Tag: classical mechanics

The Kapitza pendulum

Rarely does a ‘problem’ come along that makes you think more than casually about the question of mathematics’s reality, and problems in mathematical physics are full of them. I came across one such problem for the first time yesterday, and given its simplicity, thought I should make note of it.

I spotted a paper yesterday with the title ‘The Inverted Pendulum as a Classical Analog of the EFT Paradigm’. I’ve never understood the contents of such papers without assistance from a physicist, but I like to go through them in case a familiar idea or name jumps up that warrants a more thorough follow-up or I do understand something and that helps me understand something else even better.

In this instance, the latter happened, and I discovered the Kapitza pendulum. In 1908, a British mathematician named Andrew Stephenson described the problem but wasn’t able to explain it. That happened at the hands of the Russian scientist Pyotr Kapitsa, for whom the pendulum is named, who worked it out in the 1950s.

You are familiar with the conventional pendulum:

Here, the swinging bob is completely stable when it is suspended directly below the pivot, and is unmoving. The Kapitza pendulum is a conventional pendulum whose pivot is rapidly moved up and down. This gives rise to an unusual stable state: when the bob is directly above the pivot! Here’s a demonstration:

As you can see, the stable state isn’t a perfect one: the bob still vibrates on either side of a point above the pivot, yet it doesn’t move beyond a particular distance, much less drop downward under the force of gravity. If you push the bob just a little, it swings across a greater distance for some time before returning to the narrow range. How does this behaviour arise?

I’m fascinated by the question of the character of mathematics because of its ability to make predictions about reality – to build a bridge between something that we know to be physically true (like how a conventional pendulum would swing when dropped from a certain height, etc.) and something that we don’t, at least not yet.

If this sounds wrong, please make sure you’re thinking of the very first instantiation of some system whose behaviour is defying your expectations, like the very first Kapitza pendulum. How do you know what you’re looking at isn’t due to a flaw in the system or some other confounding factor? A Kapitza pendulum is relatively simple to build, so one way out of this question is to build multiple units and check if the same behaviour exists in all of them. If you can be reasonably certain that the same flaw is unlikely to manifest in all of them, you’ll know that you’re observing an implicit, but non-intuitive, property of the system.

But in some cases, building multiple units isn’t an option – such as a particle-smasher like the Large Hadron Collider or the observation of a gravitational wave from outer space. Instead, researchers use mathematics to check the likelihood of alternate possibilities and to explain something new in terms of something we already know.

Many theoretical physicists have even articulated that while string theory lacks experimental proof, it has as many exponents as it does because of its mathematical robustness and the deep connections they have found between its precepts and other, distant branches of physics.

In the case of the Kapitza pendulum, based on Newton’s laws and the principles of simple harmonic motion, it is possible to elucidate the rules, or equations, that govern the motion of the bob under the influence of its mass, the length of the rod connecting the bob to the pivot, the angle between the line straight up from the pivot and the rod (θ), acceleration due to gravity, the length of the pivot’s up-down motion, and how fast this motion happens (i.e. its frequency).

From this, we can derive an equation that relates θ to the distance of the up-down motion, the frequency, and the length of the rod. Finally, plotting this equation on a graph, with θ on one axis and time on the other, and keeping the values of the other variables fixed, we have our answer:

When the value of θ is 0º, the bob is pointing straight up. When θ = 90º, the bob is pointing sideways and continues to fall down, to become a conventional pendulum, under the influence of gravity. But when the frequency is increased from 10 arbitrary units in this case to 200 units, the setup becomes a Kapitza pendulum as the value of θ keeps shifting but only between 6º on one side and some 3º on the other.

The thing I’m curious about here is whether mathematics is purely descriptive or if it’s real in the way a book, a chair or a planet is real. Either way, this ‘problem’ should remind us of the place and importance of mathematics in modern life – by virtue of the fact that it opens paths to understanding, and then building on, parts of reality that experiences based on our senses alone can’t access.

Featured image: A portrait of Pyotr Kapitsa (left) in conversation with the chemist Nikolai Semyonov, by Boris Kustodiev, 1921. Credit: Kapitsa Collection, public domain.

