Posted on March 10,
Updated May 18, 2003
Different interpretations for photons and material particles?
Diffraction of particles on slits
The main thesis
The slits experiment revisited
In the course of my career as a physicist and a university teacher of physics I have more than once taught quantum mechanics and some special courses based on the application of quantum-mechanical concepts, both on the undergraduate and graduate levels. In doing so, I have never evinced ideas in any way close to what I intend to say in this essay. I did not do so because I realized that there was nary a chance my colleagues would approve an iconoclastic view for which I had no empirical evidence of my own and no supporting data from experiments or theoretical calculations designed specifically to test that view. Moreover, these problems belonged in fact outside my specific expertise as a researcher, so endeavoring to discuss these problems would most likely be judged an amateurish attempt.
Now when my laboratory research is behind me and I choose the topics of my essays simply as sources of fun, I thought that I might embark on this speculative journey, and should it only invoke shrugs, so be it. I can stand it with a smile. If the common judgment will be that I am dead wrong, it may at least prevent another curious mind from wasting time on a similar vain effort and this would justify my effort.
In every college textbook of general physics we find chapters describing what is referred to as the wave-particle duality. The empirical evidence that gave rise to this concept was originally observed in experiments with visible light. The concept was subsequently extended to the entire range of electromagnetic radiation from gamma rays and X-rays to long radio waves. Finally, after de Broglie's seminal theoretical work  of 1923, Davisson and Germer's discovery of the diffraction of electrons reflected from a nickel crystal  (in 1927), and later the discovery by Thomson  of the diffraction of electrons passing through a platinum foil, it was extended to all particles including those with a rest mass ("material particles").
For example, in the photoelectric effect (discovered by Hertz in the last quarter of the 19th century, explained by Einstein in 1905  and thoroughly studied by Millican  in 1914-1916) light behaves as a stream of discrete portions ("quanta") of energy (named photons). Likewise, electromagnetic radiation clearly displays its granular character in the Compton effect and in the thermal radiation from solid bodies (e.g. from "black bodies") . On the other hand, if a beam of light illuminates a diffraction grating, which is a set of transparent slits separated by equally spaced non-transparent strips, a diffraction pattern is formed which is a typical wave phenomenon.
The common thesis is that the full description of the physical reality requires accounting for both wave and particle behaviors which complement each other. In other words, it is often said that every particle, regardless of whether it is "material" (such as electrons, protons, neutrons, or even whole atoms and molecules) or has no rest mass (photons) is both a particle and a wave.
The thesis of this essay is based on the assumption that the phenomena attributed to the wave-particle duality of material particles (MPs) may have an alternative explanation requiring no assumption that material particles also are waves.
In this thesis I distinguish between MPs on the one hand and photons on the other, so that my alternative explanation is not extended to photons, for which the wave-particle duality is preserved as the most plausible interpretation of their behavior.
The reasons for viewing photons as inherently different entities from MPs seem to have foundation in the substantial differences in their behavior.
The term "photons" was introduced for quanta of energy of the electromagnetic (EM) radiation. There is no doubt that electromagnetic radiation is emitted in portions, as was discovered by Planck in 1900. In 1905 Einstein, in his paper on the photo effect, suggested that electromagnetic energy is inherently granular so it is not only emitted in quanta but also propagates in quanta and is absorbed by materials in quanta. This led to the concept of photons as real particles and subsequently to the concept of the "wave-particle duality." The question of whether the granularity of electromagnetic radiation is its intrinsic property (i.e. whether or not the EM field always exists only as photons) or it is only the result of the properties of oscillators that emit the EM radiation was extensively investigated. The behavior of quantum oscillators has been well understood and explained by quantum mechanics. If we assumed that the granularity of the EM field is simply the result of the properties of atomic oscillators, which ensure the emission of energy in portions, then the view of photons as real particles would have no foundation.
If such a view were taken, photons would be considered not real particles but rather what used to be referred to as classic "wave trains." This view has been rejected in physics because of the evidence showing that a photon of energy hf (where h is Planck's constant and f is the frequency of the EM wave associated with the photon) cannot be "split."
