In the first phase of the Universe’s existence, matter and energy were subjected to such extreme pressures and temperatures that we can state with certainty that the laws of classical physics didn’t hold. To properly understand the physical conditions of the primordial Universe and the phenomenon that led to its origin, the laws of quantum physics are necessary for investigating the creation of energy and matter in the first place and nuclear physics to know how the first elements came into being. The former approach is that of quantum cosmology, while the latter developed into the theory of Big Bang nucleosynthesis (BBN).

Because the standard model (SM) of particle physics is a rock-solid theory which could not be superseded by a theory of quantum gravity (QG), likewise, while present quantum cosmology remains speculative, the foundations of modern cosmology are still rooted in very successful cosmological basics which were developed in the 1950s, on top of the discoveries of quantum physics and nuclear physics. Particularly successful in the microphysical domain became the theory of ‘Big Bang nucleosynthesis’, a well-established theoretical framework which describes the creation of the first light elements and which became the primordial bricks from which all the elements of the periodic table come. It received strong observational validation.

Nowadays, we know that most of the elements that built up the stuff of which we are made were not synthesized during the Big Bang phase. Rather, they were synthesized in the stellar interior through nuclear fusion processes. However, stars could come into existence only because of the elements created during the early phases of the Universe – first and foremost, hydrogen and helium and, to a much lesser degree, lithium and beryllium.

How do we know? Nuclear physics is a well established branch of science and its research is also facilitated by investigation with modern particle accelerators that can recreate, with few particles, the temperatures and pressures to which matter was subjected until about a few microseconds after the Big Bang. That’s another reason why particle accelerators are considered such important experimental tools. When particles are smashed against each other, they allow us to literally recreate ‘small bangs’. Moreover, a lot of evidence coming from astronomical observations made with powerful telescopes (on the surface as in space) has allowed us to collect a set of data which confirmed the theoretical predictions.

Immediately after the Big Bang, the Universe was still too hot to allow for the formation of not only atoms but also particles such as protons and neutrons. The Universe must have still been a ‘soup’ of even more fundamental constituents of matter – namely, a quark-gluon plasma. However, things changed quickly at those times: After about one microsecond, the Universe cooled down sufficiently to allow the quarks to bind together and form protons and neutrons (and other baryons – that is, particles made up of two or three quarks, which, however, decay quickly in protons or neutrons themselves). However, the temperature was still much too high to go beyond that; still no atoms could form. It took only another hundredths of a second of the Universe’s expansion to diminish the pressure and temperatures to about 10^9 K, which led to the formation of the first elements. At that point in time, the protons and neutrons (electrons, photons, and neutrinos as well, which we will not consider here) could form the first nuclei of the light elements through nuclear fusion reactions.

Fig. 1 provides a simple scheme of how the primordial elements were formed. This simplified Big Bang nucleosynthesis scheme can give you a glimpse at how the stuff that makes up stars was built from the bottom-up. For a more technical explanation see note1.

Right after the Big Bang, the universe was extremely hot and dense. During the first few minutes, particles were moving so fast that they constantly collided and sometimes stuck together. At first, there were mostly protons (hydrogen nuclei) and neutrons. These particles began combining through nuclear fusion, creating the first atomic nuclei.

The process happened roughly like this:

1. Neutrons and protons combined

   Some neutrons joined with protons to make deuterium, a heavier form of hydrogen.

2. Deuterium fused into helium

   Deuterium nuclei collided with other protons or deuterium nuclei, producing helium nuclei and releasing energy as gamma rays.

3. Small amounts of heavier elements formed

   Tiny quantities of helium-3, lithium, and unstable tritium and beryllium

   were also produced.

However, the universe cooled very quickly. After only about three minutes, it was no longer hot enough for these nuclear reactions to continue.

The final result of this early “cosmic nuclear factory” was that about three quarters hydrogen (mostly protons), about one quarter helium, and only very small trace amounts of Deuterium, helium-3, lithium-7.

The heavier elements that make up planets, rocks, and living organisms (carbon, oxygen, iron, etc.) were not made in the Big Bang. They were created much later inside stars through stellar nuclear fusion and supernova explosions.

Thus, the first three minutes of the universe produced the simplest ingredients of the cosmos: mostly hydrogen and helium, with tiny amounts of lithium — the raw material from which stars later built all heavier elements.

This is what, in its essence, the Big Bang nucleosynthesis predicts. It is the result of a study first pioneered in 1948 by the Russian-American cosmologist George Gamow. He was also the first to predict the existence of a cosmic microwave background (CMB) with a blackbody spectrum. (His guess was 5 K temperature, against the observed 2.7 K.) However, Gamow did not realize that all the elements could be produced in the early Universe.

His co-workers, Ralph Alpher and Robert Herman, completed the theory by pointing out how there couldn’t be enough time to synthesize heavier elements. These could come only afterward by means of nuclear fusion reactions inside the stars. That’s why you might have heard scientists say that we are ‘stardust’. The elements heavier than lithium of which our bodies are made (especially carbon, oxygen, iron, and other building blocks essential for the emergence of life) were synthesized in a stellar furnace billions of years ago by a progenitor of our Sun and then were ejected into the solar system through a supernovae explosion.

The Big Bang nucleosynthesis is not just a theory. Nowadays, it is backed by solid observational evidence. In fact, it makes clear cut predictions which can be tested observationally—namely, that once the average density of matter in the universe is known, the model fixes a precise abundance of deuterium, helium, and lithium relative to hydrogen. The result can be summarized as what is known as the ‘Schramm’s plot’ of the primordial abundances (see Fig. 3—notice that the axes are drawn in logarithmic scale). We already knew of these relative abundances from astronomical observations coupled with spectroscopic analysis of the light coming from throughout the Universe, which indirectly furnished the universal density of matter and radiation. The matter density of the Universe, being defined as the ratio between the number of all baryons (essentially, protons plus neutrons) and the number of photons, could be measured directly. In fact, this is what the Wilkinson Microwave Anisotropy Probe (WMAP) satellite did by measuring the CMB. The CMB that we can observe in the sky today is the microwave radiation left from the Big Bang ‘echo’ corresponding to the ‘recombination epoch’, in which the electrons began to combine with the nuclei synthesized earlier and created the lighter chemical element atoms. This recombination epoch dates back to about 380,000 years after the Big Bang. The analysis of the CMB, such as the density of radiation, also furnishes the density of matter.

WMAP furnished a value of a ratio of about 5x10^-10 baryon to photon density (the vertical line in Fig. 3). Therefore, on average, there are two billion photons for every proton or neutron in the Universe. When one plugs this into the Big Bang nucleosynthesis theory (the curves in Fig. 3), the relative light element abundances are fixed. The dashed rectangles represent the observed abundances (obtained with optical and/or radio telescopes.)

It turns out that under these conditions, the mass fraction of is roughly 24-25% that of hydrogen. Thus, shortly after the Big Bang, about ¾ of matter was hydrogen and ¼ helium. Only as little as of the D/H fraction (0.0015%) is left for deuterium, with one part over for (0.001%). Notice how the observed primordial abundances—the dashed rectangles–contain the predicted and observed data.

Unfortunately, this is not so for lithium-7. The observed Li/H relative abundance is about 1-2 x 10^-10, but what is predicted is 5 x 10^-10 — that is, lithium appears to be about three to four times less than predicted by the BBN model. This inconsistency is called the ‘cosmological lithium problem’. So far, different hypotheses have been developed to explain it. One possible explanation is that some yet-to-be-discovered nuclear mechanism, which goes beyond the presently known standard model of particle physics, was at work.

However it is, for all other elements produced shortly after the Big Bang, the theory conforms well to the measured data. This can’t be just a coincidence and is taken as a reassuring sign. This neat correspondence between observational data and theory (together with the neat match we could observe with the black body spectrum of the CMB) assures scientists that the basic mechanisms with which elements were synthesized during the first three minutes after the Big Bang theory are understood. Moreover, it is another nice example of the history of science in which particle physicists studying the microcosm collaborated successfully with astrophysicists investigating the macrocosm.

The history of the Universe could be graphically summed up as shown in Fig. 4.

During the time between 380,000 years and 200 million years after the Big Bang, the Universe must have been a boring and dark place, as photons were all high-energy light particles invisible to the human eye. However, the temperature after recombination was still about 3000 K. Only at the end of these 200 million years did the Universe cool down sufficiently to form giant hydrogen clouds, which collapsed due to the gravitational pull forming galaxy clusters, galaxies, and stars and where, later on, planets could form.

The BBN model works remarkably well back in time to about one second after the Big Bang. Before that point, different physics becomes dominant, and we can only speculate about what happened. To describe this earliest phase of the Universe, we would need a theory of quantum gravity, which still doesn’t exist. In the next article, I will discuss what might have occurred even earlier—before the “era” described above—namely, the phase of cosmological inflation, which is thought to have taken place almost instantaneously after the Universe’s beginning.

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A philosophical problem has plagued philosophers since the 17^th century, the time when René Descartes stated, “cogito, ergo sum” – namely, a ‘mind-body problem’ which ultimately boils down to the problem of consciousness. Nowadays, modern neuroscience dismisses Descartes’ ideas about the mind and consciousness. Thinking, being, and perceiving are separate categories that can’t be summarized by a slogan. However, what is particularly noteworthy is that the concept of ‘consciousness’ continues to elude a definition and clear scientific categorization and explanation. Despite the huge amount of data and discoveries of neurophysiology and neuropsychology, driven mainly by technological advances that allowed for the mapping and monitoring of the brain’s activity, when it comes to the so-called ‘hard problem of consciousness’, one can safely say that not much progress has been made in the last four centuries. In modern philosophy, the debate over consciousness was summarized by the Australian philosopher David Chalmers, who distinguished between the easy and hard problems of consciousness. The relatively easy problem of consciousness is that which concerns the explanation of how the brain processes information and environmental stimuli or focuses attention. To specific cognitive functions, one correlates a specific neuronal activity in the brain. This is, so to speak, an ‘easy’ problem because the correlation has been extensively investigated using modern brain measuring and imaging techniques (e.g., electric and magnetic encephalography, neuroimaging with different computer tomography and magnetic resonance methods, etc.).

However, the hard problem has more of a philosophical nature and relates to the question of why a biochemical activity, however complex it may be, should give rise to an experiencing subject that perceives qualities. It looks like a complicated interaction of neurons does give rise to an individual that seems to be a separate “I”, having subjective sensorial perceptions of pain and pleasure and registering colors, smells, tastes, touch, and sounds, which philosophers call ‘qualia’. The information from the environment is not experienced just as a stream of data of abstract bits, as it is encoded in our neural networks, but also as a qualitative phenomenal lived experience. Also, we do not perceive our mental activity as a program running on a set of logical circuits but, rather, as a content of insights that can span a spectrum beyond representations in bits and bytes and that find no parallel in a computer program or any algorithmic structure. Strictly speaking, a materialistic physicalist science should completely reject conscious perception as pseudo-scientific woo-woo and give it no more credence than the existence of unicorns or goblins. This would certainly be the case if not for the fact that scientists can verify by themselves that they are more than a data-crunching robotic zombies. For some unknown reason, we are living organisms that have subjective experiences. If modern science accepts the existence of consciousness, as characterized above by the ability to perceive qualities, it is not because of any empiric external evidence, which is instead completely missing, but only because of an ‘inter-subjective consensus’.

Further, nobody knows how the brain manages to bind its perceptions of thoughts, feelings, and the environment into a unique and undivided whole to which it associates a meaning. This is another problem, called the ‘binding problem’. For instance, when we look at an image, we do not consciously register each pixel one by one and begin to make calculations or ponder on it separately. Instead, we just see, almost instantly, an image to which we associate a unified meaning, being itself a conglomerate of meanings and perceptions at once. For example, when we see the image of a giraffe, we know at once that it is an animal (a categorial meaning) with a long neck (a morphological meaning) and a dark-yellow leopard-like coloring (a chromatic experience). Even if not displayed in the image, we eventually append to it (more or less unconsciously) the environment in which it lives, such as the savanna or a zoo (a mental figure). This is more than the sum of the parts and much more than pattern recognition (something artificial neural networks can already do). Instead, it is about the unified lived experience of a subject that, from a batch of seemingly disordered pixels and other data, suddenly ‘collapses’ it in the perception of the emergence of a meaning which, however, inherently retains a holistic cognition—not just a number or vector in a multidimensional space. The fact that meaning is not inherent in the objects ‘out there’ but must emerge due to the binding skill of our consciousness, is easily recognized through the famous Gestalt figures, in which different meanings could be ‘entangled’, such as those in which we see, at first glance, two faces and then, shortly thereafter, a vase.

Is Quantum Consciousness the Answer?

The fact that these holistic aspects of our cognition are reminiscent of quantum effects, such as quantum entanglement, has led some to conjecture that perhaps quantum coherence, with its indeterministic and non-local aspects, might be secretly at work in our brains. Maybe our brains are, at least partially, like a quantum computer able to process some information all at once.

It was almost inevitable that the conjecture surrounding quantum biology (QB) would sooner or later have been applied to the function of our brains. The human brain is a biological processing unit made of about 100 billion neurons. If QM plays a role in biological cells, and because neurons are nothing other than cells specialized for cognitive functions, it is hard to escape the temptation to extend the potential role of QB to neurons as well.

Moreover, there are also some peculiar aspects that characterize consciousness. For example, it behaves as something which is quite indeterministic and unpredictable. Is our free will real or only an illusion that arises from a complicated biological machine? Or, to put it in the quantum parlance: Are there hidden variables—underlying, unobserved variables that determine the results of measurements—in our brain processes that determine our behavior according to a deterministic view, or are there none and are we really just as free as an electron in the double slit experiment is ‘free’ to choose its interference fringe? The so-called “superdeterminists” such as Stewart Bell or the supporters of Bohmian mechanics would probably tend to believe that there is no free will because, in their interpretation of QM, even the electron’s behavior is predetermined by hidden variables. Others, who accept an indeterministic and non-reductionist view of reality, might be more in favor of the existence of free will. In this latter case, if QM is a truly indeterministic theory without hidden variables, it would naturally explain the unpredictability of conscious behavior beyond a mere manifestation of complexity.

This isn’t a new idea. It is almost a century old. Foundational figures in quantum physics, such as Arthur Eddington, Arthur Compton, and Pascual Jordan, speculated that microscopic quantum stochastic phenomena could be amplified to a macroscopic scale, influencing brain processes.

Jordan was the first to speculate that the unpredictability of life, our behavior, the mind’s apparent indeterminacy could be reconducted to a macroscopic amplification of the chance and unpredictability of the quantum world that is somehow scaled up inside living organisms. He called this the ‘amplifier theory’. The idea is that microscopic quantum indeterminism could also influence macroscopic biology by amplifying the effects of Heisenberg’s uncertainty principle and, thereby, could influence the entire organism.

Nowadays, this amplification theory could be compared to the famous ‘butterfly effect’ in non-linear dynamical systems in which only a tiny difference in the initial conditions of a physical system can cause entirely different behaviors and outcomes in the long term. The typical analogy people use to exemplify this is in weather forecasting and atmosphere dynamics. Say, for example, that a butterfly flaps its wing in one part of the world today; this would determine, in the long term, a completely different weather condition on the other side of the world than that what it would have been if the butterfly had not flapped its wings. This is a well-known extreme sensitivity to the slight modification of the initial conditions of non-linear systems—that is, it is an amplification effect. It is without question that the brain is a strongly non-linear system that inevitably must be sensitive to ‘neural butterfly effects’ as well.

The question is: Does this sensitivity go so far as to also incorporate the quantum effects? In a certain sense, we can be reasonably confident that highly sensitive nonlinear systems are capable of rapidly amplifying microscopic uncertainties, implying that quantum indeterminacy persists across scales. For example, it has been shown that a driven, damped pendulum may exhibit sensitivity to quantum fluctuations down to the Planck scale. While this does not constitute proof that an analogous process occurs in the brain, it nevertheless suggests that quantum-level indeterminacies cannot be ruled out a priori as potentially relevant to neural dynamics.

Why, when discussing the nature and origin of consciousness and mental processes, do some feel compelled to reference quantum indeterminacy? I believe they intuitively recognized that the inherent indeterminacy of the quantum world could provide an answer to escape the idea that we are “lumbering robots,” as Richard Dawkins put it.

