Last Christmas, my father and one of my brothers wondered how a spacecraft’s planetary slingshot works. It’s a well-known manoeuvre to get a big swing forward by passing by a planet. They realized that if the craft falls toward the planet due to its gravitational pull, it will gain momentum. However, as soon as it has flung around the planet, wouldn’t that same gravitation slow down the craft just as much? Great question, of course, so, let’s dive into this gravity assist short and simple.
First things first, you are absolutely right in thinking that the gravitational pull of the planet is a symmetrical situation. The same ‘force’ pulling the craft toward the planet making it accelerate, will also slow it down once it has reached its maximum point around the planet when it tries to escape the planet’s influence again, continuing its journey in space.
So, the whole planet’s gravity system is entirely symmetrical. What isn’t symmetrical though… is that the planet has an orbital momentum(beginfootnote)Momentum is mass times velocity in a specific direction(endfootnote) around the Sun! The planets in our solar system all orbit around the Sun. Though at different rates, their position relative to the Sun changes every single second. This means they have a great deal of momentum and we, the rather clever clumps of human cells that we sometimes are, can exploit this. So, by using the gravitational pull of the planet to get closer to it, the craft gains an additional tug by the simple fact that the planet itself is also moving through space. And so, while the momentum the spacecraft gained by the planet’s gravitational pull was lost again while escaping the planet’s gravitation, it did gain a bit of the planet’s orbital momentum!
It’s like catching a train while it’s hurling past the platform. Suppose you’re Batman and you possess a grappling hook. As soon as the train is coming through the station, you start running along with it on the platform. You pull out your grappling hook gun and grapple onto the train. Then you press the button so that the retractable cable attached to the hook pulls you toward the train (still speeding along its trajectory). At the right moment, you release the hook, spread out your batwings, and away you go, having parasitized off of the train’s momentum for a little bit.
And you’re right, this also means that the train lost a bit of its momentum due to you ‘pushing off of it’. However, due to the huge difference between your mass and the train’s mass, in combination with either velocities, the train’s momentum loss is negligible while your momentum increase is significant.
The same goes for the planet. A spacecraft doing a gravity slingshot, or, enjoying a gravity assist, as it’s also called, will decrease the orbital velocity of the planet, meaning that the planet’s orbit will get closer to the Sun. Of course, given the ratio between the spacecraft’s mass and the planet’s mass, this amount is negligible.
Voyager
Two of the most famous instances of gravity assists are the voyages of Voyager 1 and 2. In the animation below, you can see the trajectory of Voyager 2, where it gained assists from several of our solar system’s planets. Earth is the fast-orbiting blue dot. Jupiter is green, Saturn is cyan, Uranus is yellow, and orange is Neptune. And while the Voyager spacecrafts flew by the planets, they sent some of the best postcards back to Earth.
Of course, what goes around, may come around too. If you’d like the spacecraft to decelerate, you make it fly opposite the planet’s orbital motion. This way it slows down while donating a (negligible) bit to a planet’s momentum.
So, what Christmas message can we take from this? That’s right: even the tiniest entity in the Universe is capable of changing an entire planet’s momentum. No matter how small, in one way or another, its effects are significant.
Imagine transparent materials didn’t exist. What would cars look like? How would you be able to look at the cold Winter Moon from your bedroom window without getting cold yourself? Would air be opaque too? What about the lenses in our eyes? And if all materials consist of molecules, how is it that the molecules of glass are transparent while others aren’t? This last question was asked by the oldest(beginfootnote)If I’m not mistaken, thirteen at the time I’m writing this.(endfootnote) son of one of my best friends last week.
Firstly, I’ll try to give an answer as short as possible. If you’d like to know more, you can read on. Be warned, however: the article is quite possibly a bit long. It’s just that the answer to this seemingly easy question asks for quite some background knowledge. On the other hand, with a brilliant question like this one—let’s just say, you’re asking for it.
Short answer
Sometimes the composition of a molecule is such that its electrons will hardly respond to passing photons. For all intents and purposes, they will leave them be. At most, they will change their course a little. To our eyes, the material is then transparent.
If electrons do react to incoming photons, they might reflect them or absorb all of their energy, making the photons disappear. Sometimes they just vibrate a little bit and make the entire atoms vibrate a little bit too (phonon), but nothing much else happens. The material just increases temperature for a minuscule amount. Sometimes the electrons vibrate so much that they’ll radiate that energy very soon after, causing new photons to be created, which then move on to the rest of the universe. To us, that material is then opaque; the photons radiated from the material end up in our eyes.
So, this was the short version. If you’d like to know more, do read on!
Molecules, atoms, elementary particles
Perhaps you know this but just to be sure: all solids, liquids, and gasses consist of molecules. Here you see a photo of a bunch of so-called pentacene molecules made by Canadian scientists[1]. These molecules aren’t present in glass, however, it does give an impression of what molecules can look like.
Every caterpillar-like thingy is a molecule. When they stick close enough together, they form a solid. If they are capable of sliding past each other, it’s a liquid. And if they’re capable of jiggling a lot more away from each other, it’s a gas.
It’s possible to look more closely. Those molecules consist of atoms. Do have a look at this cool photo of one such molecule which Swiss physicists and one physicist at Utrecht University were able to snap in 2009[2].
You might be able to distinguish five hexagonal shapes with protrusions. At every corner and every protrusion, an atom is present. You don’t see the individual atoms – they’re too small for that. However, you can see the structure formed by the chain of atoms, thereby shaping the molecule into existence.
Back to glass. The window in your bedroom is composed of different types of molecules. There are a lot of silicon dioxide molecules, sodium carbonate molecules, calcium oxide molecules, magnesium oxide molecules, and some aluminium oxide molecules. Here you see a drawing of one such silicon dioxide molecule.
Do they look like that for real? No, absolutely not. It’s just a conceptual model. In science, a model is meant to be a tool and never an exact copy of reality. And yet, we use models as they are quite helpful for imagining what we’re working with and for doing calculations on them. You have to keep in mind, though, it’s not what it really looks like.
