Quantum mechanics has given scientists and engineers a new and deeper understanding of physical reality. It explains the behaviour of electrons, atoms and molecules, the nature of chemical reactions, how light interacts with matter, the evolution of stars, the bio-chemistry of life and the evolution of mankind itself. Despite its successes it remains an intensely controversial theory. It suggests that very small objects such as electrons or photons behave in ways that contradict the common sense ideas. Yet many scientists to this day refuse to accept the fact that contradiction is an essential part of all matter.
Quantum mechanics is the part of science that deals with the motion of matter on the scale of atoms and sub-atomic particles. It was the experimental and theoretical response of scientists in the first half of the 20 th century to a series of contradictions that had emerged in 19 th century physics.
Quantum mechanics has given scientists and engineers a new and deeper understanding of physical reality. It explains the behaviour of electrons, atoms and molecules, the nature of chemical reactions, how light interacts with matter, the evolution of stars, the bio-chemistry of life and the evolution of mankind itself. Semiconductors, transistors, computers, lasers, plastics, all result from insights gained from this part of physics. When tested by experiment, the predictions of quantum mechanics have been successful to extraordinary levels of precision. In conjunction with the other great breakthrough of 20 th century physics, Einstein’s theory of relativity, it leads to the possibility of enormous advances in human society through the limitless supplies of energy from nuclear fusion – or the possibility of humankind’s destruction by atomic weapons.
Yet despite its successes it remains an intensely controversial theory. It suggests that very small objects such as electrons or photons (particles of light) behave in ways that contradict the common sense ideas and physical intuition that derive from the world of objects that we see around us. Very small objects appear to behave very differently from large objects - from things we can see and hold. Light spreads out through a diffraction grating like a wave, then thuds into a detecting screen as if it was a particle. Strange effects occur when electrons are scattered by crystals that seem to suggest an electron is not a particle – but a wave – but not always.
Worryingly for many physicists, quantum mechanics seems to fail exactly where it should be strong – in describing the motion of individual small particles of matter. It describes only the relative probability of, for example, a moving particle arriving at one place or another, or of an electron in an atom having one energy level or another. It has nothing to say about why or how the particle arrives here but not there, why the electron has this energy but not that, why an atom of a radioactive substance decays at this time but not another. This is acceptable, and very useful, when there are many particles, as in a transistor for example, when the probability that an individual particle can behave in various ways translates into the predictable overall behaviour and an observable and useable effect. But physicists would like to know more, and several generations of physicists have had to live with an uneasy feeling about quantum mechanics – that something is missing, that the theory is in some sense not complete.
Why should Marxists concern themselves with this part of science? Best leave it to the scientists, perhaps, those experts who know best. But bourgeois ideology permeates every aspect of life under capitalism. Scientists claim to be objective, simply dealing with the facts. There are countless examples that prove the opposite, from the cover-up for decades of the health-effects of smoking to the Nazi experiments in eugenics. Anyway, how can a scientist be objective when under capitalism science and technique are the key to vast profits?
Those most conservative academics who developed quantum mechanics inserted into the subject a direct attack on the philosophical basis of Marxism – dialectical materialism - at the most fundamental level. This was their chosen response to the incompleteness of quantum theory. Almost unbelievably perhaps, they chose to interpret the strangeness of quantum behaviour by denying the existence of physical reality. And as a standard textbook interpretation of quantum mechanics, physicists have been taught for the last 80 years that physical reality therefore only exists as a result of the act of observation. This is the “ Copenhagen interpretation” of quantum mechanics, developed in the late 1920’s by Niels Bohr and Werner Heisenberg. To quote Heisenberg: “I believe that the existence of the classical ‘path’ can be pregnantly formulated as follows: The ‘path’ comes into existence only when we observe it” .
If ideas are weapons, then, like religion, this is another weapon in the armoury of the bourgeois, another part of the defences that surround the unmentionable - the private ownership of the means of production. But there is nothing particularly new in this. The bourgeois are consistently obliged to deny reality to justify their rule. Bush and Blair pray together to the Almighty for guidance (for their “precision” bombing of civilian targets, perhaps?). The educated elite in the universities and government research laboratories meekly fall into line with their discussions on whether “the notion of physical reality is an ambiguous one” and that “quantum mechanics is a discussion of measurement phenomena without any relevance to a reality that is not observed”. The more astute may have other ideas, but they keep these to themselves. Like the old Soviet bureaucracy, they think one thing, say another, and do a third.
