Quantum Theory and Wave/Particle Duality
A work in Progress; modified: 7-March-2001
Many interpretations of quantum physics incorporate the
idea that particles (or some property associated with particles) propagate
as waves.
The object of this article is to examine the core assumptions
behind this wave/particle idea with a view to developing a different model
that is consistent with relativity, observation and the mathematical formalism.
Copyright © 1998, 1999, 2000 John
K. N. Murphy, Kohimarama, Auckland, New Zealand.
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1.0
Introduction [Next]
Essentially, wave/particle duality employs the
notion that an entity simultaneously possesses localized (particle) and
distributed (wave) properties.
The idea has been introduced into modern physics to account
for observations in which particles of matter interact to produce effects
that appear to be identical to the effects that occur when waves diffract
and interfere.
However, the concept of rests on an assumption. It is
assumed that wave propagation mechanisms can provide the only possible
explanation for scattering effects observed in experiments such as the
Twin Slit experiment.
At face value, the assumption looks convincing. The French
physicist, Louis de Broglie introduced the wave/particle concept into physics
in the 1920s, with a brilliant prediction that particles of matter possess
wave properties and act as though they were composed of propagating waves.
Experimental confirmation of de Broglie's prediction (electron diffraction
in crystals) has led to the theory being cemented into the foundations
of modern physics.
Later, Danish physicist Neils Bohr used the wave-particle
duality concept as the foundation for his well known "Copenhagen Interpretation"
of quantum mechanical effects.
At the time the theories were developed, wave propagation
effects were the only mechanism that people were aware of that could produce
those kinds of phenomena. Consequently, the ideas became fixed and
that fundamental assumption has since persisted.
My contention is that even though the assumption looks
reasonable, it may not necessarily be correct.
The object of this article is to re-examine the observations
and core mathematical relationships with the intention of opening up the
possibility of a different mechanism that is both consistent with relativity
and quantum theory.
For more detailed background and an illustrative example
->[Historicc Summary] [Twin
Slit Example].
1.1 Basic Issues
On examination, the fundamental rationale for wave/particle
duality lies in the assumption that the only possible mechanism for the
observed phenomena is the wave propagation mechanisms of diffraction and
interference.
By making the assumption, we can obtain a simple explanation
for particle diffraction and interference but we lose the ability to make
sense of particle behavior from a relativistic perspective.
Acceptance of that assumption also requires a set of "supernatural"
mechanisms and behaviors that are needed to reconcile the distributed mechanism
of wave propagation with the localization of particle interactions.
Two factors stand out, the need to:
-
Propose ad-hoc mechanisms that have no representation. (To
the best of my knowledge, nobody has ever written a mathematical
equation that explicitly represents a particle undergoing any such process
as wave function collapse.)
-
Accept the existence of distributed mechanisms that act in
ways that contradict relativity.
The alternative is that the assumption could be mistaken
and there is a different mechanism operating. The intention of the rest
of this article is to explore that possibility.
Fig 1. Single slit diffraction, artificially modeled without wave
propagation effects
1.2 Generating a New Approach
Some background to taking a fresh look at the mechanism are:
-
During engineering studies, I came across systems in which
the interference like patterns resulted, without wave propagation mechanisms,
leaving me to wonder whether the same could apply to quantum theory.
-
On examining quantum theory, I saw that there are two distinct
"wave" models in quantum theory, Schrödinger and De Broglie. Physical
wave propagation (that is, in time and space) is a feature of the De Broglie
model only. From that, I wondered if it were possible that the De Broglie
model can be shown to be co-incidental or spurious, while still retaining
the core relationships contained in the Schrödinger representation.
-
I could see that eliminating the wave propagation model could
allow for the reconciliation of relativity and QED.
With regard to this last point, there is a conceptual
problem in relativity if you regard a photon as a propagating wave.
When you look at the "existence" of a single photon from
a photon's "frame of reference", then you find that the photon retains
the same phase across its whole existence, as though it is "frozen" in
stasis as it propagates. In other words, the photon experiences no
time when it moves from its start point to its end point.
This could mean that a photon does not oscillate when
it propagates from one place to another, and may be inconsistent with wave
propagation models.
Certainly, you can plot the appearance of this constant
phase across an observer's reference frame and see that it projects as
oscillations, (consistent with Maxwell's equations) but this is not the
same thing as wave propagation. Rather, it looks to be an illusion that
arises because that which is simultaneous in the photon's frame is not
simultaneous in any other observer's frame of reference. It is an artifact
of the way time occurs across space.
