Quantum Theory and Wave/Particle Duality

A work in Progress; Revised 11th-January-2000
_Master copy at John's site - please look at it - it is much more up to date.

The popular concept of wave/particle duality rests on an assumption that wave propagation processes must be responsible for quantum effects observed in experiments such as the Twin Slit experiment and Multi-Path interferometers.

This article explores the possibility that a different kind of mechanism could be consistent with both observation and the mathematical representations used in quantum theory.

Copyright © 1998, 1999, 2000 John K. N. Murphy, Kohimarama, Auckland, New Zealand.

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1.0 Introduction [Next]

1.1 Background

The concept of Wave/Particle duality appears counter-intuitive because it employs the notion that an entity simultaneously possesses localized (particle) and distributed (wave) properties. The theory, introduced by Louis de Broglie, holds that particles of matter posess wave properties and act as though they were composed of propagating waves. It has been introduced into modern physics to account for the way that particle interactions produce effects that appear to be identical to the effects that occur when waves diffract and interfere. I began to question the acceptance of the duality model as a result of a change of course at University.  I first completed a bachelor's degree in physics then switched courses and took up electronic engineering, graduating with a master's degree. During my engineering course, I became exposed to signal processing methods and noticed something striking.

I noticed that the behaviour of some types of signal filter mirrored patterns that occur in quantum experiments. The interesting point about the similarity is that the patterns and relationships observed in the signal filters look exactly like those produced by wave diffraction and interference, and yet the mechanism that generated the patterns did not directly involve wave propagation and interference.

That led me to look more closely at the role of interference and diffraction in quantum theory. It became evident to me that there are really two distinct "wave" models in quantum theory and that interference was a feature of only one of those models.

The descriptions of interference effects are based on wave models introduced by Louis De Broglie whereas the models used for precise descriptions of electronic behaviour (the core mathematical model that underpins quantum electrodynamics QED) is based (in the non relativistic case) on Schrödinger's wave mechanics.

From there, I began to question the basis of De Broglie's interpretation of the relationships that he had discovered. I wondered whether, instead of being fundamental, that those relationships could arise as a side effect of a single basic model.

In addition, I had another reason to feel that there are difficulties in the wave propagation interpretation. I have never been comfortable with the notion that photons propagate as waves because of certain features of relativity. In particular, when you project the "existence" of a photon into a "photon's frame of reference" then what you see is that the photon experiences no time when it moves from its start point to its end point.

In wave terms, the photon retains the same phase across its whole existence. 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 artefact 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.

1.2 The Basis For a New Approach

In short, some features of wave/particle duality have no explicit theoretical description and others appear to directly contradict relativity. This leads me to question the assumption that wave propagation effects are fundamental to the relationships in quantum theory, at least, in the way that is conventionally understood.

While proponents of wave/particle duality insist that experiments that show diffraction and interference patterns are proof that direct wave effects are responsible, this "proof" depends on fundamental assumptions. While it may be reasonable to form such an interpretation, it is also possible that direct wave propagation processes are not responsible.

This article begins by exploring the consequences that arise from addressing the issue of twin slit interference as scattering problem. This approach is normally omitted from textbook analyses which typically progress to a discussion of wave propagation effects and dismiss scattering by appealing to conventional assumptions.

2.0 Analysing Twin Slit Interference and Diffraction [Next][Prev][Top](More background & diagrams)


Computer generated image of a Twin slit pattern created using a scattering model. 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:-
    1. The pattern has exactly the form that would occur if light were made up of waves.
    2. The pattern is "speckled", being made up from a myriad of individual particle impacts.

2.1 Reviewing The Orthodox Interpretation [Next]

To the best of my knowledge, all orthodox interpretations of these observations begin with an acceptance of the hypothesis established by Victor Louis De Broglie. L=h/P Where h = Planck's constant and P = the momentum of the particle/photon. This hypothesis has been tremendously successful. If wave propagation effects are responsible, then it yields 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). It is quite possible that a different type of process could be responsible for the observed patterns. All that is required is that the alternative model produces predictions consistent with the existing mathematical representations and experimental observations.

2.2 A Scattering Analysis [Next]

When you allow the possibility that a scattering process could be responsible then an omission in the wave model becomes apparent. The wave model treats the slit screen as an "amorphous" classical wave absorber and fails to deal with the momentum exchanges that occur between the particle and the screen, ignoring the quantum properties of that exchange process.

Examine the basic features of a scattering process that would produce the observed patterns.
From a scattering perspective:


This means that for each deflected particle (by conservation of momentum), an impulse of momentum would have been required 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.

Now examine the momentum impulse required to deflect the particles that are scattered to the first off centre bright line in the pattern. The solution is given by a simple formula calculated by taking angle of deflection and the incident momentum to solve for the transverse momentum impulse.

 The deflection 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 with the screen, we see that there are no variables that depend on the magnitude of the momentum of the incoming particles.