If an electromagnetic wave is emitted from a point source, it propagates radially in all directions. As the wave travels away from the source, it spreads over a spherical area of a gradually increasing radius, so the field's intensity decreases inversely proportional to the squared distance. In the photon model this is interpreted as the decrease of the number of photons per unit area, since the same wave which is carried by a constant number of photons spreads over the increasing spherical area. What, however, if the EM radiation is very feeble, so that single photons are emitted one by one? Is the photon's energy hf spreading over the spherical area of gradually increasing radius or does it propagate as an indivisible particle? If the model of a photon as a classical wave train is adopted, we can expect that the photon's energy is spreading over the spherical area. Experimental evidence is contrary to such a model. Photons behave as indivisible portions of energy hf and never split into fractions of hf. These data are interpreted as evidence that photons are indeed particles which however also possess the properties of a wave. Of course, an entity which is both a particle and a wave cannot be visualized because there is no analogy to such entities in our macroscopic world.
When the diffraction of material particles was discovered, the concept of wave-particle duality was extended to the material particles as well.
The wave-particle duality of photons seems to be established beyond doubt. Does this mean it must be adopted for the MPs as well? In this essay I will suggest an approach wherein the behavior of MPs, in particular the diffraction of MPs, is explained without assuming that the wave-particle duality is intrinsic to them as it seems to be to photons.
Let us review some evidence. Diffraction was observed not only for electrons, neutrons  or other subatomic particles, but also for whole helium atoms, hydrogen molecules , and even for the so-called fullerenes  which are spherically symmetric molecules comprising tens of carbon atom (in the quoted study it were C60 and C70 ). Does this mean that helium atoms, hydrogen molecules and fullerenes are waves? Possibly, but remember also that atoms of various elements have been “observed” and photographed using that wonderful device, the tunneling electron microscope. It is sufficient to look at these pictures (see for example STM Gallery) to realize that atoms indeed are lumps of matter occupying a definite volume in space, each having a specific shape. They have been manipulated by a human hand and arranged in various patterns. Among the atoms shown in these pictures we see gadolinium, cesium, copper, iron, iodine, nickel, platinum, sodium and xenon (plus molecules of carbon monoxide). Looking at these images, we realize that atoms are not waves but indeed are particles. Although helium atoms or hydrogen molecules have not yet been seen in the described way, there is no reason to assume that they are different entities than atoms of, say, nickel or iodine. Nevertheless, those helium atoms and hydrogen molecules which undoubtedly are real particles have been observed to produce diffraction.
Since it seems obvious that atoms and molecules are not waves but indeed lumps of matter, yet they display wave behavior (in particular they form diffraction patterns), this gives rise to the idea that subatomic particles which cannot be "seen" may not be waves either. Then their wave behavior has to be explained not by attributing to them intrinsic wave properties, but by looking for some other factors. In this essay I will suggest one such possible explanation.
The wave function, which is a mathematical device invented to describe the behavior of particles, is not an analog of electromagnetic fields which are emitted, propagate and are absorbed in discrete portions named photons. While the electromagnetic field "within" a photon is an actual oscillating physical entity, the wave function found from solving Schrödinger's equation does not represent oscillations of any physical entity, either "material" or as a physical field. As was first suggested by Born  (and promptly supported by Bohr and Jordan) and as is commonly accepted in science [6, 9], the wave function is a mathematical construct providing a probabilistic description of particles' behavior. I submit that the concept of a wave packet for particles other than the photon is a mathematical abstraction having no tangible physical meaning (although instrumental for solving many problems) and therefore the very concept of "wave properties" of "material" particles, unlike of photons, has no straightforward meaning. I will suggest (in rather general terms, comprehensible to a layman) a more parsimonious interpretation of the experimental data which shows that particles under certain conditions only seem to behave as waves.
The distinction between a mathematical construct, which is very useful as a tool for the description of an entity's behavior, and the real entities it describes can perhaps be illustrated by an example which is in a certain sense opposite to the situation with the MPs. In the physics of solids the concept of particles is commonly applied to situations where no real particles exist. Imaginary particles such as phonons, polarons, polaritons, excitons, and, in a little different vein, "holes" in the band theory of electric conductivity, are the staple of fruitful theories despite our knowledge that no such real particles exist. For example, phonons (which are imaginary particles associated with sound waves), are treated in a way very similar to photons in the physics of EM radiation . Moreover, the interactions of phonons with photons, electrons and other real particles are treated mathematically as if phonons were real particles possessing real physical characteristics, such as, for example, momentum, although phonons in fact posses no physical momenta.
If a mathematical apparatus turns out to be a potent tool for the description of physical reality, it does not necessarily mean that a mathematical construct is a physical reality.
