign or use
the multiplicative inverse. The former choice seems to lead to a conceptual dead end, so
we will try the latter. Substituting these into the equation gives:
The symbol, [=], means that only the dimensions are being considered in the equation.
We see that energy does indeed have the space/time dimensions of t/s. But will this
work in the Hamiltonian? To find out, we will have to re-write it slightly. The p will be
expressed as mechanical momentum, mv, and so kinetic energy will be 1/2
mv2 ; potential energy is force through a distance and will be expressed as
(ma)x. So we have:
Note that the units of the two terms are consistent, as they must be before they can be
added together. Note also that they have the space/time dimensions of energy!
This is really an astounding result. Quantum phenomena are normally
studied at very low energies (room temperature or even cryogenic temperatures) and at very
tiny dimensions. Relativistic phenomena occur at high speeds or high temperatures and can
involve astronomical distances. You would not expect one to have much in common with the
other. But both types of phenomena require their activity to be described in the
space/time dimensions of energy. Quantum phenomena occur because of a unit space
limitation and relativistic phenomena occur because of a unit speed limitation.
Either of these limitations force the phenomena to be temporal, rather than spatial. The
measure of speed in the spatial system is s/t. In the temporal system, as seen from the
spatial, the ratio inverts and becomes t/s. It is still speed, but temporal speed is seen
as energy from our spatial standpoint. The Hamiltonian describes a system in terms of its
energy, and so its use in quantum mechanics is a natural application.
As I have said above, a correct paradigm makes understanding these
mysteries easy. Almost trivial . . .
"Faith is . . . the conviction of things not seen. . . .
By faith we understand that the worlds were prepared by the word of God,
so that what is seen was not made out of things which are visible." (Hebrews
11:1,3)
Commutation
and Angular Momentum Note: this new section is being edited and
reviewed.
Commutation and angular momentum are
very important topics in quantum mechanics. But to discuss them, I have to first say a bit
about operators, non-commutative mathematics, and commutators.
An operator is a mathematical
symbol that specifies an operation (some kind of action) on its operands, which can be a
simple number, a variable, or a function. For example, the + symbol is the “addition
operator” as in 2 + 3 = 5. The x symbol is the “multiplication operator” as in 2 x 3 = 6. The symbol d( )/dx is
the “differentiation operator” as in d(x2)/dx = 2x.
In quantum mechanics, the so-called
“observables” are represented by operators.
“In classical mechanics one is
accustomed to working with the distance x, the momentum p, the total
energy W, etc. These are examples of quantities called dynamical variables. In
the solution of practical problems one finds expressions involving these variables, which
will give numerical values under any specified conditions.
In quantum mechanics the dynamical
variables play a completely new role. They are converted by a set of rules into
mathematical operators which then operate on the wave function Y" (Introduction to
Quantum Mechanics, C. W. Sherwin, 1959, p. 13)
To create a quantum mechanical
expression, we write the applicable classical expression and then convert it into the
quantum mechanical version by making certain substitutions. For instance, we may have a
situation where the expression for total energy (kinetic + potential) would be useful.
Classically, it is expressed as:
To convert it into a quantum mechanical
expression, all we need to do is make the appropriate substitutions. Usually, we can find
these substitutions in any introductory book on quantum mechanics. They are often given in
tabular form:
Dynamical variable:
becomes this operator:
x
x
f(x)
f(x)
px
h/(2pi){ ¶ /¶x }
W
- h/(2 pi){ ¶ /¶t }
These operators will
“operate” on a wave function, Y, which is assumed to exist, and which is assumed to contain all
that is knowable about the quantum mechanical system. We would then solve the resulting
equation to get the particulars.
And so, in this example, we can write
the substitutions into the classical expression for total energy. Wherever we find
“x” (which classically means “x coordinate position” we put in “
x x
” (which in operator terminology, means “multiply by x”. The multiplication
symbol is always omitted however, and so in the resulting equation, you will see just
“x”; make a mental note that this is now an operator, not the usual
kind of algebraic variable. The same applies to constants like h and i.
However, time, in quantum mechanics is a parameter, not an operator, not
an observable.) When we make the substitution for px (momentum
along the x axis) we note that this term is squared, and so we have to square the
expression that is substituted for it. We will assume that Y is some function of position
and time and will write that aspect in the usual notation as Y(x, t) It all comes out as:
How did physicists arrive at these
particular substitutions? Historically, it came down to noticing some strong clues, and to
a lot of intuition and effort. The substitutions that worked were retained and the ones
that did not were dropped. (There are also other substitutions applicable to relativistic
systems, electrical and magnetic systems, and so forth.)