Similar, but somewhat more sophisticated ideas were developed later by Australian neurophysiologist, philosopher and Nobel laureate John Eccles, who suggested that quantum theory plays a role in the workings of the brain as an alternative to a rigid deterministic neurophysiology. In 1994, Eccles claimed that quantum theory enters brain dynamics in connection to cerebral exocytosis–that is, the transport of vesicles filled with neurotransmitters from a nerve terminal into a synapse. Exocytosis is triggered by an action potential build-up caused by the flow of calcium ions through ion channels into the nerve cells. These ion channels are the size of a nanometer and thereby are small enough to be subjected to quantum phenomena. If so, quantum effects are amplified through the exocytosis processes, which determines the workings of the 80-100 billion neurons of all the brain circuitry.

During the same period of time, Henry Stapp, an American mathematical physicist, argued that consciousness might precede matter in being fundamental to the Universe. Our mental processes are, in themselves, due to quantum collapse, while free will is the manifestation of the quantum mechanical effects in the brain. He invoked the quantum Zeno effect—the phenomenon whereby a quantum system's evolution can be slowed or even halted by sufficiently frequent measurements—as evidence that processes can be delayed or modulated in their temporal occurrence by a conscious act. Stapp’s worldview is that of a panpsychist – namely, that consciousness and mind are a primordial property of matter.

An approach not too different from the wavefunction collapse was taken by the Nobel laureate Roger Penrose and Stuart Hameroff. The theory suggests that consciousness does not arise solely from classical neural computation in the brain but also involves quantum processes occurring within microtubules, cylindrical protein structures found inside neurons. According to Orch-OR, quantum superpositions can develop within microtubules and persist long enough to influence brain activity. These quantum states eventually undergo a fundamental, gravity-related collapse process called objective reduction, as proposed by Penrose. Hameroff and Penrose argue that the orchestrated sequence of such quantum state reductions generates discrete moments of conscious experience, linking the physical processes of the brain with the emergence of subjective awareness. However, the attempts to address the problem of the environmental-induced decoherence, remain largely contested.

More recently, some have proposed that a mechanism similar to that of photosynthesis or magnetoreception and that we saw in the last post, might occur also in and between neurons. There is convincing evidence in support of the fact that some chemical reactions inside cells, with neurons being no exception, are capable of emitting ‘biophotons’ (with the mitochondria being likely sources). We know that the neuron’s activity is mainly of an electric and biochemical nature but light emission might also play a role. We know that light-sensitive proteins exist in the brain and could potentially function as single-photon detectors. Where photons are emitted or absorbed inside living tissue, the same or a similar QB of photosynthesis and magnetoreception might be at work. For instance, some imagine a neural complex entangled state of spins generated through the exchange of photons. Entangled spins and photons are somewhat less susceptible to the thermal decoherence effects which could potentially be a partial solution to the problem posed by the ever-present environmental quantum noise. If Nature is able to provide a mechanism by which the coherence can be preserved for time ranges of a subjective experience (milliseconds to a second), that could potentially furnish an indication.

The Current State of the Debate

These represent only a small selection of the many speculations surrounding the possible connection between QM and consciousness. Many other ideas deserve consideration, but the scope of this short article is simply to introduce the topic and set the stage for further exploration.

All these hypotheses raise many interesting questions in neuroscience, biology, biophysics, and philosophy. However, these are even further than QB from being accepted and established scientific facts. It is not at all clear whether QM and consciousness have something to do with each other, and several scientists dismiss this hypothesis altogether. Again, it is difficult to imagine how that warm grey matter could contain and sustain, even in principle, long-lived phenomena of that extremely fragile phenomenon that is quantum coherence. Few believe that decoherence at room temperatures in such a mushy macroscopic object could be prevented from settling in almost instantly. And the idea that quantum indeterminacy could be amplified to brain-sized scales and play a functional role is regarded as even less plausible by most scientists and philosophers.1

Moreover, most do not feel that it is necessary to invoke QM in the brain because they are interested only in solving the easy problem of consciousness. They see no reason to believe that some cognitive functions couldn’t be explained by the classical laws of physics.

However, the divide is more ideological than scientific in nature. Those who lean toward some sort of dualist, metaphysical, or at least to some non-strictly materialist worldview that doesn’t reduce everything to matter, are generally more willing to consider the possibility that quantum phenomena play a role in brain activity. It is no coincidence that spiritualist New Age movements have extended quantum terminology to almost every aspect of reality. Instead, those who are more inclined toward a physicalist and deterministic view of life, are typically much more skeptical. Some dismiss it altogether as pseudo-science.

We must be aware that the role of QM in the brain is still more a topic discussed in popular magazines than a scientifically established and actively pursued field of research. It is a line of inquiry that receives very limited funding and is therefore often constrained to theoretical and speculative investigations.

Times are changing, however. In recent years, research into quantum consciousness and, in particular, findings from quantum biology suggesting that quantum effects may play a role in biological microstructures have attracted renewed interest. An increasing number of scientists are beginning to take these possibilities seriously. We should hope that this trend continues.

Let us not forget that the history of science has shown that even highly foundational research, which initially appeared to have no practical applications, can eventually prove revolutionary. Not only by reshaping existing paradigms but also by leading to unforeseen practical developments.

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However, Nature has surprised us so many times in the past and may do so this time, too. Quantum biology (QB) is a research field in its infancy but there are several indications that this may change in the not-too-distant future, though the speculations that QM may have something to say in biology are not new. Already in 1944, Schrödinger wrote a famous booklet entitled “What is Life?” in which similar conjectures were advanced for the first time. He asked if the ingredients that seem to make life so special and biology so different from chemistry might be traced to QP.

Also, the German physicist Pascual Jordan, who, among many others, contributed to the development of QM, speculated about the possible connections between quantum indeterminism and life. If the indeterminism of QP acting in cells could be amplified by some unknown mechanism at macroscopic scales, that could potentially explain the unpredictability of the complex biological structures that we are. Despite his achievements in theoretical physics, after WWII, Jordan’s reputation became discredited because of his connections to Hitler’s national socialistic ideology, as he had joined the Nazi party and its paramilitary unit. Probably because of this past political association, his works have been long dismissed and his conjectures about the possible connections between QP and biology have been forgotten.

In 1963, the Swedish physicist Per-Olov Löwdin suggested that the tunneling of protons between atomic nuclei might be a mechanism that could induce DNA mutations, and introduced the term ‘quantum biology’. This hypothesis must still be proven but it sounds plausible to conjecture that quantum tunneling is at work between nucleotides, i.e., the building blocks that make up DNA and that carry genetic information. The two DNA strands are held together by hydrogen bonds. In the process of replication, in which each strand makes a copy of itself, there is a small probability that a proton might ‘jump’ via a quantum tunneling process from one hydrogen atom to the other, thereby inducing a mutation. Computer simulations indicate that proton quantum tunneling in DNA is large enough to cause point mutations in the genome faster than the biological timescales.

In the 1980s-90s, British physicist Roger Penrose and the anaesthesiologist Stuart Hameroff advanced the hypothesis that QM could be at the base of the emergence of consciousness in the activity of neurons in our brains. Tiny microstructures in the cytoplasm of cells, called ‘microtubules’, are small enough to potentially support some quantum phenomenon. No evidence backs such a claim so far. We will return to this in the next article.

Not until 2014 did the British theoretical physicist Jim Al-Khalili publish a popular science book on QB with the suggestive title “Life on the Edge: The Coming of Age of Quantum Biology”, which made the subject known to the broad public.

All these attempts to look for a possible interface between the quantum world and biology rest on the fact that nowadays we know how molecular biology, genetics, and organic chemistry have been very successful in describing the processes of the functions of life in the microphysical domain. These are all about molecules essential to living organisms, many of which are made up of only a few atoms and stick together through chemical bonds which, ultimately, could be described only by resorting to the atomic quantum orbital models that are based on the laws of QM.

In fact, in the last decade, several instances of (though not completely conclusive) experimental evidence have emerged to indicate that QM might, indeed, play a role in biochemistry. For example, there is nowadays some consensus that enzymatic activity can’t be explained without resorting to QP. Enzymes are proteins that act as catalysts in chemical reactions, accelerating biological processes in cells. Without enzymes, almost no living being could survive; they are among the most fundamental ingredients for life, at least as we know it. Some enzymes’ chemical kinetics are faster than what classical chemistry can explain. Studies show how electron transfer (redox reaction) over long distances from one molecule to another (redox centers) could be achieved via quantum tunneling effects. This quantum mechanism plays an important role in enzymatic activity.

One of the strongest indications that quantum effects are at work in cells is photosynthesis. As everyone learns in school, photosynthesis is the chemical reaction in plants that converts the Sun’s light into chemical energy. When photons hit the green-pigmented chlorophyll molecules, these induce, through an electron excitation mechanism, a charge separation triggering a complex chain of chemical reactions that convert them into chemical energy and transform carbon dioxide and water into glucose and oxygen. This energy transfer from the chlorophyll molecules to the cell’s internal reaction sites has an efficiency of almost 100%, which can hardly be explained by classical chemical or physical principles. It looks like photosynthesis resorts to QM in order to achieve such efficiency. In fact, some studies suggest that long-lived quantum coherence combined with quantum tunneling may furnish the needed mechanism. Only by this mechanism can the packets of energy be transported to the cell by simultaneously following all possible paths to the reaction center.

Currently, research is seeking to determine whether QM could explain olfaction – that is, the biochemical reactions we use for the sense of smell. We know that olfactory receptors in the nose are quite efficient at binding odorant molecules. However, it is not entirely clear how the receptor molecules bind and recognize the different chemical substances. One theory states that the shape and size of the molecules are what determine the types of smells. These molecules are detected by the olfactory receptors onto which they must fit according to their shape and, by special signaling to the brain, decode it as different smells. In other words, according to this hypothesis, the recognition of smells is related to the form of the molecules carrying that smell. However, more recent tests suggest that molecules with very different shapes can lead to the same smell. Therefore, a new theory suggests that olfactory experience is not mediated by the shape of molecules but, rather, by their frequency of mechanical vibration, which is triggered by the quantum tunneling of an electron between the olfactory molecule and specific amino acids within the olfactory receptor. The latter theory is still somewhat controversial but experiments suggest that both are not mutually exclusive: The chemical composition, shape, and size of the molecules determine whether they are ‘locked’ into the receptors; their vibrational frequency then determines what we perceive as smell.

Another domain in which, quite surprisingly, QB might play an important role, and which has recently attracted attention, is migratory birds. It has been shown that at least some species are sensitive to the Earth’s magnetic field – an ability called ‘magnetoreception’. The most striking example is the European Robin, which migrates in the winter from northern Europe to the Mediterranean Sea, and then back in the summer.

For a long time, scientists wondered what the reference system of these birds could be such that they always find their way along their thousand-mile journey. The Earth’s magnetic field seems to be the answer. However, it is a mystery as to how they can perceive such a weak magnetic field. Research on the European Robin suggests that the directional information about the Earth’s magnetic field lines (more precisely, its inclination with respect to the surface) might be conveyed to the bird’s brain by a mechanism relying on QM – in particular, on quantum entanglement. It is triggered by the light shining on the bird’s retina, where magnetically sensitive protein radical-pairs molecules, called ‘cryptochromes’, reside.

A radical-pair is made of two molecules with an odd number of unpaired electrons due to an ionization process photo-induced by light radiation. These two electrons are separated because each resides on its respective molecule but they are an entangled singlet. Of course, each of these electrons has a spin, which gives magnetic momentum to each separate molecule. And, just as the electron’s dynamic can be influenced by its own magnetic moment, here, also, the Earth’s magnetic field can change the spin state. Therefore, the radical-pair mechanism explains magnetoreception insofar as a magnetic field can affect the radical-pair’s chemical reactivity, which, in turn, determines the speed at which other chemical reactions and products are formed. In short: The direction of the Earth’s magnetic field can, through electron entanglement, catalyze the biochemical reactions in the robin’s eye. These reactions are then signaled to its brain. We will never know unless we become a European Robin but perhaps, thanks to QM, migrating birds might literally ‘see’ magnetic fields and orient themselves along these field lines.

QB is far from being an established and accepted science. So far, the evidence supporting it has gone beyond mere speculation but it isn’t conclusive. Many physicists and biologists remain skeptical. As with quantum computing (QC), QB isn’t really a new field of research. However, the decisive difference between the two research lines is that, while QC has been intensively pursued and heavily funded by billions in investment in the last couple of decades due to its potential practical and commercial applications, QB remains a largely ignored science. It has never received serious attention apart from that of a few scientists and a handful of research institutions. Thus, it has remained very limited in terms of scope, funds, and time. One of the reasons for this is that nobody has been able to come up with a convincing mechanism that could prevent quantum decoherence due to thermal noise and environmental interference.

On the other hand, even if one day we discover quantum phenomena influencing or regulating life functions, it would not be clear what this knowledge could be good for. We live in a society driven by pragmatism and utilitarianism. If a relatively small field of research, however interesting and original it might be, isn’t seen as useful for some practical and commercial application, it rarely gets funded. Pure science for the sake of knowledge is out of fashion.

One can only hope that QB will receive more attention in the coming years. It would be interesting to know for sure whether QM does, or does not, play a role in the development of life. If it does, this could open an entirely new line of research leading to a paradigm shift in biology and potentially also to applications. The history of science shows that pure science can – and, indeed, in most cases does – lead to unexpected applications. Galileo, Newton, Einstein, and Planck, like many other great physicists, were driven primarily by a thirst for knowledge and not by the prospect of applying their discoveries. And yet, without their discoveries, we would still be stuck in the technological middle ages. Despite its inception half a century ago, QB remains a non-mainstream science. One can only hope that this will change soon.

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In the previous section I discussed the how a state vector represents the quantum state of a system, the Born rule, and the measurement postulate.

Note how the sum of all the probabilities adding up to unity is represented in Dirac’s notation neatly as:

If we want to deal with real-world measurements, we must specify which information from this mathematical construct (the wave function or state vector) we want to extract – that is, which physical quantity, and more precisely which ‘dynamical variable’, we want to measure (including, for example, position, momentum, angular momentum, energy, etc.). Mathematicians and physicists define an ‘operator’, which, as the word says, ‘operates’ on the wave function or state vector to obtain a specific piece of information associated with the observable.

Examples of mathematical operators that every one of us knows are the addition (+), subtraction (-), multiplication (*), and division (:) operators. The multiplication operator acts on two quantities, a and b, to obtain another quantity c as a*b=c. Similarly, one can conceive of an operator O acting on a state vector |Ψ⟩. If we consider this operation to be a formal equivalence to an act of measurement of a dynamical variable, the observable of a system, then the output of this operation should tell us something about the outcome of this measurement, a scalar quantity lambda, a number, which represents the expected value from this measurement, the so-called ‘eigenvalue’:

Lambda is the outcome of a measurement or, more broadly speaking, simply an ‘observation’. This equation is therefore called the ‘eigenequation’. It is the mathematical procedure we must perform to make predictions once we know the experimental setup and boundary conditions.

Stated a bit more precisely, one should say that each dynamical variable (e.g., position, momentum, angular momentum, energy, etc.) is associated with an (self-adjoint) operator O, the observable, which acts on the state vector of the quantum system |Ψ⟩ and which furnishes a scalar quantity, the measurement value, called the eigenvalue. Self-adjoint means that the complex conjugate of the operator is again the operator itself; however, these are mathematical aspects that are not very important for our purposes. Simply remember the connection between the classical notion of dynamical variables and its counterpart in QM as operators, the observables.

To avoid a possible source of confusion, let us state this again from another perspective. Operators are not numbers. 1,2,3,4,5… are numbers, whereas ‘+’, ‘-‘, ‘*’, or ‘:’ are binary operators. They are binary because they combine two numbers (called ‘operands’) to obtain another number (for example, 1+2=3). Therefore, it makes no sense to state that if O|Ψ⟩ = lambda*|Ψ⟩, then O = lambda, because you are then equating an operator with a number—that is, mistaking apples for bananas. Moreover, the operators used in QM operate on functions rather than on numbers. Perhaps the simplest non-trivial unary operator is the derivation operator:

It is unary because it operates on one function to obtain another function. Quantum operators—that is, observables—are also called ‘Qnumbers’ because they have a formal analogy with numbers. However, remember, they aren’t numbers at all.

Once that has become clear, here are some examples of observables. The simplest one is the position operator:

If we want to know the position of a particle, we multiply the wave function with a position variable x and then take the modulus square to obtain, as usual, the probability density in function of x. This tells us something about its whereabouts.

The momentum operator, that observable which tells us something about the momentum of the particle, has the form of a derivative operator.