The molecule that was also a wave

According to the principles of quantum mechanics, you’re a wave – just like light is both a particle and a wave. It’s just that your wavelength is so small that your wave nature doesn’t matter, and you’re treated like a particle. The larger an object is, the smaller its wavelength, and vice versa. We’re confused about whether light is a particle or a wave because photons, the particles of light, are so small and have a measurable wavelength as a result. Scientists know that electrons, protons, neutrons, even neutrinos have the properties of a wave.

But while the math of quantum mechanics says you’re a wave, how can we know for sure if we can’t measure it? There are two ways. One, we don’t have any evidence to the contrary. Two, scientists have been checking if larger and larger particles, as far as they can go, exhibit the properties of a wave – and at every step of the way, they’ve come up with positive results. Both together, we have no reason to believe that we’re not also waves.

Such tests reaffirm the need for quantum mechanics to understand the nature of reality because the rules of classical mechanics alone don’t explain wave-particle duality.

On September 23, scientists from Austria, China, Germany and Switzerland reported that they had measured the wavelength of a group of molecules called oligoporphyrins. Specifically, they used “oligo-tetraphenylporphyrins enriched by a library of up to 60 fluoroalkylsulphanyl chains”. Altogether, they consisted “of up to 2,000 atoms”, becoming the heaviest object directly known to exhibit wave-like properties.

The molecule in question. DOI: 10.1038/s41567-019-0663-9

According to the scientists’ peer-reviewed paper, the molecules had a wavelength of around 53 femtometers, about 100,000-times smaller than the molecules themselves.

* * *

We have known since at least the 11th century, through the work of the Arab scholar Ibn al-Haytham, that light is a wave. In 1670, Isaac Newton propounded that light is made up of small particles, and spent three decades supplying evidence for his argument. His push birthed a conflict: was light wave-like or made up of particles?

The British polymath Thomas Young built on the 17th century Dutch physicist Christiaan Huygens to devise an experiment in 1801 that definitively proved light was a wave. It is known widely today as the Young’s double-slit experiment. It is so simple even as its outcomes are so immutable that it has become a mainstay of modern tests of quantum mechanics. Physicists use upgraded versions of the experiment to this day to study the nature and properties matter-waves.

(If you would like to know more, I highly recommend Anil Ananthaswamy’s biography of this experiment, Through Two Doors At Once; here’s an excerpt.)

In the experiment, light from a common source – such as a candle – is allowed to pass through two fine slits separated by a short distance. A sheet of paper sufficiently behind the slits then shows a strange pattern of alternating light and dark bands instead of just two patches of light. This is because light waves passing through the two slits interfere with each other, producing the famous interference pattern. Since only waves can interfere, the experiment shows that light has to be a wave.

An illustration of the double-slit experiment from ‘Though Two Doors At Once’ (2019).

The particulate nature of light would get its proper due only in 1900, when Max Planck stumbled upon a mathematical inconsistency that forced him to conclude light had to be made up of smaller packets of energy. It was the birth of quantum mechanics.

* * *

The international group’s test went roughly as follows: the scientists pulsed a laser onto a glass plate coated with the oligoporphyrins to release a stream of the molecules; collected them into a beam using collimators; randomly chopped the beam into smaller bits; passed each bit through diffraction gratings to split it up; then had the two little beams interfere with each other. Finally, they counted the number of molecules striking the detector while the detector registered the interference pattern.

They had insulated the whole device, about 2m long, from extremely small disturbances, like vibrations, to prevent the results from being corrupted. In their paper, the scientists even write that the final interference pattern was blurred thanks to Earth’s rotation, and which they were able to “compensate for” using effects due to Earth’s gravity.

A schematic diagram of the experimental setup. The oligoporphyrins move from left to right as the experiment progresses. The results of the counter are visible in a diagram above the right-most component. DOI: 10.1038/s41567-019-0663-9

To ascertain that the pattern they were seeing on the detector was in fact due to interference, the scientists performed a variety of checks each of which established a relationship between the shapes on the detector with the properties of the components of the interferometer according to the rules of quantum mechanics. They were also able to rule out alternative, i.e. classical, explanations this way.

For example, the scientists fired a laser through the cloud of molecules post-interference. Each molecule split the laser light into two separate beams, which recombined to produce an interference pattern of their own. This way, scientists could elicit the molecules’ interference pattern by studying the laser’s interference pattern. As they varied the laser power, they found that the visibility distribution of the molecules more closely matched with quantum mechanical models than with classical models, confirming interference.

