Tag: Erwin Schrodinger

• From the Heisenberg cut to the Copenhagen interpretation

  The following post was motivated by this exchange (on X.com), which prompted me to write out my understanding of the Copenhagen interpretation of quantum mechanics and the part the Heisenberg cut plays in it. I haven’t gone into the variants of the interpretation that Maria Violaris brings up; I only focus on understanding what the interpretation does and doesn’t say to begin with, and its history.

  There are many interpretations of what quantum mechanics says about reality. This is unlike classical physics, where theory and reality converge almost perfectly. If using Newton’s laws of motion you determine that a ball flying through the air will have some speed at some point, you’ll find that to be the case when you take measurements. Quantum mechanics on the other hand has some uncertainty baked into the outcomes of certain measurements; there’s no escaping it. That means the mathematical formalism describes only the probability of the outcomes of measurement rather than the event itself, creating a fundamental gap between the theory and observations that different interpretations have tried to bridge with competing philosophical explanations.

  Perhaps the most popular among them is the Copenhagen interpretation: a small 2016 survey found it enjoys the most agreement among physicists; it also holds sway in the popular imagination thanks to Erwin Schrödinger’s thought experiment involving a cat that’s both dead and alive. However, Schrödinger came up with that idea to illustrate his belief that the Copenhagen interpretation of quantum mechanics paints an absurd picture of reality. The interpretation has been refined over time and is more complicated than that, and certainly not absurd.

  In Schrödinger’s thought experiment, the cat is a metaphor for an observable property of a quantum system. That the cat is both dead and alive — a statement that the wavefunction of the property is in a superposition of two (or more) states. When you open the box to see if the cat is dead or alive (but not both) in the metaphor, the description of the system updates from a superposition to a single outcome.

  Note that this is a simplified picture. For a more thoroughgoing account, I recommend Jim Baggott’s post ‘The Copenhagen Confusion’. Here’s a line from the operative passage: “The ‘collapse of the wavefunction’ was never part of the Copenhagen interpretation because the wavefunction isn’t interpreted realistically. The only thing that happens when an electron is detected on a screen in the context of Copenhagen is that we gain knowledge of the position of the electron.” In this post, however, I’m going to flatten these details for simplicity’s sake where necessary.

  

  A useful entry point to the interpretation is the Heisenberg cut, which is a conceptual boundary within the interpretation. It draws the line between the quantum system, i.e. the wavefunction and probabilistic laws, and the measuring apparatus or the observer, described by classical mechanics and deterministic laws. And these two parts of the overall system share a foundational relationship: the Copenhagen interpretation uses this cut to bridge the gap between the mathematical formalism of quantum mechanics and the empirical reality of what scientists observe in a lab.

  In Niels Bohr’s view, the cut is required because humans are macroscopic entities who communicate using classical language. (“It’s very hard to talk quantum using a language originally designed to tell other monkeys where the ripe fruit is”: Terry Pratchett.) Bohr argued that we don’t have a choice but to describe experiments in terms of everyday physics, including positions, momenta, and times, because these concepts also define our cognitive and linguistic capabilities. This means even though the subatomic world is quantum mechanical, the instruments we use to measure it, like photographic plates and our eyes, must be treated as classical objects. The Heisenberg cut is an imaginary boundary in our description of experiments where we stop using quantum concepts and start using classical ones.

  An important feature of the cut is its mobility, i.e. that a person can draw it anywhere in their description of the thought experiment: when a photon of light hits the cat, when a photon reflected by the cat reaches your eye, when you first open the box or somewhere else. According to the Copenhagen interpretation, the physical predictions of quantum mechanics don’t change based on where you make the cut, as long as it is placed somewhere along the chain of measurement. And the cut must exist if you’re to be able to ‘measure’ the system.

  The Heisenberg cut is also intimately tied to the measurement problem. On the quantum side of the cut, the system will evolve according to the Schrödinger equation, which is deterministic and preserves superpositions, i.e. it allows a particle to be in two states at once. On the classical side of the cut, you observe definite outcomes: the particle is either here or there.

  In effect the cut marks the point where multiple possible outcomes give way to a single recorded result. And in the Copenhagen interpretation, this transition isn’t a physical process that can be derived from the Schrödinger equation itself; instead it’s a non-dynamical event that occurs whenever a quantum system interacts with a classical measuring device. This leads to the somewhat paradoxical conclusion that quantum mechanics is a complete theory of the microscopic universe yet it banks on classical concepts (that it can’t make sense of) to make sense of its predictions.