The heart of quantum mechanics: the double-slit experiment
Quantum mechanics is often associated with advanced mathematics, and mathematics can be used to develop the ideas of quantum mechanics to applications in complex situations. The mathematics, however, is only a vehicle for the physical ideas. The central ideas of quantum mechanics – the wave-like behaviour of matter and the particle-like behaviour of light – can be accurately described without the need to use any mathematics. The essence of the subject lies, however, in a description of physical reality and behaviour at small scales, which is very different from that of everyday objects that we are familiar with.
One of the clearest, and more consistently materialist, introductions to quantum mechanics is that given by the physicist Richard Feynman in his small book “Six easy pieces” and, in a slightly more mathematical presentation, in the first few chapters of volume 3 of his “Lectures on Physics”. Feynman introduces the subject by describing the “double slit experiment”, which he says in a famous quote “is absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality it contains the only mystery”. This is an experiment from classical optics that explicitly demonstrates the contradictory behaviour of matter at small scales – that matter can behave simultaneously as both particles and waves. It also reveals the roots of the idealism of the Copenhagen interpretation, and that the denial of physical reality was Heisenberg and Bohr’s response to this contradiction.
Waves are a process of energy transport, as we can see from the motion of the sand and pebbles on a beach when a wave breaks on the shore. Waves on the surface of a body of water disturb the surface as they pass by, moving it up and down. If two waves from different directions come together at some point on the surface the movement adds up – locally there can be a bigger peak or a deeper trough. If one wave is moving the surface upwards while another is moving it down then the total movement will be less than from each individual wave. At a place where the disturbances from different waves cancel each other out, the total movement will be zero.
These patterns of motion and interference between different waves are characteristic of how waves behave; particles – lumps of matter – do not do this. If two moving particles, say two pieces of rock, happen to meet they do not normally “add up” in some way. They collide, and depending on the force of the collision they might break into smaller pieces of rock or they might bounce off each other and continue moving in new directions. A bullet hits a target. Another bullet might hit the target in the same place. It never “cancels out” the first – there are simply two bullets where there previously was one.
From the beginning of the 19 th century it was accepted that light had the properties of a wave. Thomas Young presented experimental evidence to the Royal Society of London that appeared to demonstrate this conclusively. In this classic experiment he showed that if light is passed through two slits in an otherwise opaque barrier and then allowed to fall on a screen, the screen will show a pattern of light and dark bands. The prevailing view prior to that time, due to Newton , was that light consisted of small particles of matter. But the pattern that Young observed could only be explained by waves from each slit adding up – not by particles. “...It will not be denied by the most prejudiced that the fringes [which are observed] are produced by the interference of two portions of light. ” 
Water waves passing through two gaps in a
Young’s perspective that light was waves and not particles was accepted for over 100 years. It was extended by the experimental work of Michael Faraday and the theoretical work of James Clark Maxwell, who showed that light waves were a form of electromagnetic radiation. In the same way that water waves are a disturbance in the surface of water, light, they said, was the result of disturbances in electrical and magnetic fields. In 1887 these results were confirmed by the physicist Heinrich Hertz, who produced electromagnetic radiation at lower frequencies than light, in the form of radio waves. The wave theory of light seemed firmly established.
At the end of the 19 th century, however, this solid piece of classical physics came apart. Several scientists showed that when light shines on certain metals it can cause an electric current. Classical physics said that the strength of the current should depend on the intensity of the light but not its frequency. It didn’t. When the frequency was increased, the current increased. When the frequency was decreased, below a certain frequency the current stopped, no matter how strong the light was. Electromagnetic waves should not do this, but light did.
In 1905 Einstein showed that this could be explained by assuming that light was not waves, but small particles – photons. He suggested that when light shines on a metal the photons collide with electrons in the metal and produce an electric current. Each particle of light - each photon - has an energy that is proportional to its frequency. If the photon has enough energy – if its frequency is high enough - it can knock an electron out of an atom and then the electron can move freely throughout the metal.
Then i n 1909 the physicist Geoffrey Ingram Taylor reported the results of an experiment in which interference fringes were produced with a very weak light source. The light was so weak that only one photon at a time passed th rough the apparatus. Yet interference fringes were still observed. The experiment has been repeated many times since. With the development of sensitive photo-detectors in the second half of the 20th century it has become possible to perform interference experiments that actually observe the arrival of individual photons. The images shown here are the results from one such experiment by Robert Austin and Lyman Page of Princeton University . (See http://ophelia.princeton.edu/~page/single_photon.html )
The photons arrive at positions which appear initially to be completely random. With time, more photons arrive, but mainly at the strong parts of the interference pattern and never at the completely dark parts. Eventually when thousands of photons have arrived (and in normal light intensities, there would be trillions of photons) we see the interference pattern created by the arrival of individual photons.