This could mean that a photon does not propagate as a
wave. It may well interact in an oscillatory manner and, as a result,
scatter in patterns that resemble wave propagation effects. However, this
is quite distinct from the fundamental processes of interference and diffraction
that require a particle to distribute its presence across a large volume
of space.
The object of this section is to explore the
relationship between the form of a scattering pattern and the spatial pattern
in the scattering object (the slit screen). By analyzing the situation
in this manner, it can be seen that the properties of the final pattern
depend only on the spatial pattern of holes in the screen and do not depend
Fig. 2 Computer generated image of a Twin slit pattern.
The image above is typical of a twin slit pattern as
would be produced by a laser (or electron) beam illuminating a pair of
slits. It has two key characteristics:-
-
The pattern has exactly the form that would occur if light
were made up of waves.
-
The pattern is "speckled", being made up from a myriad of
individual particle impacts.
2.1 The Basic Interference Relationship [Next]
To the best of my knowledge, all orthodox interpretations
of these observations begin with a set of relationships developed
by Victor Louis De Broglie.
-
That particles of matter, as with light (optical interference
patterns were well known at the time), inherently possess wave properties
characterised by a wavelength (L) given by the relationship:
L=h/P
Where h = Planck's constant and P = the momentum
of the particle/photon.
This hypothesis has been tremendously successful. The
relationships yield the correct wavelengths and frequencies required to
produce the observed diffraction and interference patterns for an impressive
range of "particles" (including, light, gamma rays, x-rays, neutrons, electrons
and atoms) in a variety of situations (optics, crystal diffraction of x-rays
and electrons as well as neutron diffraction and interference).
However, the question is, despite the success of the relationship,
can we eliminate the wave propagation aspects and still obtain the correct
answers?
2.2 A Scattering Analysis [Next]
The object of this section is to examine the basic features
of a twin slit diffraction process and identify the basic characteristics
that would be required of a scattering process that could produce the observed
patterns.
Fig. 3 below shows a typical optical interference pattern
as it develops. At the left, the particle distribution is well defined,
it goes through some transient phases as the pattern emerges, then the
distribution stabilizes into a set of well defined "rays".
What I wan to look at here, is examine the relationship
between the initial pattern in the slit screen and the final "stable" pattern
that the beam settles into in what is conventionally described as, the
"far field".
Fig 3. Emergence of an optical twin slit diffraction pattern.
From a scattering perspective:
-
A beam of particles spreads into a pattern of bright and
dark lines.
-
If we increase the momentum of the particles in the beam,
the pattern narrows.
-
Conversely, if we decrease the momentum of the particles
then the pattern widens.
-
Primarily, the particles are scattered elastically. That
is, they just change direction and retain their original speed and kinetic
energy. In the case of photons, they retain their original energy and magnitude
of momentum. That is, the particles exchange momentum and not energy.
If particles undergo a scattering process then each
deflected particle receives an impulse of momentum to produce its change
of direction. Further, that impulse would need to be directed transversely,
across the path of the beam, in the plane of the slits.
Starting with the first off-center line, examine the transveres
momentum impulse imparted to the particles arriving at the screen.
The momentum is given by a simple formula calculated by taking angle of
deflection and the incident momentum to solve for the transverse momentum
impulse.
The cross-beam component of the momentum, P,
is then given by:
P=h/D where
h=Planck's
constant and D is the separation between the slits.
Straight away, we see something interesting:
-
The magnitude of the impulse depends only on Planck's
constant and the separation of the slits. It is completely independent
of the type of particle and the momentum of the incident particles in the
beam.
-
The De Broglie wavelength of the incident particle is
eliminated.
The same feature characterizes the rest of the bars
in the interference pattern. The deflecting momentum for the peak is always
in multiples of h/D.
That is to say, that for all lines on the screen, the
equation for the deflecting momentum for that line is a simple function
of the slit separation and Planck's constant.
By looking only at the deflecting momentum exchanged during
the scattering process, we see that there are no variables that depend
on the magnitude of the momentum of the incoming particles.
If particles are considered to follow a trajectory
then the incident particles are acquiring a cross-wise component of momentum
that is taken from a set or 'spectrum' of possible deflections that is
determined by the pattern of holes in the screen.