  While both Bohr and Werner Heisenberg, for whom the cut is named, agreed that this cut should exist, they arrived at it for different reasons. Heisenberg treated the cut as a moveable mathematical boundary that separated the object from the subject, highlighting the subjective nature of observation. He was interested in how the observer’s knowledge changed the state of the system. Bohr on the other hand viewed the cut as an epistemological necessity fixed by the experimental arrangement. In other words for Bohr the cut wasn’t about a subjective observer disrupting nature but about the objective impossibility of separating the observer from the observed in the quantum realm (a.k.a. the uncertainty implicit to quantum mechanics).

  Second, let’s look at how the Copenhagen interpretation treats the maths of quantum mechanics. The theory postulates that a quantum system evolves according to the Schrödinger equation. However, our human experience is obviously discontinuous: we see definite outcomes, not superpositions. The ‘collapse’ is the instant when the system switches from its smooth quantum evolution to a single, definite state.

  Without the Heisenberg cut, on the other hand, there’s no logical place for the wavefunction to collapse. If you treated the entire universe — including a subatomic particle, a microscope, a scientist, and the scientist’s brain — as one giant quantum system, everything would just keep evolving according to the Schrödinger equation forever. Eventually you’d end up with a universe in a massive, complex superposition but you’d never arrive at a specific measurement or result. This is actually the premise of the many-worlds interpretation of quantum mechanics, which removes the collapse and thus removes the need for a cut.

  In the Copenhagen interpretation, however, because you eventually arrive at a definite result (and which you need to do for science to be science), you’re forced to draw a line: “Everything on this side is quantum and describes probabilities and everything on that side is classical and describes facts”. The wavefunction ‘collapse’ is defined as the point at which the quantum description gives way to a single, definite experimental outcome. When the quantum system crosses the Heisenberg cut and interacts with the classical side, the wavefunction is said to have collapsed.

  Thus to discuss the Heisenberg cut is essentially to discuss the mechanism of collapse and highlights the implicit dualism of the Copenhagen interpretation: the universe is divided into the observer and the observed. The wavefunction describes what’s being observed and the collapse ensures the observed entity matches the observer’s reality.

  The concept of the cut originated in a few intense months leading up to Heisenberg’s publication of a paper in March 1927. At the time, Heisenberg had been working at Bohr’s institute in Copenhagen on rescuing the concept of particle trajectories, e.g. the tracks of particles recorded in a cloud chamber, which seemed to contradict the (then) new quantum mechanics.

  In 1925, Heisenberg formulated matrix mechanics, the first logically consistent mathematical framework for quantum mechanics. (This invention was an important first step of the ‘new’ quantum mechanics, whose centenary physicists celebrated worldwide last year.) Among other things, matrix mechanics predicted that certain physical quantities, such as energy, take on discrete values. However, this raised questions about reconciling the theory with physicists observing apparently smooth, continuous particle tracks in cloud chambers.

  

  Heisenberg resolved this contradiction by redefining what a ‘path’ actually is in a cloud chamber. This is a device filled with alcohol vapour that’s supersaturated, meaning it’s cooled to the point where it’s just about ready to turn into liquid. When a charged particle moves through this gas, it knocks electrons out of the alcohol molecules, creating a trail of ions. The vapour rapidly turns into liquid droplets around these ions, forming a visible white track that traces the exact path of the subatomic particle through the chamber.

  But Heisenberg argued that we never actually see a continuous path in a cloud chamber — only the sequence of individual droplets formed by ionisation. Solving the problem of the particle’s trajectory in matrix mechanics would never spit out a continuous path but it could determine the probability of an electron’s state transitioning from one discrete droplet to the next.

  When we say an object transitions from point A to point B in everyday life, we mean it moved through the space in between them. But in matrix mechanics, an electron state transitioning between droplets means a discontinuous update of reality rather than movement. In the context of this post, the state of the electron is a mathematical list of properties the electron possesses at the exact moment it hits a gas molecule and creates a droplet.

  So say when it hits droplet 1, the electron has energy E_high, momentum P₁, and is roughly at position X₁. At droplet 2, scientists find the same electron has energy E_low (because it lost some energy when it smashed into the first atom), momentum P₂, and is roughly at position X₂. In Heisenberg’s telling, the laws of physics don’t describe this journey so much as the probability of state 2 happening given state 1 just happened.

  This description resolved Heisenberg’s problem because his maths only handled the energy levels and transitions; it had no variable for the particle’s location at each instant in time. In other words by looking at the cloud chamber and saying, “Aha! This track is just a pile of separate water droplets”, he could claim that the physical world also works like his maths. Which means the path we see in the cloud chamber is just our human brains drawing a line between the dots. The electron itself only becomes classically describable when it hits something.