How can this happen? Interference is a wave phenomenon, but the localised dots imply that light is made up of small particles, not waves. Why are there dots on the screen at the bright parts of the interference pattern and none at the dark? We can’t explain this by saying that the photons interfere with each other - the same happens even when there is only one photon in the apparatus. Does the photon split into two and go through both slits? Or perhaps, as the quantum physicist Paul Dirac mystically asserts “each photon interferes only with itself” . (Dirac is one of the leading physicists of the 20th century, but his philosophical statements are symptomatic of the idealism that has infected modern physics; take for example the quote: “This result is too beautiful to be false; it is more important to have beauty in one's equations than to have them fit experiment.” )
One hundred years later physicists are still asking how it is that a single particle can show interference, and are repeating these basic experiments, as in the Princeton examples, to see if there is something new to learn. The problem posed by Einstein in 1938 still has no answer for them: “But what is light really? Is it a wave or a shower of photons? …. It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.” 
Single electron events building up an
The situation becomes even more puzzling if instead of light we shoot electrons through the two slits. J. J. Thomson in 1897 performed experiments that showed that electrons were small charged particles of matter. This was the prevailing view in physics for the next 30 years. But in 1927 Clinton Davisson and Lester Germer observed diffraction (wave) effects when electron beams were scattered by crystals; George (G P) Thompson saw the same effect with thin films of celluloid and other materials shortly afterwards. These experiments alerted physicists once more to the strange behaviour of waves and small particles – that not only can light waves behave like particles, but sub-atomic particles can behave like waves. The double-slit experiment with electrons was not technically possible at that time, but it was nonetheless proposed as a “thought-experiment” used by early quantum physicists to explore their ideas about the wave-like behaviour of matter. The electron double-slit experiment was eventually performed in 1961, by Claus Jönsson of Tübingen; the single electron double slit experiment was performed by Pier Giorgio Merli, GianFranco Missiroli and Giulio Pozzi in Bologna in 1974, and repeated by Akira Tonomura and co-workers at Hitachi in 1989. The results of these experiments were as anticipated by the early quantum physicists; electrons, even single electrons, can interfere like waves even when they are detected like particles. An image from the Hitachi experiment is shown here and a film of the Bologna results can be obtained from http://lotto.bo.imm.cnr.it/educational/main_educational.php. The soundtrack of the film includes a rephrase of Dirac’s statement when it says that “each electron interferes only with itself” and fades out to the triumphant sound of baroque violins and flutes.
The Copenhagen interpretation
Attempts to explain the particle-like behaviour of light or the wave-like behaviour of electrons in terms of the classical ideas of particles and waves, which derive from observations of the behaviour of matter on a large scale, appeared to be impossible. The essential contradiction was between the localised behaviour of particles and the non-local behaviour of waves, summarised by early physicists in the phrase “the “wave-particle duality of matter”.
When science encounters a paradox or a contradiction this can be an opportunity to learn something new. With more work, more experimental results perhaps, science can progress. But for university academics, particularly in the Old World universities of Europe , it can be difficult to admit a contradiction, a possible error, in their results, and even more difficult to admit to being unable to solve a problem. The training of an academic scientist revolves around a competitive and individualist approach to solving problems, in which rewards are given for having a better answer compared to the next person. The laboratory scientist whose equipment is the product of the collective labour of a thousand hands is often unconscious of this basic fact and talks about “my” work, “my” results, “my” breakthrough. In the discussions in the seminar rooms what is at stake is not just the defence of a person’s ideas but the person themselves – not to mention the question of salaries, grants and promotion .
For Heisenberg the contradiction of wave-particle duality must have been intolerable – even more so, his inability to explain it. But if the great professor does not know the answer it is clearly unknowable.
Bohr (left) and Heisenberg (centre)
This most conservative scientist was the son of a classic languages professor. He had taken part in the suppression of the Bavarian soviet forces in 1918 (he later wrote: “I was a boy of 17 and I considered it a kind of adventure. It was like playing cops and robbers ...” ). In the second world war he was the head of the Nazi atomic weapons program. His upbringing and his training in classical philosophy not only made him adverse to accepting the blurring of boundaries and the contradictions implied by wave-particle duality. It also provided him with the philosophical weapons for a rejection not only of a dialectical but also a materialistic interpretation of quantum mechanics.