But let’s not get sidetracked. I
just want you to know what an operator is at this point, and to get an inkling about how
they are used in the mathematics of quantum mechanics. Our ultimate topic is commutation
and angular momentum.
Operators, are, in general, not
commutative. That means that when operators are applied to, say, a function, the final
result will depend on the sequence of the operations. Applying operator A and
then operator B, will give a result different from applying operator B
and then operator A.
Let’s perform two identical
operations on some term like “x2 +1” as an example and see what
happens when we apply the operators in different sequence. We will use multiplication and
differentiation because these are commonly used in quantum mechanics:
multiply by x, then
differentiate by x:
x2 +1 gives x3 +x which then gives 3x2 +1
differentiate by x
then multiply by x:
x2 +1 gives 2x which then gives 2x2
Obviously 3x2 +1 is not equal
to 2x2. The order or sequence of the operations was important.
Mathematicians have a special word for
the difference in these results. It is called a “commutator”:
“A commutator of
two operators A and B is denoted [A,B], and is the
difference AB-BA. The symbol AB means that operation B is
performed first, and is followed by operation A; BA implies that A
precedes B.
Two operators are said to commute if
their commutator is zero. If a commutator is nonzero, the final result of performing two
operations depends on the order in which the operations are done: operation A
followed by operation B results in a different outcome from operation B
followed by A. . . .
The importance of commutators in quantum
mechanics comes from the identification of physical observables with operators, not all of
which commute with each other. It turns out that if two operators do not commute with each
other, then the observables they represent cannot be determined simultaneously. . . . The
fact that some operators do not commute with each other is a principal factor underlying
the differences between quantum and classical mechanics." (Quanta: A Handbook of
Concepts, P. W. Atkins, 2nd ed.,1994, p. 60-61)
Angular momentum in quantum mechanics, it turns out, is defined by a
set of commutation relations:
“In quantum mechanics, an angular
momentum is most generally defined in terms of a set of commutation relations. Any
set of observables represented by three operators that satisfy the commutation rule [jx,jy]
= ihjz (and cyclic permutations of the subscripts) is called an
angular momentum. (Quanta: A Handbook of Concepts, P.W. Atkins, 2nd ed.,1994, p.
8)
Customarily, the
symbol j is used to denote total angular momentum, l is used
for orbital angular momentum, and s is used for spin angular momentum. For our
purposes we will use the following cyclic permutations:
[lx,ly]
= ihlz [ly,lz] = ihlx [lz,lx] = ihly
For completeness I'll
include [l2, lq] = 0
The former can be written compactly
in vector notation as:
l
xl=ihl
But be careful here. Anyone who knows
elementary vector algebra will tell you that the cross product of a vector with itself is
always zero, not something like ihl. Again, remember, that
despite the appearances, l is a vector operator, not a
classical vector.
Ok. So what is the meaning of something
like [lx,ly] = ihlz ? And why
should we care? Consider the following diagram which shows two results of two
rotational displacements that are performed in different order. The illustration is
greatly exaggerated to show the effect, but the results are equally valid with small
rotations. In the first sequence, point P on the surface is rotated around the x
axis, say 30 degrees. This result (P1) is followed by a rotation around the y
axis, say 60 degrees. Its final position is P2.
For the second sequence we start all
over again at the original P. The second sequence rotates around the y axis first by
60 degrees (resulting in P3) and this is followed by a 30 degree rotation about
the x axis (resulting in P4).
Then we evaluate what has happened.
We see that the two final points do not coincide. In other words, rotational
displacements are non-commutative. Furthermore, because the two endpoints end up at the
same "latitude", the difference in the locations (P4 and P2)
could be expressed as a small single rotation about the z axis. This small rotational
displacement is the geometric equivalent of a commutator. It is the analogue of [lx,ly]
= ihlzin quantum mechanics.
". . . the difference between two
infinitesimal rotations is equivalent to a single infinitesimal rotation . . . about
the z axis, which is geometrically plausible. . . . The reverse
argument, that it is geometrically obvious that the difference is a single rotation,
therefore implies that [lx,ly] = ihlz .
Hence, the angular momentum commutation relations can be regarded as a direct consequence
of the geometrical properties of composite rotations." (Molecular Quantum
Mechanics, Peter Atkins, Ronald Friedman, 4th ed. p. 162)
This also seems like a convenient place to introduce the Pauli spin
operators, which express the same idea:
Let's see what [Sx, Sy] evaluates
to:
For this one you'll have to use the rule for matrix
multiplication:
Hence:
Hence, [Sx, Sy]
evaluates to 2iSz .