In effect, it looks as though the far-field pattern we
see is an image of a "momentum spectrum" that depends on the slit dimensions
and not on the momentum of incoming particles. In other words, the form
and scale of this spectrum does not change with the magnitude of the incoming
particle momenta. The beam appears to passively act like a image projector
for a spectrum that is inherent in the screen.
So far, we have only calculated the location of the peaks
in the pattern and ignored the continuous oscillation in image density
from light to dark. Even so, this result strongly indicates that
the overall pattern is characterised by a cross-wise momentum spectrum
that is related to the structure of the screen and is independent of the
magnitude of the momentum of incoming particles.
The illustration below illustrates the extent to which
the final pattern does not depend of the supposed "wavelength" of the incident
particles.
The image is a composite of two twin slit diffraction
patterns. The top half of the image is constructed with a wave-length that
is five times longer than the lower half. To match the patterns, the horizontal
axis of the lower half of the image is reduced in scale by a factor of
five.
Fig 4. Combined images of two diffraction patterns. Upper half shows
pattern with a wave-length five times longer than the lower half.
So what of the exact form of the final pattern of transverse
momenta?
In the far field, the exact form of the pattern is a result
familiar in optics, it is obtained by taking the Fourier transform of the
spatial pattern made by the slits in the screen. This result is general
and covers both the one and two slit cases, in fact, it holds for any pattern
of slits or holes in the screen.
So how does this Fourier pattern in the scattering momentum
relate to quantum theory? The answer is quite literally a textbook case.
For example:
[Cassels, Basic Quantum Mechanics, 1970, McGraw-Hill
(London) p33] "The required expansion (for the Momentum Distribution)
amounts simply to a Fourier analysis of Psi." (The text shows the Fourier
transform of the expectation function for position multiplied by (2*Pi*Hbar)-1/2.
Note
that since Hbar = h/(2*Pi) the factor of 2*Pi is eliminated if h is
substituted for Hbar ).
Applying Schrödinger's equation in this manner
(to the spatial pattern of holes in the screen, rather than the incident
particles) gives the exact result for the required scattering
momenta in both the single and twin slit cases.
Interpreting Schrödinger's equation this way opens
up the possibility that the relationships discovered by De Broglie do not
correspond to wave propagation effects and are no more than a curious side
effect of the characteristics of a scattering process.
When one examines the context in which Schrödinger
developed his mathematical model (atomic states), it becomes apparent that
the equation very precisely characterizes the states that particles can
adopt with respect to one another. Furthermore, in doing so, the equation
characterizes the spectrum of energies that the system can exchange with
other systems (e.g. atomic spectra).
In the twin slit context, the equation seems to characterise
the spectrum of crosswise momenta that can be transferred to the incident
particles as the space in which the particles exist opens up from a constrained
pattern to a smooth state (far-field). This model has some
very different features compared to the interference model:
-
It is understandable why different types and classes of particle
(e.g.. photon/electron/atom) produce the same effect. - The particles
are partners in the scattering process rather than the vehicle that "brings"
waves that are instrumental in the process.
-
There is no longer a need to find a mechanism to "collapse"
the wave. The structure of the holes in the target screen is the "engine"
that produces the effect. The particles remain localized (at least, within
relativistic limits).
It also raises questions:
-
What kind of scattering would be consistent with observations
of the optics of interference patterns?
-
If not waves, then what are quantum interactions about?
-
How does the model relate to the mathematical description
of quantum theory?
-
Can a model like this be used to explain the behavior of
multi-path interferometers such as the Michelson/Morely and Mach/Zehnder
interferometers?
-
What of secondary effects, such as the Ahranhov/Bohm effect?
I say that all of these questions can be clearly and
simply addressed and that it leads to an astounding conclusion: Neither
photons nor particles propagate in space as waves and that, in a sense,
electromagnetic waves as we usually envisage them, don't exist.
Additionally, we no longer need to classify all particles
(photons especially) under the same, restrictive, umbrella of "quantum
wave/particle".
2.3 The Optics of Pattern Development [Next]
This section explores an important question. If the observed
patterns are due to scattering, then how does the evolution of the scattering
pattern look compared to wave diffraction.
The image below illustrates the evolution of an optical
twin slit interference pattern. In the far field, (to the right) the pattern
settles into a set of "rays" that match those that one would expect from
a random scattering of particles into a Fourier transform of the slit pattern.
However, when one examines the part of the pattern when
the beam begins to emerge, one sees a fine pattern that is not consistent
with an instantaneous scattering in to straight-line trajectories.