From (the model of) the silicon dioxide molecule you can see that it is comprised of three atoms: one silicon atom (grey) and two oxygen atoms(beginfootnote)‘Oxygen’ and ‘oxide’ stem from the ancient-Greek words for ‘sharp’ (ὀξύς, oxús), and ‘birth’ (γένος, génos), and the Latin word for ‘acid’, which is acidus. Lastly, the word ‘di’ stems from the ancient-Greek δίς (dís), meaning ‘twice’. As the molecule consists of two oxygen atoms, the official chemical name of the molecule is thus silicon dioxide.(endfootnote) (red). You may also wonder what these two little bars on each side are supposed to be. They symbolise the electrons which are shared by all the atoms amongst each other. Atoms can stick together when they share electrons with each other. In other words, it refers to how strong the atomic bond is. More bars equals a stronger bond.
Every atom consists of yet smaller parts. Apart from one(beginfootnote)the hydrogen atom(endfootnote), atoms are made up of three types of particles: electrons, protons, and neutrons. At the core of the atom are all the protons and neutrons. The electrons kind of swirl around them in a cloud-like type of existence. As electrons don’t themselves consist of smaller things, they are said to be elementary(beginfootnote)‘Elementary’ stems from the Latin word elementum, carrying a meaning like ‘first principle’. There is nothing that goes further down than what is elementary. Elements always form the basis for other things.(endfootnote) particles. Here you see a model of an atom.
The core (or ‘nucleus’) with all the protons and neutrons is so small that it’s usually drawn as a point or a little ball. However, if you were to zoom in, you’d see a lump of protons and neutrons. Around it there is the cloud-like electron or multiple electrons. (If you’d like to know exactly why a cloud is the model for one or more electrons, you can read This is not an atom.)
It’s quite possible that an electron is further removed from the nucleus than shown here. If an electron receives energy, it’ll jump further away from the nucleus. After a very short period, the electron might jump back to its old position. If it does so, its energy leaves the atom again in the form of light. One of the ways in which the electron receives energy is light.
Light
What is light? Is it a wave, does it consist of particles? For a long time, physicists had no idea what light was exactly. Since the seventeenth century, great debates went on between physicists supporting Sir Isaac Newton and physicists supporting Christiaan Huygens. I fear a little that some science teachers at high schools still think it’s a big mystery. One of my science teachers in high school told us he was still on the fence whether it’s particles or waves. Unfortunately for him, since slightly less than a hundred years ago, we know.
Isaac Newton (left): light = particles (tiny balls). Christiaan Huygens (right): light = waves. The correct quantum mechanical answer is: light = a disturbance in the Universe-pervading electromagnetic field. Depending on what is practical, one uses the mathematics of classical waves or the mathematics of photons (wave packets, not balls!) to work with. Physics students learn to use both approaches.
What I’m about to tell you is not something you’ll likely learn in high school. I’m not sure why but it may have to do with textbook authors finding the mathematics too complicated. So, what you’re about to read is more or less what you’ll learn at university as a physics student, only without the mathematics.
The problem with the question ‘wave or particle’ is that it suggests there’s only one choice. This is incorrect. The question should be: What is light? The answer is: a field(beginfootnote)In mathematics and physics, we call this a gauge field. It’s quite abstract mathematics. However, no matter how abstract, it has proven to be highly applicable in practice. Mobile phones would not have existed without these abstract mathematics.(endfootnote), one of the many Universe-pervading fields present, in this case the electromagnetic field. And to be even more precise: light is a disturbance of this electromagnetic field. One can describe this disturbance as either a wave or a particle, depending on what is more practical for the matter at hand.
Besides, in modern physics the meaning of the word ‘particle’ differs from what you’d normally expect. In physics a particle is actually a packet, a wave packet. It’s not a pellet, it’s not a tiny ball or even a point. It’s a tiny packet of information which we mathematically describe as a tiny wave (a disturbance).
Richard Feynman was an important, Nobel Prize-winning physicist who made enormous contributions to quantum electrodynamics.
According to one of the best theories we have of our Universe to date, so-called quantum electrodynamics(beginfootnote)‘Quantum’ is Latin for ‘how much’. Physicists have been using the word as a synonym for ‘particle’. Plural is quanta. ‘Dynamics’ stems from the ancient-Greek δυναμικός, dunamikós, ‘powerful’ en refers to the theory describing forces and change of forces.(endfootnote) (QED), the Universe is pervaded by a mostly invisible – yet sometimes visible! – electromagnetic field. In most cases, that field does nothing at all. You can’t smell it, you can’t touch it, you can’t see it.
However, when the electromagnetic field is being disturbed at a specific place in the Universe – e.g. on the inside of the LED lamp in your lavatory – then that disturbance will propagate in all directions, from that specific spot in the Universe towards the very rest of the Universe – i.e. the space of your lavatory. This disturbance you can see! I’m not sure how your pets might call this disturbance, however, humans call it light.
Albert Einstein
If your eyes were able to zoom in immensely, you would see that light actually consists of billions and billions and billions of tiny disturbances. Light is a bundle of tiny disturbances in the omnipresent electromagnetic field. Those tiny disturbances used to be called ‘light quanta’ by Albert Einstein and others. However, since 1928, we call them photons(beginfootnote)This stems from the ancient-Greek φῶς, phôs, which ironically means ‘light’.(endfootnote). In popular books and magazines and even by physicists they are called ‘particles’. Again, they’re not pellets or tiny balls or anything. The word ‘particle’ refers to them being very tiny but it doesn’t say anything about what they look like. As model, tiny pellets or points are sometimes used, however, it’s not what they are. Photons are, just like electrons, elementary, however.
In high school and at university, to do calculations on light, the wave model of light is used rather often. The great mathematician and physicist James Clerk Maxwell was one of the founders of the mathematical framework of the wave model of light. He and others before him are responsible for us still talking about ‘light waves’ instead of photons. The classical electromagnetic theory of Maxwell works so well that it’s compulsory for physics students to study this wonderful theory. So, it’s not at all wrong to speak of light waves.