  In other words, in classical physics, the particle has a path regardless of whether we look at it, and the droplets merely reveal it. In Heisenberg’s view, the particle has no defined position or path in the empty space between the droplets. Instead a path as such comes into view only because the cloud chamber is performing a rapid series of measurements: each droplet represents an observation that forces the electron to take a stand on its position while the eventual smooth line is a mental construct we create by connecting these dots.

  Continuing from this idea, in a famous letter to Wolfgang Pauli and subsequently in his March 1927 paper, The Actual Content of Quantum Theoretical Kinematics and Mechanics, Heisenberg introduced a thought experiment involving a gamma-ray microscope. He argued that to observe an electron, one must hit it with a photon. This interaction would disturb the electron. He initially framed the measurement problem as a physical interaction between the electron (the system) and the photon (the probe), where the act of measurement mechanically disturbed the system.

  Bohr’s critique of Heisenberg’s draft then reforged the cut as a central tenet of the Copenhagen interpretation. When Heisenberg showed Bohr his paper, Bohr tore into it arguing that Heisenberg was wrong to focus on the disturbance because he assumed the electron had a definite position and momentum before the measurement and which the measurement then messed up. Bohr insisted on the more radical view that the properties of the electron aren’t well-defined until the experimental arrangement itself is fixed. For Bohr, the cut wasn’t just where a disturbance happened but the line where the observer switched from using quantum concepts to classical concepts to describe the experiment.

  The conversations on this point between the two men in February and March 1927 were intense, protracted, and emotionally exhausting. Heisenberg was 25 years old at the time and convinced he had solved the riddle of quantum mechanics with his paper whereas Bohr was relentless in his criticism, insisting Heisenberg’s fundamental premise was logically flawed.

  According to historical accounts, including Heisenberg’s own recollections later in life, the discussions would go on for hours, often late into the night. At one point, the combination of mental exhaustion and Bohr’s stubborn refusal to accept Heisenberg’s interpretation caused Heisenberg to break down in tears of frustration. But Heisenberg eventually capitulated, though not entirely: he didn’t rewrite the entire body of his paper but he did add a postscript to the end of the published version where he acknowledged that his explanation of the gamma-ray microscope had been too simplistic and that Bohr’s view regarding the electron’s indefiniteness was the deeper truth.

  The tears were the physical manifestation of the painful process of aligning the two different viewpoints into what became the Copenhagen interpretation. In fact, and at the risk of repetition, let’s treat this interpretation as the peace treaty that reconciled Heisenberg’s idea of uncertainty with Bohr’s idea of complementarity. Heisenberg’s view was initially very mechanical and focused on the observer’s limitations; he held that the fuzziness of the quantum world was a result of our clumsiness: i.e. the reality existed but our clumsy hands destroyed the data every time we tried to touch it. To him the Heisenberg cut was the place where this mechanical disturbance happened.

  Bohr however worked with the concept of complementarity: that the electron has a dual nature, wave and particle, and that these two natures are mutually exclusive, meaning we can’t see both at the same time. And the uncertainty isn’t because we hit the particle but because the electron literally doesn’t have a defined position and momentum at the same time. If you build an experiment to measure its position, the wave nature would vanish, and vice versa. He was saying in effect that the experiment itself defined what reality was allowed to exist at all in that moment.

  The Copenhagen interpretation loosely synthesised these two views, though it leaned heavily toward Bohr’s. It stated that we must accept two contradictory truths: the mathematical formalism (Heisenberg’s matrix mechanics and the Schrödinger equation) that predicts probabilities and the classical world of our measuring devices. The interpretation is the agreement that we can’t speak about what the electron is doing when we aren’t looking. We can only speak about the results of the interaction between the electron and the machine.

  In effect, the Copenhagen interpretation asserts that physics isn’t about the ontological nature of the electron, i.e. what it is, but about the epistemological nature of our knowledge, or what we can say. And the Heisenberg cut is the necessary border where the indefinite, contradictory quantum world based on Bohr’s idea of complementarity is forced to collapse into a single, definite fact.

  If Bohr and Heisenberg provided the philosophical foundation for the Copenhagen interpretation, the Hungarian-American physicist John von Neumann gave it its formal mathematical form in his 1932 book Mathematical Foundations of Quantum Mechanics. Von Neumann was also the one to show that the mathematics of quantum mechanics allowed the cut to be placed anywhere in this chain without changing the final calculated probabilities.

  Where’s Schrödinger’s cat in all of this, then? As it happens, the famous thought experiment in which the cat is both dead and alive is often misunderstood as a quirk of quantum physics; it was actually a scathing piece of satire Schrödinger designed to show that the Copenhagen interpretation was absurd. Schrödinger in fact didn’t believe a cat could be simultaneously dead and alive. His point was that if you followed Bohr and Heisenberg’s logic to its ultimate conclusion, you’d end up with such a nonsensical reality.