Taking the derivative in space of the wave function (times the Planck constant bar and an imaginary number) will furnish a mathematical expression that tells us something about the speed (times mass) of the particle that the wave function describes.

Another important quantity in QM is the ‘expectation value’ of an operator – that is, the average value one obtains by measuring an observable many times. It is operatively a fundamental notion in QM because in the laboratory, what one usually must tackle in the real world are averaged measured values. For example, for the position operator X = x of a particle, one integrates its position x, weighting it by the probability of finding it in x or, in analytical terms (recall what we saw here):

It is the average value of the position x of a particle that one expects to obtain after a large number of measurements. More generally, this can be written in a more compact form by using Dirac’s notation, stating that the expectation value of an operator A in state Ψ is defined as:

with right-hand side Dirac bra-kets as defined in the integral above.

Another operator of paramount importance in QM is the energy operator, the ‘Hamiltonian’, which has a bit more complex structure. It is given by a second derivative in space (times a constant involving Planck’s constant and the mass of the particle), plus a function V(x), which represents the particle’s potential energy. By ‘potential energy‘, physicists mean the energy content of an object that can arise due to its position inside a force field (such as an object in a gravitational field or an electric charge in an electric force field) or stresses within itself (for example, an expanded spring) or in the form of chemical energy (the chemical bonds between atoms and molecules), etc. Without us getting into additional rigorous definitions and details, just imagine it as a sort of ‘stored energy’ (here, V(X) can be the function which tells us the stored energy an electron has at a certain distance x from the nucleus), which, however, can be transformed into another form of energy for practical purposes. The archetypal example is the transformation of potential gravitational energy (say, the potential energy of a stone at the top of a hill) into kinetic energy (the stone rolling down the hill and acquiring a certain speed.) The Hamiltonian operator is written as follows:

It is this latter observable in particular which leads to the famous ‘time-independent Schrödinger equation’:

This is, however, not the most general equation because, as the name indicates, it is time-independent, while physical systems are, in many cases, time-dependent. Therefore, the time-dependent energy operator is called ‘evolution operator’ and is given by:

That’s why you will frequently find the ‘time-dependent Schrödinger equation’ in this form:

An important property of Schrödinger’s equation is that it is linear. A differential equation is said to be ‘linear’ when, given two different allowed solutions, their sum is also an allowed solution. As an example, you might recall the superposition of two waves forming a new wave, as discussed with the interference phenomena. This will be crucial later, when we discuss the quantum superposition principle.

The Hamiltonian operator also defines the ‘time evolution’ of the wave function from a time 0 to a time t. In particular, if H is independent of time, given a state at an initial time t=0, the state of the quantum system at any subsequent time t is ‘generated’ by the so-called ‘unitary operator’

as

Its main physical interpretation is that it is an operator that does not describe an observable (that is, a measurement), but the evolution in time of a quantum system. It changes the phase of the wave function or state vector but not its length. This ‘length conservation’ reflects nothing else than the necessity to conserve the sum of all the probabilities associated to the eigenstates before and after the state evolution, that is, it must hold:

Therefore, its name ‘unitary operator’. Moreover, note that if U is the unitary operator that evolves the quantum state forward in time, its complex conjugate

sends it backwards in time by an interval t. Then it becomes obvious that applying first and (or vice versa) consecutively it preserves the state, that is, an equivalent formal definition for a unitary operator is that it satisfies the general relation:

With one being a scalar (just the number ‘1’) or, more generally, an identity operator. This reflects a fundamental property of QM, that of ‘unitarity’, which must always hold.

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This first volume on the basics of quantum physics is a guide bridging the gap between oversimplified pop-science and dense academic textbooks by offering a rigorous yet accessible A-Z overview of quantum physics’ conceptual foundations, including topics like wave-particle duality, entanglement, Bell’s theorem, and quantum erasers. It targets students, physicists, philosophers, and others who want a deep, misconception-free understanding, requiring only high school math while avoiding the dry, formal approaches typical of university courses. The first volume concludes with a discussion of philosophical idealism, emphasizing conceptual clarity over material physicalism or “quantum woo.

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The second volume extends the first volume’s focus on 20th-century foundations into 21st-century principles and research, covering advanced topics such as quantum ontology tests, interpretations of quantum mechanics, quantum field theory, quantum computing, the black hole information paradox, holographic principle, quantum cosmology, and even speculative areas like quantum biology and consciousness. Written in a concise but rigorous intermediate style, it serves as a guide for those seeking a serious understanding of modern quantum theory, helping them distinguish scientifically established facts from media-driven pseudoscience and unfounded speculation. The final chapter again connects quantum physics to philosophical idealism, offering readers an unparalleled conceptual background to navigate contemporary debates with clarity.

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We have seen that QM is a purely probabilistic theory. Accordingly, events occur only with a certain probability but the single event can never be predicted with certainty unless the system is already prepared into a state called an eigenstate—that is, that kind of quantum state in which repeated measurements of the quantum system will always lead to the same result.

Yet, in general, quantum events are purely random and we must resort to statistical reasoning and predictions as well as a statistical formalism and math. This is essentially what distinguishes CP from QM, the purely probabilistic and random nature of the latter.

As we already mentioned with Laplace’s demon, in CP, if you fix the boundary conditions, a system will always evolve in the same manner, and it is deterministic and predictable, at least in principle. In QT, even if you shoot the particles from the same place, with the same speed toward the very same slits, you will always see them lighting up a different spot on the detection screen.

However, this does not mean that nothing can be said about the average behavior of a system, say, a system of particles, or its dynamic and temporal evolution. In this case, it is perfectly possible to describe it according to some statistical rules—for example, the statistics that describe the distribution of a large number of particles on a screen according to an interference fringes pattern. This also means that, in QM, the math describing the dynamics of a physical system must be very different. Instead of ascribing to a particle a single final possible state, one must resort to the wave function, which describes several possible measurement outcomes probabilistically, though the particle’s initial condition is known. Let us, therefore, go through a (not at all rigorous) brief introduction to some formal aspects of QM. I will try to keep it a intuitive-friendly introduction in contrast to a complete course on QM but it is nevertheless necessary to digest some simple yet very important formalism and symbolism, as it will continue to emerge frequently here and there, and it is that kind of representation that you will encounter throughout all textbooks on QM. Therefore, this section assumes familiarity with basic mathematical concepts such as vector spaces, complex numbers, and elementary algebra.

So, let us turn our attention back to the concept of the wave function. We considered it to be continuous for the position in space of a particle, and its value, the probability density or the integrated probability function of the particle in space, does not change abruptly from one point to another. That is, the wave function ψ(x) is a continuous function in x.

We saw, however, that there can be other behaviors of quantum systems—for example, the energy state of atoms is quantized. Here, the wave function can no longer be a continuous function but must be represented by a set of discrete possible states of the system—for instance, as we have already seen in the case of the hydrogen spectrum or in the Franck-Hertz experiment. In these cases, we are dealing with a different physical context than that of a continuous spectrum of possible outcomes, such as that of a particle moving smoothly (i.e., continuously) along a spatial axis, but we must represent the quantum system’s states with a discrete spectrum—that is, the wave function ψ is not a continuous function in space but is a collection of a more general set of N distinct abstract states, so-called eigenstates (from the German “eigen,” which means “self”) e1, e2, .., eN , each standing for the system’s possible discrete states, here the energy levels E1, ..., EN . The same fits for other discrete dynamical variables, such as the position, the momentum, the angular momentum, the spin, etc., or other dynamical variables that are a combination of these physical quantities.

Notice that N , the total number of possible states a system can assume, may be either finite or infinite. For example, as we shall see, the quantum spin of a particle can take only two possible states (N = 2), whereas an unconstrained particle in a continuum can, in principle, occupy infinitely many position states (N = ∞). However, keep in mind that an infinite number of possible states does not necessarily imply a continuous spectrum. For instance, as illustrated, the energy levels of the hydrogen atom are infinite in number, yet the spectrum is discrete. This is because the higher energy levels accumulate, with the spacing between successive levels becoming arbitrarily small.

In QM, these eigenstates are defined on a Hilbert space. Without going too much into mathematical details, it suffices to say that a Hilbert space is an abstract vector space that is an extension of the real Euclidian space to a multidimensional space of complex numbers and complex vectors. If the Hilbert space has a finite dimension, it is the state space of a quantum system with a finite number of eigenstates. Otherwise, it is infinite—as infinite as the number of its degrees of freedom. Another important difference in a “space” as we conceive of it in school geometry is that the elements of a Hilbert space are no longer points and coordinates but, rather, vectors or functions, with the wave function being the primary object on which to act. A Hilbert space is essentially an extension of the Euclidean space from a real to complex vector space and with a well-defined notion of distance and angles.

The eigenstates e1, e2, .., eN , form the eigenbasis of the Hilbert space, and the overall possible states of a quantum system are represented by a vector defined on this eigenbasis, called the state eigenvector, or just eigenvector, or simply the state vector (in Fig. 1 a simple case for a system with three possible eigenstates is shown):

where c1, c2, .., cN , are complex coefficients which are calculated and interpreted as follows.

Consider the orthonormal eigenstate vectors |e1⟩ , |e2⟩ , .. |eN ⟩. Geometrically, this is displayed in Fig. 1 as being all these eigenstate vectors perpendicular (with unity length) to each other.1 Physically, this means that one or the other state can be measured, though never more than one at a time. Orthogonality implies mutually exclusive measurement outcomes.

Because we are no longer dealing with a continuous function but, instead, with a state vector in which each element corresponds to discrete eigenstates (think again of the eigenenergy spectrum of the hydrogen atom), the probability density function (Eq. 1 here) must also be replaced by discrete probabilities. In general, the probability of finding the system in one or another eigenstate is not the same. For example, the probability P1 of finding the Hydrogen atom in an energy state E1 is not the same probability P2 of finding it in a state E2, etc. That’s why the spectral lines have different thicknesses, that is, intensities. Nonetheless, the set of all probabilities P1, P2, .., PN must sum up to unity—that is, certainty that the system will be found in at least one among all the possible states.

Formally this information regarding with what probability each of the eigenstates will be observed after a measurement is encoded in the complex coefficients c1, c2, .., cN , which determine the length of each eigenvector, as illustrated in Fig. 1.

This can be summarized in the Born rule, which still holds: The probability P associated with the i-th eigenstate ei is the squared modulus of the i-th complex coefficient ci :

The state vector notation |ψ⟩ of Eq. 1 is a common notation in QT. State vectors can be written according to the Dirac-notation. P. Dirac was a British physicist who made several contributions to QT.

We already saw how the kind of notation for integrals we encountered as brackets, like ⟨a, b⟩, turns out to be a useful convention. Each bra-ket is a product of a “bra-vector” ⟨a| (a row vector whose elements are the complex conjugates of the vector elements of a) and a “ket-vector” |b⟩ (a column vector with the vector elements of b). As you can see, Dirac wasn’t lacking a bit of weird lexical fantasy, though this turned out to be extremely useful. For example, the general “bra-a, ket-b” is:

This is called an inner product and is just an extension of the familiar scalar product or dot product and that some might recall from their high school algebra, but extended to Hilbert spaces where vectors have complex entries. Never forget that if two vectors |a⟩ and |b⟩ are perpendicular (orthogonal), then ⟨a|b⟩ = 0, always.

With Dirac’s notation, many calculations are made formally simpler.

For example, consider the Hilbert space representation of the eigenvectors |ei⟩ relative to the eigenbasis (e1, e2, ..., eN ) and the state vector |ψ⟩ of Eq. 2, that is:

respectively.

Then the probability of finding the i-th eigenstate can also be written in the

bra-ket notation as follows:

Again, with the right-hand-side sum of all the modulus squared complex coefficients, each representing the probability of the i-th eigenstate to be measured, must obviously sum up to certainty (or, to put it another way, the sum of the probabilities of all possible events gives unity). Note also how the sum of all the probabilities adding up to unity is represented in Dirac’s notation neatly as:

It is a symbolic formalism introduced by Dirac, and later developed further by others, such as the mathematician John von Neumann, with which one represents the probability of observing a system with a number of possible discrete states.

It is also customary to state that Eq. 5 is the modulus square of the projection of the state vector ψ onto eigenstate ei. In fact, the act of measurement, “projects” the state vector onto one of its eigenbasis vectors (one can interpret the dashed lines in Fig. 1 as “projections.”) This is analogous to the wave function collapse we saw in the case of a continuous wave function. There, the act of measurement “collapses” the wave function describing the particle position in a continuous space on one of the infinitely possible states. After all, the position of a particle is also a “state.” Here, it projects the state vector onto a discrete spectrum of possible eigenstates.

This description of quantum measurements derives directly from experimental observation and was first formalized by von Neumann, also known as von Neumann’s projection postulate, or von Neumann’s collapse postulate, which expresses the discontinuous, instantaneous and irreversible nature of the measurement process in QM.

In its modern version, it is also called the measurement postulate of QM, which states that:

i) For each measurement of a dynamical variable (or observable, see later on), the probability of finding it in one of its possible eigenstates is given by Eq. 5.

ii) After the measurement, the system instantly “collapses” from state |ψ⟩ to an eigenstate |ei⟩; (i = 1, .., N ).

iii) From that moment on, any new measurement will always show the system to be in the same eigenstate.

While this postulate has always been employed with success as a powerful predictive tool in QM, later on, we will see how it leads to some still-unsolved issues, summarized by the so-called measurement problem and famously represented by the Schrödinger’s cat experiment or paradox.

Moreover, once the quantum system has been placed into one of these eigenstates, in QM jargon one says it has been prepared into a specific state |ei⟩, and if it is not perturbed—that is, it does not interact with the environment suffering a state change—every subsequent measurement will always deliver the same result: the system being in state |ei⟩.

The next section will deal with operators and the Schrödinger equation.

At this point, the question is whether the wave function is simply an abstract mathematical description of the statistical behavior of a system that simply updates our knowledge, or whether it represents a real object in the physical world. And, if it must be considered real, how can it be something that behaves like a wave, say while traveling from the double slits to the detector, and then, once detected on the screen as a point-like pixel, turns out to describe a particle? This seemingly abrupt wave-to-particle change in behavior is the so-called wave function collapse , also called the state reduction or state projection. It is one of the typical quantum philosophical problems and will lead to the more general measurement problem we will discuss later. It can be illustrated with what we already saw in the double slit experiment and what was depicted here, and is summarized in Fig. 1.

First, a plane wave is diffracted by the slits. Then the two expanding concentric circular wavefronts emerging from each of the apertures travel towards the screen. These wavefronts can be described by the wave function Ψ. Once they interact with the screen, the interference fringes form but as a pattern of dots, each representing a point-like interaction at some specific position. Nothing is left of any wave.

This suggests an interpretation that tries (a bit desperately) to reconcile the two apparently opposite points of view. First, before the measurement, we must conceive of the single photon (or electron, or whatever particle we may be talking about) as a wave, or a wave packet which (still before the measurement, that is, before it hits the screen) continues to behave according to the diffraction and interference principles. Nothing should be conceived of as a particle. When the wavefront goes through the slits, it is still a wave and no particle is supposed to go through one or another or both slits.1 Whereas, at the instant of measurement, in our case, at the instant when the wave interacts with the screen, it suddenly reduces, or “collapses,” to a point on the screen, manifesting itself again as a particle. However, before that instant when the act of measurement occurs, we must still consider it to be a wave with an extension in space, not as a single point.

It was first the Austrian-Ungarian mathematician John Von Neumann who introduced the wave function collapse as a postulate and the description of a physical measurement in QM as a discontinuous, non-causal, irreversible and instantaneous process.

The Collapse Postulate: When a measurement is performed, the system’s wave function—previously evolving smoothly and deterministically according to the Schrödinger equation—undergoes a sudden, non-unitary reduction to a single eigenstate corresponding to the measured observable.2

This postulate formalizes the intuition that measurement is fundamentally different from ordinary physical evolution. While it successfully accounts for definite outcomes in experiments, it introduces a conceptual tension at the heart of quantum theory: the laws governing measurement are not derived from the theory’s dynamical equations but are added as a separate rule.