James Clerk Maxwell
In the twentieth century, however, physicists found that quantum electrodynamics was able to predict and describe more phenomena than Maxwell’s classical electromagnetic theory, so the first kind of replaced the latter. Put differently, Maxwell’s theory is still highly useful in industrial applications, however, with QED, you can do what Maxwell’s theory can do plus a lot more.
This is the way it usually goes in physics. The law of universal gravitation of Sir Isaac Newton works brilliantly. You can even apply it to Mars landings. However, the theory of gravity by Albert Einstein, so-called general relativity, can do what Newton’s theory does and a lot more, more precisely. So, general relativity has kind of replaced Newton’s law of universal gravitation. And yet, the latter is compulsory in high school and at university. It’s not wrong. It’s very useful, even! However, it does have its limitations. That’s why we first learn about Newton’s gravity and only later do physics students have to learn about Einstein’s gravity. Without Einstein’s general relativity, Google Maps and GPS-systems inside cars would not have worked properly.
Hence, physics students learn everything about Maxwell’s wave theory and only later do they learn about quantum electrodynamics. And without quantum electrodynamics you would not have had computer processors, there would have been no internet, no mobile phones, no touchscreens.
Light and energy
The great physicist Max Planck came up with the idea that every photon has a specific energy level. He also showed that with every energy level comes a particular light colour. Bright blue light carries more energy than deep-dark red light. Sometimes light (photons) has (have) so much energy that it has (they have) become invisible to our human eyes. High-energy ultraviolet(beginfootnote)‘Ultra’ is Latin for ‘beyond’. So, ultraviolet means beyond violet.(endfootnote) light (UV light) is invisible to us. However, if your eyes were much more sensitive than they are now, you would see a very bright ‘more violet than violet-coloured’ light. Conversely, light can have very little energy. So little even, we won’t be able to see it anymore. Hence, infrared(beginfootnote)‘Infra’ is Latin for ‘below’. So, infrared is ‘below’ or ‘less than’ red.(endfootnote) light is invisible to us. However, if our eyes were slightly more sensitive, we would see ‘less red than red-coloured’ light.
This cheery looking fellow was a physicist and a genius. His name was Max Planck. A photograph from 1933.
WiFi and 4/5G are light too. The photons have very little energy compared to the photons in your lavatory. If our eyes had been thousands of times more sensitive than they are now, you would have seen that the antennas of the WiFi router and the mobile phones are basically lamps radiating ‘less than less than less than (thousands of times ‘less than’) red-coloured’ light.
The electromagnetic field pervading our Universe can thus be disturbed at various energy levels. Depending on that, light looks differently. It has varying colours or is invisible – which it is most of the time. Our eyes aren’t the best instruments to look around with. Of all possible energy levels the electromagnetic field can be at, we can only discern just a few. That energy portion is what we call visible light.
Another word for disturbances of the electromagnetic field is electromagnetic radiation. Depending on the energy level of the radiation, we have different terms for it, such as ‘radioactive radiation’ or ‘gamma radiation’. However, all these things – the lavatory light, the WiFi, 4/5G for the mobile phone, the head lights of the car, the radio waves from the neighbour, the Bluetooth speaker in the kitchen, the x-ray images at the dentist, the microwave – are all light, are all electromagnetic radiation, are all disturbances of the one and the same electromagnetic field. The only difference is the energy level of that disturbance.
The correct order going from very little to deadly amounts of energy is the following: radio, WiFi, microwave(beginfootnote)If you want to know whether microwave radiation is deadly or not, do give my article Is microwave oven radiation unhealthy? a read.(endfootnote), 4/5G >> infrared light (TV remote) >> visible light (lavatory light, club lights) >> UV light (take care, apply sunscreen) >> x-rays (only operated by professional medical workers) >> gamma radiation (deadly, except for Bruce Banner) >> cosmic radiation (deadly, except for Captain Marvel).
As most electrons inside of walls of houses won’t respond much to electromagnetic disturbances (photons) at the energy level of WiFi (very little energy), to WiFi photons, the walls are almost transparent. This is why you can receive WiFi straight through the walls. If your eyes were sensitive enough, you would be able to see the light coming from the router, straight through the walls. Those same electrons, however, do react to photons at the much higher energy level corresponding to visible light. This is why those photons do not fly through the wall. And this is why we find walls to be quite the opaque type objects. Nevertheless, the electrons do not respond again to photons at the even higher – much higher – energy levels of x-rays. This is exactly why walls are perfectly transparent to Superman.
Depending on the composition of the molecules and atoms do electrons more or less react to the presence of photons at varying energy levels. If electrons of the material do not respond to photons at the energy level corresponding to visible light, then the material is transparent to us.
Below you see a diagram of the full spectrum(beginfootnote)‘Spectrum’ is Latin for ‘appearance’. So, if you speak of the spectrum of something, such as electromagnetism, then you’re referring to all of its appearances.(endfootnote) of electromagnetic radiation (click to enlarge). As you can see, only a small portion is visible to us.
A diagram (not to scale) of electromagnetic radiation, or photons, if you will. The mentioned values are the frequencies of the photons, expressed in gigahertz (GHz). The higher the frequency, the higher the energy of the photon.
GHz refers to the frequency of the photon and is a measure for the photon’s energy level. The higher the frequency, the higher the energy level. Ultraviolet radiation is where it’s starting to become dangerous to us. This is where our cells become damaged (‘DNA damage’). As long as you’re not exposed to the Sun for too long and x-ray photography is done in very short amounts of time, it’s going to be fine. But be careful. Again, gamma radiation and cosmic radiation are deadly. I understand, it doesn’t feel comfortable at all, but please, please do listen to your parents when you’re going out for a space walk. Put on that spacesuit.
Impressionable electrons
In the previous century, physicists such as Albert Einstein discovered that electrons can be influenced by incoming photons(beginfootnote)This is what he received the Nobel Prize for. You can read more about that in my article The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time.(endfootnote). It very much depends on the way the electrons are captured inside the molecules – which depends on the type of atoms – at which energy level photons they will start reacting.