  In fact, the thought experiment, published in 1935, targeted the concept of the Heisenberg cut. In the Copenhagen view, a quantum particle like an atom doesn’t have a defined state: it exists in a superposition of all possible states until an observer measures. Schrödinger could accept this for atoms but couldn’t digest the prospect of applying the idea to macroscopic objects.

  In his mental argument, Schrödinger described a radioactive atom placed in a sealed steel box. If the atom decays in a random quantum event, a Geiger counter nearby would push a hammer, which would smash a vial of cyanide and kill a cat. If the atom doesn’t decay, the cat would live. According to the strict logic of the Copenhagen interpretation, this system remains in a superposition until an observer opens the box to check the cat’s existential status. But until the measurement itself, because the atom is both decayed and not decayed, the Geiger counter is both triggered and not triggered, and the cat is simultaneously dead and alive. Schrödinger’s question was about where the quantum ends and the classical world begins. In other words, where’s the Heisenberg cut?

  

  If we make the cut at the Geiger counter, the cat would be a classical object and thus either dead or alive, not both. However, Bohr, Heisenberg, and von Neumann had shown that the cut was mobile. If we moved it to the human observer opening the box, the cat itself would become part of the system’s overall wavefunction — and Schrödinger had contended that treating a living organism as a probability wave was ridiculous. He used the cat to argue that there must be something missing in the theory, some hidden variables or physical reality, that would determine the state of the cat before an observer looks at it.

  For Schrödinger, the cat proved that the Copenhagen interpretation’s refusal to define objective reality between measurements was a philosophical failure. It showed that while the cut could work mathematically, as von Neumann had proved, it led to macroscopic impossibilities in the physical domain.

  The Copenhagen interpretation in turn didn’t surmount Schrödinger’s critique by answering the riddle but by dismissing Schrödinger’s question as unscientific. Bohr argued that Schrödinger was ‘illegally’ extending quantum concepts beyond the point where a classical description would be required. In his view a Geiger counter is a macroscopic measuring device so the cut between the quantum and classical worlds would occur the moment the particle interacts with the Geiger counter. And by the time the signal reaches the hammer, let alone the cat, the quantum description would already have yielded a definite outcome at the measuring device, so the cat would never have had to be described as being in superposition.

  There was also a powerful sociological narrative at the time that painted Schrödinger and Albert Einstein as an ‘old guard’ that was too stuck in classical determinism to accept the radical new truths quantum mechanics was throwing up. By 1935, the Copenhagen interpretation was the dominant orthodoxy among the younger, more productive generation of physicists like Pauli and (to a lesser extent) Paul Dirac, who viewed the cat and the Einstein-Podolsky-Rosen paradox not as genuine physical problems but as the confusion of men who couldn’t let go of the past. The proponents of the interpretation essentially declared that if the theory predicted the results of experiments correctly, then any philosophical discomfort about cats that were both dead and alive was the philosopher’s problem, not the physicist’s. And quantum mechanics perfectly predicted the results of experiments.

  Historical timing also played an important part in cementing the Copenhagen interpretation’s dominance. Shortly after Schrödinger published his paper, physics shifted dramatically from the philosophical debates of the 1920s to the pragmatic urgency of the 1930s and 1940s. The rise of fascism and World War II turned the focus of the community towards nuclear energy and The Bomb. In this environment, the “shut up and calculate” approach — a phrase coined later to describe this attitude — took over and physicists shelved questions about the reality of the cat as irrelevant metaphysics.

  The interpretation was also shielded by von Neumann’s mathematical authority. His 1932 book also claimed to show that ‘hidden variable’ theories, i.e. which would restore a specific reality to the cat independent of observation, were mathematically impossible. While Grete Hermann and John Bell later found this proof to be circular, for decades it served as a brick wall that convinced the physics community that there was literally no alternative to the Copenhagen interpretation.

  2026.01.31

• Chasing solitons

  Every once in a while, I dive into a topic in science for no reason other than that I find it interesting. This is how I learnt about Titan, laser-cooling, and random walks. This post is about the fourth topic in this series: solitons.

  A soliton is a stable wave that maintains its shape and characteristics as it moves around. In 1834, a civil engineer named John Scott Russell spotted a single wave moving through the Edinburgh and Glasgow Union Canal in Scotland. He described it thus in a report to the British Association for the Advancement of Science in 1844 (pp. 319-320):

  I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [30−45 cm] in height. Its height gradually diminished, and after a chase of one or two miles [2–3 km] I lost it in the windings of the channel.

  Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears; which I have since found to be an important element in almost every case of fluid resistance, and ascertained to be the type of that great moving elevation in the sea, which, with the regularity of a planet, ascends our rivers and rolls along our shores.