In his classic paper on the “uncertainty principle” Heisenberg showed that an experiment that attempts to measure which slit the object goes through (to localise the object, and therefore see it behave like a particle) will disturb it just enough to destroy its wave-like behaviour. If a microscope is used to observe the particle going through the slits then the wavelength of the light used by the microscope has to be small enough to distinguish one slit from the other. But if the wavelength of the light is short enough to do this then the light will have enough momentum to change the direction of the object and destroy the interference pattern. From this Heisenberg derived his uncertainty principle: “the more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” 
A short pulse is the sum of many
This result is a consequence of the geometry of the experiment, and the particle behaviour of light (that light of a certain wavelength is equivalent to photons with a certain momentum). Niels Bohr, working with Heisenberg in the University of Copenhagen, preferred to derive the result in a different way. Suppose, he said, that we are able to produce a short pulse of light by switching a light source on and then off very quickly. From a particle perspective, the source sends out a large number of photons that all travel in the same relatively small region of space. But if we try to understand the experiment from a wave perspective, we find that that a large number of waves of different wavelengths are needed in order to produce a short pulse. Wavelength is equivalent to momentum, because of wave-particle duality, so in a short pulse there is a large range of momentum, even though the position of each photon is known accurately. For a long pulse we have the opposite result - the momentum is known accurately, but the position of the photons is known less accurately. This is the uncertainty principle again – but without the microscopes, the disturbance produced by an observer, or any of the other paraphernalia introduced by Heisenberg. For Bohr, this approach put the wave-particle duality of matter at the centre, with the uncertainty principle as an inherent consequence, whilst for Heisenberg it was the act of observation that was more important.
For both Heisenberg and Bohr, however, the uncertainty principle became the opportunity to construct the mathematical and philosophical edifice of the Copenhagen interpretation of quantum behaviour – the mathematical expression (or more accurately, the mathematical alibi) for a rejection of material reality. They asserted that Heisenberg’s analysis of the double slit experiment, and similarly Bohr’s analysis of the properties of a pulse of light, were examples of a general law which became known as Bohr’s “principle of complementarity”: it is impossible ever to observe both wave and particle behaviour simultaneously. It is possible to either observe particles – a localised piece of matter detected on its way through a slit, with no interference pattern – or observe waves – a non-localised disturbance going through both slits, with an interference pattern. But any attempt to observe both simultaneously must fail. Thus the contradiction was resolved, by asserting that the question is simply not to be asked, for if asked it can never be answered. “You will see either waves or particles but never both”.
Furthermore - all that can be done is to make observations; physics must be viewed as the science of outcomes of measurement processes, and speculation beyond that cannot be justified. The question of where the particle was before its position was measured is meaningless. The particle materialises as a result of the act of observation. In the jargon – “the act of measurement causes an instantaneous collapse of the wave function”. What happens, you see, is that the measurement process randomly picks out exactly one of the many possibilities allowed for, and the wave instantaneously changes into a localised event to reflect that pick.
Caustically, and quite accurately, Feynman was known to refer to this as “the magic of the wave-function collapse” . That the observer can influence what is observed is not a new idea. But this is something far different. For Heisenberg and Bohr the observer not only affects what is observed – the observer creates it.
This is little more than an attempt to shore up the inadequacies of formal logic in the face of the evidence for the combined wave and particle aspects of matter. The alternative would have been to accept wave-particle duality as a deep-seated example of the union and interpenetration of opposites in motion at small scales; in other words, to accept that in motion rigid concepts are not adequate. Such an approach could be the starting point for a deeper investigation, for more experimental observations and more theory. What fundamental aspect of the interaction of matter with matter or with light is it that leads simultaneously to both particle and wave behaviour? What assumptions, observations, mathematical tools, should we revisit to obtain a deeper understanding of this phenomenon? But instead for political reasons – because dialectical materialism (Marxism) has been outlawed from the bourgeois professor’s study - we arrive at a dead end, where all further enquiry is deemed impossible in the face of the unknowable:
“The whole point is that the laws of formal logic break down beyond certain limits. This most certainly applies to the phenomena of the subatomic world, where the laws of identity, contradiction and the excluded middle cannot be applied. Heisenberg defends the standpoint of formal logic and idealism, and therefore, inevitably arrives at the conclusion that the contradictory phenomena at the subatomic level cannot be comprehended by human thought at all. The contradiction, however, is not in the observed phenomena at the subatomic level, but in the hopelessly antiquated and inadequate mental schema of formal logic. The so-called "paradoxes of quantum mechanics" are precisely this. Heisenberg cannot accept the existence of dialectical contradictions, and therefore prefers to revert to philosophical mysticism— ‘we cannot know’, and all the rest of it.” 