In the case of glass, the electrons do feel electromagnetic disturbances slightly. This is why they do start to jiggle differently just a notch. That jiggling causes changes in the part of the electromagnetic field that is inside of the glass. And these changes will influence the photons (disturbances in that same electromagnetic field) in such a way that they’ll change course slightly.
This is why the image behind glass can seem to be slightly warped. The same happens when light goes from air to water (as water, too, contains electrons which react to incoming photons). The electrons don’t do too much so that photons can just pass through, however, they do enough so that the photons do change course slightly. Or a lot as you can see by the water in the photo above. In my article Why, exactly, do glass and liquids refract light? we take a deep dive into this phenomenon.
Why is glass transparent?
And so, glass is transparent as the electrons in glass molecules aren’t capable of reacting very much to incoming electromagnetic disturbances (photons). Just a little. So, they do bend the original trajectory of the photons slightly.
There are materials containing electrons responding to all energy levels except those corresponding to blue light, for example. This means that blue light can just pass through while the rest is being absorbed. To us, this material seems to be a blue filter.
It’s also possible to produce materials carrying electrons which react to all photons in the visible part of the spectrum. They do this so strongly that photons will be reflected completely. We call that a mirror.
Note that we’re talking mostly about photons we can see. To us most glass is transparent. However, we can also produce glass which seems transparent as it lets visible light pass through, while they are much less transparent to birds at the same time.
We are incapable of seeing UV light. Birds can, however. So, if glass is produced in such a way that they will let visible light pass through but not UV light, they seem less transparent to birds, preventing them to bump into it.
So, the answer to the question, ‘And if all materials consist of molecules, how is it that the molecules of glass are transparent while others aren’t?’, should rather be: ‘Transparent to whom? To birds? Or to humans?’
References
[1] Dinca, L. E. et al. (2015) “Pentacene on Ni(111): Room-Temperature Molecular Packing and Temperature-Activated Conversion to Graphene,” Nanoscale, 7(7), pp. 3263–3269. doi: 10.1039/C4NR07057G.
[2] Gross, L. et al. (2009) “The Chemical Structure of a Molecule Resolved by Atomic Force Microscopy,” Science, 325(5944), pp. 1110–1114. doi: 10.1126/science.1176210.
As is usually the case with scientific buzz words in everyday parlance, in books, in the cinema, on TV, and on the internet – like energy – the meaning of the word dimension rarely aligns with what mathematicians and physicists understand it to be. This day and age it’s rather uncommon to not have been exposed to phrases such as ‘higher’ or ‘other dimensions’. It’s likely you’ve used them yourself once or twice in your life. In this episode, we’ll explore what mathematicians and physicists mean when they talk about dimensions, and, more interestingly, the spaces they yield.
Dimensions are not Universes
In science-fiction or even everyday lingo, the word ‘dimension’ is often synonymous with entire worlds, or realms or (pocket) Universes. For instance, aliens may have come from another dimension. Or souls or ‘essences’ dwelling on a ‘higher plane of existence’ in another ‘dimension of reality’ are spoken about.
On a regular basis, portals to other dimensions are opened from which exotic forms of matter and energy are extracted to benefit either the hero or the bad guy of the story.
It’s also a favourite way to travel great distances within our reality. Just hop through a dimensional portal and out you come, back into our reality, only thousand kilometres away from where you started. Occasionally, you may also travel in time by flying through other dimensions.
And, of course, other dimensions can be summoned into our own reality or, if the story goes that they have always been present inside our reality, they can be made visible by powerful minds. This is, again, alluding to dimensions being whole separate realms within our realm.
Figure 1. Doctor Stephen Strange (Benedict Cumberbatch) is about to step into the Mirror Dimension as summoned within (or next to) our reality by his mentor, the Ancient One (Tilda Swinton), in the 2016 film Doctor Strange of the wildly popular Marvel Cinematic Universe (MCU). License note. (Click to enlarge.)
This whole section was just to let you know that what is meant by dimensions in most science-fiction stories is not what is meant in mathematics and physics. They are not realms, realities, worlds or pocket Universes. If we were to refer to realms, realities, worlds, and Universes, we would just say realms, realities, worlds, and Universes, but not dimensions.
Ordinary spaces and dimensions
So, what do mathematicians and physicists mean when they talk about dimensions?
In many cases, they pertain to the actual directions you and I are able to travel in ordinary space. I prefer to think of birds and fish as gorgeous examples of being able to travel in all directions of space all by their own.
They can fly from your left to your right and vice versa (first direction). They can fly head-on towards you and whizz by over your head and fly further behind you and vice versa (second direction). And, obviously, they can fly up from underneath you and they can keep on flying to way above your face. And vice versa (third direction).
In many cases, all three directions are oriented perpendicularly with respect to each other. To use another word, they are orthogonal. All motion can be described as some combination of moving in these three orthogonal directions, i.e. orthogonal dimensions.
In high school we have gotten all too familiar with these three dimensions. We were tortured with finding distances between vertices of a cube along the edges, the sides, and straight through the block. Of course, this is what modern gadgets and cinematography refer to when they use the term 3D, three-dimensional. In some way or form, all three orthogonal dimensions are either taken advantage of or simulated in a virtual way.
Mathematically, the capability of travelling (or ‘transporting’) along these three orthogonal directions automatically give rise to a space, a topology, of some shape or form. Ordinary space is the space you and I are born in and have grown very much accustomed to.
So, while dimensions may give rise to spaces, they are definitely not the same. Besides, while one dimension by itself technically yields a topology, a space, it’s still a one-dimensional space, meaning, no three-dimensional bodies are able to traverse this without being torn apart.
Figure 2. In ordinary space, we have three dimensions in the $x$-direction, the $y$-direction, and the $z$-direction. In high school, we were to calculate the distance between points $O$ and $F,$ for instance.