The solid blue line indicates the quantum mechanical model and the dashed red line is a classical model, both scaled vertically by a factor of 0.93. The shaded areas on the curves represent uncertainty in the model parameters, and the dotted lines indicate unscaled theory curves. DOI: 10.1038/s41567-019-0663-9

What these scientists have achieved isn’t only a feat of measurement. Their findings also help refine the border between the classical and the quantum. The force of gravity governs the laws of classical mechanics, which deals with macroscopic objects, while the electromagnetic, strong nuclear and weak nuclear forces rule the microscopic world. Although macroscopic and microscopic objects occupy the same universe, physicists haven’t yet understood how classical and quantum mechanics can be combined into a single theory.

One of the problems standing in the way of this union is knowing where – and how – the macroscopic world ends and the microscopic world begins. So by observing quantum mechanical effects at the scale of thousands of atoms, scientists have quite literally pushed the boundaries of what we know about how the universe works.

The calculus of creative discipline

Every moment of a science fiction story must represent the triumph of writing over world-building. World-building is dull. World-building literalises the urge to invent. World-building gives an unnecessary permission for acts of writing (indeed, for acts of reading). World-building numbs the reader’s ability to fulfil their part of the bargain, because it believes that it has to do everything around here if anything is going to get done. Above all, world-building is not technically necessary. It is the great clomping foot of nerdism.

Once I’m awake and have had my mug of tea, and once I’m done checking Twitter, I can quote these words of M. John Harrison from memory: not because they’re true – I don’t believe they are – but because they rankle. I haven’t read any writing of Harrison’s, I can’t remember the names of any of his books. Sometimes I don’t remember his name even, only that there was this man who uttered these words. Perhaps it is to Harrison’s credit that he’s clearly touched a nerve but I’m reluctant to concede anymore than this.

His (partial) quote reflects a narrow view of a wider world, and it bothers me because I remain unable to extend the conviction that he’s seeing only a part of the picture to the conclusion that he lacks imagination; as a writer of not inconsiderable repute, at least according to Wikipedia, I doubt he has any trouble imagining things.

I’ve written about the virtues of world-building before (notably here), and I intend to make another attempt in this post; I should mention what both attempts, both defences, have in common is that they’re not prescriptive. They’re not recommendations to others, they’re non-generalisable. They’re my personal reasons to champion the act, even art, of world-building; my specific loci of resistance to Harrison’s contention. But at the same time, I don’t view them – and neither should you – as inviolable or as immune to criticism, although I suspect this display of a willingness to reason may not go far in terms of eliminating subjective positions from this exercise, so make of it what you will.

There’s an idea in mathematical analysis called smoothness. Let’s say you’ve got a curve drawn on a graph, between the x- and y-axes, shaped like the letter ‘S’. Let’s say you’ve got another curve drawn on a second graph, shaped like the letter ‘Z’. According to one definition, the S-curve is smoother than the Z-curve because it has fewer sharp edges. A diligent high-schooler might take recourse through differential calculus to explain the idea. Say the Z-curve on the graph is the result of a function Z(x) = y. If you differentiate Z(x) where ‘x’ is the point on the x-axis where the Z-curve makes a sharp turn, the derivative Z'(x) has a value of zero. Such points are called critical points. The S-curve doesn’t have any critical points (except at the ends, but let’s ignore them); L-, and T-curves have one critical point each; P- and D-curves have two critical points each; and an E-curve has three critical points.

With the help of a loose analogy, you could say a well-written story is smooth à la an S-curve (excluding the terminal points): it it has an unambiguous beginning and an ending, and it flows smoothly in between the two. While I admire Steven Erikson’s Malazan Book of the Fallen series for many reasons, its first instalment is like a T-curve, where three broad plot-lines abruptly end at a point in the climax that the reader has been given no reason to expect. The curves of the first three books of J.K. Rowling’s Harry Potter series resemble the tangent function (from trigonometry: tan(x) = sin(x)/cosine(x)): they’re individually somewhat self-consistent but the reader is resigned to the hope that their beginnings and endings must be connected at infinity.

You could even say Donald Trump’s presidency hasn’t been smooth at all because there have been so many critical points.