In fact, from the mathematical and formal abstract point of view, everything is clear: You start from the wave function and then take its squared modulus and obtain the probability density where you will find your particle. Then you integrate over a region of interest and obtain a statement about the probability of observing the system in some specific range of possible states (here, the particle to be detected inside as a spatial interval along the screen). The wave function collapse, or state reduction, is simply conceived of as an update of the information we have about the state of a system. Previously, we had uncertainty over a possible range of values that can turn out before a measurement and represented this lack of information with a statistical function. However, once the measurement is made, we know the exact value; therefore, there is no need to use the wave function. One gets only one measurement outcome and updates the knowledge one has about the state of the system. It is just like tossing a dice without looking at it. Before controlling the outcome, one has six possibilities to be realized. However, after one looks at the dice, only one of the six possible outcomes appears to be the truth. Therefore, the problem is not mathematical; we may say it is not even scientific. It is an interpretational problem.

Yet if that is a common interpretation of the probability notion in classical physics, it is not so straightforward in quantum physics because, as the double slit experiment tends to suggest, the wave function seems to be more than a probability wave but, rather, something real propagating from the slits to the detection screen. Otherwise, how can interference come into being in the first place?

Therefore, a question arises in the domain of philosophical speculation. From a strictly practical standpoint, the mathematics alone provides all the predictive power needed to assign probabilities to observations, and for the pragmatist, as most physicists are, that may be enough. However, from the ontological point of view, there is no consensus as to how we should interpret this “collapse” and what kind of objective reality quantum mechanics is describing after all. What kind of thing behaves as a wave unless one looks at it, and then collapses to a point instantly, in an infinitesimal interval of time? What is really “out there”? Before the measurement, it seems that we have only a sort of ghostly entity whose wavy nature we will never observe directly. However, we can infer it from the interference phenomena, and at the measurement act it nevertheless manifests as a completely different entity, a point-like localized interaction. And there is no way to intercept a morphing, a transformation from one entity to another. After all, we are talking about a probability wave travelling through space.

Yet the notion of probability is an abstract mathematical concept; it is just a mathematical function that mathematicians write with pencil on a piece of paper—a concept that we do not think of as an object having a concrete existence moving somewhere. One can see water waves on a lake, hear sound waves, perceive light, and see birds flying, but nobody has ever seen a mathematical function or the square root of -1 moving throughout space.

And yet, this is what quantum physics seems to be about. It seems that not only must we resort to very abstract notions to describe the world but the world itself seems to be a sort of mathematical abstraction, reminiscent of a Platonic realm. Is anything physical “out there”, represented by Ψ? Is the collapse of Ψ only a statistical occurrence, or also a real physical one?

So far, nobody has been able to furnish a convincing and generally accepted answer to these questions. Most physicists simply do not bother and will tell you that they are too busy making calculations. “Shut up and calculate” is their motto. They are happy that the math works. And, indeed, the mathematical foundations of quantum mechanics were extremely powerful in furnishing predictions that could be measured and confirmed, and that led to the tremendously successful standard model (SM) of particle physics, atomic physics, solid state physics, etc. But, from the interpretational point of view, not much progress has been made.

This suggests that we still have not gone deep enough in our understanding of the world. For many years, physicists delighted themselves with calculations but with recent attempts to build quantum computers which require more or less direct answers to these questions, and after decades of unsuccessful attempts to discover a general theory of quantum gravity, that kind of theory that reconciles the gravity force with the electromagnetic and nuclear forces, these foundational issues have gotten more attention again.

The next post will discuss an extension of the wave function, and what determines it: the state vector and Schrödinger’s equation.

Let us return briefly to the de Broglie hypothesis. Here we had to deal with an abstract object, with a sort of conceptual synthesis that resembles quite well both a wave and the appearance of a particle. Every particle in quantum theory, like the electrons, protons, photons, etc., can be represented by a wave packet, that is, a small localized wave that travels throughout space and that has a frequency (or wavelength, if you prefer), and a maximum central value that decays quickly on both sides (recall Fig. 2). It is a way to visualize a wave that nevertheless can interact locally as a particle and conceptually synthesizes the wave-particle duality, where the quantum objects seem to behave as waves and yet as particles too.

Then, starting from Section I.2 we have also seen that in quantum mechancis we must deal with interference phenomena, which bear some similarities to that of the waves of classical mechanics. We introduced the concept of the phase between two waves, or a phase relative to an origin, and we saw how this is reflected in the formalism of quantum theory.

Now, the mathematical function which describes a wave packet or any quantum mechanical description of the state of a system is called the wave function and is usually written with the Greek letter ψ (read “Psai” for uppercase, “psi” for lowercase). The wave function contains all the information about a particle or a quantum system, for example, in space coordinates x and in time t as: ψ(x, t). This is, in fact, the conventional and standard mathematical description of waves in quantum mechanics.

In section I.3 we also found that the intensity of a wave—that is, the physical quantity that we really measure, say, the number of photons we measure arriving in a specific area on a detection screen—is given by the squared modulus of the wave function as,

where, here, we consider only the change in the space, variable x, leaving out the change in time, variable t. Therefore, it is convention to depict a wave packet as in Fig. 1.

You see the real part of a wave function (the wave with positive and negative values) and its squared modulus (the wave with only positive oscillations), which represents the effectively measured intensity.1 The ∆x is taken as measuring the width of the wave packet and corresponds to the uncertainty over the position of a particle due to Heisenberg’s inequality.

If you think of the density of dots, for instance, those forming the interference fringes on the screen of the double slit experiment, the number of photons hitting per unit area the photographic plate (or a ccd camera, or whatever kind of detector), the intensity is just proportional to the density of dots. The registered number of particles on the screen per unit area (or, as it is represented graphically in this picture, like the transparency of the particle around an origin) can be taken to be proportional to the probability of finding the particle in that specific area.

Recall that while the overall disposition of many photons on the screen resembles that of the interference fringes, the site where the single particle will hit the screen is nevertheless a completely probabilistic process. The appearance of one photon as a dot on the screen is purely a statistical phenomenon.

Therefore, one can regard the intensity—that is, the squared modulus of the wave function—as something proportional to a probability per unit area, that is, a probability density. The squared modulus of the wave function is the master tool with which physicists describe probabilities in quantum mechanics, that which connects theory with experiments and observations. It is what allows an abstract complex mathematical function to correlate with reality: all the measurement processes that physicists perform in the laboratory.

We can introduce the following interpretation, which is due to the German physicist Max Born, and which is also called the Born rule :2

The squared modulus of the wave function according to Eq. 1 is a probability density p(x) to observe a quantum system in a specific state x.

To clarify what this rule means, let us consider the graph of Fig. 3. Let us suppose that the curve represents the probability density to find a particle around an origin.

If the horizontal axis represents the position x where a particle can be found, the vertical axis is the probability density p(x) to find it at x. You will frequently find such kinds of bell-shaped curves in several textbooks; it is the normal distribution or Gaussian distribution.

However, a probability density is still not a quantity that can be measured because it represents only an infinitesimal quantity. It makes no sense to ask what the probability is of finding a particle in a specific position, as the interval on the space axis, in such a case, would be zero. One can only say with what probability a particle can be localized in a specific spatial interval around the origin (say, the x we worked with in Heisenberg’s uncertainty principle). The name “density” is no coincidence. Just think of the density of matter. The material density function of a body describes how the density varies inside its volume. Yet this still isn’t enough to tell us something about the amount of matter if we don’t specify how large the volume is and how its density function varies inside that volume. Similarly, the wave function and its associated squared modulus, the probability density, describes all the possible states of a system but still does not tell us the probability of finding it in a range of possible states. What we must consider is the sum of the probability density function over a specific range, which tells us something about the effective probability P to find it in that range.

For example, for normal distributions, the spatial x-interval of ±σ (“sigma”) is per definition the range that determines a 68.27% probability of finding a particle. σ is called the standard deviation and tells us something about the dispersion of a set of data values. The probability of finding the particle far from the center drops off quickly. At ±3σ, it is almost zero.

In general, to obtain a probability P of finding the particle inside some spatial range between x = a and x = b, we must take the integral of the probability density over that interval—that is, the integral over the squared modulus of Ψ(x). An integral is the area under the graph’s function which can be obtained as the sum over all the function’s values at each point along the x-axis, in our case:3

And, if you consider the probability of finding a particle anywhere, throughout the Universe, the spatial range must be infinite, taking a = −∞, b = +∞ as the domain of integration. Then, obviously, you will get 100% probability that is, the certainty of finding it somewhere, which mathematically translates into the integral giving unity:

The expression in the brackets on the left-hand side of the equation is simply a more compact form to express the integral. It is quite common shorthand, the so-called Dirac notation, that we will take up in more detail soon.

More generally, the wave function applies not only to a particle’s position but to any possible state. It is a calculation tool with which to also find the probability of finding a particle having a momentum or angular momentum (the rotational speed), an atom or a nucleus being in some energetic state, etc.

The takeaway message of all this is that quantum mechanics is a purely probabilistic theory. This is due to its wavy nature. There is no way of knowing how a single particle will behave, not even in principle (except when the particle is in what is called an eigenstate, as we will see later). That is why, in quantum theory, statistical tools play such an important role. We can compute the probability only to measure a value in some interval. In quantum mechanics, it makes no sense to speak of a system that is in some state. It makes sense only to say that there is a certain probability that a measurement will furnish an outcome that we associate with some state at the instant of measurement. However, we are not allowed to say anything about its state before and after that measurement. The wave function is a state function only in the sense that it is a mathematical tool that tells us about the statistics and probability of finding a specific system in a specific state or range of possible states.

The next section will tackle with the question is whether the wave function is simply an abstract mathematical description of the statistical behavior of a system that simply updates our knowledge, or whether it represents a real object in the physical world.

Now let us analyze what interpretation Heisenberg gave to his own principle. He illustrated this with a famous “thought experiment”. 1

Heisenberg’s thought experiment elucidates what nowadays is remembered as Heisenberg’s microscope interpretation. It is an interpretation that, unfortunately, several professional physicists have also adopted (probably to make it more understandable to the public or, being busy only with calculations, to avoid digressions in the conceptual foundations that might trigger annoying questions from too-curious students). However, if you have carefully followed what we have said so far, you will be able to recognize that Heisenberg’s interpretation is somewhat misleading. It went as follows.

Imagine (see Fig. 1) that we want to detect the position and momentum of a particle in a region x by using an incoming photon which interacts with the particle. This photon is then scattered toward some direction. We have already described something similar with the Compton scattering effect.

Then, as was Heisenberg’s reasoning, we can look to where the photon has been scattered to deduce the electron’s position.

However, according to classical optics, the larger the angular aperture θ of the microscope, the better it will resolve the position of the scattered photon and, therefore, deduce the position of the electron. Yet this angular aperture depends not only on the size of the microscope lens but also on the wavelength of the photon. To have high precision in determining from which direction the photon comes—that is, in determining the position of the electron—we must use a photon with a short wavelength. However, if the photon has a short wavelength (high frequency), it will carry more energy (recall Planck’s relation Eq. 1) and, thereby, more momentum (recall Eq. 2), and a smaller de Broglie wavelength (recall Eq. 1). Therefore, the “kick” that it imparts to the electron will also be larger. That is, if I want to observe the precise position of the electron, I must use photons which will produce a large recoil on the electron itself. This makes its position again indeterminate. On the other hand, if I want to more precisely determine the momentum of a particle, I’m forced to use photons with large wavelengths because only they have a small momentum and will not displace the electron too much through Compton scattering. Yet if I use long wavelength photons, the microscope optics will resolve their position with less accuracy. So, again, we must search for a tradeoff between the wavelength we use (the precision over the momentum p) and the microscope aperture (the ability to look at a precise position x).

In the macroscopic world of our daily experience, this effect does not play any role for practical purposes because objects are huge compared to elementary particles, and are not disturbed by tiny photons. However, for atoms and even more for particles like electrons, a single photon introduces a non-negligible interaction scattering them away and, thereby, preventing us from determining their position. That’s why Heisenberg’s uncertainty principle is invoked to explain that in the microscopic world, a measurement always perturbs a system in such a way that the scale of the measurement’s perturbation is about the same magnitude of the effect it produces and, therefore, renders the result imprecise if not entirely useless.

Heisenberg was able to show that, by reasoning in such a manner, making the appropriate calculations which resort to classical optics, he could indeed reobtain his famous inequality over momentum and position of Eq. 1. Heisenberg’s microscope thought experiment boils down to the following argument: We cannot determine the position and momentum of a particle at the same time because when we make an observation, we must inevitably interact with the object we want to observe, such as sending other particles of comparable mass or energy to the particle being observed. This inevitably disturbs the system, which then loses its original position and momentum that we sought to determine.

This argument is correct insofar that every interaction with microscopic particles inevitably causes them to scatter and prevents us from knowing their precise position and momentum. In spite of that, it becomes wrong when it is invoked as a restatement that is supposed to explain the origin and cause of quantum uncertainty. That would be in stark contrast to the wave-particle nature of matter itself, which doesn’t need any interactions with the outside world to be described consistently. It is an interpretation that does not stand up to the tests of modern QT. The popular belief that Heisenberg’s uncertainty principle is about the interaction between the observer and the observed object is wrong. Heisenberg’s uncertainty principle is a fundamental law of Nature whose roots are in the wavy nature of matter and has nothing to do with the interaction between the measuring apparatus and the measured objects.

Of course, the interactions and imperfections of measurement devices must always be taken into account in a real laboratory but these add further uncertainty to the intrinsic “quantum fuzziness” that is present a priori. In fact, we shall see later that, as strange as it might sound, it is nevertheless possible to set up experiments which make measurements without necessarily interacting with the system, and yet the Heisenberg uncertainty principle remains inescapable. It also holds in the case in which we are able to reduce to zero any interaction with the observed object. It is an inherent law of Nature that is independent of observational interactions.

At the time of Heisenberg, it was still legitimate to think of particles in this way. Heisenberg can be excused because several experiments which showed how his own interpretation must be revised came much later—some not even until before the 1990s, with the development of sophisticated laser and quantum optics devices. So, while the historical context and the lack of more experimental evidence justifies Heisenberg, it does not do the same with modern physicists. Nowadays, we can no longer stick with the microscope interpretation as a correct understanding of the workings of the uncertainty principle. This is no less fundamental than Bohr’s atomic model, the geocentric model or a flat earth theory.

We should accept that Nature is telling us that we should never forget the wave-particle duality. When a particle goes through a pinhole, we must think of it as a physical process described by a transverse planar matter wave, which is diffracted like any other wave and produces a spherical wavefront—neither more nor less than in Fig. 2.

Forget about classical understandings of particles which possess definite properties such as a position, and which move along precise deterministic trajectories and possess a definite momentum. There are no positions or momenta which describe a particle. Instead, we must seriously consider these to be emergent properties (or qualities?), not intrinsic properties.

Particles do not at all have a clear and precise position and momentum as we imagine them to have at our macroscopic level and as our naive intuitive understanding wants to make us believe.

QP seems to suggest that the physical objects we imagine as point-like or hard material billiard balls are, instead, entities which are intrinsically somewhat fuzzy, with no sharp and well-defined boundaries and that travel throughout space as waves that eventually “collapse.” This nature of particles is independent of the precision of our measurement apparatus and independent of the fact that we interact (or do not interact) by observing it.

Thus, quantum indeterminacy is not epistemic—that is, it is not a matter of ignorance—it is ontological. Heisenberg’s uncertainty principle is not a principle about a lack of information. It is not about particles whose whereabouts and speed we cannot determine. It is about physical entities that simply don’t have anything to do with our anthropocentric imagination.

This is, ultimately, what truly distinguishes classical from quantum physics.

But what is then this “matter wave” we are talking about? This will be the topic of the next section on the wave function.

We have seen in this section, especially with Bragg diffraction, how diffraction and interference phenomena determine the wave-particle duality not only of massless light particles, photons, but more generally of all particles—including material particles that have a mass and that behave like waves, even though we think of them as localized pieces of solid matter.

A question we might ask is: What happens with a particle if we want to know its precise whereabouts in space? For example, let us determine the precise position of a particle by letting it go through a tiny pinhole, as Isaac Newton did with photons in his investigations of the nature of light.

If a particle goes through that single little hole on a piece of paper, we are authorized to say that we can determine its precise position in space. In fact, this can be done, but at a cost. Because, on the other hand, we know that due to the wave-particle duality, we can’t forget the particle’s wavy aspect. When a particle, also a material particle, goes through this pinhole, it will likewise be diffracted and afterward position itself on the detector screen according to an interference pattern.

If, instead of dealing with slits, we take a tiny round hole of a size comparable to that of a few multiples of the wavelength, we obtain circular interference fringes, like in Fig. 1. This is an intrinsic and unavoidable effect for all types of waves.