Challenges to the Copenhagen interpretation
A bubble chamber photograph showing the
paths of charged particles in a magnetic field.
Unfortunately for Heisenberg, developments in modern technology have allowed scientists to show that the path of a sub-atomic particle is very real. It is common to observe particle paths in high-energy physics experiments, where both the position and the velocity can be determined to within less than the uncertainty limit. Heisenberg defended his position against such evidence by saying that his uncertainty principle was only relevant to predicting the future. But he also said that “this knowledge of the past is of a purely speculative nature…It is a matter of personal belief whether such a calculation concerning the past history of the electron can be ascribed any physical reality or not.”  This let’s the cat out of the bag, to use an English expression – “it is a matter of personal belief”. Heisenberg himself is admitting here that his idealistic interpretation of quantum behaviour is an ideological choice. And his alternative escape route - that the uncertainty principle is only relevant to predicting the future – is a distinctly bland statement. If the momentum is only known to a certain degree of accuracy, we can only predict the future position to a certain degree of accuracy. There is nothing new or particularly profound in this.
The physicist Max Born  developed an alternative interpretation of wave-particle duality that avoided the idealism of the Copenhagen interpretation. Erwin Schrodinger had shown how to compute the “quantum mechanical wave-function” of a system; Born interpreted Schrodinger’s wave functions not as physical objects but as a way of describing the probability that a particle is at a particular location. For example, in the double slit experiment there is a wave-function for arriving by one slit and there is a wave-function for arriving from the other slit. The probability of arriving there is the magnitude of the superposition of the wave-functions for that position, in much the same way that the amplitude of a water wave is the sum of the different waves at a point on the surface of the water. Einstein explained the idea like this:
“…. it proved impossible to associate with these Schrodinger waves definite motions of the mass points - and that, after all, had been the original purpose of the whole construction. The difficulty appeared insurmountable until it was overcome by Born in a way as simple as it was unexpected. The de Broglie-Schrodinger wave fields were not to be interpreted as a mathematical description of how an event actually takes place in time and space, though, of course, they have reference to such an event. Rather they are a mathematical description of what we can actually know about the system. They serve only to make statistical statements and predictions of the results of all measurements which we can carry out upon the system.” 
As Einstein points out, an important aspect of this view of quantum behaviour is that the wave-functions are not assumed to have a physical existence. Particles of matter exist, they interact, pass through slits, move around in atoms. But the wave functions associated with them are a means to an end, a mathematical device that allows the physicist to compute the probability of a state or a combination of states – the probability that an electron in a hydrogen atom has a particular energy, or the probability of a particle of light arriving at a detector by a variety of different possible paths. When there are many particles, the probabilities become translated into the densities of the arrivals – more at the bright peak in a double-slit experiment, none at the dark peak.
This insight into quantum mechanical behaviour is essentially the approach taken in all practical applications of quantum mechanics. It has sometimes been described as the “shut up and calculate” method (an expression often credited, probably wrongly, to Richard Feynman) as an understandable reaction to the idealism and mysticism of other interpretations. When, for example, a scientist in industry sets about designing a TV screen, it is this approach that he will use. The electrons leave the heated filament here with this probability, giving rise to this current; they are accelerated by the magnetic field there, and deflected to that position on the screen. (If asked by the research department manager, however, it is clear of course that the path does not exist.)
Feynman himself used this approach – particles plus probabilities – in his work on quantum electrodynamics, described in his very readable and accessible book “QED - The Strange Theory of Light and Matter”. Quantum electrodynamics is itself an extremely successful theory, with predictions that match experimental observations to a very high level of accuracy.
A different type of double slit experiment has been performed recently by the scientist Shahriar Afshar, at Rowan and Harvard Universities . Results from these experiments, published on the web, directly contradict Bohr’s principle of complementarity. The complementarity principle asserts that it is not possible to observe both wave and particle behaviour simultaneously. But Afshar’s results suggest otherwise. His experiments are the subject of a detailed discussion on weblogs at http://irims.org/blog/index.php/questions (a good example of how the internet can open up the discussion of new scientific results to a wider audience, in contrast to the secretive review process used by traditional scientific journals). A copy of a paper describing some of his results is available at http://irims.bluemirror.net/quant-ph/030503/ .