Where world-building “literalises the urge to invent” to Harrison, it spatialises the narrative to me, and automatically spotlights the importance of the narrative smoothness it harbours. World-building can be just as susceptible to non-sequiturs and deus ex machinae as writing itself, all the way to the hubris Harrison noticed, of assuming it gives the reader anything to do, even enjoy themselves. Where he sees the “clomping foot of nerdism”, I see critical points in a curve some clumsy world-builder invented as they went along. World-building can be “dull” – or it can choose to reveal the hand-prints of a cave-dwelling people preserved for thousands of years, and the now-dry channels of once-heaving rivers that nurtured an ancient civilisation.

My principal objection to Harrison’s view is directed at the false dichotomy of writing and world-building, and which he seems to want to impose instead of the more fundamental and more consequential need for creative discipline. Let me borrow here from philosophy of science 101, specifically of the particular importance of contending with contradictory experimental results. You’ve probably heard of the replication crisis: when researchers tried to reproduce the results of older psychology studies, their efforts came a cropper. Many – if not most – studies didn’t replicate, and scientists are currently grappling with the consequences of overturning decades’ worth of research and research practices.

This is on the face of it an important reality check but to a philosopher with a deeper view of the history of science, the replication crisis also recalls the different ways in which the practitioners of science have responded to evidence their theories aren’t prepared to accommodate. The stories of Niels Bohr v. classical mechanics, Dan Shechtman v. Linus Pauling and the EPR paradox come first to mind. Heck, the philosophers Karl Popper, Thomas Kuhn, Imre Lakatos and Paul Feyerabend are known for their criticisms of each other’s ideas on different ways to rationalise the transition from one moment containing multiple answers to the moment where one emerges as the favourite.

In much the same way, the disciplined writer should challenge themself instead of presuming the liberty to totter over the landscape of possibilities, zig-zagging between one critical point and the next until they topple over the edge. And if they can’t, they should – like the practitioners of good science – ask for help from others, pressing the conflict between competing results into the service of scouring the rust away to expose the metal.

For example, since June this year, I’ve been participating on my friend Thomas Manuel’s initiative in his effort to compose an underwater ‘monsters’ manual’. It’s effectively a collaborative world-building exercise where we take turns to populate different parts of a large planet with sizeable oceans, seas, lakes and numerous rivers with creatures, habitats and ecosystems. We broadly follow the same laws of physics and harbour substantially overlapping views of magic, but we enjoy the things we invent because they’re forced through the grinding wheels of each other’s doubts and curiosities, and the implicit expectation of one creator to make adequate room for the creations of the other.

I see it as the intersection of two functions: at first, their curves will criss-cross at a point, and the writers must then fashion a blending curve so a particle moving along one can switch to the other without any abruptness, without any of the tired melodrama often used to mask criticality. So the Kularu people are reminded by their oral traditions to fight for their rivers, so the archaeologists see through the invading Gezmin’s benevolence and into the heart of their imperialist ambitions.

Scientists make video of molecule rotating

A research group in Germany has captured images of what a rotating molecule looks like. This is a significant feat because it is very difficult to observe individual atoms and molecules, which are very small as well as very fragile. Scientists often have to employ ingenious techniques that can probe their small scale but without destroying them in the act of doing so.

The researchers studied carbonyl sulphide (OCS) molecules, which has a cylindrical shape. To perform their feat, they went through three steps. First, the researchers precisely calibrated two laser pulses and fired them repeatedly – ~26.3 billion times per second – at the molecules to set them spinning.

Next, they shot a third laser at the molecules. The purpose of this laser was to excite the valence electrons forming the chemical bonds between the O, C and S atoms. These electrons absorb energy from the laser’s photons, become excited and quit the bonds. This leaves the positively charged atoms close to each other. Since like charges repel, the atoms vigorously push themselves apart and break the molecule up. This process is called a Coulomb explosion.

At the moment of disintegration, an instrument called a velocity map imaging (VMI) spectrometer records the orientation and direction of motion of the oxygen atom’s positive charge in space. Scientists can work backwards from this reading to determine how the molecule might have been oriented just before it broke up.

In the third step, the researchers restart the experiment with another set of OCS molecules.

By going through these steps repeatedly, they were able to capture 651 photos of the OCS molecule in different stages of its rotation.

These images cannot be interpreted in a straightforward way – the way we interpret images of, say, a rotating ball.