In the case of phonons the theory introduces the concept of particles to describe the behavior of sound waves whereas in fact only the waves are real and the particles are imaginary. In the physics of subatomic particles an opposite mathematical construct is utilized: the behavior of real particles is successfully described by attributing to the particles wave properties. I suggest that wave properties have to be attributed to photons which have to be indeed considered as both particles and waves, despite the impossibility of forming a tangible image of such an entity. However, I suggest a different approach to the MPs. I suggest viewing them as particles only and looking for the explanation of their apparent wave behavior in factors which are external to particles.
This thesis may be countered by pointing to the similarity of the diffraction patterns formed by photons and MPs.
However, the similarity between some aspects of the behavior of photons on the one hand and the MPs on the other may have a simple and natural explanation. To this end, let us look first at the differences between photons and MPs.
The main difference is in that the electromagnetic field is a real physical entity whose behavior is described by Maxwell's equations . On the other hand, no real physical entity is represented by a de Broglie wave. Solving Schrödinger's equation for a material particle (for example, for an electron) under given boundary conditions results in finding a wave function which is a function of coordinates and time. This function, unlike an electromagnetic field, is not a measurable quantity. It does not represent any physical entity, either "material" or a field. It only reflects the spatial and temporal distribution of probabilities of the particle's "states." Maxwell's equations play the same role for photons as Schrödinger's equation plays for MPs possessing rest mass .
Although Maxwell's equations differ substantially from Schrödinger's equation, both describe a wave. In the case of EM radiation this is a wave carrying a real physical entity, the oscillating electromagnetic field. In the case of a "wave function" it is a wave carrying no physical entity as it is just a mathematical construct reflecting the probabilities of a particle's state. Since, however, both waves have practically identical mathematical structures and the behavior of MPs obeys the probabilities' distribution as dictated by the wave function, there is little wonder that the diffraction patterns formed by MPs look like replicas of the diffraction patterns formed by photons. This does not signify, though, that both diffraction patterns are necessarily due to the same mechanism.
With some limitations which are not essential for this discussion, in a certain way a photon may legitimately be viewed as a "wave packet," which is tantamount to a supposition that an oscillatory behavior of electric and magnetic fields occurs "within" each individual photon (as Einstein suggested). Indeed, such a description is often viewed as being in accord with the experimentally observed interference of light, even when supposedly a single photon was in the interference device (as, for example in the experiments by Taylor  in 1909 and much later in more precise experiments, like those by Janosi and Narai ). (I am not sure that the experiments in question can be unequivocally interpreted that way, but this is not important for my thesis here).
There are other differences between photons and MPs. No particle with a rest mass can be accelerated to the speed of light in vacuum. Photons, on the other hand, do not exist in any state other than in motion with the speed of light. For photons there are no frames of reference wherein they are at rest. In other words, it is impossible to choose a frame of reference attached to photons. If such a frame of reference existed, photons would be at rest in it and this is impossible. In such a frame of reference (non-existent) time would stop and any two events, say the departure of signal from any source and its arrival in any destination, however remote, would occur simultaneously. Photons move with the same speed in all frames of reference.
MPs possess no such properties. If photons are so drastically different from MPs, no wonder their behaviors in diffraction experiments may stem from different mechanisms. The similarity of EM waves (described by Maxwell's equations) to the waves of probability (described by the wave function found from Schrödinger's equation) is then just a consequence of the similarity between the mathematical structures of both waves. (Why such similarity exists is a metaphysical question which in all likelihood cannot be answered using the tools of science).
I submit that while for photons both terms – "wave behavior" and "wave properties" are equally meaningful, for particles with a rest mass it is more reasonable to distinguish between these two terms. I suggest that for a "material" particle the first term may be construed as reasonable while the second is misleading.
Note also, that no diffraction experiments have been conducted so far with particles of zero spin (such, as, for example, the short-living pions or Kaons) which do not possess a magnetic moment (perhaps because there are no sources of such particles readily available and the particles themselves are highly unstable). Until such patterns have been observed, no evidence contradicts the assumption that all material particles that show a wave behavior do possess magnetic moments.
One of the examples demonstrating the supposed "wave-particle duality" is what happens when a stream of particles is passing through slits in a wall which everywhere outside the slits is impervious (opaque) to the particles. If the slits have a certain size and are situated at a certain distance from each other, a diffraction pattern is formed on a screen placed behind the wall. This pattern consists of fringes wherein the locations that are hit by a multitude of particles are interspersed with locations the particles tend to avoid. The exact configuration of the fringes is easily predicted in all details based on the wave model of the beam. This is interpreted as a manifestation of the particles' dual nature, their being not just discrete lumps of matter but also waves.