I hope this helps educate your intuition
about commutators. But what is the significance?
". . . we shall say that an
observable is an angular momentum if its operators satisfy these commutation relations.
[main text] Because all the properties of the observables are the same, this seems to be
an appropriate course of action. However the procedure does capture some strange
bed-fellows. The electric charge of fundamental particles is described by operators that
satisfy the same set of communication [sic] relations, but should we regard
it—or imagine it—as an angular momentum? Electron spin is also described by the
same set of communication [sic] relations, but should we regard it—or
imagine it—as an angular momentum? [footnote]" (Molecular Quantum Mechanics,
Peter Atkins, Ronald Friedman, 4th ed. p. 100-101)
What I hope you will get from this
presentation:
1. That spin and angular
momentum are very important in quantum mechanics. This reinforces my suspicion that
composite rotations are the basis for the existence of all particles (photons, electrons,
atoms, etc., —everything).
2. That complex numbers are
used in spin operator representations, and that although spin "is a rigorous feature of real Dirac theory . . . it remains hidden
in the matrix formulation". (Hestenes; See
Effects of Spin )
3. That commutators for angular momentum
have an intuitive geometric interpretation.
4. That quantum mechanical
wave equations can be constructed from classical analogues.
5. The notion that the
mathematical machinery of quantum mechanics is “too classical”, too indirect,
and unnecessarily cumbersome.
The last point needs some elaboration.
In The Origin of Intrinsic Spin I pointed out that
rotations result in a difference with the linear progression of space/time, and that in
turn creates a "thing" that may be observable. In the example given there,
gravitation evolved out of the concept of fundamental rotations. The implication is that
electrical, magnetic, and gravitational forces can actually arise from the
"geometrical properties of composite rotations" (Atkins, as above). And
within the “quantum realm” the forces are orientable (due to spin) and have an
actual three-dimensional geometry (but it is in coordinate time, not space). These
characteristics result in the existence of the whole Periodic Table and all the chemical
relationships that are implied by it. It should be possible to calculate dipole and
quadrupole moments, bond angles, bond energies, bond distances, transition and decay
probabilities, and so on from very fundamental principles based on composite rotations.
The methodology would state the problem directly in terms of s/t ratios), work out the
solution with the operations inherent to that realm, and then translate the results back
into the spatial reference system. This would seem to be a more direct approach than that
currently used in quantum mechanics.
Fun fact: In the Stern-Gerlach experiment if spin up
(z+)and spin down (z-) beams are combined, they produce a spin polarization in the x-y
plane not simply "spinlessness". ( Unfortunately, I cannot find the
reference for this statement)
For the
mathematically inclined, see also "How to Derive the Schrödinger Equation" by
David W. Ward, Feb 2008 at: http://arxiv.org/PS_cache/physics/pdf/0610/0610121v1.pdf
(One thing that I realized when reading this article is that the Schrödinger equation actually has the form of a diffusion
equation, not a wave equation. A diffusion equation uses the second derivative of
space, and the first derivative of time, whereas the classical wave equation uses
the second derivative of space with the second derivative of time.
Ref: Engineering Mathematics Handbook, Jan J. Tuma, 2nd ed., (1979)
p. 210-211)
html 5/07
The
Problem of Quantum Reality
An issue among physicists, and
the subject of much debate in both popular and scientific publications, is about the
picture of reality presented by quantum mechanics.
Reality is defined as
something that is factual, objective, actual, not merely seeming, pretended, or imagined.