In contrast to Heisenberg’s thought experiment about how to detect which slit the particle passes through, Afshar uses a lens and photodetectors positioned behind the interference fringes in order to observe photons passing through the slits. In the single photon form of his experiment (described verbally on the web, but results from which are not yet publically available) a light flash at the position of the image of a slit shows unambiguously that the photon passed through that slit. The photon is localised at that slit and is behaving like a particle. According to Bohr’s complementarity principle, an interference pattern –wave-like behaviour – should then not be observed.
Afshar checks to see whether or not interference is still present by placing thin wires at previously measured positions of the dark parts of the interference pattern. Even when he observes photons going through the slits, he can show that the wires are still in the dark parts of an interference pattern; the photon has been observed behaving both as a particle and a wave. The results of all of Afshar’s experiments are not yet available publically, and his experiments have not yet been repeated by others, which will be an important test of their accuracy. But if Afshar is correct, Bohr’s complementarity principle is dead.
Order from chaos
“Dialectics is a method of thinking and interpreting the world of both nature and society. It is a way of looking at the universe, which sets out from the axiom that everything is in a constant state of change and flux. But not only that. Dialectics explains that change and motion involve contradiction and can only take place through contradictions. So instead of a smooth, uninterrupted line of progress, we have a line which is interrupted by sudden and explosive periods in which slow, accumulated changes (quantitative change) undergoes a rapid acceleration, in which quantity is transformed into quality. Dialectics is the logic of contradiction. ” 
The picture of reality that has emerged from quantum mechanics and modern science is one of restless continual motion and change on an atomic and sub-atomic level. Atoms are bound together by a continuous interchange of particles between particles; electrons in molecules move from atom to atom; energy and matter interchange; particles turn into their opposite and then recombine. A central, distinguishing feature of this theory is change through steps, not as a continuum.
The developments of modern science in this sense confirm and deepen dialectical materialism. Yet, slowly decaying in the basement of modern physics, there is an absurdity – a logical, not a dialectical, contradiction. Without a dialectical approach to motion and change there is no way out of this contradiction.
Modern physicists have been forced to accept that concepts which had previously been considered separate must be linked, that they can not be thought of as separate but are different yet interconnected aspects of the physical world. In particular, the physicist’s concept of motion has to be extended to acknowledge the simultaneous wave and particle aspects of matter. When matter moves, a physicist can describe the process by momentum, which is the mass of the moving body times its velocity. A wave, on the other hand, is a different type of physical process. It is a disturbance, of the surface of a body of water or of an electrical field for example, and is a process in which energy moves. A physicist might describe a wave by its wavelength, the distance from one peak of the disturbance to the next. Momentum and wavelength are two quite distinct abstractions used to describe two different processes. Yet after Einstein’s work on the photoelectric effect, and after the theoretical work of the founders of quantum mechanics, physicists were forced to accept that momentum, a characteristic of matter behaving like a particle, is directly related to wavelength, a characteristic of matter behaving like a wave.
Much of the confusion surrounding quantum mechanics, added to and propagated by Bohr and Heisenberg, relates to the insistence that concepts such as wave and particle, or momentum and wavelength, must be kept separate - “we have two contradictory pictures of reality” as Einstein put it. This confusion is deeply rooted in the rejection – or the lack of awareness - of dialectics by modern scientists. “On the one hand, but then on the other” says the academic as he agonises over his choice between apparently contradictory options, wondering why the world is always like this. That apparently contradictory properties can be present simultaneously is not only possible but also universal. Light and dark, hot and cold, north and south, wave and particle, an inevitable and unavoidable combination, the existence of one being impossible without the other, and out of which comes change and motion:
“Whereas traditional formal logic seeks to banish contradiction, dialectical thought embraces it. Contradiction is an essential feature of all being. It lies at the heart of matter itself. It is the source of all motion, change, life and development. The dialectical law which expresses this idea is the law of the unity and interpenetration of opposites.” 