Let us briefly review the history of the idea of the wave-particle duality.
The idea of particles having wave properties was a brainchild of de Broglie  originally born not from empirical evidence but from his imagination. In his early attempts to interpret "waves of matter," de Broglie suggested the concept of a "pilot wave" which guides the particle's motion. This concept entails the assumption that in fact there are two different waves associated with each particle. As evidence consistently contradicted that model, de Broglie vacillated between defending and dropping his concept, which was ultimately largely abandoned by science. With the advent of quantum mechanics, initiated by Heisenberg  and Schrödinger , de Broglie's idea of "waves of matter" was given a new interpretation via the concept of the wave function usually denoted ψ. It can be found from a differential equation (Schrödinger's equation) for any specified boundary conditions (although solving Schrödinger's equation may sometimes pose serious mathematical difficulty, powerful methods have been developed to resolve the problem). The meaning of the wave function was explained in 1928 by Born as reflecting the behavior of particles in a probabilistic way. According to this interpretation, which is universally accepted, the quantity ψ2, which is a function of coordinates and time, describes the distribution of probabilities (in space and time) of the "states" of a particle in question. The "state," for example, may be the combination of particle's coordinates, momentum, and possibly other properties, like magnetic moment.
Note that the probabilistic interpretation of the wave function, which is not disputed, does not depend on the assertion that a particle is also a wave. The experimental data indicate that the particles' behavior displays certain wave features. This is not equivalent to the statement that a particle is also a wave.
To see what I mean, consider a wave in water. If we drop a stone into a pool, waves will propagate radially from the spot where the stone hit the water. What is that wave? It is a propagation of disturbance without propagation of matter. Molecules of water are forced to perform an oscillatory motion, and this oscillatory motion is gradually encompassing an ever increasing circular area. However, the molecules do not move in the radial direction, they oscillate about stationary points remaining at the same radius from the source of disturbance. The water molecules participate in a wave, but we don't say that therefore each molecule is also a wave.
However, when subatomic particles are discussed, then, despite the universally accepted probabilistic interpretation of the wave function, a rather standard assertion is that every particle is also a wave. Of course we can't visualize an entity which is both a particle and a wave. To shed light on the concept of wave-particle duality, which sounds paradoxical from our common-sense viewpoint, the idea of "complementarity" of wave and particle properties was suggested (by Niels Bohr).
Besides being discussed in science, the concept of the particle-wave duality found its way into crank science wherein, along with such concepts as the collapse of the wave function, the alleged role of a conscious observer in that collapse, and some other speculative ideas, it has been misused in attempts to prove the supposed compatibility of scientific data with the biblical story. These attempts, sometimes pursued by bona-fide scientists who, when out of their labs, apparently indulge in reminiscences of their childhood with its sweet religious emotions, are usually not based on evidence but gain easy popularity among a gullible readership. Although extensions of some not universally accepted ideas in science into esoteric areas of pseudo-science have no substantiation, their rebuttal may sometimes be more successful if the underlying ideas in science are first reconsidered to see if they may be replaced by a more parsimonious interpretation.
The real question is not whether a particle is also a wave. I suggest that it is not. It seems to be just a particle to all intents and purposes. The question then is: why is the distribution of particles' paths described by a wave function?
Since it is commonly accepted that the wave function reflects the probabilistic character of events - in the case of diffraction on slits it is the spatial distribution of particles' paths - the question may be reformulated as: Why is the probability of a particle's choosing this or that path described by a wave-like regularity?
The full answer to that question can be no more given than a full answer to the question, say, of why the electron's charge is what it is or why there is the first law of mechanics, etc. However, a reasonable explanation of what makes the particles choose various paths can be attempted based on known facts about particles and solid bodies.
Perhaps the best starting point for my following discussion is that, possibly except for photons, no single particle has ever been observed to create a diffraction pattern. For example, in a slit experiment, when particles pass the slits one by one, each of them hits the screen at a definite location, creating a spark of light if the screen is of a phosphorescent material or a dark spot if the screen is a photo-emulsion, etc. Only when a multitude of particles have passed the slits do diffraction fringes form gradually as a result of the particles' collective behavior. The wave behavior of a particle manifests itself in the fact that each particle, while following a specific path, never "chooses" a path "forbidden" by the wave function; the larger is the value of ψ2 for a certain location on the screen, the larger is the number of particles hitting that location. No particle hits a location for which ψ2=0.