If you find it hard to believe that physicists,
of all people, are confused about what constitutes reality, then consider the following
quotations:
"Questions like these [about
elementary particles] raise doubts as to whether the conceptual scheme of nuclear physics
is a ‘real’ account of the structure of the universe." (Modern Science
and Modern Man, J. B. Conant, (1952), p. 46)
"The physicist thus finds himself in a
world from which the bottom has dropped clean out; as he penetrates deeper and deeper it
eludes him and fades away by the highly unsportsmanlike device of just becoming
meaningless. No refinement of measurement will avail to carry him beyond the portals of
this shadowy domain which he cannot even mention without logical inconsistency. A bound is
thus forever set to the curiosity of the physicist. . . . The world is not a world of
reason, understandable by the intellect of man, but as we penetrate ever deeper, the very
law of cause and effect . . . ceases to have meaning. The world is not intrinsically
reasonable or understandable; it acquires these properties in ever-increasing degree as we
ascend from the realm of the very little to the realm of everyday things; here we may
eventually hope for an understanding sufficiently good for all practical purposes, but no
more." (Reflections of a Physicist, P.W. Bridgman, (1955), pp. 185-186)
"When we thought we were studying an external world
our data were still simply our observations; the world was an inference from them. Until
this century it was possible to make such an inference intelligibly . . . But now we find
that . . . we can no longer express them as the structure of an external world unless we
accept a world which is arbitrary, irrational and largely unknowable." (The
Scientific Adventure, Herbert Dingle, (1952), p.260)
"The ‘real’ world is not only unknown and
unknowable, but inconceivable—that is to say, contradictory or absurd." (A
Century of Science, Herbert Dingle, (1951), p. 315)
"Some physicists would prefer
to come back to the idea of an objective real world whose smallest parts exist objectively
in the same sense as stones or trees exist independently of whether we observe them. This
however is impossible." —Werner Heisenberg
Physicists
arrived at these distasteful conclusions only after decades of debate and examination of
perplexing experimental facts. In the single momentous year of 1925 three quantum theories
had appeared on the scene: Schrödinger's wave mechanics, Heisenberg's matrix mechanics, and Dirac's
transformation theory. Although they seemed quite different, they were found to be all
mathematically equivalent. (A forth one, by Feynman, and fundamentally different, would
appear in the late 1940s). The picture of reality implied by these theories was simply
unbelievable and was the focus of much debate. The famous Bohr - Einstein debates over
quantum theory and reality took place from about 1925 to 1935. Eventually the views of
Niels Bohr, Werner Heisenberg, and Max Born prevailed and by the mid 1930s the
"Copenhagen interpretation", as it came to be called, emerged as the generally
accepted doctrine. This doctrine holds, among other things, that there is no deep reality
to our world. This view is easily seen in the above quotations.
Mind you, there were no
debates about whether an electron exists or that it possesses innate attributes like mass,
charge, and spin. But it could not be denied that other attributes, like
position/momentum, seemed to be inextricably linked to the measurement process. The former
were called static attributes and the latter, dynamic attributes. The electron seemed to
have no innate position, or even to possess an infinite number of simutaneously existing
positions, until a position measurement was actually made. If you took away the
measuring apparatus, you would actually take away the position attribute.
A rather loose illustration would be like
asking "What is the color of a chameleon?" and the answer is that "It
depends on the color of what it is sitting on." Well, how can that be? Doesn't a
chameleon have a color in the same sense that trees and stones have color? In the thinking
patterns of a quantum physicist, a chameleon is a creature that simutaneously exists in
the form of multiple colors, and its "color" becomes actualized only when an
observation is made. The "color" of a chameleon is thus a dynamic, rather than
static, attribute.
We think of color as a real property
of an object. but we also know it is not an innate attribute. Several years ago, many
people here in Arizona wanted low pressure sodium vapor lights required for outdoor
commercial lighting. They wanted these lights used because they would cause less
interference with the sensitive telescopes at Kitt Peak. But many other people did not
want them because they made everything look weird. I drove to a local Kmart store one
night where the parking lot was illuminated with these lights. My white car appeared to be
yellow. Two cars next to me were both black, until the headlights of a passing car
illuminated them and one turned red and the other blue. If you had been there, would
you have said that color is a "real and objective" property? Or rather that
color depends somehow on external conditions and a "coupling to the observer"?
Let's make this example a
little more extreme. Take a close look at a yellow spot on a color TV display or a color
monitor on a computer. If you look at the pixels with a 15x jewler's loupe, you can see
that there is no yellow phosphor on an RGB (Red, Green, Blue) display. Yellow is made by
exciting the red and green phosphors and the combination is perceived as yellow. Ask
yourself, using Heisenberg's words about trees and stones, whether Yellow exists
objectively on a color monitor in the same sense that Red and Green exist. You could say
that Yellow has "no deep reality" but that in the words of Bridgman, "it acquires these properties in ever-increasing degree as we ascend from the
realm of the very little to the realm of everyday things."
And what would you think if
one day your monitor created Yellow from Red and Green, and the next day from White and
Black, and the next day from Purple and Orange, and so on? Let's say that you study this
for years and conclude, reluctantly, that a color monitor can make Yellow from any color,
just ANY color! Wouldn't you conclude that the world of Yellow on color monitors is, in
the words of Dingle, "a world which is arbitrary, irrational and largely
unknowable"? Or that the answer to this question "fades
away by the highly unsportsmanlike device of just becoming meaningless"?
(Fortunately, color monitors are not this extreme, nor is color itself.)