Instantaneous scattering would produce a multitude of crossing trajectories
that would hide the fine detail apparent in the observed pattern.
To be consistent with the observed pattern, the particle
trajectories would need to emerge downstream of the slits in some sort
of process similar to the way a toy boat may be diverted from it's course
when crossing the wake of larger craft.
The idea that particles may follow a definite trajectory
through such a pattern is not new. David Bohm and Basil Hiley present
an example in their book "The Undivided Universe" 1993 showing a diagram
(Fig 3.1) of a set of possible electron trajectories for a situation where
the particles emerge from the each slit with a Gaussian distribution. The
example shows the trajectories not crossing, and flowing through the pattern
rather like a laminar flow of a fluid.
Bohm and Hiley's choice of "Gaussian" slits may be deliberately
chosen to produce a "laminar" pattern. The Fourier transform of a Gaussian
curve is a Gaussian curve. Hence the single slit component of the pattern
will remain static from the the slits out to the far field. In the
text, Bohm and Hiley use the example to illustrate a concept of a wave-like
quantum field that accompanies each particle and determines its path by
providing "information" that guides the particle according to the "form"
of the field rather than it's strength.
Examination of the early stages of an optical interference
pattern generated by "rectangular" slits shows beams of particles coming
off either side of each of the two main beam (as in the fig 5 below) and
judging from appearance, parts of each beam do look to cross each other.
Fig. 5 Early stages of an optical interference pattern.
[I apologize, this Segment
of the article is being re-constructed and is temporarily unavailable]
3.0 Characteristics of Quantum
Behavior [Next][Prev][Top]
At this point, I feel it is important to recap on some
of the very basic features of quantum behavior. These effects were discovered
and described early this century by Earnest Rutherford, Max Planck and
Niels Bohr.
First, Rutherford, for discovering that atoms were largely
"empty" space; each consisting of a small positively charged nucleus "orbited"
by one or more negatively charged electrons.
In classical physics such an arrangement for the components
of an atom is problematical. The positive and negative charges are
powerfully attracted to each other and should, according to classical theory,
collapse together with the attendant release of a large amount of electromagnetic
radiation.
Second, Planck, for finding a feature of the thermal radiation
that emanates from matter;
Matter does not emit a continuum of radiation; Planck
discovered that the thermal excitations in matter are characterised by
a form of quantisation. Particle systems emit photons with discrete energies.
The frequency (f) of the emitted radiation is related to
the energy (E) of the state change by the formula:
E=hf Where his
a constant, called Planck's constant.
Note: That by speaking of quantisation I want to be clear
that Planck's relationship does not imply that energy comes in packets
of certain fixed sizes.
All it says is that there is a relationship between
the frequency of a photon (as it interacts) and its energy. That is to
say, you can have a photon of any energy, you are not restricted to any
steps or intervals, it is just that you will always find that a photon
interaction of a certain energy has a periodic behavior with a frequency
given by Planck's formula.
Thirdly, Bohr, for tying the two together.
Bohr found that Planck's constant applied to electrons
"orbiting" atoms; The electrons and nucleus did not "collapse" when
the electrons adopted "orbits" at certain specific angular momenta that
were an exact multiple of Planck's constant.
From this point, I suggest a diversion from the orthodox
interpretation of this behavior. Most orthodox interpretations involve
making the assumption that the observed frequency is a property
of the photons being emitted because it was assumed that light is a wave
and displays interference. This may be an unwarranted assumption;
It is possible that interference does not occur
in the propagation of light. That the frequency referred to in Planck's
formula could arise when photons interact with particles or systems of
particles.
As a result, I propose that the implications of Planck
and Bohr's discoveries can be restated, along the lines of:
Matter is composed of particles of opposing charges
that are in constant motion with respect to one another. By rights, they
should collapse in a blaze of radiation. However, whenever opposing charges
adopt periodic states of motion where the frequency and energy of that
state of motion fit criteria characterised by the relationships in QED
then no energy is lost.
In terms of the classical electromagnetic fields, when
particles adopt states that fulfill certain constraints then the combined
field pattern of the charges becomes 'stable' so that no radiation occurs.
Precisely why, I have no idea. And, as far as I can tell, neither does
anyone else.
The effect of this behavior is that certain specific
states of motion are the only ones that can persist in matter. Furthermore,
anything that perturbs such states will almost immediately resolve itself
into radiation (photons) plus a new set of 'stable' states.