That is: It is possible that the incident particles are acquiring, apparently at random, a deflection that is taken from a set or 'spectrum' of possible deflections that is determined by the structure of the screen.

In effect, it looks as though the 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 momentum spectrum that is related to the structure of the screen and is independent of the magnitude of the momentum of incoming particles.

So what of the exact form of the pattern? 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.

In the case of the computer generated image shown above, the pattern is composed of a fine "twin slit" pattern superimposed on a broader "single slit" background as would be observed in a real experiment.

So how does this Fourier pattern in the scattering momentum relate to quantum theory? The answer is, quite literally, (provided that you alter your viewpoint as to the interpretation of Schrödinger's equation) 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 gives the exact result for the required scattering spectrum in both the single and twin slit cases.

I do not think that this is a coincidence. Instead I think this relationship opens up alternative ways of interpreting the nature of quantum interactions and the mathematical relationships contained in quantum electrodynamics.

Given the exact and striking nature of this result, I find it remarkable that the features of a scattering model are never discussed in conventional texts.

This provides evidence that it is reasonable to clearly distinguish the relationships discovered by De Broglie from the relationships expressed in the Schrödinger equation.  The result strongly suggests that the equations of quantum electrodynamics relate to states that arise when particles interact and exchange energy or momentum and are not wave propagation equations.

Furthermore, the result shows that it is possible 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 the 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 experiment, a beam of particles interacts with a structure that is composed of particles that are constrained into states that are characterised by the Schrödinger equation. The equation characterizes those constraints and allows us to calculate the set or spectrum of momenta that the system will exchange with the incident beam. However, unlike the Hydrogen atom (two particles and a discrete spectrum), the slit screen is composed of many particles and the available states form a continuous spectrum of varying density.

This approach to the twin slit experiment has some very different features compared to the interference model. Notably:

 It also raises questions: 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. 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 following series of images records the progress of an emerging interference pattern from a point close to the slits through to the "far field". It was computer generated without resorting to the processes of interference and diffraction and contains the same "photons" in each image.







Successive stages in the development of a combined "diffraction" and "interference" pattern as it develops from near the pair of slits to the far field. The images were created by a Monte Carlo method in which:
    1. Calculating the momentum "spectrum" for the slits.
    2. Assigning photons a random arrival position at the slits.
    3. Making a random selection of a deflection for the photon from the required weighted momentum spectrum.
    4. Plotting the position of the photon at the different stages along its (straight-line) path.
    5. Repeat steps 3 & 4 until a picture emerges.
It turns out that in the far field, the results are indistinguishable from patterns produced by 'wave' optics as well as being in accordance with observation.

The same is also true of "single slit" diffraction (not shown here). A ray optic treatment of the developing pattern looks exactly like the patterns that we observe. Effectively, a scattering mechanism together with a "ray optic" propagation model produces predictions that are identical to the wave model and exactly matches experimental observations.
 
Aside: In the wave model, much significance is often attached to the idea that determining the position of a particle close to the slits "destroys" the "interference" because the "wave function collapses".

I suggest that this effect can be accounted for in a scattering scenario and does not provide secondary evidence for the existence of an interference mechanism or wave/particle duality.

My argument is that when you calculate the minimum momentum that you need to apply to a particle to determine its position (relative to the slits) then you find that exactly the same limitations apply in the 'ray optic' context. That is, if you can measure the position of the photon/particle then you destroy the visibility of the bars in the twin slit pattern.

I propose that the limitation is due to the quantum properties of the states that photons create when they are absorbed in matter.

For example, imagine performing the experiment with a beam of particles, electrons perhaps. You try to determine which slit an electron passes through by illuminating the slits with photons.

There are limitations on the energy of the photons that you could use. When you calculate the momentum imparted by photons of the minimum required energy then you find that they always add enough of a deflection to destroy the visibility of the twin slit bars in the pattern.

The pattern is destroyed even when you try using a process that minimizes the deflection. For example, illuminating a beam of atoms such that they absorb, then re-radiate the photon, you still find that you "kink" the path of the particle enough to destroy the visibility of the bars in the pattern.

 In summary, a "ray optic" approach to diffraction and interference provides a model for the evolution of the patterns that does not involve the process of interference and exhibits the same properties as those predicted by the wave scenario.

3.0 Characteristics of Quantum Behaviour [Next][Prev][Top]

At this point, I feel it is important to recap on some of the very basic features of quantum behaviour. 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 behaviour 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 behaviour. 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 fulfil 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 behaviour 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 propagated 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:-
    1. Crystal diffraction of X-rays and Electrons.
    2. 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:-
    1. Calculate momentum/energy spectrum by taking a (usually 3-D) Fourier transform of the crystal/electronic spatial structure.
    2. 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 travelling 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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Copyright © 1998, 1999, 2000 John K. N. Murphy, Kohimarama, Auckland, New Zealand.