Not only that, but in their insistence on reductionism – one particle, one photon - scientists unwittingly and unconsciously destroy the living reality that they originally set out to investigate. In the images from the Hitachi electron double-slit experiment, at what stage does the wave behaviour of matter become visible? After 8 electrons? Definitely not – the electrons appear to have arrived at random, with no obvious pattern. After 270? After 2000? Even after 6000, the pattern is still blurred. Born’s probability interpretation allows the physicist to compute the relative probability of the particle arriving at a certain position. But the probability, or the wave-function, is only a statistical property of the system, and each individual arrival can be (almost) anywhere. We become aware of the wave behaviour of matter only when we have many particles. Similarly, in a gas we observe the laws that connect temperature, volume and pressure only when we have many molecules. Wave-like qualities emerge in a transition from quantity to quality; one particle or molecule is unpredictable, but many obey well-defined laws conforming to their statistical properties. Both waves and particles are observed – individual particles, which in large groups have the properties (interference patterns) of waves.
In that sense, single particle experiments and images of the sort obtained from the Hitachi experiment also directly contradict Bohr’s principle of complementarity. In the face of this evidence, supporters of the Copenhagen interpretation, like Dirac, have to wriggle around, imitating the photons they describe, saying that a photon goes through both slits and interferes with itself and then – in a puff of smoke, when the magician waves his wand - the wave function collapses.
One of the possible wave-functions
for the single electron in
a hydrogen atom.
It is common to draw pictures, as here, of an atom surrounded by an electron “cloud”. An interpretation of this image that is common among physicists is that the electron is somehow stretched over the region occupied by the cloud. True, the electron is moving very fast. A cloud is perhaps one way of representing the rapidity of the motion, and the fact that the electron could be anywhere in the shaded region. But there is only one electron in the hydrogen atom. During any small instance of time the electron is moving through a definite small, localised, region of space. It is no more stretched over space than a single photon is stretched through both slits in a double-slit experiment. To assume otherwise would be to arrive back once more at Dirac’s mystical “the photon interferes only with itself” and the magic of the wave-function collapse.
If we had many atoms and superimposed a picture of each, then we would see a cloud; we would see the wave function and its magnitude, the relative probability of the electron being at a particular position. The wave function describes the behaviour over many atoms, but says little about the motion of the electron associated with an individual atom. There lies both the strength and the weakness of quantum mechanics.
But does the path exist? Yes, providing that motion is understood dialectically. The path is the trajectory along which the particle moves. When the particle is in motion, it is not at any one position; it is in the process of moving from position to position. It moves along a definite trajectory. But to say it is here, or there, at some point in time means nothing. It is moving from here to there. It is the confusion that comes from an undialectical understanding of motion, the attempt to say that the particle is here at a particular point in time, that is exploited by Heisenberg to develop the mysticism that “the path does not exist”.
In the two slit experiment it is not possible to predict where the particle will go after the slits, other than on average. There is an indeterminancy, in the sense that the precise trajectory cannot be predicted in advance. But this is different from acausality. The particle arrives where it does as a causal chain of events. The apparatus fires the particle at the slits; it passes through one of them; it arrives at the detecting screen. And there are many examples in nature of causal but non-deterministic systems. A toboggan sliding down a bumpy hill arrives at a position at the bottom which is impossible to predict beforehand. If it starts from a slightly different position at the top it will arrive at a widely different position at the bottom. Unpredictability does not preclude causality. In fact modern science is beginning to understand that often causality is expressed through unpredictability – that necessity is expressed through chance:
“At first sight, we seem to be lost in a vast number of accidents. But this confusion is only apparent. The accidental phenomena which constantly flash in and out of existence, like the waves on the face of an ocean, express a deeper process, which is not accidental but necessary. At a decisive point, this necessity reveals itself through accident.” 
Quantum mechanics, the new physics, incorporated large elements of the old physics into its mathematical description. The mathematics of wave theory, techniques for solution of integral equations, and also the matrix representation of wave functions (which has been revisited and developed in recent years due to the applicability of matrix and vector formulations to digital signal theory) are elements of the mathematical methods of classical physics which are an essential component of quantum mechanical theory. The old is present in the new. It was a powerful boost to the development of quantum mechanics that a large range of mathematical tools of this sort were available that could be incorporated from classical physics. To develop further, however, perhaps quantum mechanics needs to overcome the limitations of the old – in particular its dependence on linear and low order differential equations.