  Russell was able to reproduce a similar wave in a water tank and study its properties. American physicists later called this wave a ‘soliton’ because of its solitary nature as well as to recall the name of particles like protons and electrons (to which waves are related by particle-wave duality).

  Solitons are unusual in many ways. They are very stable, for one: Russell was able to follow his soliton for almost 3 km before it vanished completely. Solitons are able to collide with each other and still come away intact. There are types of solitons with still more peculiar properties.

  These entities are not easy to find: they arise due to the confluence of unusual circumstances. For example, Russell’s “wave of translation” was born when a boat moving in a canal suddenly stopped, pushing a single wave of in front that kept going. The top speed at which a wave can move on the surface of a water body is limited by the depth of the body. This is why a tsunami generated in the middle of the ocean can travel rapidly towards the shore, but as it gets closer and the water becomes shallower, it slows down. (Since it must also conserve energy, the kinetic energy it must shed goes into increasing its amplitude, so the tsunami becomes enormous when it strikes land.)

  In fluid dynamics, the ratio of the speed of a vessel to the square root of the depth of the water it is moving in is called the Froude number. If the vessel was moving at the maximum speed of a wave in the Union Canal, the Froude number would have been 1.

  If the Froude number had been 0.7, the vessel would have generated V-shaped pairs of waves about its prow, reminiscent of the common sight of a ship cutting through water.

  

  Then the vessel started to speed up and its Froude number approached 1. This would have caused the waves generated off the sides to bend away from the prow and straighten at the front. This is the genesis of a soliton. Since the Union Canal has a fixed width, waves forming at the front of the vessel will have had fewer opportunities to dissipate and thus keep moving forward.

  Since the boat stopped, it produced the single soliton that won Russell’s attention. If it had kept moving, it would have produced a series of solitons in the water, and at the same have acquired a gentle up and down oscillating motion of its own as the Froude number exceeded 1.

  Waves occur in a wide variety of contexts in the real world — and in the right conditions, scientists expect to find solitons in almost all of them. For example they have been spotted in optical fibres that carry light waves, in materials carrying a moving wave of magnetisation, and in water currents at the bottom of the ocean.

  In the wave physics used to understand these various phenomena, a soliton is said to emerge as a solution to non-linear partial differential equations.

  The behaviour of some systems can be described using partial differential equations. The plucked guitar string is a classic example. The string is fixed at both ends; when it is plucked, a wave travels along its length producing the characteristic sound. The corresponding equation goes like this: ∂²u/∂t² = c² • ∂u²/∂x², where u is the string’s displacement, x is where it was plucked, c is the maximum speed the wave can have, and t is of course the time lapsed.

  The equation itself is not important. The point is that there’s a left-hand side and a right-hand side, and one side can equal the other for different combinations of u, x, and t. Each such combination is called a solution. One particular solution is called the soliton when the corresponding wave meets three conditions: it’s localised, preserves its shape and speed, and doesn’t lose energy when interacting with other solitons.

  The “non-linear” part of “non-linear partial differential equations” means that these equations describe ways whose properties that have different properties depending on their amplitude. The guitar string equation is an example of a linear system because u, the string’s displacement, has a power of 1 (i.e. it isn’t squared or cubed) nor are the other terms of the equation multiplied with each other. Another famous example of a non-linear partial differential equation is the Schrödinger equation, which describes how the wave function of a quantum system will change over time given a set of initial conditions.

  (The Austrian-Irish physicist Erwin Schrödinger postulated it in 1925, which is one of the reasons the UN has designated our current year — a century later — the International Year of Quantum Science & Technology.)

  An example of a non-linear partial differential equation is the Korteweg-de Vries equation, which predicts how waves behave in shallow water: u_t + 6u • u_x + u_xxx = 0. The second term is the problem: u_x is a way to write ∂u/∂x and since it is multiplied by 6u the equation is non-linear, i.e. a change in its character induces changes in itself that may also change its character.

  But for better or for worse, this is the only milieu in which a soliton will emerge.

  (If you’re really interested: for example a soliton solution of the Korteweg-de Vries equation looks like this: u = A sech² [ k (x – v_t – x₀​) ], where A is the soliton’s amplitude or maximum height, k is a term related to its width, x₀ is its initial position, and v_t is its velocity over time. ‘sech’ is the hyperbolic secant function.)

  Physicists are more interested in particular types of soliton than others because they closely mimic specific phenomena in the real world. Sometimes it’s a good idea to understand these phenomena as if they were solitons because the mathematics of the latter may be easier to work with. This lucid Institute of Physics video starring theoretical physicist David Tong sets out the quirky case of quarks.

  I myself was more piqued by the Peregrine and breather solitons.