If particles must be linked to a wave, according to the de Broglie relation, conceiving of it as a wave packet, we will always have the interference fringes, even with only one hole or slit and even with only one particle. (Recall the single photon experiment.)

Fig. 2 compares the two cases in which the pinhole determines the position of the particle with an aperture of a = 2λ (high precision) and a = 20λ (low precision), with , as usual, the wavelength of the photon or, in case of a matter wave, the de Broglie wavelength.

If the pinhole is small, while it will determine the position of a particle with higher accuracy, it will also produce a relatively broad diffraction pattern that renders the position of the photon on the screen uncertain. We can know in a relatively precise manner where the photon or matter particle went through the piece of paper, inside the space constrained by the pinhole’s aperture. But it will be displaced laterally on the screen anyway due to diffraction and interference phenomena.

Of course, with a single particle producing a single spot on the screen, no interference figure is visible. However, as we have learned with the single photon diffraction at the double slit, the probability of finding this spot is in one-to-one correspondence with the intensity of the interference fringes that many particles produce. Recall also that we can’t predict where precisely this spot will appear, other than resorting to probabilistic estimates.

If, instead, the pinhole is large, fringes will become less pronounced. We will know where the photon will hit the screen with relatively good accuracy (the spike for a = 20λ in Fig. 2, which means it “felt” only a tiny displacement along the screen.

However, by doing so, we will lose our capacity to determine where exactly the particle went through the pinhole, as it is no longer a pinhole at all, but a large hole. There is no way, never ever, not even in principle, to obtain the precise measurement of the particle position and at the same time avoid the production of interference fringes (or interference circles, as in the case of a circular aperture or large diffraction effects). One will always obtain a more or less pronounced bell-shaped or peaked distribution of white spots on the screen. This is not because we don’t have a sufficiently precise measurement apparatus but because it is a consequence of the intrinsic wave-nature of particles. It is a universal law of Nature, according to which it is hopeless to believe that we can pass a wave through a slit and not observe any interference and diffraction phenomena.

So, if you have followed thus far, it should be easy at this point to understand the essence of Heisenberg’s uncertainty principle. Werner von Heisenberg was a German physicist who, around the 1930s, asked himself if and how there was a way to precisely determine the position of a particle.

This was one of the most pressing questions for which to find an answer because, after all, in CP everything is expressed, analyzed and described in terms of space, time, forces and particles following accordingly precise trajectories. If we are no longer able to describe the position in space of a particle, we are substantially blowing up almost all known physics which, until then, relied completely on particles, interacting through forces with each other moving along precise trajectories in space and time. Heisenberg reasoned as follows.

Let us suppose that we want to determine, with high precision, the position of a particle, as in Fig. 4. (Notice that in this figure, the x-axis is the vertical one.) Imagine it comes from the left with a specific amount of momentum p. Recall that the momentum of a particle, its quantity of motion, in classical non-relativistic mechanics is simply given by the product of its mass times its velocity. 1

Again, from the de Broglie relation of Eq. 6, we know that, once a momentum p is given, in QP we can also relate to every particle a wavelength: λ = h/p . This particle, which we must intend as a transverse plane wave, is then incident on the diffraction plane with one slit (or pinhole) of size x. Once it has passed through the slit, it will be diffracted and, shortly after, interact on the detection screen at a position according to a probabilistic law which the interference pattern determines, as you certainly know by now.

So, the particle, even without interaction from outside forces, even with no physical disturbance from the environment, and even by constructing an experimental setup which is perfect and ideal without any imprecision, will nevertheless be displaced on the screen (the vertical x-axis). This happens because of our attempt to localize the particle in that tiny region x. We can arrange our experiment as we wish, but if the size of the slit is comparable to the wavelength of the photon or to the de Broglie wavelength of the material particle, we will always and inevitably note that the particle, after it passed the slit of size x, will be displaced on the screen according to the laws of diffraction and interference. There is no way to avoid this because it is a phenomenon intrinsic an inherent to waves.

Now, this could also be interpreted as follows: The particle, once it has gone through the slit, will acquire an extra momentum, p, along the vertical axis. This does not happen because of the interaction of an outside force or, as we might imagine naively, by an interaction, deviation, or bouncing effect of the particle with the slit’s edges because, in that case, we would observe a random distribution but not an interference pattern. This extra momentum p which displaces the particle along the detection screen is due only and exclusively to the wave nature of matter and light. We might interpret this also as a scattering of the particle but we should keep in mind that this is misleading terminology, as there is no scattering force at all. No scattering interaction or forces from the outside are necessary to make this happen.

Where then does this extra amount of momentum p come from?

It is the uncertainty on the outcome of a measurement of the particle’s momentum in the first place. It is an inherent indefiniteness of the particle’s properties due to its wave nature.2 This is the only possible conclusion if we want to avoid violating the principles of the conservation of momentum and energy.

The point is, we will never be able to determine with extreme precision—that is, with an infinitely small slit of size x=0—without blurring the momentum because, by doing so, we will inevitably diffract the plane wavefront, the wavelength of which is given by the de Broglie’s relation. This will inevitably displace it according to a statistical law which reflects the diffraction and interference laws.

So, we must conclude that the smaller our uncertainty in determining the particle position (the size x of the slit), the larger the diffraction effects and, therefore, the larger the uncertainty ∆p over the momentum—p becomes large in the vertical direction (see Fig.4). On the other hand, if I want to know the particle’s momentum with small uncertainty (∆p small), we will have to open up the slit’s aperture (∆x large) to reduce the diffraction. However, I will never be able to determine with precision both the momentum and the position of a particle at the same time. We have to choose whether we want to keep focused on one or the other; never ever are we allowed to obtain both. Again, this is not because we are perturbing the system but because we are dealing with waves.

Since we have interference and diffraction phenomena, the particle can displace itself very far from the center of the screen, especially if the slit becomes very small. However, this also happens with a decreasing probability. It is much more probable to find the particle hitting the screen at the central white peaks than at the little ones far from the center. For practical reasons, one must set a limit as a convention. Due to statistical reasons (which will be clarified in section III.3), one takes as convention the normal distribution which states that the uncertainty ∆p is the width of the central fringe corresponding to 68.3% of all the particles it contains. That is because this is the central region with higher intensity and, therefore, that region where we have the highest probability of detecting a single particle. In fact, as we have seen with the single slit, the secondary interference fringes beyond the central one are usually quite small (smaller than in Fig. 4).

So, this means that, to a specific amount of ∆x, we must expect to find also a specific amount of ∆p—always and inevitably, by the laws of Nature. Heisenberg was able to show that the product of these two uncertain quantities is a constant and he summed this up with his very famous and extremely important inequality, which encapsulates Heisenberg’s uncertainty principle:

where the barred Planck’s constant (read “h-bar”), is Planck’s constant divided by 2π, also called the reduced Planck constant.

This is a mathematical way of stating what we have said so far: If you force the value of the position of a particle to be tight, it will always have an uncertainty over the momentum of ∆p , that is, a precise measurement of the particle position will imply an imprecise state over its momentum. A small ∆x induces ∆p to become large:

Vice versa, if you force the value of the momentum of a particle into a narrow range of values, it will always have an uncertainty over the position of:

This was only one of the possible ways to obtain Heisenberg’s inequality (1). One closely related to this is via the Fourier transforms, that is, a mathematical tool developed by French mathematician Joseph Fourier (1768–1830) who showed how every time series (a signal in time, such as a complicated traveling wave) can be decomposed into the sine and cosine functions of different frequencies. In modern QT, Heisenberg’s inequality is obtained through a mathematical operator theory approach, which is mathematically more rigorous but somehow obscures the deeper meaning of Heisenberg’s principle related to the wave nature of matter.

At any rate, always keep in mind the little but very important inequality of Eq. 1, as it will turn up more or less everywhere in QM.

The next section will discuss an interpretation of Heisenberg’s uncertainty principle: the (misleading) Heisenberg’s microscope.

Another famous principle that almost everyone has heard of at least once in their lifetime is Heisenberg’s uncertainty principle (or indetermination principle). Usually it is explained with a few sentences or with wrong analogies and people (sometimes even professional physicists) have a misled understanding of what it is about. Therefore, to get a clear picture and avoid misconceptions, it is instructive to further deepen diffraction and interference phenomena for the one, two, and many slits cases.

So far, we concentrated our attention on Young’s double slit case. In that occasion we saw how, when a couple of slits diffract a transverse plane wave, the typical interference pattern is produced, made of white maxima and dark minima fringes on a detector screen. This is equally true if we send only one (massless) photon (or massive particle) at a time through the slits. After a sufficient number of photons have excited the single photosensitive molecules of the photographic plate or the CCD camera sensor, the interference fringes reappear again.

The question at this point is: What happens if we use only one slit? Or vice versa: What happens when we use many slits?

By the end of the seventeenth century and well before the birth of QM, the single and N-slits interference phenomena were well known and extensively studied. As every good textbook on electromagnetism will tell you, a single slit, or a pinhole, will produce an interference pattern as well, though a less pronounced one than a larger slit produces. It is an easy experiment that you can do by yourself—just take a piece of paper, poke a little round hole with a pin or needle, and look towards a light source. You will observe the image of the light source surrounded by several concentric colored fringes (the colors appear only due to the fact that, fortunately, the world we live in is not monochromatic).

For example, Fig. 1 shows how a plane wave behaves when one aperture, a single slit of size a, is about five times the size of the light’s wavelength, that is, a = 5λ. Due to the diffraction that occurs whenever a wave encounters an object or a slit, especially when its size is comparable to the wavelength, the plane wavefront is converted to a spherical or a distorted wavefront, which then travels towards the detector screen.

In this case you see that, even though we are dealing with only one slit, some weak but still clearly visible secondary minima and maxima fringes can be observed.

An important point to keep in mind is to avoid a common misconception (which is frequently promulgated in some popular science books) which states that only two or more slits can produce interference fringes, whereas, for the single slit, interference effects disappear. This is not entirely correct.

True, it is easier to produce more pronounced interference patterns with more than one slit (or pinhole); and for most applications, especially when the wavelength of the incident wave is much smaller than the size of the aperture, these effects can be neglected.

However, strictly speaking, a single slit also produces small diffraction and interference phenomena.

An elegant explanation of how interference comes into being, also for a single slit, dates back to the French physicist A. J. Fresnel. He borrowed an idea from Huygens—hence the name Huygens-Fresnel principle—according to which every single point on a wavefront should itself be considered a point-source of a spherical wave.

Therefore, once the incoming plane wave reaches all the (virtually infinite) points along the aperture, every single point becomes a spherical wave point-source, as shown in Fig. 3.

All these point-like sources along the slit’s aperture emit at the same time their own spherical wavefronts which, however, when seen from a position on the screen, add up, interfering each with its own phase, to produce an interference pattern. The reason for this is not so difficult to visualize. Since all fronts are initiated in different locations along the aperture they will also travel a different path length, which implies that they have different phase shifts when they overlap on the screen.

For instance, consider Fig. 1 once more, where we saw the two paths of the two sources from the edges. As in the case of the double slit, these two rays have a relative phase shift by an amount ∆ϕ and, when superimposed together on the screen, they form a resultant intensity according to the interference laws. In this case, however, this holds not only for two waves, but for an infinite number of point-sources along the aperture. Fresnel, by making the appropriate calculations, was able to show that if one sums up all of the spherical wavefronts coming from the points of the aperture of the single slit and projects these onto all the points along the detector screen, then one obtains indeed the known diffraction and interference patterns.

If we repeat the same experiment with a slit that has a size close to the wavelength of our incoming wavefront, then we see that the interference fringes disappear (see Fig. 4).

Only when the size of the slit is equal to or smaller than the wavelength are the fringes definitely absent. This is because the slit is so small that only a single point-source can form a spherical wavefront with a wavelength equal to the slit size, and there can be no path difference and phase shift with some other source which could produce the interference pattern. However, diffraction has become very large instead, so that the photons will displace themselves on a relatively large area on the detector screen, according to a bell-shaped distribution called the diffraction envelope.

Thus, an interplay between interference and diffraction occurs: for apertures somewhat larger than the wavelength, interference effects dominate while diffraction is negligible; for apertures comparable to the wavelength, interference tends to disappear while diffraction becomes significant. Both effects can be made negligibly small only in the limit of an infinitely large slit (or pinhole). However, in doing so, one loses any precise information about the position at which the photon traverses the aperture.

Let us take also a quick look at what happens if we change the number of slits maintaining a fixed size. Fig. 5 compares three different cases. On the abscissa we have the angular position on the screen (angle of incidence θ between the center line of the diffraction plane and a point on the detector screen), while on the ordinate axis we have the usual intensity.

The parameters that determine the angular dependence of the interference pattern are: first, the size of the aperture relative to the wavelength (here: a = 3λ); second, the spacing d between the slits (here: d = 3a); and, of course, the number of slits. The three curves represent respectively the 1, 2, and 10 slits diffraction cases. The intensities have been normalized in all the cases to unity.

For the one slit case you see there are some weak but clearly discernible secondary lateral peaks. They reduce almost to the diffraction envelope. For the two slits, as in the case of Young’s double slit experiment, we obtain more pronounced fringes. You can see how the one slit pattern “envelops” the two slits pattern. The interference figure of the single slit pattern is split into several more fringes. However, notice again how it would be incorrect to say, as you might hear frequently, that when we switch from the double slit to the single slit case, interference phenomena disappear. That is, in general, not the case. What happens is that we return to the envelope of the single slit which contains many fewer fringes, but still might have some other interference fringes too (and in this case it does). Interference is not a phenomenon specific to the double (or more) slits experiment. Interference does not disappear if one slit is covered; it merely becomes weaker than it is with more slits. And, at any rate, diffraction phenomena become dominant. Finally, in the case of 10 slits, the two slits curve turns out to be the envelope of the 10 slits curve. So, you can observe how this is a more general trend and phenomenon which results from the interplay between diffraction and interference. In fact, generally, the N-slits fringes and their spacings arise due to this combined effect between diffraction and interference.

These were only a few examples to outline, at least intuitively, how wave interference works.1 What does all of this have to do with Heisenberg’s uncertainty principle? Everything! I invite you to continue with the next essay.

If you have followed what we learned about the electron and neutron Bragg diffraction, it should not come as a surprise that the wave-particle duality is something that holds for material particles as well. In fact, the double slit experiment (link & link) which Young performed with light produces exactly the same results if we use electrons or other material particles instead of photons.

Therefore, the only thing we can say for sure is what can be represented in Fig. 1, which is a different attempt to model the observed phenomena. Before the detection of anything, we should only imagine waves or matter waves in the case of material particles: first, a plane wave hitting the screen and, after the two slits, the two concentric wavefronts. Once these hit the screen they interact with it according to a probability amplitude dictated by the interference amplitudes and suddenly “collapse” onto a single spot at the detection site.

This is what the wave-particle duality is about. It is about “something” that propagates and spreads out throughout space behaving and interfering like a wave, but when asked about its whereabouts, answers as a point-like interaction. More than that, we can’t say.

The question might arise: Does all this apply only to tiny elementary particles, such as electrons, or does it also hold for molecules or even larger material objects? There is no limit in the principles of QM that prevents the de Broglie wavelength relation, Eq. 1, from being equally valid for any object with any mass m. In fact, it could be shown that this is, indeed, the case. Interference patterns arising from a beam of large organic molecules diffracting at single, double, and triple slit material masks could be observed experimentally and quantum superposition at mesoscopic scales scales (a scale between micro- and macroscopic) of 2000 atoms were reported.

In principle, even an elephant could be diffracted by a double-slit device and interfere with itself. Why do we (fortunately) not observe such weird phenomena? Because the aperture of even the smallest slit we can manufacture is still huge compared to the extremely tiny de Broglie wavelength of macroscopic bodies.1

This should also make it clear that the de Broglie wavelength must not be confused with the measure of the size of an object. If the size of a material object grows with its mass and becomes macroscopic, its de Broglie wavelength decreases to sub-microscopic scales.

So, at this point we should be getting accustomed to the idea that there is neither a particle, molecule a “chunk of matter” nor a wave at all, but only a physical phenomenon which we should not even attempt to visualize with our human intuitive macroscopic understanding of the world. The nature of matter can be described only in terms of probability waves.