This is because a ball, even though it is composed of millions of molecules, has enough mass for the force of gravity to dominate proceedings. So scientists can understand why a ball rotates the way it does using just the laws of classical mechanics.

But at the level of individual atoms and molecules, gravity becomes negligibly weak whereas the other three fundamental forces – including the electromagnetic force – become more prominent. To understand the interactions between these forces and the particles, scientists use the rules of quantum mechanics.

This is why the images of the rotating molecules look like this:

Steps of the molecule’s rotation. Credit: DESY, Evangelos Karamatskos

These are images of the OCS molecule as deduced by the VMI spectrometer. Based on them, the researchers were also able to determine how long the molecule took to make one full rotation.

As a spinning ball drifts around on the floor, we can tell exactly where it is and how fast it is spinning. However, when studying particles, quantum mechanics prohibits observers from knowing these two things with the same precision at the same time. You probably know this better as Heisenberg’s uncertainty principle.

So if you have a fix on where the molecule is, that measurement prohibits you from knowing exactly how fast it is spinning. Confronted with this dilemma, scientists used the data obtained by the VMI spectrometer together with the rules of quantum mechanics to calculate the probability that the molecule’s O, C and S atoms were arranged a certain way at a given point of time.

The images above visualise these probabilities as a colour-coded map. With the position of the central atom (presumably C) fixed, the probability of finding the other two atoms at a certain position is represented on a blue-red scale. The redder a pixel is, the higher the probability of finding an atom there.

Rotational clock depicting the molecular movie of the observed quantum dynamics of OCS. Credit: doi.org/10.1038/s41467-019-11122-y

For example, consider the images at 12 o’clock and 6 o’clock: the OCS molecule is clearly oriented horizontally and vertically, resp. Compare this to the measurement corresponding to the image at 9 o’clock: the molecule appears to exist in two configurations at the same time. This is because, approximately speaking, there is a 50% probability that it is oriented from bottom-left to top-right and a 50% probability that it is oriented from bottom-right to top-left. The 10 o’clock figure represents the probabilities split four different ways. The ones at 4 o’clock and 8 o’clock are even more messy.

But despite the messiness, the researchers found that the image corresponding to 12 o’clock repeated itself once every 82 picoseconds. Ergo, the molecule completed one rotation every 82 picoseconds.

This is equal to 731.7 billion rpm. If your car’s engine operated this fast, the resulting centrifugal force, together with the force of gravity, would tear its mechanical joints apart and destroy the machine. The OCS molecule doesn’t come apart this way because gravity is 100 million trillion trillion times weaker than the weakest of the three subatomic forces.

The researchers’ study was published in the journal Nature Communications on July 29, 2019.

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 center 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 quantized 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 behavior 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 recognized this and insisted on building on what they had.

A way ahead

To this end, a German named Arnold Sommerfeld provided a generalization 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, even before 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 successful, 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, generalized 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 pathbreaking work were important because they planted the seeds of ingenuity in our minds, and led us to think outside of convention.

—

This article, as written by me, originally appeared in The Copernican science blog on May 19, 2013.

Bohr and the breakaway from classical mechanics

Niels Bohr, 1950.
Niels Bohr, 1950. Photo: Blogspot

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.

Thinking quantum

In quantum physics, every metric is conceived as a vector. But that’s where its relation with classical physics ends, makes teaching a pain.

Teaching classical mechanics is easy because we engage with it every day in many ways. Enough successful visualization tools exist to do that.

Just wondering why quantum mechanics has to be so hard. All I need is to find a smart way to make visualizing it easier.

Analogizing quantum physics with classical physics creates more problems than it solves. More than anything, the practice creates a need to nip cognitive inconsistencies in the bud.

If quantum mechanics is the way the world works at its most fundamental levels, why is it taught in continuation of classical physics?

Is or isn’t it easier to teach mathematics and experiments relating to quantum mechanics and then present the classical scenario as an idealized, macroscopic state?

After all, isn’t that the real physics of the times? We completely understand classical mechanics; we need more people who can “think quantum” today.