Euclid and Descartes
A very informal definition of dimensions is the number of coordinates needed to locate an object (in a space of some kind). So, on a flat surface (a plane), such as a ceiling, you need two coordinates to locate a fly. A fly can be 2 metres away from the left wall (the first direction) and 3 metres away from the back wall (the second direction, perpendicular to the first direction). Its coordinates are therefore (2,3). Hence, a plane is two-dimensional.
In ordinary, three-dimensional space, we need three coordinates to locate a fly in a room. A fly can be 2 metres away from the left wall, 3 metres away from the back wall, and 1.5 metres up from the floor. Its coordinates are therefore (2,3,1.5).
This was one of René Descartes’s great insights while lying in bed late in the afternoon or so the story goes. Hence, these numbers are called Cartesian coordinates. Descartes was pivotal to the development of what we now call the Cartesian coordinate system.
The space to which these type of coordinates belong is called Euclidean space as the great Greek mathematician Euclid was the father of Euclidean or classical geometry.
I think I can safely say that Euclid and Descartes enabled mathematics teachers to torment us with a whole slew of homework in order for us to fully explore the realm of Euclidean space in both two- and three-dimensional Cartesian coordinate systems.
Figure 3. Home of Descartes in Utrecht, the Netherlands, where he wrote parts of his famous Discours de la Méthode. The house has been demolished. Nowadays, the place looks very different. (Click on the image for a link to the original Instagram post where you can also swipe for the photo of what is looks like today. Opens a new tab.)
Space and time
In real life, besides a position in ordinary space, you also need to specify when. Getting the coordinates to be inside an office located on the corner of two streets on the 24th floor (that’s the three dimensions of ordinary space right there) just isn’t enough. You also need a time-coordinate. When are you supposed to be there?
Einstein called the fact that you’re inside an office at a certain time an event. In other words, where, in ordinary, Cartesian coordinates, we talked about some thing being somewhere, Einstein had the insight to now only start talking about events taking place in terms of space and time, space-time – using space-time coordinates.
When Einstein introduced the special theory of relativity, the German mathematician Hermann Minkowski realised this theory could also be understood geometrically in a four-dimensional space-time, where time is taken to be the fourth dimension. We now call this space Minkowski space. Note that we’re using the word ‘space’ in a broader sense: it doesn’t just encompass ordinary spatial dimensions but it now also includes a dimension of time (and, for technical reasons, isn’t Euclidean).
By the way, another word mathematicians and physicists like to use is manifold. A manifold is a topological object which can take many shapes – such as a two-dimensional plane, a three-dimensional Euclidean space, four-dimensional Minkowski space or any other space you can mathematically think of.
In Einstein’s general theory of relativity (gravity), we still work with four-dimensional space-time, except the shape of the space isn’t Minkowskian any more. The shape of the space is warped, curved, and stretched. In the best theory of gravity we have to date, we work on a so-called pseudo-Riemannian manifold, named after the great German mathematician Bernhard Riemann. The dimensions are still all the directions you can take on this manifold, i.e. the minimum amount of coordinates you need to locate an event. However, in this case, they are not necessarily oriented perpendicularly with respect to one another.
Figure 4. In the film Interstellar (2014), director Christopher Nolan featured an object which had something to do with space and time. (Click to enlarge.) If you haven’t seen the film and still intend to, do not read this footnote:(beginfootnote)Astronaut Joseph Cooper (Matthew McConaughey) finds himself in this spatial representation of space-time. All four dimensions of particular events in the past, present, and future of a room in his house, chopped up into manageable time chunks, are mapped onto an object (called a Tesseract) inside of a black hole (where the roles of space and time are reversed) through which Cooper can transport himself freely. This enables him to trickle information into the events of his choosing. In the still image above, you see many instances of the same room of his house with his daughter at different positions in time (which is the equivalent of different positions in space for Cooper).(endfootnote). License note.
Four ordinary space dimensions
Imagine a Pac-Man living on the surface of a sphere. To them, the world is flat. If they were to travel straight on – and on and on and on – eventually, they would be quite surprised to find themselves returning to the point where they started.
Figure 5. Imagine being as flat as a Pac-Man, travelling on what seems to be a flat surface. You might be surprised to find you’d eventually end up where you started. If you had no knowledge of the three-dimensional concept of a sphere, that is. We do. We know that you’d return because that’s what a sphere – or a circle, for that matter – does to your path. But what about our Universe? What if we would travel billions and billions of years in a straight line through the Universe? Would we end up where we started? Could our Universe be some kind of hypersphere(beginfootnote)Yes, technically, it’s a glome, or an n-sphere, where $n=3,$ and the space it’s embedded in is $n=1$, not a hypersphere. Apologies to the mathematicians and physicists.(endfootnote)?
We, the three-dimensional beings most of us are, see them as a little surface, a shape, because we can see them ‘from above’, from the third dimension. We can also see how they’re travelling around the surface of a sphere. They don’t know what a sphere is. They only think of flat surfaces. To us, however, it’s quite logical they would eventually return to their point of origin.
Okay, so, back to our 3D world. Imagine we travelled in a spaceship, always in a straight line through the Universe. Now imagine, after billions of years, we end up where we started: Earth. What happened? Could our Universe be some kind of sphere, only four-dimensional?
While no experiment has proven the existence of a fourth spatial dimension (let alone five or six etc.), it is a wonderfully entertaining world for the mind to ponder about.
Just to be absolutely sure: time is not the fourth dimension we’re talking about here. We were talking space – spatial dimensions. Quite often these two get confused: four-dimensional space-time is three spatial dimensions plus one time-dimension while four-dimensional space is four spatial dimensions without time.
Abstract spaces
There’s another way in which dimensions and spaces are used by mathematicians and physicists. Imagine an object having several properties at once: a position (in ordinary space), motion, direction of that motion, temperature, colour. To describe the state of this object, you need more than just four space-time coordinates. Suppose, its space-time coordinates are (0,1,1,1), in other words, it exists at time $t=0$ at position $(x=1; y=1; z=1)$.