Note also how we coincidentally are talking about the wave nature of photons or matter without detecting and measuring any wave at all but inferring its existence only indirectly, a posteriori—that is, after the measurement of its particle nature, by looking afterwards at the interference pattern, which consists of several white spots. We talk of matter waves without seeing waves at all, quite the contrary. Paradoxically, we deduce the wave nature of matter only when several measurements of particle-like objects have been performed. As we shall see later on, this is a prelude to a typical situation which occurs frequently in QP: The kind of situation where we tend to reconstruct in our mind an objective reality, a model of something which is already in the past and no longer in the present, by looking at the effects it produced. We tend to believe there was a wave in the past, despite the fact that we observe only particles in the present time. It is just this logical reverse engineering, and the question of the extent to which we are allowed to resort to this kind of reasoning in QP, that has given so many headaches to generations of philosophers. Honestly, we really don’t know what these objects really are which we call “photons,” “electrons,” “protons,” and “neutrons.” We even are no longer sure if it makes sense at all to ask for a description of reality that is supposed to have an objective existence and ontology independent from our mental models.

The fact is that physical reality must be a bit more complicated and counterintuitive than we thought. We have to deal with phenomena that refuse to be forced into an objective ontological worldview made of classical objects independent from the act of measurement—at least, not in the classical Newtonian physics sense. In the world “out there,” there is something that produces effects which in turn become the causes of future events, and that tells us this: “I am a particle when you look at me but, as long as you don’t look at me, I will continue to behave like a wave”.

From the abstract mathematical point of view there is no problem with this. We just make all of our calculations with our nice wave packets and wave propagation and interference laws, and we obtain as an answer which is confirmed by the experimental results. From the ontological point of view, things are not so straightforward. It is not clear what QP is trying to really tell us about the physical world and whether it makes sense to speak about objective realities in the first place. This aspect has puzzled the best minds of science and philosophy in history and continues to do so into the modern day.

The next post will deepen further our uncertainty… with Heisenberg’s uncertainty principle (Part I).

In the previous post we have seen that “wave–particle duality” does not mean a photon literally switches between being a wave and a particle, rather, quantum objects behave in ways that resemble both concepts depending on the experimental setup. The double-slit experiment shows this most clearly: what propagates through the slits behaves like a wave, producing an interference pattern, but what is actually detected are always localized, particle-like events (individual photon hits). Even when photons are sent one at a time, the interference pattern gradually emerges, implying that each photon somehow contributes to a wave-like distribution while being detected as a discrete point. This challenges classical intuition and suggests that quantum entities are neither simple particles nor waves, but something fundamentally different that is only described probabilistically and revealed through measurement.

An important aspect always to keep in mind is that the process is completely random in the following sense. Were we to look at this interference pattern only at a small local scale (again, with a microscope as in Fig. 1) and without knowing how the rest of the interference picture is made, there would be no way to know if the single white dot on the detector screen is the result of an interference phenomenon through the two slits or a single photon coming from a source with or without the slits in between. It is only when we look at the global picture that we can infer that interference must have taken place. In this sense we say that QP is governed by purely probabilistic or stochastic laws. We can say that there is a certain probability that a photon is detected in a specific position, but we will never be able to tell in advance what this single event will turn out to be. We can only predict what will happen statistically, after a high (statistically significant) number of events have taken place. When such a number is collected, the overall disposition of the photons on the detection plane always obeys the classical interference laws, with its typical geometrical disposition of the dark and white fringe pattern. The single event, however—the displacement of a single photon on one or the other fringe—is a purely probabilistic event.

There is a higher probability that the single photon will be detected on the brighter fringes, but there is no way, no method at all, not even in principle, to predict in advance which of the fringes the single photon will be detected on. Despite its name, quantum mechanics does not provide a deterministic, mechanistic account of causal processes specifying where and when an event takes place; rather, it furnishes a probabilistic formalism that yields statistical predictions for measurement outcomes, without specifying the underlying dynamical mechanism by which a particular outcome is realized at a given time and location. As we shall see, there are good reasons to conclude that no such underlying mechanism exists in the first place—or, in the language of quantum foundations, no hidden variables exist.

This hints at a quite different conception than what we intend in classical physics with the words “randomness,” “chance,” and “coincidence.” In classical physics the side of the die that faces up after having been tossed is said to be a “random result,” in the sense that the physical process which determines how the die rolls on a table is so complicated and practically impossible to calculate that the outcome is unpredictable. But in principle, if we would know everything about the die, the forces that act on the die, how it is thrown on the table, every detail about the table itself, and the precise initial conditions of the system, we could ideally predict the outcome with a supercomputer that knows everything about the laws of classical mechanics. In principle, that would not be impossible.

The throw of a die manifests a form of randomness, or indeterminism, that arises from our inability to specify and predict the exact state of a system and thus reflects a limitation of our knowledge. For this reason, it is referred to as epistemic randomness or epistemic indeterminism. Classical mechanics is about the description of exact physical states quantified by exact quantities, for example, the precise position of a particle at a given time. It conceives only of epistemic randomness or indeterminism, there is no conception of an inherent uncertainty independent from our observations and intrinsic in the things themselves, otherwise called ontological or ontic randomness or indeterminism.1

This classical determinism has its roots in a way of thinking which is nicely summarized by Laplace’s demon. P. S. Laplace was a French mathematician and astronomer of the nineteenth century who reasoned as follows.

“We may regard the present state of the Universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set Nature in motion, and all positions of all items of which Nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the Universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”

This determinism is (more or less consciously) still the leading paradigm in science. If someone—here, the intellect of Laplace’s demon—knows the precise position and momentum of every particle in the Universe, its dynamical evolution for any given time could be calculated with certainty from the laws of physics. But we will see repeatedly during this journey through the weird world of quantum physics that this is an assumption that can no longer be considered true in the quantum domain.

QP suggests that, at the microscopic level, ontic indeterminism governs physical events: individual outcomes lack a determinate cause, they are, so to speak, ``a-causal”, and only the probability distribution constraints them. In the double-slit experiment, this distribution is given by the interference pattern, where the nodes (the regions with no particle count between the fringes) correspond to points of vanishing detection probability.

Thus, when and where an individual photon passing through the two slits will strike the detection screen—and therefore which interference fringe it will contribute to—is fundamentally unpredictable. Quantum mechanics does not provide a rule that determines the exact position of each photon. Instead, it describes the situation in terms of probabilities: it tells us how likely it is for a photon to be detected at each point on the screen. Only after many photons have been detected does a clear interference pattern emerge, reflecting this underlying probability distribution.

Another important point, often overlooked and seldom mentioned, is that quantum randomness manifests not only spatially (in the double-slit experiment in the spatial distribution of dots on the detection screen) but also temporally. The temporal sequence of detection events on the screen is also entirely random (see the animation of Fig.1). For example, one possible time-series might be: photon 1 hits pixel 1 at time t1​, photon 2 hits pixel 2 at time t2​, photon 3 hits pixel 3 at time t3​, and so on. However, this particular order is purely stochastic. It could just as easily have occurred in any other permutation, for instance: photon 1 hits pixel 3 at time t1​, photon 2 hits pixel 1 at time t2​, photon 3 hits pixel 2 at time t3​, etc.

Even if we know everything about the system and the initial conditions of a single particle, the single event will remain forever unpredictable. Not because of our ignorance, as Laplace believed, even if we have a “demon” which knows everything and is able to calculate quickly and precisely enough. It seems that in QM things are not random in the classical sense, but this randomness is an inherent aspect of Nature. We shall take up this issue again in the following chapters.

Almost everyone has heard of the famous “wave-particle duality” , but there are unfortunately many misconceptions surrounding it. One of the most frequent sounds more or less like this: “A particle can be sometimes a wave or a particle, depending on the observer.” This is a misleading waxy to present a much more subtler and more interesting phenomenon. Firstly, one should speaks instead of a particle that acts like a wave. Secondly, it is high time to debunk the “observer myth” in QP. No quantum phenomenon needs a human mind or consciousness observing it to validate its existence, and there is no “wave-to-particle” or “particle-to-wave transmutation” in the sense that our naive intuition tends to suggest. It is no coincidence that we are presenting wave-particle duality to the reader after dwelling first on a historical and conceptual introduction in order to furnish the context with which to grasp the deeper meaning and implications of the theory. We will do the same for Heisenberg’s uncertainty principle in the next section.

We have seen that photons can behave either as waves or as particles, according to the context and the experimental arrangement. For instance, the photoelectric effect or the Compton effect gives one answer, and the double slit experiment and Bragg diffraction give the opposite one. In the latter case we have also seen how the wave character fits with material particles too, like electrons or neutrons.

However, we might ask at this point: When, how, and under which circumstances does the one or the other aspect arise? Is this a property that quantum objects acquire switching from one state to another? Is there somehow a moment in time where a somehow ill-defined entity is or behaves like a particle and afterwards morphs into a wave? If so, does this happen in a continuous fashion, or instantly? Or is there another possible interpretation and understanding which unites the two aspects of photons, as waves and as matter and energy?

In some sense we have already seen how, with the de Broglie hypothesis, it is possible to unify this disparity with the concept of the wave packet. Therefore, so far we have shed some light (pun intended) on this wave-particle duality, but the picture is still somewhat incomplete. Because, if we think about it carefully, nobody has observed a light wave in the way we can observe a water wave. What we always observe are the effects of something that we imagine, by deduction after the fact, to be something that acts like a wave. The several experiments that hinted at the particle- or wave-character of light did not show that light is a particle or is a wave, but that it behaves like a particle or a wave. We should not confuse our human mental (sometimes even ideological) projections of how reality appears, with reality itself.

It is here where Young’s famous double slit experiment comes into play again in its modern quantum version. It clarifies further some very important aspects and clarifies the nature of physical objects in quantum physics.

So, let us turn back to the transverse plane wavefront of light (see Fig. 1), say, just a beam of light coming from a source which ideally is highly monochromatic, that is, having a single or very narrow range of wavelengths and colors (nowadays an easy task to accomplish with laser light).

We know that the slits should have a size and separation which should be comparable with that of the wavelength of the incoming light (something which, even for small wavelengths, can be achieved with high precision using modern lithographic technologies). When the plane wave goes through these slits it is diffracted and is detected with a screen at some distance, and the usual resulting interference pattern appears. These fringes can be easily photographed with a photographic plate or a CCD camera or whatever kind of sensor sensitive to the wavelength we are working with.

Through Young’s experiment, the question about the nature of light seemed to receive a clear answer. We conceive of light as a wave since it behaves like a wave, due to displaying the interference phenomenon typical to waves.

The (more or less unconscious) logical background of this conclusion relies on what is called the duck test of abductive or retro-ductive reasoning: “If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck”. This is a funny way to describe what philosophers mean with the technical term of counterfactual definiteness—that is, the ability to assume the existence of an object and its properties even when they have not been measured directly.

However, upon closer inspection at the microscopic level, the double slit experiment turns out to be just an upside-down view of the larger story. How so? Because what we detect are always and only interactions between the light and the detector screen in tiny localized spots. What one observes locally is not some nice continuous shading of the intensity between the black and white fringes, but instead a random distribution of points, that is, of point-like localized interactions on the detector screen, just like pixels on a monitor.

For instance, if we use an old conventional photographic plate as our detector screen, develop it, and then look at the photograph under a microscope (magnifying the region of the interference pattern in Fig. 1), we see many tiny white spots which correspond to the grains that, due to a chemical reaction, become white if a photon is absorbed.

After all, when we make a measurement, what we ultimately observe are particles, not waves. Be it a photographic plate, a modern CCD-sensor, or any other kind of detector which responds to an electric signal or just “clicks” (like a photodiode or a photomultiplier), what we really measure are one or many dots, local point-like interactions which form a granular interference pattern, never a continuous pattern. Even if we were to build a detector with the highest resolution possible—say, every pixel is an atom—the same dotted interference pattern would emerge at smaller scales. This should not come as a surprise, since from what we have already learned about the blackbody radiation theory and atomic transition spectra, we know that matter, and therefore atoms and molecules, absorb “quanta” not in a continuous fashion but rather in the form of discrete amounts of energy.

Therefore, also in Young’s experiment, we recover and find again the corpuscular nature of light: What seems to be diffracted at the slits and interferes on the screen behaves like a wave, but what is finally observed are photons hitting the screen. On one side, we have something we imagine (we don’t see it) to be a wave that goes through the double slits; but when we attempt to detect it, it inevitably shows up as a localized interaction.

In a certain sense we find ourselves again at the starting point. What is a photon really? If it is a wave, why, when, and how does it become a particle in order to be detected as a point-like structure?

Let us further inspect this state of affairs. If we would strictly maintain the particle picture, then we must assume that each particle must go through one or the other slit (see Fig. 43 left), and then travel further in direction of the detector screen and show up at one or the other fringe, that is, in two sets of piled-up particles detected only in two locations on the screen.

However, we know now that this is not the case, since we observe other lines appearing at the interference fringes as well. If we still believe that there are only particles, we are forced to assume that something like in Fig. 2 right must occur.

One has to assume the existence of some ill-defined “wave” or “force” or “field” that “guides” the particles along its path, so that they match the interference fringes.1

In fact, let us take a further step in our reasoning and ask ourselves what happens if only one photon is sent towards the double slits. Do interference fringes arise also with a single photon? Or are they the result of the collective interaction of many photons? Or, which seems to be another possible alternative interpretation, does the single particle interfere with itself?

The latter question may seem at first meaningless, because we might object that a single photon could not be subjected to interference, since to observe these interference phenomena we need a collection of photons to build up as fringes on the screen. However, there is a way to test this hypothesis: We could try to deceive Nature by seeing whether and how the interference pattern builds up if we send only one photon after another and then wait to see how things develop over time. One after another means that we shoot a single photon at the two slits, wait until the single photon hits the screen and record its position, and only after that, we send the next photon, wait again until it is detected on the screen, and so on.2

Experimentally this is not easy, but nevertheless possible to accomplish by making the light source extremely dim (for example, take a luminous source and dim it with some almost black foil), in such a manner that it emits only one photon at a time. Then the single photon is forced to go through one or the other slit. If it is a particle, as we imagine it being in our naive intuition, then it cannot go through both slits. If this particle travels through the first slit, then one expects to find it colliding with the detector screen in the region of the first fringe of Fig. 2 left. And when another particle goes through the second slit, then it is expected to hit the second fringe. So, since we are sending only one photon at a time, this seems to be a method to force Nature to manifest its corpuscular aspect.

This experiment proceeds as follows. We produce only very few photons at a time and then look at the detector screen after a short time interval. In Fig. 3a you can observe how only few photons are distributed in an apparently random manner all over the area where previously we observed the interference fringes.

So, there is no hint that interference occurs, and we are happy with that: We believe to have sent through the slits only one photon at a time and we observe only tiny white spots, as expected (even though the random displacement of the photons seems already to disprove the existence of only two lines). If we wait a little longer and collect more events on the detector screen, we see something like in Fig. 3b: Things look still clumpy, but now there definitely seems to be no trace of the two expected white lines. At this stage of the experiment, however, we still cannot discern clearly what the truth is. Then, after collecting a sufficient number of events on the screen, we begin to observe indeed the interference fringes, as in Fig. 3c. And, as we wait even longer and collect larger numbers of spots on the screen, the whole interference structure finally emerges (Fig. 3d and 3e).

Thus, we must conclude that, despite having dimmed our light source to such a degree that it emits only one photon at a time, the wave-particle duality does not disappear. The interference phenomenon is not a collective phenomenon whereby we might imagine many light particles somehow interacting among themselves to produce the interference fringes. Single photons obey the interference law. We might interpret the overall result as if the single particle goes through both slits at the same time, as a sort of “ghostly wave” that is subjected to the laws of diffraction and interference, but then, when its position is measured, suddenly collapses to a single point at the detector screen. There it appears as a little localized spot. The kind of classical model of reality we imagined in Fig. 2 left is no longer tenable at all. It is simply wrong.

What this shows instead is that as long as light is not observed, it travels and expands throughout space as a wave; but when the act of measurement is performed, it chooses to place itself on a specific location. We see that something that seems to behave as a wave, suddenly “collapses” to a point. We will never see the wave and particle nature at the same time.

An important aspect always to keep in mind is that nevertheless, the process is completely random in a sense I will outline in the next post.

The de Broglie hypothesis unifies the wave and particle nature of both light and material particles, proposing that all particles can be described as localized wave packets whose wavelength is inversely proportional to their momentum. This concept explains phenomena from X-ray diffraction in crystals to the dual behavior observed in electrons and photons, laying the foundation for understanding atoms as matter waves.