While this observation certainly calls for an explanation, it does not require the introduction of the incomprehensible notion that every particle is also a wave. A collective of particles displays a "wave behavior," and each particle "knows" that it belongs to that collective, so its individual behavior conforms to the overall wave pattern. To say that each particle is also a wave is however not any more substantiated than to say that each molecule participating in a wave propagating in water is in itself also a wave. (Of course, as with every analogy, the situation with water waves differs in many respects from the wave behavior of particles).
Another assertion which has found its way even into introductory courses of general physics is that the experimental data force us to assume that a particle somehow passes through more than one slit at the same time. For example, here is a quotation from Serway's textbook of physics (page 1178 in the 1990 edition ): "Somehow the electron must be simultaneously present at both slits to exhibit interference." Similar quotations can be found in many other textbooks. In fact, no such assumption is necessary since there are alternative assumptions which are more parsimonious. A couple of lines further Serway says, "In order to interpret these results, one is forced to conclude that an electron interacts with both slits simultaneously." The second statement is indeed a very reasonable conclusion from the observed facts. However, interaction is not necessarily tantamount to being "simultaneously present at both slits." I am afraid that Serway's first assertion, which can also be seen in many other sources including textbooks, is arbitrary, as it is not based on direct factual evidence but is rather a supposition made without considering more mundane interpretations. In the following paragraphs I will suggest such more mundane interpretations. My suggested interpretation will not entail any problems with non-locality of quantum effects, superluminal transfer of information, Bell's theorem, etc . It will be limited only to the well-established non-esoteric concepts without assuming any paradoxes or mysterious "hidden variables."
It is worth mentioning that the famous physicist Richard Feynman, in his acclaimed lectures titled "The Character of Physical Law" , when describing in detail the slit experiment, speaks about wave behavior of particles, but never asserts that a particle is also a wave (although such an interpretation is rather common; see, for example [6, 16]).
My main thesis can be briefly evinced as follows:
I submit that the assertion of the dual particle-wave nature of the matter is a misnomer. Particles are exactly that – particles. They are no more waves than are the molecules of water participating in a wave's propagation. First, no single particle has ever been observed to display a wave property unless it does so as a part of a collective of fellow particles. Second, the so-called wave behavior has been observed only in the particles' interaction with macroscopic devices. (In individual encounters between any two particles they always behave as discrete entities without any signs of wave behavior).
I submit (I do not pretend to be the first to assert that albeit I also cannot provide a specific reference) that ensembles of particles display a wave-like behavior because it is induced by forces external to the particles and stemming from the macroscopic bodies necessarily present in any experimental setup.
Since we always observe the behavior of subatomic particles not directly but only by using a macroscopic intermediary, the observed "wave behavior" of particles can be attributed either to the particles' intrinsically "wavy" nature or to their behavior being affected by the macroscopic intermediary. The attribution of "wave behavior" to the particles' intrinsic dual nature is common but entails serious problems in that we have no analogy to such a combination of granularity with wave behavior in our macroscopic world, so, if we use Feynman's words, with such an interpretation we simply don't understand the quantum mechanical phenomena. Therefore it seems attractive to explore the alternative explanation, i.e. to look for the source of the apparent wave behavior of particles in the effect of macroscopic intermediaries.
Let us review the slit experiment. A single particle never forms a diffraction pattern after passing through slits in an opaque wall. It always hits the screen at a definite location. After many particles have passed through the openings, regardless of whether they arrive one by one or in a beam of a high density, their collective behavior results in the emergence of a diffraction pattern. The shape of the diffraction pattern depends on the number of open slits. In other words, when passing through a certain slit each particle, although it creates no diffraction spectrum by itself, "chooses" its unique path after the slit as though it "were aware" of the presence of other open slits. Thus the diffraction pattern formed if only one slit is open differs from the diffraction pattern formed when more than one slit is open. For each configuration of open slits the diffraction pattern is specific and exactly as it would be if a wave were passing through the open slits.
If a particle passes through a slit there is no reason to necessarily attribute the particle's choice of a path after the slit to the mysterious "wave properties" of that particle. A more parsimonious assumption is that the material of the wall in which there is a slit, i.e. the electromagnetic fields generated in the slit by the material's ions and electrons, force the particle to "choose" this or that path.
For the purposes of this paper, the situation with particles moving through slits in an opaque wall is a convenient example for the clarification of my thesis, so I will discuss this situation in detail.
I will discuss four particular situations:
Although many other configurations are possible, my thesis can be explained by discussing only the four listed situations.Before delving into the details of the listed situations, a few preliminary notions seem to be in order.