The nature of reality is an issue
that Scriptural Physics cannot dismiss. God surely knew, and even intended, for man to
understand the realm of the microphysical. Things there should be no less real, no less
objective, no more bizzare, than anything else in the physical universe. (see What is Scriptural Physics? for more on that view) The
strange behavior of the quantum world must have a very reasonable and satisfying
explanation. Scriptural Physics is, again, not a theory with the answers, but rather
a methodology that will point out the path that leads to such explanations.
html 2/02
The Problem of Quantum Locality
"Locality" is defined as:
"The condition that two events at spatially
separated locations are entirely independent of each other, provided that the time
interval between the events is less than that required for a light signal to travel from
one location to the other." (McGraw-Hill Dictionary of Scientific and Technical
Terms, Sybil P. Parker (editor), 1994, 5th ed., under "locality")
Let's say that I have two
cigars and a cigarette lighter. One cigar is located here on Earth and is in my immediate
possession and control. But the other one is located out on Mars in a top-secret
laboratory there. I claim that these cigars have been "correlated" by a special
process that "entangles the phase of their matter waves" and that when I light
the one here on earth, the one on Mars will light up too, and at the very same time. You
claim that this idea is ridiculous, that no such thing can possibly happen, that it defies
common sense, that the idea is "voodoo physics", "magic", "spooky
action-at-a-distance" and so forth. We devise a careful experiment which we both
agree will settle this claim in a definite, unambiguous way. Then we perform the
experiment. It turns out that the cigar on Mars lights up exactly when I light up the
cigar on Earth. We repeat the experiment and even retry it with several different
variations. But we always get the same result. We agree that there is no question at all
about what actually happens to the cigars. But we now start wondering about just what the
word "location" means. How can two things be located far away from each other,
and yet—somehow—also act like they are located in the same place?
This is essentially the situation that arose
after Einstein, Podolsky, and Rosen jointly wrote a scientific paper in 1935 pointing out
that quantum mechanics predicted such an effect (the so-called "EPR
Paradox"). While it cannot be done with cigars, it can be demonstrated with
specially prepared atoms or photons, and today has a solid experimental basis. It
thus raises the same sort of questions at the atomic level: What is locality? What is
reality? You will see comments like the following in journals and textbooks:
From Quantum Reality, Nick
Herbert, 1985, p. 214:
"Non-local influences, if they
existed, would not be mediated by fields or by anything else. When A connects to B
non-locally, nothing crosses the intervening space, hence no amount of interposed matter
can shield this interaction.
Non-local influences do not diminish with distance. They
are as potent at a million miles as at a millimeter.
Non-local influences act instantaneously. The speed of
their transmission is not limited by the velocity of light.
A non-local interaction links up one location with another
without crossing space, without decay, and without delay. A non-local interaction is, in
short, unmediated, unmitigated, and immediate."
From Quantum
Chemistry, Levine, op. cit. p. 196:
"Further analysis by Bell and
others shows that the results of these experiments and the predictions of quantum
mechanics are incompatible with a view of the world in which both realism and locality
hold. Realism (also called objectivity) is the doctrine that external reality exists and
has definite properties independent of whether or not this reality is observed by us.
Locality excludes instantaneous action-at-a-distance and asserts that any influence from
one system to another must travel at a speed that does not exceed the speed of
light. Clauser and Shimony stated that quantum mechanics leads to the
"philosophically startling" conclusion that we must either "totally abandon
the realistic philosophy of most working scientists, or dramatically revise our concept of
space-time" to permit "some kind of action-at-a-distance." (Clauser and
Shimony, Rep. Prog. Phys., 41, 1881; see also B. d'Espagnat, Scientific
American, Nov. 1979, p. 158.)
Quantum theory predicts and experiments confirm that when
measurements are made on two particles that once interacted but now are separated by an
unlimited distance the results obtained in the measurement on one particle depend on the
results obtained from the measurement on the second particle and depend on which property
of the second particle is measured. Such instantaneous "spooky actions at a
distance" (Einstein's phrase) have led one physicist to remark that "quantum
mechanics is magic" (D. Greenberger, quoted in N.D. Mermin, Physics Today,
April 1985, p .38)."