In fact, if we understand electrons not as tiny localized point-like particles which orbit around the atomic nucleus, but as matter waves, that is, waves distributed concentrically around the orbit, it becomes quite natural and intuitive to understand why the energy levels of atoms must be quantized.

Recall the standing wave we already discussed for the blackbody cavity. Here the principle is the same, but instead of reflections of a wave on a cavity wall, the wave circulates along an atomic orbit and for the same reason allows only for specific oscillating modes for constructive interference to exist. If this matter wave has a wavelength which does not match and reconnect to itself again at the same point along the orbital path (see the diagram of Fig. 1) and if the circumference of the atomic orbit of the electron (the circle) is not exactly an integer multiple of the electron’s wavelength (the wavy line), then constructive interference with itself is no longer possible, destructive interference disallows stable patterns, and the wave packet distributed around the atomic nucleus cannot build up as a standing wave structure.

The classical notion of orbital velocity of the electrons around the nucleus is fixed by the fact that the electrostatic attractive forces between the positive charge of the nucleus and the negative one of the electron must be balanced by the centrifugal forces, and which keep the electron in orbit and which are given by its distance from the nucleus. But this distance cannot be an arbitrary one; the velocity of the electron, which determines the wavelength of the electron (due to de Broglie’s relation relation ), must match just that orbital distance where constructive interference occurs and is equal to just that speed which is enough to keep the electron in a stable circular orbit.

It is possible to calculate this, and indeed it turns out that the electron will distribute itself naturally only on well-defined, discrete orbits. For the simple case of the hydrogen atom, this can be calculated with relatively simple algebra and CP concepts plus the de Broglie relation.

And voilà, it turns out that only those matter standing waves are allowed to exist which match the quantized energy levels experimentally observed in the hydrogen atomic spectrum (see Fig. 6). Bohr’s atomic model added with the de Broglie hypothesis seemed therefore to gain credibility, since it explained naturally why atomic levels are quantized. Still, as we already pointed out, it does not explain the spectra of all the other elements.

So, let us turn back to Bragg diffraction and clarify what we really mean by “matter waves”. Since we can conceive both material particles, like electrons, and massless particles, like photons, as waves, and since we observed the Bragg diffraction for EM waves, as was shown for photons in the X-ray spectrum, the question at this point is: Does the Bragg diffraction hold equally also for material particles? Is it conceivable that tiny chunks of matter like electrons, neutrons, protons, or whichever massive particles can be diffracted by the lattice of a crystalline substance and display the nice interference patterns as has been observed with light waves and X-rays?

To check this idea one must send no longer photons, but instead material particles—say, electrons or neutrons—into the crystal lattice by modulating its energy, that is, its momentum (its speed). In fact, by doing so, one modulates, according to the de Broglie relation, its wavelength as well. If the wavelength of the wave packet matches the size of the crystallographic planes’ separation, one should expect the interference patterns.

This experiment was conducted in 1927 by the American physicists L. Germer and C. Davisson, and independently also by the British physicist G.P. Thomson. Passing a beam of electrons through a thin film of metal, they observed how electrons are subjected to the exact same interference laws that photons are. What they found were indeed the Bragg diffraction patterns, like those of Fig. 3.

So, when subjected to appropriate physical conditions, what was imagined to be rock-solid chunks of matter displayed a wave-like nature. This was an experimental verification of the de Broglie hypothesis, which at this point was no longer a hypothesis at all but an experimental fact: The observed interference pattern corresponded exactly to those expected if we adopt the de Broglie relation. This is a universal property of Nature—which, however, is not at all intuitive.

From the abstract mathematical point of view everything is clear and is nicely quantified and experimentally measured as described by theory. But from the more ontological perspective things are not that clear. The question is: What are these particles or matter waves really, at the microscopic level? Are they particles, waves, something which should be conceived as a synthesis of both, or something else entirely? It is here that the philosophical journey of QM began, and this led then to the famous wave-particle duality, Heisenberg’s uncertainty principle, the concept of the wave function, and so forth, as we will explore in the following sections.

This concludes chapter II of my book. The table of contents you can see here.

The next chapter will deal with the first foundations of quantum physics, starting from the wave-particle duality, then proceeding with Heisenberg’s uncertainty principle, the wave function and its ‘collapse’, the state vector and Schrödinger’s equation, and the particle in a box as the basis for atomic physics. Stay tuned and subscribe!

In the last article we saw how thin-film interference arises when light reflecting from different layers travels unequal path lengths, creating phase differences that produce the characteristic color patterns seen in soap bubbles and oil films. The same principle applies at the atomic scale in crystals, where X-ray interference—described by Bragg’s law—reveals diffraction patterns that allow scientists to infer the structure of crystal lattices. That is, we see how light, and even material particles, can be described in terms of wave behavior.

On the other hand, while phenomena such as the double-slit experiment reveal the wave nature of light at macroscopic scales, the photoelectric effect and Compton scattering demonstrate its particle nature at the microscopic level. How can we reconcile such apparent dichotomy?

One might be tempted to believe that, microscopically, light is in fact a particle—in the sense that it might appear and behave like a wave only at human scales, while close observation at the scale of molecules, atoms, and particles reveals it to be a particle and just like a water wave to be microscopically made up of tiny H20 water molecules.

Not so. Bragg diffraction and interference phenomena act at the size of an atomic crystal lattice layer and cannot be explained with classical particle behavior. Considering light merely as a wave, Bragg father and son could explain the observed phenomenon, just as Young did using the double-slit experiment. In fact, Young would have been delighted to see the wave nature of light confirmed also at microscopic scales.

But there is much more. One question that must arise at this point is whether this ambiguous twofold wave-particle manifestation is something inherent only to light. Since light seems to present itself as a wave or a flux of particles according to whichever kind of experimental setup one chooses, does this duality also hold true for material particles with a mass, like electrons or protons?

It might sound unreasonable and counterintuitive to think of material particles like an electron, which we imagine as a tiny chunk of matter, to behave like a wave as photons do. But an attempt to categorize them under the same umbrella came from the French physicist Louise de Broglie, who was able to synthesize in a unique theoretical concept the corpuscular and wavy nature of material particles.

What de Broglie realized was that there is no logical and physical reason to believe that material particles with a mass could not have this double corpuscular and wavy nature as well as photons do. He imagined all the particles—including those which make up objects like atoms, molecules, and every material body in the Universe—no longer as particles or waves but as something which contains both, conceptually speaking: the so-called wave
packet.

This was the famous de Broglie hypothesis, framed in 1924. The idea was to imagine every object being a traveling wave, but with the special property of being localized in space. If we see things in this perspective, we can consider any particle (with or without a mass) as having a specific wavelength, with the amplitude of the wave having a maximum in a region of space and which we intuitively think of as being classically the position of the particle, but with this amplitude decaying quickly from the center of the peak in all directions.

In Fig. 2 two arbitrary examples of wave packets are shown: on the left a broader wave packet and on the right a smaller one. The two differ in their wavelengths.

If you look at these wave packets in Fig. 1 from a large distance compared to their wavelengths (say, from a distance of ten meters from you monitor), you may approximately consider them particles, since their wavy nature is no longer discernible to your eye. But if you look at the wave packets at a distance of the order of their wavelengths or shorter, they appear to be waves. In this sense we may speak of particle wavelength: an expression which otherwise, in the classical context, would have seemed an oxymoron.

A gentle warning is compulsory here. One might be tempted to interpret the size of the wave packet as the size of the particle or the object under consideration. However, we shall see that this intuitive understanding is not correct. For the time being, just take the wave packet for what it is: a mathematical abstract construct that helpfully explains the observed phenomena of the experiments where the particle- and wave-nature can coexist.

L. de Broglie quantified all this with a very simple and neat formula, which relates the wavelength λ of a particle, or more precisely the wavelength of its corresponding wave packet, as being inversely proportional to its momentum p, which in classical mechanics is defined as the product of its mass m times its velocity v: p = m v. De Broglie’s wavelength is then defined as:

where h is, as usual, Planck’s constant. The right-hand side of the equation holds true only for material particles with a nonzero mass and is correct only if we ignore relativistic effects. On the other hand, the term h*p is more general and has universal validity, for photons as well as for material particles, and is exact also in relativity. It associates a wavelength with any kind of particle, independently if it has a mass (say an electron, proton, or a neutron), which moves with speeds less than that of light, or if it is massless, as in the case of photons, and which, in vacuum, always has no other possible speed than c.

This can be shown with a (not too rigorous) proof which highlights where this equation comes from. From Einstein’s mass-energy relation E=mc^2 we know that the mass m can be rewritten as m = E/c^2 . If we take seriously the idea to put photons and material particles on the same footing, we can replace the mass in Eq. 1 with the latter expression and the speed with that of light (v = c) to obtain:

This expresses the momentum a photon of energy E carries. Photons, as all other particles, are able to carry and transmit certain amounts of momentum to other particles. Recall how photons, even when not visualized as material particles themselves, are able to give a “kick,” so to speak, to material particles, as we already saw in the case of Compton scattering.

Now, recall also the ever-present Planck’s relation: E = hν = h c/λ . Rewriting all of this in terms of λ, we obtain:

with the last replacement due to Eq. 2. This proves de Broglie’s relation.

So, what de Broglie did was to put together the concept of the wave and particle into a single entity: the wave packet, the wavelength of which is determined by its momentum. In the case of material particles—not photons but particles with a mass moving slower than the speed of light—it is customary to speak also of so-called matter waves.

In the next article we will see how the matter wave of the de Broglie hypothesis also sheds further light on Bohr’s atomic model.

We are now in a position to analyze wave interference in real-world physical phenomena. We can begin with a classical example—what one observes in thin film interference. This will open up the way for us to understand Bragg diffraction and the wave-particle duality.

In Fig. 1 you can see a couple of light rays with a certain angle of incidence reflected by a thin film surface. This thin film surface could be for example made of water, or the mixing of oil and water on the road after a rainstorm, or, as in this case here, that of a soap bubble. The nice coloring of oil films or, as we enjoyed as children, of soap bubbles, is an interference phenomenon that can be explained as follows.

These films have a very small but non-negligible thickness limited by an upper and lower layer. The reflection of light onto these layers can be visualized as a first beam, beam A, that goes unperturbed through the upper layer and will be reflected by the lower layer of the film, and a second beam B which is reflected by the upper layer. If we examine the situation carefully, it is easy to recognize that there is a difference in the travelled path between the two light rays. Even if both light rays are emitted at the same time from the same source and arrive with the same angle—that is, they are parallel to each other—when they reach the two thin film layers, they are nevertheless reflected at different depths and therefore acquire a path difference l, which implies a phase difference relative to each other (the little segment of length l, in Fig. 1).

The two beams then sum up and superimpose after the reflection, interfering according to their phase difference. This phase difference is determined only by the incident angle once we fix the thickness of the film and its refractive index (a dimensionless number which tells how much light is refracted and reflected or, equivalently, how fast it travels in a medium). This implies also that the outgoing light can interfere constructively, destructively, or somewhere in between, for every specific reflection angle. We will observe the resultant waves varying in intensity and wavelength along the different angular positions where we place our eyes. This is why we can observe the beautiful plurality of color patterns on the surface of a soap bubble.

The same thin film interference works not only with water layers in visible light but also for atomic crystal lattice layers at X-ray wavelengths. In the latter case we need very short EM wavelengths because the distances that separate the different atomic crystalline layers (also called crystallographic planes) are very small, of the order of few atomic sizes. Obviously, at these wavelengths, we will no longer deal with color patterns since X-rays are well beyond the human-visible light spectrum, but the underlying principle remains the same.

In 1913, W. H. Bragg and his son W. L. Bragg, two British physicists, discovered how crystalline solids produce patterns of reflected X-rays that can be interpreted only as interference patterns, nowadays called Bragg diffraction patterns. They observed intense maxima and minima of reflected radiation, depending on the wavelength and angle of incidence—that is, they could not be observed for reflected radiation of longer wavelengths.

In fact, more precisely, at the atomic scale, we can no longer speak of classical reflections on a surface layer; we must conceive of the photons (or waves) as absorbed by the atom and re-emitted in all directions in the form of spherical waves. All these spherical wavefronts then sum up and form an interference pattern somewhere, for example on a screen. Let us take a typical example of an atomic lattice: a salt crystal, which is nothing other than sodium chloride, a crystal made by sodium and chlorine atoms (depicted in Fig. 3 with larger and smaller atoms, respectively).

The function of the thin film we considered in the example of the soap
bubble with liquid layers can now be taken up by the atomic layers of the
salt crystal lattice. Here again, two incident and parallel light beams will be
reflected by two underlying crystallographic planes.

There are essentially three physical variables that determine how the beams will be diffracted and will interfere—that is, how the outgoing waves will display their relative phase shift which gives rise to the interference pattern. First, all depends from the wavelength λ of the EM wave. Second, what determines the phase shift induced by the path difference is also the distance d which separates the atomic layers. This causes a path difference (the short wavy segment that connects the two atomic layers in Fig. 1), and consequently a phase difference between the two emerging beams. Finally, we must also choose the angle of incidence, or equivalently the scattering angle θ.

The appropriate combination of these three parameters will give rise to constructive or destructive interference. Once we have chosen some material with some specific chemical composition, the crystal lattice layers distance d is fixed. If we maintain the wavelength λ of our X-ray source as constant, the wavelength is fixed too.

This means that, if we observe the outgoing interfering waves at different angles, then we will see constructive interference (Fig. 4 left) or destructive interference (Fig. 4 right), due to the fact that they correspond to different paths of the light beams, giving rise to the respective type of interference.

This can be summarized by Bragg’s law, which tells us the conditions for constructive interference from successive crystallographic planes:

For each integer (n=1, 2, 3,...) we are able to calculate each angle where the maxima will occur.

So, if we look at different scattering angles at the same time, for example on the screen of a photographic plate or a CCD sensor placed in such a manner that several beams with several angles are captured throughout the sensor’s surface, then we will observe a very interesting diffraction pattern, as those in Fig. 5.
Here we have a beautiful symmetric pattern obtained by X-ray interference through a crystal. From these patterns, physicists and chemists are able to infer the crystalline atomic structure of substances. Therefore, Bragg diffraction allows us to reconstruct the geometrical displacement of atoms in the lattices, in some sense; we might say that it is like a microscope that indirectly furnishes images of the chemical structure of substances.

Admittedly, this is a simplified model. In practice, other planes also contribute to the overall effect, though to a lesser degree, and these will cause many more small peaks and complex patterns. But for our purposes here, this model suffices to illustrate the principle of Bragg’s diffraction.

But what does this have to do with QM? Well, a lot, as we will see in the next essay.

Another nice and clear-cut piece of experimental evidence that atoms must have discrete energy levels came only a year later, from the Franck-Hertz experiment. G. L. Hertz was the nephew of H. R. Hertz, the German physicist mentioned as the discoverer of the photoelectric effect. While H. R. Hertz, among other things, proved the existence of EM waves, G. L. Hertz and James Franck were awarded the 1925 Nobel Prize in physics “for their discovery of the laws governing the impact of an electron upon an atom,” using the experiment we are going to describe.

The experimental setup (see Fig. 2) consists of mercury (Hg) gas atoms
inside a low-pressure bulb. An electric cathode—that is, something like the
heated filament of a light bulb—emits electrons. Therefore, this part of the
device emits not only light but also negatively charged particles. An electric
field is applied between the electron emitting cathode and a positively poled
grating, with a battery or some other electric source, which builds up an
electric potential.

Due to their negative charge, this difference in the electric potential field leads to the electrons’ acceleration and conveys to them some kinetic energy (as you might recall from school, charges with the same polarity repel each other whereas those with opposite polarity attract each other). When the electrons reach the grating, most of them will fly through because the mesh of the grating is kept sufficiently wide to allow for that. This first part functioned as a little electron accelerator. Then, between the grating and a collecting plate on the right side, another field is applied.

However, in this second part of their journey, they will experience an inversely polarized electric field as, after passing the grating, they will be repelled because they will begin to “feel” the negatively charged collecting plate. This means that, through use of an Amperemeter, a device which measures electric current (the number of electrons), one can measure the flux of electrons which flow between the grating and the collecting plate. While the electrons initial energy is proportional to the applied electric field intensity (the voltage) between the emitting cathode and the grating, in this second part, they are decelerated due to the opposite polarity. The measurement of the current, therefore, allows one to determine the number of electrons that make it through to the collecting plate, which provides information about how their energy is affected by the atoms while flying through the gas in this second part of the bulb. This is done by varying that field, step by step, for several voltages.