  The Peregrine soliton isn’t a soliton that travels. Its name comes from its discoverer, a British mathematician named Howell Peregrine. In fact, one of the things that distinguish a Peregrine soliton is that it’s stuck in one place. More specifically it emerges from pre-existing waves, has a much greater amplitude than the background, and appears at and disappears from a single location in a blip.

  Peregrine solitons are interesting because they have been used to explain killer waves: freakish waves in the open sea that have no discernible cause and tower over all the other waves. One famous example is the Draupner wave, which was the first killer wave to also be measured by an instrument as it happened. It occurred on January 1, 1995, near the Draupner platform, a natural-gas rig in the Norwegian part of the North Sea. This is the wave’s sounding chart:

  

  That’s one heck of a soliton.

  The breather soliton is equally remarkable. It’s a regular soliton that also has an oscillating amplitude, frequency or something else as it moves around. Imagine a breather soliton to be a soliton in water: it might look like a wave with an undulating shape, its surface heaving one moment and sagging the other like the head of a strange sea monster breathing as it glides along. This is exactly the spirit in which the breathing soliton was named. Here’s an animation of a particular variety called the sine-Gordon breather soliton:

  

  The Peregrine soliton is a particular instance of a breather soliton. Breathers have also been found in an exotic state of matter called a Bose-Einstein condensate (which physicists are studying with the expectation that it will inspire technologies of the future), in plasmas in outer space, in the operational parameters of short-pulse lasers, and in fibre optics. Some researchers also think entities analogous to breather solitons could help proteins inside the cells in our bodies transport energy.

  If you’re interested in jumping down this rabbit hole, you could also look up the Akhmediev and the Kuznetsov-Ma breathers.

  At first blush, solitons seem like monastic wanderers of a world otherwise replete with waves travelling as if loath to be separated from another. Recall that one wave in 1834 gliding ever so placidly for over half a league, followed by a curious man on a horse galloping along the canal’s bank. But for this venerable image, solitons are the children of a world far too sophisticated to admit waves crashing into each other with little more consequence than an enlivening spray of water and the formidable mathematics they demand to be understood.

  Featured image: A scan of a print of Hokusai’s ‘The Great Wave off Kanagawa’. Credit: H. O. Havemeyer Collection, 1929.

  2025.04.28

• Roundup of missed stories – February 8, 2016

  Previous editions of such roundups are here and here. Basically, the following are developments I’d have liked to cover but haven’t been able to for lack of time. You’re free to dig into them.

  1. Cross-cultural studies of toddler self-awareness have been using an unfair test – “There’s a simple and fun way to test a toddler’s self-awareness. You make a red mark (or place a red sticker) on their forehead discreetly, and then you see what happens when they look in a mirror. If they have a sense of self – that is, if they recognise themselves as a distinct entity in the world – then they will see that there is a strange red mark on their face and attempt to touch it or remove it. This is called the “mirror self-recognition test” and by age two most kids “pass” the test, at least in Western countries. Several studies have suggested that the ability to pass the test is delayed, sometimes by years, in non-Western cultures, such as rural India and Cameroon, Fiji and Peru. But now a study in Developmental Science says this may be because the mirror test is culturally biased.”

  2. Quantum Physics came from the Vedas: Schrödinger, Einstein and Tesla were all Vedantists – If you know me, you know I always suspect such explorations: “In the 1920’s quantum mechanics was created by the three great minds mentioned above: Heisenberg, Bohr and Schrödinger, who all read from and greatly respected the Vedas. They elaborated upon these ancient books of wisdom in their own language and with modern mathematical formulas in order to try to understand the ideas that are to be found throughout the Vedas, referred to in the ancient Sanskrit as “Brahman,” “Paramatma,” “Akasha” and “Atman.” As Schrödinger said, “some blood transfusion from the East to the West to save Western science from spiritual anemia.””

  3. Evaluation of the global impacts of mitigation on persistent, bioaccumulative and toxic pollutants in marine fish – “The lack of standardized monitoring approaches, coupled with the globalization of seafood imports and exports, makes estimating the likely exposure to individual consumers based on market choices challenging. However, this analysis reveals the widespread and pervasive nature of persistent, bioaccumulative and toxic chemicals in seafood and the need to tackle these challenges. In terms of human health, standards are developed in a singular fashion, evaluating risks for only one pollutant at a time. In reality, fish often contain multiple classes of PBTs simultaneously. Understanding additive effects of multiple exposures to PBTs is the next step in determining the “real” exposure risk to consumers, in all kinds of food.”

