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Wave-particle duality demystified?
alternative interpretation of certain quantum phenomena
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
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
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
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
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
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
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
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
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
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
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
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
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:
- The particles encounter a wall without any slits in it;
- The particles encounter a wall that has one slit;
- The particles encounter a wall that has two slits situated close to each other;
- The particles encounter a wall that has a set of slits spaced at equal
intervals (i.e. a diffraction grating).
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.
- What actually is a slit? To obtain a good diffraction pattern, the slit width has to be in a certain range, often much smaller than the wavelength. The wavelength of a particle (which for all particles having a rest mass, i.e. for all except photons, is their de Broglie wavelength) depends on the particle’s momentum.
It is expressed by de Broglie's formula:
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
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
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 distance between the slits in case there is more than one slit? To obtain a well formed diffraction pattern the distance between slits is also one of the determining factors. In experiments with visible light the distance between the slit is usually larger than the slit width, but still very small. (This may not always be true for a Fraunhofer diffraction of light; see, for example The Pinhole Camera). It has been true though in all experiments with diffraction of material particles conducted until now. For example, diffraction of electrons occurs on crystalline lattices where the inter-atomic distance (which in this case is not only an analog of the slit’s width’s, but also an analog of the inter-slit distance) is about only 0.1 nm thus being close to the electrons’ de Broglie wavelength. In the experiments with the fullerenes  the slit width was about 50 nm, while the inter-slit distance about 100 nm, which is of course a very small distance from our macroscopic standpoint.
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
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
(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
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.
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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