Did we describe the state of the whole object? No, we’re still missing some key properties here. It is in motion, so, it has a speed, say 10 m/s. That speed has a direction – this is why we say it has a velocity, which is speed and direction. Let’s say its velocity $v = -10 \text{ m/s},$ in other words, it has a speed of $10 \text{ m/s}$ to the left.
Let’s say its temperature is 273.15 Kelvin, which is 0 ℃ and 32 ℉. And its colour is pure white. So, how many numbers do we need to describe the object’s state fully? Exactly, seven numbers (we count ‘white’ as a number).
The coordinates (0,1,1,1,-10,273.15,white) are said to live in phase space, an abstract space where the properties of the object form the dimensions of that space. This particular phase space is seven-dimensional. Of course, that’s impossible to imagine, but mathematically, you can work very well with it.
We gave an unusual example to emphasise that dimensions needn’t be related to spatial and temporal positions. However, usually, phase spaces are indeed used in the context of position and momentum.
Figure 6. A sample trajectory through phase space is plotted near a so-called Lorenz attractor, a solution to the Lorenz system, which Edward Lorenz developed to model atmospheric convection. The colour of the solution fades from black to blue as time progresses, and the black dot shows a particle moving along the solution in time. The three-dimensional trajectory in phase space is shown from different angles to demonstrate its structure.
Another example of an abstract space is a so-called vector space where each coordinate does not just occupy a point in that space but that point also has a direction. An example of such a space is the velocity of wind. Each point in that space does not just have a value pertaining to the speed of the air and its location in ordinary space, it has a direction too.
In the previous post, Complex numbers: an introduction, an entirely new kind of number line was introduced. All the spaces we just mentioned could very well contain complex dimensions. In fact, most of the time, they do. Especially in quantum mechanics. Complex numbers make up abstract complex vector spaces where wave functions thrive. Hilbert space is where it’s at, most of the time.
The Standard model of quantum physics is based on groups of symmetrical transformations in complex space, called SU(3) $\times$ SU(2) $\times$ U(1). The S stands for special and denotes all possible transformations in complex space except for one particular kind. U(1) refers to a one-dimensional unitary circle group in the complex plane. The numbers indicate the number of dimensions in which these transformations take place. The number of dimensions of the entire system is much higher, though! The dimensionality of the abstract complex space which follows from a symmetry group such as SU(3) is $3^2-1=8.$ As you can see, compared to street corner vernacular, dimensions are very different in scientific context.
In general, we can say that every manifold is a space. This needn’t pertain to spatial space. The minimum amount of dimensions needed to construct a path to a point on that manifold is the dimensionality of that space.
There are so many more types of mathematical spaces, they’re too many to mention. Suffice to say, while they have nothing to do with our ordinary space – our real-world one, which we dwell in – all these abstract spaces are brilliant mathematical tools enabling us to do predictive calculations pertaining to phenomena taking place in our ordinary, real-world space.
String theories
An interesting beast among all of this is string theory. If you accept the premise that an elementary particle such as an electron is actually a spatially one-dimensional string vibrating in specific ways corresponding to the collection of properties of an electron, then more dimensions are automatically needed in order to describe all the particles in this way. Strings need a sufficient amount of freedom, degrees of freedom, to vibrate in unique ways to be able to encompass the entire zoo of elementary particles and their properties.
Figure 7. The basic building blocks of the entire Universe, according to string theory. Unfortunately, while the theory is mathematically consistent, it cannot yet be (and hasn’t been) proven to be correct in this Universe.
In various versions of the string theories, a varying number of dimensions are needed. These dimensions are spatial and invisible. Since we don’t experience these dimensions, it is hypothesised that they are extremely small and curled up. They’re not stretched out like our ordinary three spatial dimensions.
Or they are so large that to us they don’t affect us in any way noticeable. Just as the curvature of Earth did not affect us when we were little as the Earth is so big compared to our movements.
Unfortunately, the theory cannot be tested yet. For now it’s purely a mathematical exercise. Although many discoveries have been made in pure mathematics, no experiment has proven string theory to be true (string theory in all its variety, and I’m including superstring theories and M-theory here even though the hierarchy is the other way around). No extra dimensions have been found yet.
There’s one honourable mention that I’d like to make. It’s the Calabi-Yau manifold, or the Calabi-Yau space. In superstring theory the manifold is hypothesised to encompass six invisible extra dimensions for the theory to work. The manifold is three-complex-dimensional or six-real-dimensional. I like it because it looks cool.
None of this is proven; we seem to be stuck in this three-dimensional space with one direction of time. And, if you ask me, it’s likely that our three-dimensional space turns out to be a side product of something quantum.
Figure 8. A Calabi-Yau manifold, named after Eugenio Calabi and Shing-Tung Yau. This is a complex space with complex dimensions. It yields applications in theoretical physics, most notably in superstring theory, where the manifold has six dimensions. Though not experimentally proven to be existing in our world, they do yield fascinating mathematical possibilities and puzzles.
Spaces and dimensions
There are so many different spaces with a variety of dimensions that you’d need a whole slew of posts to describe them all properly.
What can we take away from all of this? Dimensions are not realms. In ordinary space, they are the directions in which objects can freely be transported. That’s three for our world.
If you model time as a dimension, then we live in a four-dimensional space-time world. Except that you can’t freely move in time as there’s only one direction(beginfootnote)Time is definitely going to be a whole separate set of posts. Can’t wait.(endfootnote).
Though many had hoped to find extra spatial dimensions, the largest experiment humankind has undertaken, the Large Hadron Collider at CERN, has not found a shred of evidence for them. Instead, it delivered convincing evidence that the current Standard Model of particle physicswithout extra dimensions is still correct.
Nevertheless, to describe and predict phenomena in our Universe, it is almost always helpful to model their properties as extra dimensions. This has nothing to do with there actually being extra dimensions – this is probably where popular and esoteric culture get their inspiration from – but has everything to do with being able to do calculations in the abstract world of mathematics.
In a previous post, for example, we assumed imaginary time as an extra dimension to mathematically derive a set of equations in the special theory of relativity. It doesn’t mean imaginary time is an actual extra dimension you can dip appendages or your consciousness into.