  4a. Universal decoherence due to gravitational time dilation – “Here we consider low-energy quantum mechanics in the presence of gravitational time dilation and show that the latter leads to the decoherence of quantum superpositions. Time dilation induces a universal coupling between the internal degrees of freedom and the centre of mass of a composite particle. The resulting correlations lead to decoherence in the particle position, even without any external environment.”

  4b. Questioning universal decoherence due to gravitational time dilation – “A striking example in this regard is provided by the work of Pikovski et al., in which it is claimed that gravitational effects generically produce a novel form of decoherence for systems with internal degrees of freedom, which would account for the emergence of classicality. The effect is supposed to arise from the different gravitational redshifts suffered by such systems when placed in superpositions of positions along the direction of the gravitational field. There are, however, serious issues with the arguments of the paper.”

  5. Fractality à la carte: a general particle aggregation model – “In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. … Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.”

  6. In retrospect: Dawkins’s ideas on evolution – “Books about science tend to fall into two categories: those that explain it to lay people in the hope of cultivating a wide readership, and those that try to persuade fellow scientists to support a new theory, usually with equations. Books that achieve both — changing science and reaching the public — are rare. Charles Darwin’s On the Origin of Species (1859) was one. The Selfish Gene by Richard Dawkins is another. From the moment of its publication 40 years ago, it has been a sparkling best-seller and a scientific game-changer.”

  7. New insights into the properties of an atomic nucleus using ⁴⁸Ca – “Writing in Nature Physics, Gaute Hagen and colleagues push the limits of ab initio calculations to reach a benchmark medium-heavy nucleus, ⁴⁸Ca. This is an important advance because it takes ab initio calculations into the mass region where meaningful comparison with other theories, such as nuclear density-functional theory, are thought to be appropriate. Furthermore, ab initio calculations of a neutron-rich nucleus such as ⁴⁸Ca, having 20 protons and 28 neutrons, gives access to nuclear properties that are, at present, poorly established.” (Also, do we know everything about anything at all? Seems not.)

  8. An audit of scientific research? – “When it comes to enforcing compliance, there is an established method that any taxpayer or corporate accountant has a healthy fear of: the audit. We propose a systematic and independent audit of research manuscripts before they are reviewed by a journal’s panel of referees and editors. Here we outline an approach that draws on the arms of the Internal Revenue Service (IRS) and corporate auditing methods, adapting the concept for the unique needs of scientific research.”

  9. Beall took a dig at The Scholarly Kitchen. The Kitchen’s Joe Esposito interviewed him to understand why. – “Esposito: I want to be sure I understand you on this point. To an earlier question you replied that although you focus on identifying OA publishers of little or no merit, you believed that there are useful OA venues. But your response just now seems to suggest that all Gold OA is a bad thing. Can you clarify your position?

  Beall: I stand by both statements. I know some would love to catch me in a contradiction and declare victory, but some things are ambiguous, and at universities we specialize in dealing with ambiguities and uncertainties.

  You brought up the concept of self-contradiction, so I am reminded that in late 2013 you authored a mean and hurtful blog post in The Scholarly Kitchen entitled Parting Company with Jeffrey Beall. Why are you communicating with me now after so firmly declaring an intention to end contact with me?”

  2016.02.08

• Bohr and the breakaway from classical mechanics

  

  One hundred years ago, Niels Bohr developed the Bohr model of the atom, where electrons go around a nucleus at the centre like planets in the Solar System. The model and its implications brought a lot of clarity to the field of physics at a time when physicists didn’t know what was inside an atom, and how that influenced the things around it. For his work, Bohr was awarded the physics Nobel Prize in 1922.

  The Bohr model marked a transition from the world of Isaac Newton’s classical mechanics, where gravity was the dominant force and values like mass and velocity were accurately measurable, to that of quantum mechanics, where objects were too small to be seen even with powerful instruments and their exact position didn’t matter.

  Even though modern quantum mechanics is still under development, its origins can be traced to humanity’s first thinking of energy as being quantised and not randomly strewn about in nature, and the Bohr model was an important part of this thinking.

  The Bohr model

  According to the Dane, electrons orbiting the nucleus at different distances were at different energies, and an electron inside an atom – any atom – could only have specific energies. Thus, electrons could ascend or descend through these orbits by gaining or losing a certain quantum of energy, respectively. By allowing for such transitions, the model acknowledged a more discrete energy conservation policy in physics, and used it to explain many aspects of chemistry and chemical reactions.

  Unfortunately, this model couldn’t evolve continuously to become its modern equivalent because it could properly explain only the hydrogen atom, and it couldn’t account for the Zeeman effect.

  What is the Zeeman effect? When an electron jumps from a higher to a lower energy-level, it loses some energy. This can be charted using a “map” of energies like the electromagnetic spectrum, showing if the energy has been lost as infrared, UV, visible, radio, etc., radiation. In 1896, Dutch physicist Pieter Zeeman found that this map could be distorted when the energy was emitted in the presence of a magnetic field, leading to the effect named after him.