In summary, when charged particles get together to occupy
space as "matter" then that matter consists almost entirely of periodic
states of motion that fit criteria outlined in the equations of QED.
It is commonly accepted that this effect is the source
of atomic spectra. Electrons can only adopt a restricted set of "orbits"
around a nucleus; those orbits that satisfy Schrödinger's equation.
(Note: There are other criteria relating to angular momentum and degeneracy
of states that add to the complexity of the solutions, even so, the core
relationship is obeyed in all cases).
3.1 Relationship Between Momentum and Spatial Structure
As stated above, Schrödinger's equation exactly characterizes
the momentum spectrum that the slit screen exchanges with incident particles.
This situation suggests we should re-examine conventional interpretations
as to what the mathematical objects (in particular the wave function, Psi)
within standard quantum theory represent.
Because it is conventionally assumed that a particle propagates
as a wave, Psi is normally taken to be a mathematical representation of
the simultaneous superposition of states available to the particle. While
this may seem reasonable if the particle is taken to be wave-like and non
local, such an interpretation does not work in a context where a particle
does not propagate in space as a wave.
As an alternative, I suggest a different interpretation.
That
Psi represents the statistical superposition of the states present and/or
available in the system that the particle is part of/interacting with.
If that is the case then we can take the spatial structure
of the slit screen (single slit or twin slit) to represent the expectation
function for the distribution of 'position' for the 'sea' of states. From
there, as explained above (2.2), it is a textbook case to find the momentum
spectrum of such a population.
4.0 Related Situations in Quantum
Mechanics [Next][Prev][Top]
The alternate model of quantum scattering rather than
interference extends to other quantum experiments that are commonly taken
as evidence of "interference".
Two important examples are:-
-
Crystal diffraction of X-rays and Electrons.
-
Semiconductor "Band" theory.
Interestingly, in both these situations the mathematical
operations required to calculate the outcomes involve steps that are exactly
parallel to those that I propose. In both situations you get the exact
answers by:-
-
Calculate momentum/energy spectrum by taking a (usually 3-D)
Fourier transform of the crystal/electronic spatial structure.
-
Apply that spectrum to the momentum/energy of the incident/conducted
particles.
Many texts, while maintaining that the effects are due
to waves accompanying the incident particles, focus almost entirely on
the Fourier transforms of the structural properties. (See the section on
"Crystal Diffraction and the Reciprocal Lattice" in Kittel, "Introduction
to Solid State Physics" Wiley & Sons).
I acknowledge that such effects do not "prove" my case,
they do however establish that this alternate approach is viable across
a wide range of quantum effects. I also argue that the model is conceptually
simpler in that it does not require the support of "mystical" effects that
have no description or representation in the theory; For example, the purported
"wave function collapse" or perhaps the "determination of reality" via
the act of observation.
5.0 The Ahranhov-Bohm Effect
(In Brief) [Top][Prev]
The Ahranhov-Bohm effect is an effect in which an interference
pattern becomes "shifted" by the presence of a field that is completely
shielded from the particles being used to produce the pattern. Interestingly,
in a twin slit version of the experiment, the twin slit bars are shifted
far more than the single slit background.
In one twin slit version of the experiment, a shielded
magnetic field runs along the central bar between the slits. It runs parallel
to the slits and the surface of the screen so that no transmitted particles
(electrons) actually cross any magnetic potential.
In another version, the screen is made of insulating material
and the "slits" are holes made with conducting (metal) cylinders. A Voltage
is applied so that the two cylinders are at different potentials. An electron
traveling through each cylinder will not 'feel' an electric potential.
In interference based interpretations of these experiments,
the observed displacement in the pattern is taken to provide more evidence
for "quantum non-locality" because it is assumed that the particles are
propagating as waves and produce the effects by somehow "sensing" the shielded
fields.
However, if you take the approach that the pattern is
an "image" of a momentum spectrum that can be exchanged between the slits
and the incident particle then a different kind interpretation becomes
possible.
Why? The quantum exchanges between the slit screen and
the incident particles involve quantum states in the slit screen that propagate
transversely, through the "shielded magnetic/electric field" in question
and are directly affected by it. The spectrum of available excitations
that have dimensions that traverse the field, (i.e.. those responsible
for the twin slit component of the pattern), become shifted and the single
slit component does not, as observed in the experiments.
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K. N. Murphy, Kohimarama, Auckland, New Zealand.