The philosophies in physics

As a big week for physics comes up–a July 4 update by CERN on the search for the Higgs boson followed by ICHEP ’12 at Melbourne–I feel really anxious as a small-time proto-journalist and particle-physics-enthusiast. If CERN announces the discovery of evidence that rules out the existence of such a thing as the Higgs particle, not much will be lost apart from years of theoretical groundwork set in place for the post-Higgs universe. Physicists obeying the Standard Model will, to think the snowclone, scramble to their boards and come up with another hypothesis that explains mass-formation in quantum-mechanical terms.

For me… I don’t know what it means. Sure, I will have to unlearn the Higgs mechanism, which does make a lot of sense, and scour through the outpouring of scientific literature that will definitely follow to keep track of new directions and, more fascinatingly, new thought. The competing supertheories–loop quantum gravity (LQG) and string theory–will have to have their innards adjusted to make up for the change in the mechanism of mass-formation. Even then, their principle bone of contention will remain unchanged: whether there exists an absolute frame of reference. All this while, the universe, however, will have continued to witness the rise and fall of stars, galaxies and matter.

It is easier to consider the non-existence of the Higgs boson than its proven existence: the post-Higgs world is dark, riddled with problems more complex and, unsurprisingly, more philosophical. The two theories that dominated the first half of the previous century, quantum mechanics and special relativity, will still have to be reconciled. While special relativity holds causality and locality close to its heart, quantum mechanics’ tendency to violate the latter made it disagreeable at the philosophical level to A. Einstein (in a humorous and ironical turn, his attempts to illustrate this “anomaly” numerically opened up the field that further made acceptable the implications of quantum mechanics).

The theories’ impudent bickering continues with mathematical terms as well. While one prohibits travel at the speed of light, the other allows for the conclusive demonstration of superluminal communication. While one keeps all objects nailed to one place in space and time, the other allows for the occupation of multiple regions of space at a time. While one operates in a universe wherein gods don’t play with dice, the other can exist at all only if there are unseen powers that gamble on a secondly basis. If you ask me, I’d prefer one with no gods; I also have a strange feeling that that’s not a physics problem.

Speaking of causality, physicists of the Standard Model believe that the four fundamental forces–nuclear, weak, gravitational, and electromagnetic–cause everything that happens in this universe. However, they are at a loss to explain why the weak force is 1032-times stronger than the gravitational force (even the finding of the Higgs boson won’t fix this–assuming the boson exists). An attempt to explain this anomaly exists in the name of supersymmetry (SUSY) or, together with the Standard Model, MSSM. If an entity in the (hypothetical) likeness of the Higgs boson cannot exist, then MSSM will also fall with it.

Taunting physicists everywhere all the way through this mesh of intense speculation, Werner Heisenberg’s tragic formulation remains indefatigable. In a universe in which the scale at which physics is born is only hypothetical, in which energy in its fundamental form is thought to be a result of probabilistic fluctuations in a quantum field, determinism plays a dominant role in determining the future as well as, in some ways, contradicting it. The quantum field, counter-intuitively, is antecedent to human intervention: Heisenberg postulated that physical quantities such as position and particle spin come in conjugate quantities, and that making a measurement of one quantity makes the other indeterminable. In other words, one cannot simultaneously know the position and momentum of a particle, or the spins of a particle around two different axes.

To me, this seems like a problem of scale: humans are macroscopic in the sense that they can manipulate objects using the laws of classical mechanics and not the laws of quantum mechanics. However, a sense of scale is rendered incontextualizable when it is known that the dynamics of quantum mechanics affect the entire universe through a principle called the collapse postulate (i.e., collapse of the state vector): if I measure an observable physical property of a system that is in a particular state, I subject the entire system to collapse into a state that is described by the observable’s eigenstate. Even further, there exist many eigenstates for collapsing into; which eigenstate is “chosen” depends on its observation (this is an awfully close analogue to the anthropic principle).

xkcd #45

That reminds me. The greatest unsolved question in my opinion is whether the universe houses the brain or if the brain houses the universe. To be honest, I started writing this post without knowing how it would end: there were multiple eigenstates it could “collapse” into. That it would collapse into this particular one was unknown to me, too, and, in hindsight, there was no way I could have known about any aspect of its destiny. Having said that, the nature of the universe–and the brain/universe protogenesis problem–with the knowledge of deterministic causality and mensural antecedence, if the universe conceived the brain, the brain must inherit the characteristics of the universe, and therefore must not allow for freewill.

Now, I’m faintly depressed. And yes, this eigenstate did exist in the possibility-space.