In string theories, actual extra spatial dimensions are required for the theories to work. None of them can be tested as of yet (and none of them have been tested nor proven). It remains to be a beautiful, mathematical construct, but only mathematical.
In future posts, we will be exploring geometry, pseudo-Riemannian manifolds, symmetry groups, and Hilbert space for loads more bits of maths and physics.
Licenses
The featured image in the title and Figures 1 are still images of Marvel Studio’s Doctor Strange (2014) and Figure 4 of Interstellar (2014), all copyrighted films. It is believed that screenshots may be exhibited under the fair use provision of United States copyright law.
Figure 6. Lorenz attractor animation by Dan Quinn under CC BY-SA 3.0
The last few posts on quantum mechanics have been quite extensive and at times rather deep for those who are on the move. So, here are ten important ideas about particles and wave functions for when you’re en route in slightly more normal English.
To describe objects in our everyday world, such as rocks, buildings, and cars, Newton’s laws suffice. To describe subatomic, elementary particles, such as electrons, protons, neutrons, and photons, however, there is a whole different type of physics: quantum mechanics.
2: Wave functions
The most complete description of an elementary particle is called the wave function. Actually, the word ‘particles’ seems to incorrectly refer to tiny points, balls or spheres or something, which they are absolutely not. They aren’t waves either. ‘Particles’ are wave functions with wave-like properties (emphasis on ‘like’). Upon interaction with other particles and/or measurement, they exhibit particle-like behaviour, however. The wave function contains all physically possible states a ‘particle’ can be in at the moment we measure its state. The wave function can be seen as a mathematical description of the probabilities of the states that the particle will snap into as soon as you measure it. It’s often visualised as a ‘cloud’ even though that’s not what it actually looks like. It’s just a visual metaphor for a mathematical object that actually lives in complex space as it is complex valued.
I usually just draw vague spherical thingies.
3: Quantum state
Once measured, particles show one specific quantum state out of a whole range of possible quantum states prior measurement. Position is the most intuitive to understand example of a quantum state. Momentum is another (momentum is a measure of the amount of motion of a particle). Then there are states such as polarity, spin, and a bunch of others. The wave function encapsulates all these possible states and yields a probability-value for actually measuring a particular state. In other words, even before you measure it, the wave function allows you to calculate the chances of encountering this particular quantum state.
4: Measurement problem
As long you don’t measure a particle, and as long as it doesn’t interact with other particles, the particle is not in a specific quantum state yet. Instead, its wave function just describes all these possible states as though they are mathematically added on top of each other. This ‘adding of quantum states’ is what is meant when physicists talk about superposition. The term is from the mathematics of waves and linear algebra in general, not quantum mechanics in particular. While the situation is often portrayed as particles being in multiple states all at once (such as being in two positions at the same time), it’s more accurate to say that the particle does not have a specific state at all. There’s just the wave function with all the probabilities of future quantum states. As soon as you perform a measurement, the particle snaps out of its wave function full of possibilities into a single possibility. In other words, what you see is not what it was. What you observe is just a sliver of its total prior existence. How this happens, nobody knows. It’s called the measurement problem. There’s a Nobel Prize waiting for you.
This extraordinary experiment yielded a photo of the closest approximation of the wave function of an electron in a hydrogen atom we have to date. It was made by the Polish physicist Aneta Sylwia Stodolna et al. (Source: Stodolna AS et al. (2013) “Hydrogen Atoms Under Magnification: Direct Observation of the Nodal Structure of Stark States,” Physical review letters, 110(21), pp. 213001–213001.)
5: Schrödinger equation
Wave functions obey the Schrödinger equation. You could say that what Newton’s second law is for objects in our everyday world, is what the Schrödinger equation is for the subatomic world. It gives us the ability to predict how the wave function evolves in time. This is a completely classical equation; it is 100% deterministic. Where the wave function captures a range of probabilities, the Schrödinger equation tells us how this range of probabilities changes over time perfectly predictably so. In other words, it doesn’t predict the exact state of a particle once measured, but it does accurately predict the probability-value of an exact state once measured at any given time.
6: Uncertainty
There is a fundamental informational trade-off between certain possible states such as between position and momentum, energy and time, and time and frequency. The origin for this does not lie in quantum mechanics. It’s due to the way they are related to each other. Mathematically, these variables are called Fourier transform pairs or conjugate variables. To calculate one from the other, you have to execute a mathematical procedure called a Fourier transform. The trade-off is that Fourier transforming a variable whose range of possible values is smaller leads to the other variable having a larger range of possible values. And if a range of possible values becomes larger, then the exact outcome of measurement is less certain (the probability of a specific state after measurement becomes more uncertain). Heisenberg showed that this uncertainty principle also applies to the wave function in quantum mechanics, hence, there the principle is called Heisenberg’s uncertainty principle.
Fourier showed that if a sound is fairly well-defined in time (bottom), it has to be comprised of multiple frequenties (illustrated as multiple waves at multiple frequencies). That’s the fundamental uncertainty principle with waves.
7: Certainty
That same principle predicts that, while very valid at the scale of subatomic particles, this uncertainty becomes utterly meaningless at our large-scale world of everyday objects. A bowling ball whose range of possible positions is very limited (locked in a very tight enclosure with little to no leeway), will never portray any uncertainty values pertaining to its motion (momentum), for instance. By Heisenberg’s uncertainty principle, upon measurement, it might show to have the speed of $3.283 \times 10^{-35} \text{ m/s}.$ This means that after 965.9 billion years it will have travelled the distance of the width of a proton. So, no, uncertainty effects play no role in our everyday world, unless you are doing experiments with a running time of seventy times the age of our current Universe. In that case, you will have to deal with the uncertainty of the width of a proton(beginfootnote)When people state or think that everyday objects (our bodies, brains, tennis balls, animals) can exhibit quantum effects such as being at multiple places at the same time, I suspect this is because they have no well-defined idea how small subatomic particles really are and no inkling as to how large the everyday world is in those terms. Also, they didn’t do the calculations.(endfootnote). We do note that extraordinarily sensitive larger-scale equipment such as the the mirrors at the LIGO and Virgo experiments are capable of measuring quantum effects, however, this isn’t really unexpected nor is it the same as saying a human body is in a quantum superposition. Measuring quantum effects is one thing, brains supposedly being in two places on Earth (‘based on principles from quantum mechanics’) is a whole other thing.