From The New Physics, Paul Davies
(editor), 1989, p. 395 (ISBN 0-521-43831-4):
"It does not seem feasible to
interpret quantum mechanical indefiniteness, chance, probability, entanglement and
nonlocality merely as features of the observer's knowledge of a physical system. Rather,
they seem to be objective features of the systems themselves. Thus, the conceptual
innovations of quantum mechanics are likely to remain a permanent part of the physical
world view." (Abner Shimony)
A careful reading of these views,
including the definition of "locality" given above, suggests that
"non-local" simply means "non-spatial". Hence, we need another concept
of "locality" that is non-spatial, but the effects of which can still be made
manifest in a spatial reference system and which are still scientifically
accessible. The concept of a deep reality need not be abandoned, but we must
"dramatically revise our concept of space-time". ( See also the 8-10-02 Note about the Shapiro Time Delay )
html 2/02
The
Problem of Quantum Probability
Whereas classical mechanics is descriptive, quantum mechanics is predictive.
Instead of describing what is, it describes what can be. Instead of describing innate
properties, it gives us betting odds like those on the roll of a die at a gambling casino:
"Within quantum mechanics we find, in a
word, probabilities. However, the probability functions the theory uses cannot be regarded
as weighted sums of dispersion-free probability functions—that is, as weighted sums
of property ascriptions; quantum theory is irreducibly probabilistic." (The
Structure and Interpretation of Quantum Mechanics, R.I.G. Hughes,1989, p. 305)
We saw this probability aspect in the single-photon double slit interference
problem and its variations (see Counterintuitive
Quantum Mysteries ). The photons were identical in their being, but different in
their behavior. Each experimental run in each apparatus showed irreducible randomness in
the position of each diffracted photon, as well as irreducible order in the overall
pattern. The source of this probablistic nature is one of the deepest mysteries of
quantum mechanics.
html 2/02
The Problem of Quantum Uncertainty
Anyone who studies
quantum mechanics soon encounters the Heisenberg uncertainty principle:
"The relation whereby, if one
simultaneously measures values of two canonically conjugate variables, such as position
and momentum, the product of the uncertainties of their measured values cannot be less
than approximately Planck's constant divided by 2p. Also known as Heisenberg uncertainty relation." (McGraw-Hill
Dictionary of Scientific and Technical Terms, Sybil P. Parker (editor), 1994, 5th
ed., under "uncertainty relation")
Over the years this principle has
acquired many faces, and it is often confusing to sort them out:
"That this was the case is best
illustrated by the fact that Ernan McMullin, presently chairman of the Philosophy
Department of Notre Dame University, wrote a Ph.D. thesis in 1954 on the different
meanings of the "quantum principle of uncertainty." He distinguished between at
least four major classes of interpretations: Heisenberg's principle is regarded (1) as a principle
of impossibility according to which it is impossible to measure simultaneously
conjugate variables, (2) as a principle of limitations in measurement precision
according to which the accuracy of previously acquired knowledge about one variable
decreases by measuring its conjugate, (3) as a principle of statistics relating
the scatter of one sequence of measurements with that of another, and (4) as a mathematical
principle expressing the duality or complementarity of quantum phenomena." (The
Philosophy of Quantum Mechanics, Max Jammer,1974, p. 79)
Let's first consider the measurement
problem.
"We must consider light
and matter as either waves or particles. This leads to quite an awkward result. Let us
consider a measurement of the position of an electron. If we wish to locate the electron
within a distance Dx, then
we must use light with a wavelength at least that small. For the electron to be
"seen," a photon must interact or collide in some way with the electron, for
otherwise the photon will just pass right by and the electron will appear transparent. The
photon has a momentum p=h/l,
and during the collision some of this momentum will be transferred to the electron. The
very act of locating the electron leads to a change in its momentum. If we wish to locate
the electron more accurately, we must use light with a smaller wavelength. . . . Because
some of the photon's momentum must be transferred to the electron in the process of
locating it, the momentum change of the electron becomes greater.
Heisenberg . . . showed that it is not possible to
determine exactly how much momentum is transferred to the electron.
The Uncertainty Principle states that if we wish to locate
any particle to within a distance Dx, then we automatically introduce an uncertainty in the momentum
of the particle. . . . It is important to realize that this uncertainty is not due to poor
measurement or poor experimental technique but is a fundamental property of the act of
measurement itself." (Quantum Chemistry, Donald A. McQuarrie, 1983, p.
36-37)
In the above view, the Uncertainty arises
only when a measurement is performed. Contrast that with the following:
"It must not be supposed . . .
that the quantum uncertainty is somehow purely the result of an attempt to effect a
measurement—a sort of unavoidable clumsiness in probing delicate systems. The
uncertainty is inherent in the microsystem—it is there all the time, whether or not
we actually choose to measure x or p [position or momentum]." (Quantum
Mechanics, P.C.W. Davies, 1984, p. 8)
And then with this:
"In quantum physics,
uncertainty is a precise and definite thing. There are pairs of parameters, known as
conjugate variables, for which it is impossible to have a precisely determined value of
each member of the pair at the same time. The most important of these uncertain pairs are
position/momentum and energy/time.