λ = h/mv,
where h is Planck's constant, m is particle's rest mass, and v is particle's speed.
Since Planck's constant equals h ≈ 6.626×10-34 J.s, and the rest mass of an electron is 9.1×10-31 kg, then, for example, the de Broglie wavelength for an electron is
λ≈6.626×10-34 /9.31×10-31v = 0.71×10-3/v.
For example, if an electron has been accelerated by an electric filed wherein the electron passed the potential difference of 50 volt (which is a low-energy electron) its wavelength will be 0.174 nm. For electrons with a higher energy the wavelength will be even smaller.
The diffraction of electrons is easily observed on natural crystalline lattices where the inter-atomic distance (which is an analog of the slit width) is on the order of 10-10 m, or about 0.1 nm (i.e. about 1Å). Note that this quantity is minuscule from our macroscopic viewpoint, but very large from the viewpoint of a subatomic particle. For example, the size of a proton or a neutron was found to be by about five orders of magnitude smaller than the inter-atomic distance in a crystal. As for electrons, the very concept of size is ambiguous, but whatever its meaning is, obviously from electron's "viewpoint" the size of the slits ensuring a proper diffraction pattern is immense.
What is the significance of the above mentioned numbers? It is in the fact that the characteristic dimensions of the experimental setup in a slit experiment are very small from macroscopic viewpoint but large from the particles' standpoint. This ensures two things: (a) since the slit width is large from the particle’s standpoint, the slit can accommodate a multitude of various paths of passing particles; (b) since the distance between slits is very small from a macroscopic viewpoint, whatever happens to the electromagnetic field at the location of one of the slits must be felt in a measurable way in the field within the neighboring slits.
The scheme of the slit experiments normally includes several components. First, there must be a source of particles (for example, an electron gun). It is separated from the rest of the experimental setup by a partition which has an opening. The opening limits the spatial distribution of the particles so they enter the chamber where the experiment is conducted as a more or less collimated beam. (Sometimes, as, for example, in , the collimation is improved by utilizing more than one consecutive partitions with collimating openings in each of them). The beam is then aimed at a wall which is impervious to particles except for the slits in it. There can be just one slit, two slits, or many uniformly spaced slits (a grating). After passing through the slits the particles move toward a target which is capable of detecting the arrival of particles. For example, a detector may be simply a set of Geiger counters, or a plate covered with photo emulsion, or a bubble chamber, or a phosphorescent screen, or sometimes more complex arrangements (as, for example, in  where the fullerenes were detected by means of a separate multi-part setup).
It is essential to realize that the particles' beam, although crudely collimated by the ingress aperture, is wider than a slit or even than the combined widths of all slits. Various particles within the beam have various paths, differing both in their locations relative to the beam's axis of symmetry and in their exact directions. Various particles reach the slits at various distances from the slits' edges and also enter the slits in slightly different directions. Some of them pass through the slit closer to its edges while others move closer to the slit's middle, and the directions of their paths may be slightly askance relative to the slits' axis of symmetry. The distribution of the particles' paths in regard to their distance from the slit's edges and to their exact directions is unpredictable and to all intents and purposes chaotic.
Let us now consider the four situations listed above.
(a) Particles encounter a wall without slits.
Such a wall is supposed to be impervious to particles, so they could only pass through the wall if it had slits. It is easy to see that in fact this image of a solid wall impervious to particles is an abstraction rather far from reality.
Recall the famous experiment conducted in 1911 by Rutherford, Geiger, and Marsden . They aimed a beam of alpha-particles at thin metallic foils. To their surprise, many alpha-particles moved through the foil as through a sieve. The crystalline structure of a metal is indeed like a sieve. The particles falling upon the surface of a solid material encounter a sieve-like structure wherein ions perform a thermal dance about the nodes of the lattice, and the space between the ions is filled with what is referred to as electron gas. From the standpoint of the particles, the crystal offers plenty of free space through which to move, this space being filled by a periodic three-dimensional electromagnetic field. Hence the particles may either experience a collision with an ion, or, more often, only negotiate the field. The thicker is the wall the more probable are particles' encounters with ions whereby they transfer their kinetic energy to the lattice and therefore slow down. With a sufficiently thick wall the particles completely lose their kinetic energy and get absorbed by the material. This is what is actually meant by saying that the wall is impervious to the beam.
(b)Particles encounter a wall with a slit.