Non-linear systems with a sensitive dependence on initial conditions that leads to unpredictability are the subject of chaos theory. The similarities between the behaviour of chaotic systems and the unpredictability of the behaviour of matter at small scales is suggestive of a possible similar explanation, and this is currently an active subject of scientific research. That large numbers of particles exhibit a well-defined wave-like behaviour could be evidence of the nature of the underlying dynamics – much in the same way that the patterns in the “strange attractors” of non-linear systems are a symptom of an underlying causality. An individual particle is unpredictable; many particles have a precisely defined behaviour. Order comes out of chaos – quantity becomes quality - as in other complicated, many-body, non-linear systems.
With the development of computers – a direct product of the understanding of semiconductors that derives from the insights of quantum mechanics – science is now able to explore these non-linear systems which classical mathematics cannot. Perhaps it will be in this region, in the physics of chaotic non-linear systems, that a deeper understanding of wave-particle duality will become possible. Or perhaps not. Perhaps the solution lies in more experimental data. As technology advances it will become possible to perform more exact, more complete, and new experiments. We will learn more about physical reality. Some ideas will be overturned, some revised, some developed further, some incorporated into the new.
New theories that ignore the dialectical nature of material reality – that deal with rigid fixed concepts, that ignore or dismiss the contradictions of motion – will ultimately fail experimental tests. This already appears to be the case with the Copenhagen interpretation if Afshar’s experiments are confirmed. The interaction between the observer and the observed is many-sided, and to separate one from the other inevitably leads to mistakes, as in the mysticism of the Copenhagen interpretation. Cause and effect can change position, the observer can effect the observed, and the observed can effect the observer. But fundamentally, at base, reality is material, it exists, and is not created by the act of observation.
That matter has properties of both waves and particles is intriguing, but not justification for abandoning physical reality. At a macroscopic level we have developed abstractions that help us describe, understand and use the material world around us. We see a rock, a (slightly large) particle of matter, and find it can be made into a tool – or a weapon. We see waves on the sea and build boats that can travel through them. Why should it be so disastrous to find that at small scales matter sometimes has the properties of a wave and sometimes of a particle? A photon passes through a slit. It arrives at a screen, most often at places where the interference fringes are strong and never where they are dark. According to Afshar’s (as yet unpublished) results it is possible to see which slit it went through – it is a particle. Yet where many of them go, on average, is determined by a wave equation – it is a wave. Interesting. Something to think about. But please – no more wave function collapses or pregnant paths. Science and technology could advance dramatically with a deeper, dialectical, and materialist, understanding of how these phenomena come about, and with a thorough clear out of the mystical and unscientific absurdities that currently masquerade as the “philosophy of science”.
Future scientists and engineers will understand physical reality better. And with the future technology that humanity will collectively plan and develop, it will be possible raise humankind far beyond the current struggle for life’s necessities. The savage barbarity of the capitalist system, the ugly inequalities, all the brutality and the savagery, will be nothing more than a distant unpleasant memory. And that too, like an electron dot on a screen, will fade with time.
 The quote is from Heisenberg’s original paper on his uncertainty principle, published in 1927 in the German physics journal Zeitschrift für Physik, volume 43, pages 172-198. An English translation is available in Quantum Theory and Measurement, Wheeler and Zurek, 1983
 Young’s double-slit experiment, as demonstrated on November 24, 1803, to the Royal Society of London, did not actually use a double slit; instead a narrow beam of sunlight was split by the edge of a thin card, achieving the same result as a double slit.
 Paul Dirac, The principles of Quantum Mechanics, 1930
 Paul Dirac, The Evolution of the Physicist's Picture of Nature, Scientific American 208 (5) (1963)
 Albert Einstein and Leopold Infeld, The Evolution of Physics, 1938
 D C Cassidy and M Baker (eds.), Werner Heisenberg : A bibliography of his writings, 1984
 Heisenberg, uncertainty paper, as in 1 above.
 For example, in QED, the strange theory of light and matter, 1985, Feynman says on page 76, in a footnote to material explaining how to compute probabilities by summing wave functions: “Keeping this principle in mind should help the student avoid being confused by things such as the ‘reduction of a wave packet’ and similar magic.”
 Alan Woods and Ted Grant, Reason in Revolt, 1995
 Heisenberg: Physical Principles of the Quantum Theory, 1930.
 Max Born, a Jewish German and a friend of Einstein, left Germany in 1933 to escape anti-Semitism; he is the maternal grandfather of the singer/actress Olivia Newton-John.
 Albert Einstein, On Quantum Physics, 1940
 Reason in Revolt
- Reason in Revolt, Marxism and Modern Science By Alan Woods and Ted Grant