The position/momentum uncertainty is the archetypal
example, first described by Werner Heisenberg in 1927. It means that no entity can have
both a precisely determined momentum . . . and a precisely determined position at the same
time. This is not the result of the deficiencies of our measuring apparatus—it is not
just that we cannot measure both the position and momentum of, say, an electron at the
same time, but that an electron does not have both a precise position and a
precise momentum at the same time. . . . (Some reference books still tell you that quantum
uncertainty is solely a result of the difficulty of measuring position and momentum at the
same time; do not believe them!)
The uncertainty in position multiplied by the uncertainty
in momentum is always greater than the parameter ["h bar"], Planck's
constant divided by 2p. So
although you can (in principle) get as near to this limit as you like, the more precisely
one parameter is determined, the less accurately the other one is constrained. This is
related to the basic wave-particle duality of the quantum world. A particle (in the
everyday sense of the word) is capable of being precisely located at a point, but a wave
is not." (Q is for Quantum: An Encyclopedia of particle physics, John
Gribbin, 1998, under "Uncertainty")
The modern view of Uncertainty could be illustrated like
this: Consider an apparatus that passes a stream of photons through a single tiny hole and
then onto a photographic plate a short distance away from the hole. As the photons pass
through the hole, they create an exposed spot directly behind it. But photons have a wave
nature and a wave spreads out after passing through a hole, especially if the diameter of
the hole is comparable to the wavelength. As the diameter of the hole is decreased, a
diffraction pattern appears. It looks something like an archery target. The bullseye is
called the Airy disk. (You can see something like it by pricking a tiny hole in a piece of
aluminum foil and then looking at a distant street light at night through the hole.)
If the hole is made still smaller, the bullseye pattern enlarges. If made smaller still,
the pattern enlarges so much it washes out and the exposure becomes dim but uniform.
Finally, as the hole closes, the illumination is extinguished altogether.
So if the hole is made smaller in an attempt to confine the exposure to a narrower
region, the exposed region actually spreads out even more. If the hole is made larger,
there is less spreading beyond the edges, but then the position of the photons is not
known as precisely. Recall that momentum represents both a magnitude and a direction.
Changing the diameter of the hole, somehow alters the direction, and therefore momentum of
the photons. We could say that precise simultaneous knowledge of both position and
momentum are at odds with each other. We call them "conjugate variables".
The most commonly used uncertainty relations are
momentum-position and energy-time:
momentum-position: Dp
Dx
> h/4p
energy-time:
DE Dt > h/4p
Note that both
of these reduce to t2/s. That should be a strong hint about the nature and origin
of the Uncertainty relationships, as well as the wave-particle duality.
The links listed below have useful tutorial articles on:
Quantum Mechanics:
http://www.ncsu.edu/felder-public/kenny/papers/psi.html
What Do You DoWith a Wavefunction?
Gary Felder and Kenny Felder, 2003
This is a short paper on "the most basic things you should know after a quantum
mechanics class."
http://www.chembio.uoguelph.ca/educmat/chm386/chm386.htm
This is educational material for Quantum Chemistry 3860 by Dan Thomas at the
University of Guelph, Ontario,Canada. It gives historical sketches of the early developers
of quantum mechanics as well as tutorials on mathematical methods. If you want to
know about Hermitian Operators, Dirac Notation, eigenvalues, commutators, operator
algebra, etc., and how they are applied to free particles, particles in potential wells,
etc. this is a good site to visit.
http://forum.rstheory.com/ This a discussion
forum for the Reciprocal System. RS theory is a complete, monolithic theoretical system
(not just a methodology). It is very comprehensive in its coverage, but unfortunately does
not cover quantum mechanics as a specific topic. Still, forum members can speak the
language of space/time ratios. You might find some "kindred spirits" here.
Nothing But Motion, Dewey B. Larson, 1979. This book has four chapters on what
could probably be called the "chemistry of space/time ratios." It is heavy
reading, but thought provoking.
The following are books that I would recommend to those who want a more technical
understanding of quantum mechanics. A year or two of college mathematics (calculus and
differential equations) would be helpful for understanding the material in these
publications. You also might check the reviews at commercial booksellers like www.amazon.com to see if they are what you are looking
for.
Mathematics:
Elementary Linear Algebra, Howard Anton, 1994, seventh edition, ISBN
0-471-58742-7. If you want a good foundation for understanding the mathematics of quantum
mechanics, this book is a good place to start.
