1.1 Background
For the theory to be viable, the wave "component" of each particle needs to act over the full space/time envelope occupied by the whole interference pattern. This means that a semi-local compromise, such as a small localized wave packet does not suffice.
For example:
The theory needs to introduce a mechanism to account
for the change in interaction modes from distributed to local (as when
a diffracting particle is detected). In conventional interpretations, this
is accomplished by introducing an ad hoc "collapse" that occurs when the
particle is detected.
In addition, the theory conflicts with relativity in a number of ways:
On examination, we see that the fundamental rationale
that underpins wave/particle duality is an assumption that the only possible
cause for certain observed phenomena can be the wave propagation processes
of diffraction and interference.
Acceptance of that assumption leads to a conundrum in which we have to also accept that particles posses distributed non-local properties that act in ways that contradicts relativity. Aditionally, the theory also requires ad hoc elements such as wave-function collapse, that cannot be observed and do not have any explicit mathematical descriptions or representations. To the best of my knowledge, nobody has ever written a mathematical equation that explicitly represents a particle undergoing any such processes.
Single slit diffraction, modelled without wave propagation effects
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.
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.
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.
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:
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:
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.
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.
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.
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.
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;
As a result, I propose that the implications of Planck
and Bohr's discoveries can be restated, along the lines of:
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).
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.
Two important examples are:-
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.
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.