What is the difference between the solid surface and a slit? A particle hitting a solid wall encounters a grid of ions plus "electron gas" and plus an electromagnetic field in the space between the ions. A particle which happens to hit a slit encounters no grid of ions but it still has to pass through a field which, though, is somehow weaker than the field in the bulk of the wall's material and has a different configuration. Also, a cloud of thermally emitted electrons extends from the slit's edges into the slit. In the material's bulk the field is highly periodic (although far from being ideally periodic, as its regular character is distorted by lattice imperfections such as alien atoms, vacancies, dislocations, etc). Within the slit the field is expected to be smoother than in the bulk and symmetric with respect to the slit's edges. The strength of that field is non-uniform; it is expected to generally decrease from the slit's edges toward its middle.
The field in the slit affects the paths of the particles that pass through the slit. The path of moving electrically charged particles is affected in a straightforward way by the electromagnetic filed. While not all particle possess electric charge (for example, neutrons are electrically neutral), they usually possess a spin with a concomitant magnetic moment. A particle passing the slit is either both a moving charge and a moving microscopic magnet or at least just a moving micro magnet. As such, it interacts with the field in the slit; the latter affects the particle's path.
Therefore, the path of a particle after it has passed the slit depends on the exact location where it happened to pass through the slit. The particles that happen to pass the slit through its periphery experience a stronger field than those that pass closer to the slit's axis of symmetry. For various distances from the slit's edges, the strength and configuration of the field are different and varying with that distance. Therefore particles are forced to "choose" various paths after the slit and thus to hit the screen at various locations. Therefore it is expected that the stream of particles will undergo diffraction, the diffraction pattern spreading over a certain area of the screen rather than forming a sharp image of the slit.
Furthermore, if we attribute the particles' "choosing" various paths after the slit to the interaction of their magnetic moments with the EM fields in the slits, we have to remember that particles' magnetic moments as well as their projections on the axes of coordinates are quantized. Therefore it is expected that the diffraction patterns will not be continuous but rather consist of discrete fringes.
Obviously, this model fully preserves the notion of the indeterminacy of the outcomes of each particle's encounter with the slit. The distribution of particles' paths within the beam is random and unpredictable. Hence, using the words of Feynman , nature itself "does not know" where each particle will hit the screen. However, in this model we avoid any esoteric concepts of particles somehow being also waves.
According to this model a particle is a particle, period. The distribution of particles' paths is determined not by their incomprehensible wave properties but first by the random distribution of particles' paths within the beam, second by the quantization of particles' magnetic moments, and third by the regularities of the field's distribution across the slit.
An advantage of this model is that it uses a more parsimonious approach, as it does not introduce any additional concepts beyond the well known factors whose existence has been firmly established in science.
(c)Particles encounter a wall with two slits in it.
In experiments with two slits the distance between the slits has to be chosen in a certain range; as mentioned before, while it may be substantially larger than the widths of the slits, it must be still small enough for the situation at one slit to be sensed at the other.
What is the difference between the case of one slit and that of two slits? If a slit is made in a wall, the configuration of the electromagnetic field within the slit changes as compared to the bulk of the material. The particles cannot move through the bulk where they are captured by the material after having penetrated into it to a depth smaller than the wall's thickness. They can though negotiate the electromagnetic field existing within the slit. Their paths after the slit depend on where exactly they happened to cross the slit.
If, however, a second slit is made reasonably close to the first one, the configuration of the field in both slits will be different from that field's configuration when only one slit existed. The configuration in question is affected by all ions in the lattice, depending on the distances of particular ions from the slit. Since the slits are situated close to each other, the removal of a chunk of material to make the second slit means the removal of a number of ions contributing in a measurable way to the overall field in the first slit. Therefore the electromagnetic field within the width of a slit is different if there are two slits compared to the case of only one slit. If that is so, no wonder that the paths of individual particles are affected by the field in the slit in different ways depending on the number of slits.
d)Particles encounter a grating.
A grating consists of many slits situated at equally spaced locations. It is expected that the configuration and the strength of field within the multiple slits of the grating will be different from the case of either one or two slits. On the other hand, if the number of slits is increased over a certain limit, then adding more slits at the ends of the grating is expected to cause less and less difference regarding the fields in the slits. Therefore, while the diffraction pattern created on a grating is distinctively different from a pattern on one or two slits, there is no qualitative difference between patterns formed by gratings with different number of slits. There is, though, a quantitative difference in that the larger the number of slits per unit of length, the sharper are the maxima of the diffraction pattern.