  It was only in 1925 that the cause of this behaviour was found (by Wolfgang Pauli, George Uhlenbeck and Samuel Goudsmit), attributed to a property of electrons called spin.

  The Bohr model couldn’t explain spin or its effects. It wasn’t discarded for this shortcoming, however, because it had succeeded in explaining a lot more, such as the emission of light in lasers, an application developed on the basis of Bohr’s theories and still in use today.

  The model was also important for being a tangible breakaway from the principles of classical mechanics, which were useless at explaining quantum mechanical effects in atoms. Physicists recognised this and insisted on building on what they had.

  A way ahead

  To this end, a German named Arnold Sommerfeld provided a generalisation of Bohr’s model – a correction – to let it explain the Zeeman effect in ionized helium (which is a hydrogen atom with one proton and one neutron more).

  In 1924, Louis de Broglie introduced particle-wave duality into quantum mechanics, invoking that matter at its simplest could be both particulate and wave-like. As such, he was able to verify Bohr’s model mathematically from a waves’ perspective. Before him, in 1905, Albert Einstein had postulated the existence of light-particles called photons but couldn’t explain how they could be related to heat waves emanating from a gas, a problem he solved using de Broglie’s logic.

  All these developments reinforced the apparent validity of Bohr’s model. Simultaneously, new discoveries were emerging that continuously challenged its authority (and classical mechanics’, too): molecular rotation, ground-state energy, Heisenberg’s uncertainty principle, Bose-Einstein statistics, etc. One option was to fall back to classical mechanics and rework quantum theory thereon. Another was to keep moving ahead in search of a solution.

  However, this decision didn’t have to be taken because the field of physics itself had started to move ahead in different ways, ways which would become ultimately unified.

  Leaps of faith

  Between 1900 and 1925, there were a handful of people responsible for opening this floodgate to tide over the centuries old Newtonian laws. Perhaps the last among them was Niels Bohr; the first was Max Planck, who originated quantum theory when he was working on making light bulbs glow brighter. He found that the smallest bits of energy to be found in nature weren’t random, but actually came in specific amounts that he called quanta.

  It is notable that when either of these men began working on their respective contributions to quantum mechanics, they took a leap of faith that couldn’t be spanned by purely scientific reasoning, as is the dominant process today, but by faith in philosophical reasoning and, simply, hope.

  For example, Planck wasn’t fond of a class of mechanics he used to establish quantum mechanics. When asked about it, he said it was an “act of despair”, that he was “ready to sacrifice any of [his] previous convictions about physics”. Bohr, on the other hand, had relied on the intuitive philosophy of correspondence to conceive of his model. In fact, only a few years after he had received his Nobel in 1922, Bohr had begun to deviate from his most eminent finding because it disagreed with what he thought were more important, and to be preserved, foundational ideas.

  It was also through this philosophy of correspondence that the many theories were able to be unified over the course of time. According to it, a new theory should replicate the results of an older, well-established one in the domain where it worked.

  Coming a full circle

  Since humankind’s investigation into the nature of physics has proceeded from the large to the small, new attempts to investigate from the small to the large were likely to run into old theories. And when multiple new quantum theories were found to replicate the results of one classical theory, they could be translated between each other by corresponding through the old theory (thus the name).

  Because the Bohr model could successfully explain how and why energy was emitted by electrons jumping orbits in the hydrogen atom, it had a domain of applicability. So, it couldn’t be entirely wrong and would have to correspond in some way with another, possibly more succesful, theory.

  Earlier, in 1924, de Broglie’s formulation was suffering from its own inability to explain certain wave-like phenomena in particulate matter. Then, in 1926, Erwin Schrodinger built on it and, like Sommerfeld did with Bohr’s ideas, generalised them so that they could apply in experimental quantum mechanics. The end result was the famous Schrodinger’s equation.

  The Sommerfeld-Bohr theory corresponds with the equation, and this is where it comes “full circle”. After the equation became well known, the Bohr model was finally understood as being a semi-classical approximation of the Schrodinger equation. In other words, the model represented some of the simplest corrections to be made to classical mechanics for it to become quantum in any way.

  An ingenious span

  After this, the Bohr model was, rather became, a fully integrable part of the foundational ancestry of modern quantum mechanics. While its significance in the field today is great yet still one of many like it, by itself it had a special place in history: a bridge, between the older classical thinking and the newer quantum thinking.

  Even philosophically speaking, Niels Bohr and his path-breaking work were important because they planted the seeds of ingenuity in our minds, and led us to think outside of convention.

  2013.05.19