The Story of a Number, Eli Maor, 1994, ISBN 0-691-05854-7
Who is Fourier: A Mathematical Adventure, Transnational College of LEX
(translated by Alan Gleason), 1995, ISBN 0-9643504-0-8
Metric Spaces: Iteration and Application, Victor Bryant, 1985, ISBN
0-521-31897-1
Quaternions and Rotation Sequences, Jack B. Kuipers, 1998, ISBN 0-691-05872-5
Complex Analysis (university level):
An Imaginary Tale: The Story of Ö-1, Paul J. Nahin, 1998, ISBN 0-691-02795-1
Complex Variables, Stephen D. Fisher, 1990, second edition, ISBN 0-486-40679-2
Introduction to Real Analysis, R.G.
Bartle and D.R. Sherbert, 2nd ed, 1992, ISBN
0-471-51000-9
Quantum Mechanics (university level):
The Feynman Lectures on Physics, Richard, Feynman, Leighton and Sands, 1965
(Volume 3 is about quantum mechanics, but it is best to have the entire set of all three
volumes)
Modern Physics, Paul A. Tipler, 1978, second edition, ISBN 0-87901-088-6.
Molecular Quantum Mechanics, Peter Atkins, Ronald Friedman, 2005, 4th ed.
ISBN-13: 978-0-19-927498-7
The Structure and Interpretation of Quantum Mechanics, R.I.G.Hughes,1989,
(ISBN 0-674-84392-4) Quantum mechanics is essentially a mathematical theory. If you want a
brief, no-frills introduction to vector spaces and Hilbert space theory, this is a book to
consider. In the author's words, "the mathematical background it assumes is that of
high school mathematics, and the additional mathematics needed, the mathematics of vector
spaces, is presented in Chapters 1 and 5."
Quantum Chemistry, Ira N. Levine, 2000, fifth edition, ISBN 0-13-685512-1.
Quantum Chemistry, Donald A. McQuarrie, 1983, ISBN 0-935702-13-X
Quanta: A Handbook of Concepts, P.W. Atkins, 1991, second edition, ISBN
0-19-855573-3
Q is for Quantum: An Encyclopedia of particle physics, John Gribbin, 1998,
ISBN 0-684-86315-4
Quantum Physics : Of Atoms, Molecules, Solids, Nuclei, and Particles, R. M.
Eisberg, R. Resnick, 1985, ISBN 0-471-87373-X
Optics:
Optics, Eugene Hecht, 2nd ed., 1987/1990, ISBN 0-201-11609-X
Optical Physics, S.G. Lipson, H. Lipson, D.S. Tannhauser, 1995, third edition,
ISBN 0-521-43631-1
Nuclear weapons:
I had to study nuclear weapons to obtain some insights about "time pumps" and
high-density,
high-energy materials. The references below are especially informative to a general
technical audience.
Fourth Generation Nuclear Weapons, Andre Gsponer, Jean-Pierre Hurni, 1999, 6th
ed. This can be ordered by email at ianus@hrzpub.tu-darmstadt.de
. It has plenty of technical details about nuclear weapons (physics, not actual
technology), tables, illustrations, and a very valuable bibliography with 567 entries. It
is very well written and easily worth the $25 I paid for my copy.
The Making of the Atomic Bomb, Richard Rhodes, 1986. A fascinating history of
the development of the atomic bomb and its use in World War II.
Dark Sun, Richard Rhodes, 1995. Another fascinating history of the development
of the hydrogen bomb.
There is also a huge amount of useful material at the Federation of American Scientists
website and various sublinks including:
Complex numbers are used a lot in quantum mechanics. It is important to
become comfortable with their use. The following illustrations are mathematical teasers
offered for entertainment purposes.
You can also verify this math with the Google calculator. Type each of the
following threeexamples into the Google search bar and then click the Search button:
10^(0.68226*i)
i^i
i^0.5
and you will see, respectively:
10^(0.68226 * i) = -0.00016537875 + 0.999999986 i
i^i = 0.207879576
i^0.5 = 0.707106781 + 0.707106781 i
["h bar"], Planck's
constant divided by 2p. So
although you can (in principle) get as near to this limit as you like, the more precisely
one parameter is determined, the less accurately the other one is constrained. This is
related to the basic wave-particle duality of the quantum world. A particle (in the
everyday sense of the word) is capable of being precisely located at a point, but a wave
is not." (Q is for Quantum: An Encyclopedia of particle physics, John
Gribbin, 1998, under "Uncertainty")