A possible critique of the above model may refer to the haphazard distribution of the particles' paths through the slits which presumably has to result in the lack of regularity in diffraction patterns whereas in fact these patterns always fit the distribution of fringes according to the wave equation. The proper answer to that can only be given by a detailed calculation of the fields in the slits and of their effect on the particles' paths.
Another objection to my model can be offered by pointing out that diffraction was also observed in such experimental setups where the distance between the slits was too large for the EM field in one slit to be measurably affected by the removal of the material (to create the second slit). Again, the validity of that argument can only be asserted through a detailed calculation of the fields and of their effect on the magnetic moments of the passing particles.
Is the problem worth the required effort? I believe so. If the validity of the above hypothesis were confirmed, at least partially (and I would not expect anything better than that) it would provide serious advantages. It may lead to the removal of the non-parsimonious hypotheses of mysterious wave properties of what definitely are individual particles.
To summarize my thesis, I suggest that there is no real wave-particle duality for material particles. Material particles are not waves but are indeed lumps of matter localized in space whose "wave behavior" results from the interactions of their magnetic moments with electromagnetic fields created by various sources, for example by grids of ions in crystalline lattices.
Can I assert that the above model reflects the reality? No, I can't. Is there a chance it is correct, at least partially? I guess there is such a chance, although it is hard to be confident of that. How to test the above model? One way is to do so by detailed calculations of the fields' configurations in the slits and of their effect on various particles.
Another way to verify the above hypothesis would be to conduct a double-slit experiment (or, perhaps, an experiment utilizing a beam of particles interacting with a crystalline lattice) using zero-spin particles (like pions or Kaons). Since such particles do not have a magnetic moment, then, on the above hypothesis, they would be expected not to display a diffraction pattern. The problem is though that pions and Kaons are rare (they are present in cosmic rays and generated in accelerators) and unstable, so that conducting slit experiments with pions and Kaons may be not a practically viable option. If, though, diffraction patterns of zero-spin particles were observed, it would call for either a complete dismissal of the above hypothesis or at least for its substantial modification and amendment.
As far as I know, the available experimental data, while very well supporting the general thesis of a wave behavior of material particles, are in fact more qualitative, or at best semi-quantitative (as, for example, in  or ) than really quantitative and therefore unfortunately provide no direct evidence either in favor or against my hypothesis.
(The authors of  and  report a certain discrepancy observed in their experiments both with neutrons and fullerenes. For example, in their calculations they chose the values for the width of slits in a way ensuring the best fit of the theoretical curve. As Zeilinger et al report, in both cases -- neutrons and fullerenes, the width of slits which had to be chosen to achieve the best fit differed in a measurable way from the width reported by the grating’s manufacturer as well as from the data obtained by optical measurement. In  they admit having no explanation for such a discrepancy. In , referring to Grisenti et al , they offer a hypothesis that the discrepancy in question may be caused by the van der Waals interaction of particles with the material of the grating.)
In Zeilinger et al’s work these researchers, while having conducted a fine experiment, seem to have had a substantial wiggle room in their choice of quantities they plugged into the theoretical equations. Therefore their data may be viewed as semi-quantitative at best.
While Zeilinger et al’s data hardly can be used for quantitative conclusions either supporting or contradicting my thesis, we can at least note that the effect of the grating’s material on the particles’ paths, if it is indeed what has been observed, is expected from the standpoint of my thesis. The difference is though that Zeilinger et al mention the effect in question in passing only; they surmise that it is due to van der Waals forces; finally they view this possible effect as noise only. My thesis is based on the assumption that the effect of the experimental setup’s material is not noise but the very source of the diffraction effect; I suggest to view the effect in question as the interaction of the particles’ magnetic moments with the electromagnetic fields in the slits rather than a result of van der Waals interactions.
If a calculation or experiment shows that my model is too far fetched, so be it. We will then be back to where we are now, with the unenviable task of figuring out the mysterious notion of a particle as large and readily observable via an electron microscope, as a fullerene is, also being a wave. If, though, the calculation shows that indeed the fields in the slits ensure similar (albeit not identical) distributions of all the particles’ paths after the slits, or/and the experiments with zero-spin particles will reveal no diffraction pattern, or some other experiments will show quantitatively the dependence of the diffraction pattern on the material used for the experimental setup, we will have one disturbing controversy removed and the observed phenomena explained in simple parsimonious terms.
Acknowledgment. I am indebted to Matt Young and Brendan McKay for constructive remarks.
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