Franck and Hertz’s insight was that, while flying through the gas of atoms, several electrons must sooner or later hit one or more atoms and be scattered either elastically or inelastically. Elastic scattering means that when objects hit a target, they change course but maintain the same kinetic energy, while inelastic scattering implies that they lose part or all of their kinetic energy in the collision process. It follows that there must be a measurable difference between the energy of the injected electrons reaching the grating and the energy of those which flew through the gas, hitting the collecting plate. This difference is made clear to the observer by measuring the current between the grating and the collecting plate. This energy gap tells us something about the energy absorbed by the atoms in the gas.

Therefore, if atoms absorb energy only in the form of quanta, this implies that, while we slowly increase the kinetic energy of the injected electrons, we should be able to observe when and to what degree the electrons’ energy is absorbed by the gas of Hg atoms.

This is, indeed, what happens, as can be observed in the graph of Fig. 3. While the injected electrons kinetic energy is increased steadily by application of an electric potential from 0 to about 15 V between the cathode and the grating (horizontal axis), the current of the electrons measured at the collecting plate (vertical axis) increases accordingly, though not in a linear fashion. We observe that the electrons do not have a final kinetic energy which increases proportionally to the electrons’ input energy, according to what one would expect for an elastic scattering between classical objects (think, for example, of billiard balls). What we see instead is that at first (between 0 and 4 V), the relation between the input and output energy is approximately linear, which means that the electrons are scattered through the gas elastically; they do not lose considerable kinetic energy.

At about 4.5 V, the first bump appears. Between 4 and 5 V, the output energy of the electrons decreases steadily, despite their increasing initial energy. This signals an inelastic scattering: Some of the electrons’ initial energy must have suddenly been absorbed in collisions with the Hg atoms. However, this does not happen before a certain kinetic energy threshold of the electrons hitting the Hg atoms. At about 5.8 V, almost all the kinetic energy is lost and goes into the internal excitation of the atoms. There is, however, a remaining minimum energy gap which is shown in the figure with the vertical arrow. The difference between the first peak and the first minimum is the maximum amount of kinetic energy the atoms are able to absorb from the electrons. Therefore, it furnishes the first excited energy level of the Hg atom.

Then, after about 6-9 V, the energy begins to increase again, meaning that the atoms absorb only that aforementioned discrete amount of the electrons energy, but not more than that. The remaining energy goes again into elastic scattering. All this repeats regularly at about 9-10 V and about 14 V. The existence of these “bumps” at different input energies (until nowadays, experimental particle physics is all about the search for bumps appearing in graphs) means that atoms must have several different but discrete energy levels. Franck-Hertz’s was the first direct experimental proof confirming Planck’s idea that atoms absorb energy in discrete quanta. Moreover, this validated the discrete spectral lines of light spectra, as did Bohr’s idea of representing the atoms in the form of a model which resembles a tiny solar system—that is, with electrons moving only in specific orbits with their respective quantum numbers which represent different but discrete energy levels.

Not too many years later, other types of phenomena confirmed energy quantization in Nature. One of these, in 1923, was the Compton scattering or Compton effect, documented by the American physicist A.H. Compton. Compton also started from the assumption that EM waves could be considered a flux of light particles. If this were true, it would then be possible to calculate precisely the scattering process between a single photon and an electron, just like it is possible to describe the elastic collisions between two billiard balls, using the simple laws of energy and momentum conservation of CP.

Now, by using these conservation laws, Compton wrote a concise and useful formula which relates the wavelength λ of an incoming high-energy gamma-ray photon (a photon with a wavelength sufficiently small to be comparable to the size of an electron) before the scattering with the electron, and the wavelength λ′ of the scattered photon, according to a scattering angle θ (see Fig. 5). The wavelength λ′ of the scattered photon must be larger than that of the incoming one, as it loses some of its energy. This is because the higher the energy of a photon, the smaller its wavelength will be. In fact, recall that the energy of a photon is given by Planck’s relation E = hν, with ν the frequency. The wavelength of an EM wave is given by λ = c/ν , with c = 3 × 10^8 m/s the speed of light in vacuum (ca. 300.000 km/s).1 So, the wavelength we associate with a photon must be inversely proportional to its energy as: E = hν = h c/λ .

By putting all of this together, Compton was able to predict in 1923 a very strict relation between the difference in wavelength of the incoming and scattered photons and the scattering angle θ, which is the wavelength shift ∆λ radiation undergoes when it is scattered by matter. Compton’s formula was as follows:

Here λ0 is a constant and me is the mass of the scattering particle, in this case the electron. λ0 is called the Compton wavelength, which for the electron is about 2.426 × 10−12m. Notice that if θ is zero, we have no difference, which means that the photon goes straight through without scattering and frequency change. Whereas for θ=180°, for backscattering, the wavelength increases by twice the Compton wavelength. The exact relation between the input and output wavelengths (or energies) of the particles and their respective scattering angle was verified experimentally, and it turned out that Compton’s predictions came to fruition precisely. This is a great historic example of the triumph of theoretical physics confirmed by experimental verification—of how, in the history of science, we have found that sometimes first comes the math and then comes the verification in the lab (even though sometimes it goes the other way around).

Compton scattering is also an atomic scattering process. This suggests an explanation of how photons can ionize atoms. To ionize an atom means that a photon extracts an electron from an atom’s outer orbit shell around its nucleus. A light particle with sufficiently high frequency can be absorbed by one of the electrons, so that the electron acquires a certain amount of kinetic energy and can eventually be removed by overcoming the atomic force potential that keeps it bound to the nucleus. A gamma-ray photon, however, can be so penetrating that it can ionize even inner electron shells and partially or completely transfer its EM energy to the electron, which acquires the gamma ray’s momentum and kinetic energy and which for this reason is also called a “photoelectron”, something we have already seen with the photoelectric effect). Notice, however, that the photoelectric and the Compton effect are two very different physical processes. The former extracts the electrons from a metal lattice; the latter is a deep scattering process that can “kick out” the electrons which are contained in the inner shells of an atom.

In Fig. 6 you can see an illustration of this process. A single high-energy photon can also be scattered by several atoms and will lose its energy with each repeated collision. A high-energy small-wavelength (high-frequency) photon can become, via multiple scatterings whereby it loses a bit of its energy in each collision, a low-energy large-wavelength (low-frequency) photon.

This is precisely what happens in the center of the Sun. In the Sun’s core, which has a temperature of about 16 million degrees, there is a huge amount of gamma radiation. However, it takes several million years of scattering processes before the photons produced inside the Sun reach its surface, by which time they have lost most of their energy being “downgraded” to the photons that we know as visible light. The light we observe today coming from the Sun’s surface was once an intense radiation foam of gamma photons in the Sun’s interior.

What makes Compton scattering so important and interesting for our considerations here is that all this seems to suggest again that we can reasonably think of EM waves no longer as waves at all, but rather in the context of light particles, or photons, kicking around other particles like tiny billiard balls. If we were to stick to the classical idea that light is made of waves, we should observe a scattering of concentric waves by the electron (think of water waves scattered by an almost point-like object). Compton’s scattering instead indicates that the application of the conservation laws of CP, which determine the scattering between classical particles, holds also with photons. There is therefore no longer any reason to reject a “particle picture” in QM in favor of the “wave picture”.

Another effect which is worth mentioning is the pair creation effect, which can occur as an alternative to Compton scattering. Pair creation is a physical phenomenon whereby a high-energy massless photon is converted into particles with a mass. Instead of the photon and electron being scattered, what occurs in this case is the absorption of the photon (by another particle or atom nucleus) and an immediate release of its energy in the form of material particles.

The pair creation effect was discovered in 1933 by the British physicist P. Blackett. This phenomenon clearly shows that we must conceive of light not only as made of photons but also in the context that photons can transform into other material particles, like electrons. Pair creation, or even annihilation, which is the opposite process whereby particles with mass are converted into photons, is a striking example of Einstein’s mass-energy equivalence which is described by his famous formula: E = mc^2 , where m is the rest mass of a particle, and which is called also the “rest energy” (in relativity, the mass of a fast-moving object cannot be handled as the classical rest mass, but we won’t go further into this here) and, again, c is the speed of light.

The mass-energy equivalence tells us that every particle, atom, and any material object with a mass contains an energy in potential form which is given by Einstein’s formula. The fact that it factors in the speed of light squared makes the amount of energy contained in matter huge. Only a gram of matter would cause an explosion like that of the bomb of Hiroshima.

In the case of a gamma photon pair creation (Fig. 8 left), two electrons can “materialize” if a single photon interacts with matter, say a heavy atomic nucleus, and converts into matter itself. One of these material outgoing particles will be the negatively charged electron e−, while the other is a positively charged anti-electron e+, also called a positron.

Anti-matter in general is made of the same particles with the same mass—that is, of electrons, protons, neutrons, etc.—but with the opposite electric charge and opposite spin. This process, however, can only happen if the energy of the incoming photon is sufficiently large to produce the two electrons according to Einstein’s mass-energy equivalence formula. If the photon’s energy is lower than twice the rest energy of two electrons, the process cannot occur and the photon will not be absorbed but instead eventually will be scattered via Compton scattering.

The annihilation process (Fig. 8 right) is the opposite process: Two massive particles (one ordinary and the other an anti-particle) interact with each other and are converted into gamma photons, which subsequently fly apart in opposite directions. For example, if an anti-electron comes into contact with ordinary matter, it annihilates to produce two gamma-photons with an energy given by Einstein’s equation.

Fortunately, for some reason that is still not entirely clear, the Universe we live in is made almost completely of one type of particle; otherwise, we would have already “evaporated away” in a foamy Universe where particles annihilate and create themselves continuously and stars, planets, and of course life, at least as we know it, couldn’t have formed in the first place. In the laboratory, we always observe the creation or annihilation of particles in pairs. The former is always the creation of a particle and an antiparticle, whose electric charges, taken together, sum to zero (positive + negative =zero). We never observe the creation of only one type of particle. This makes sense, because otherwise electric charge would be created out of nothing (resulting in a net positive or negative charge). Yet the universe we observe is composed almost exclusively of one type of matter. Where has the other half gone? At the time of writing, this remains one of the major unsolved problems in modern particle physics.

Thus far, we have listed several phenomena which indicate how energy is absorbed and emitted in a quantized manner and which therefore led to a corpuscular interpretation of light: Blackbody radiation, the photoelectric effect, the Franck-Hertz experiment, the Compton effect, and pair creation and annihilation can all be viewed in this context. So, on one side we saw how light behaves as a wave (recall Young’s double slit experiment); on the other side, EM radiation behaves as if it were made of particles. Which of the two points of view are correct? What is Nature trying to tell us here?

This still isn’t the whole story, as we are going to see in the next article.

At this point, physicists had a sufficiently broad experimental and conceptual understanding of the physics of atoms to go a step further. The fact that matter emits and absorbs EM energy in a discrete manner had become, at that time, an established and accepted fact. What was missing, however, was a structural model which could, at least partially, explain this. Ernest Rutherford (1871-1937), a physicist from New Zealand, discovered in 1911, by performing scattering experiments with radioactive particles, that atoms are almost empty objects, as over 99.9% of the mass of an atom is concentrated inside a tiny positively charged nucleus.1 He conceived of the atom as a kind of tiny solar system, with the negative electrons circling the positively charged nucleus made of protons.

Putting things together, a couple of years later, Niels Bohr, a Danish physicist, postulated the electron’s orbits as being discrete. This is the model most of you may have encountered in school. What characterizes Bohr’s model is that it retains the idea of electrons flying along very well-defined and precise orbits. Every orbit can be occupied by a certain number of electrons. Moreover, every orbit corresponds to some energy level of the electrons in the atom. There were no intermediate orbits, only orbits corresponding to specific energy levels, which can be labeled with progressive integers, n = (1, 2, 3, ...). These numbers were called principal quantum numbers because these define the possible discrete energy states of the atom.

This was the first atomic model that could explain why atoms absorb and emit light in discrete quantities. If the orbits are fixed, the energy levels of electrons in those orbits around the nucleus must also be discrete. The energy absorption and re-emission is conceptually visualized with the transition of an electron from one orbit to another. If an electron absorbs a photon coming from the environment, it transits towards a higher orbit, but then will quickly re-emit the photon back by returning to the lower energy level orbit. Re-emission occurs because every system will always tend to return to the lowest energy level possible. Because an empty place is left by the electron, it will re-occupy it back by emission. It is not necessary that the transition be between neighboring orbits; it could also occur between orbits with very different quantum numbers. The difference in energy of the two orbits will be simply that energy which the electron absorbs and emits. This energy difference will also be that which determines the frequency (or wavelength) of the absorbed and emitted photon according to Planck’s relation (see Eq. 1).

This model could indeed explain the atomic spectra of the hydrogen atom measured experimentally. For every electron transition from one orbit to the other, in each of its possible combinations, one can calculate the energy difference and, from there, the energy and wavelength of the absorbed or emitted photon.

Fig. 2 displays a qualitative picture of some of these possible transitions. For example, the transition to the ground state (quantum number n=1) corresponding to the energy E0 from all the above discrete electron orbits (n=2, n=3, etc.) of energy E1, E2 . . . etc., emits photons of energy E1 − E0, E2 − E0, etc. This is the so-called Lyman series of hydrogen’s spectral lines. Many other possible series exist, each labeled with the name of its discoverer.

As an additional historical side note, it might be worth mentioning that Bohr’s atomic model was refined and extended by Arnold Sommerfeld, a German physicist, who also considered elliptical electron orbits, just as we know it from planetary orbits. This is sociologically interesting because it shows how we, as humans, are instinctively driven first by the desire to transfer our understanding of the macrocosm onto the microcosm. In fact, this attempt was not without success.

Sommerfeld introduced a second quantum number, the orbital (or azimuthal) quantum number l, which for each principal quantum number n specifies n allowed orbital angular momenta of the electron around the atom’s nucleus (l=0, 1, 2,. . . n-1, and designated as s, p, d, f, g,. . . ), and which characterizes the ratio between the semi-major and semi-minor axis—that is, the elongation of the corresponding electron orbit. Fig. 3 shows the first three energy levels of the hydrogen atom each with its possible elliptical elongations (the size is given in Angstrom: 1 ˚A = 10^(−10) m).

This was quite an exciting success at the time, as it seemed to indicate that physicists were after something that could lead them in the right direction. However, many inconsistencies soon became apparent. The Bohr-Sommerfeld model was able to make these predictions with only limited precision and, at any rate, only for the hydrogen spectra. Bohr and Sommerfeld were able to explain the hydrogen spectrum but failed in describing any other atomic spectra. Such a simple theory cannot explain all the other elements of the periodic table, nor can it explain the molecular energy levels and structure.

It was an attempt to explain atomic behavior and energy states. It is, indeed, an interesting historical remnant of the intellectual journey of humankind, and it made perfect sense when viewed in that historical context.

However, it is no longer tenable today. Unfortunately, it is still taught in schools as if it were the final sentence and summation of atomic physics. And it was a good example which shows that some theories can make correct new predictions and yet still be wrong. This is something that many scientists seem to forget but that we should always keep in mind when working with modern theories.

If a theoretical model correctly describes and predicts what we observe, this should not be taken as proof of the true ontology of things—that is, of how reality actually is. The history of science has already demonstrated this in other contexts.

For example, the geocentric model was able to describe and predict certain astronomical phenomena quite accurately, even though it placed the Earth rather than the Sun at the center of the universe. By extending Bohr’s model to include elliptic orbits, Sommerfeld essentially did what the Ptolemaic astronomers had done centuries earlier: they kept adding epicycles upon epicycles to the geocentric system. In doing so, they were indeed able to reproduce the observed paths of the planets with impressive accuracy. Yet this refined model was still fundamentally wrong. It was eventually replaced by the more correct heliocentric model.

You should always keep this epistemological aspect in mind, as it will become centrally important when trying to make sense of quantum mechanical phenomena. A theoretical model may work perfectly on paper and make precise predictions that are confirmed by observations, and yet turn out to be nothing more than a “shadow”, or even a bad caricature of reality.

Modern QM will explain things much more accurately, as we shall see later. It was only upon the advent of a more mature theoretical framework and its application to the atomic structure that science could correctly predict atomic spectra. The atomic model that arises from it will be completely different from that of Bohr, which could be considered only a temporary sketch, as Bohr himself had to admit.

Coming next: The Franck-Hertz Experiment, Compton Scattering and Pair Production