Missing the pins has nothing to do with practical nor theoretical quantum effects. You’re just not that good.
8: Quantum entanglement
When two or more particles can only be described by one wave function – not as separate wave functions – those particles are said to be quantum entangled, either partly or completely. A measurement performed on one particle immediately determines the measurement outcome on the other entangled particle, irrespective of the spatial distance between them. This is why this phenomenon is said to be non-local. How this happens, is unknown. This effect dissipates to zero when entangled particles interact with yet other particles. At the large scale of our everyday world, the number of particles inside an object to be interacted with is so great, quantum entanglement completely fades away. In very special conditions, however, such as in our labs, entanglement can be sustained for quite some time.
This isn’t what quantum entanglement looks like. It’s just a picture.
9: Quantum Field Theory
Over the years, the mathematical and physical theory of (quantum) wave mechanics has been extended to describe quantum fields as the fundamental building blocks of our Universe. The Universe is made of quantum fields. The most complete description of fields are wave functions. This is called Quantum Field Theory (QFT). The most successful version of QFT is called the Standard Model of quantum physics. ‘Particles’ are here some kind of disturbance in their field: an electron is a disturbance in the electron field. The particle’s description is here part of the wave function of its entire field. The challenge is now to extend this quantum field theory into its next form, encapsulating something called quantum gravity. What we don’t know yet, for example, is how to have space and time in extreme regions such as black holes, naturally appear out of a quantum theory.
My very sketchy way of showing quantum fields. Proton field is not really a thing. It’s just a shortcut for several quark fields. Besides, fields aren’t two-dimensional, they’re obviously three-dimensional.
10: Applications
While the famous physicist and Nobel Prize winner Richard Feynman is known for having said, ‘I think I can safely say that nobody understands quantum mechanics’, this is sometimes incorrectly taken to be a reason to state that, therefore, physicists don’t know what they’re talking about. Feynman alluded to the fact that there is much we don’t know about the foundations of quantum mechanics. There is still much employment in solving hard problems such as quantum gravity, the measurement problem, the strong CP problem, the interpretation of quantum mechanics, non-locality, and so forth.
On the other hand, we now have WiFi, internet, touchscreens, lasers, MRI scanners, LEDs, flash memory, solid state disks, the old crunchy hard disks, transistors, and CPUs or integrated chips (ICs) in general.
I think I can safely say that the fact that you’ve plucked this article out of the air to have it displayed on your (touch)screen is at least an indication of the level at which ‘nobody understands quantum mechanics’.
Nevertheless, we’re far from done. There is still much to discover in this Universe with a bit of maths and physics.
Complex numbers have fascinated me since high school. Usually, it’s where we are taught about natural numbers, integers, rational, irrational, and real numbers but never about complex numbers. This post is for those who might be interested in an easy introduction into the realm, or rather, plane of complex numbers. And they’re not without practical significance either: no electronic device such as the one you’re using to read this post could have been built without physicists, electrical engineers, and computer scientists knowing anything about the gift of complex numbers from sixteenth century mathematicians.
Blown away
‘There are such things as negative numbers’, explained my father to me when I must have been about six or seven years old since I was a second-year pupil in primary school. He explained the notion of negative possession when owing a certain number of marbles to someone which was greater than the number of marbles you physically carry with you. As this was one of those I-still-remember-where-I-was-when moments, like it was yesterday, I remember sitting on the floor besides the coffee table in the living room of our terraced house in the town of Emmeloord, which had been reclaimed just forty-three years earlier from the IJsselmeer, a lake formerly part of the North Sea.
I clearly remember feeling exactly the same when he had told me earlier our planet wasn’t flat and when my Mum told me in the car yet a few months earlier, that we were living on the sea floor. The cap of my mind was blown away, yet again. It took a while before I managed to fold my slow and wet brain lobes around the notion that negative numbers existed, even though you couldn’t see them in the real world like you could ‘see’ regular numbers such as in lengths or the number of marbles(beginfootnote)Inexplicably, I had never considered the fact that temperature could get below 0 ℃, which it still did, back then in The Netherlands. We used to enjoy an outdoor activity called ice skating, on frozen lakes, ponds, rivers, and ditches.(endfootnote).
I hastened to tell my primary school teacher excitedly about negative numbers. She just nodded and then told me to proceed with doing my homework on boring regular arithmetic. She had a point as I wasn’t very good at it.
Fast forward to when I must have been about fifteen or sixteen when I read about complex numbers in a popular textbook about quantum mechanics. The fact that they were called ‘complex’ may have triggered my curiosity as I assumed that term pertained to it being very difficult, but mostly because, apparently, so-called imaginary numbers are a thing! I had that exact same feeling again. The cap of my mind had melted. The whole notion seemed to radiate some kind of magical power. What sorcery was this? Could this be a doorway to extra dimensions?
The next day, I told my mathematics teacher, Mr Es – Es is not his actual name but it was his two-letter code in our high school timetable. I’ve always found it appropriate Es is also the symbol for the element Einsteinium in the periodic system. As his first name happened to be the same, my friends and I used to joke that we were on our way to the lessons of Albert Einstein.
Mr Es did what every good teacher does when a student tells you something they get enthusiastic about: he encouraged it – in his case by lending me his old textbook from when he was a first-year mathematics student in Amsterdam. It was an introductory text about complex numbers at the level of undergraduate mathematics.
The very textbook. (Click to enlarge.)
I’m ashamed to say I kept it. It was one of those instances where, after the nth time of moving house, I realised, oh my god, I still have this!? It’s also true that I treasured it. It carries a special meaning to me. It signifies how, at least once in my lifetime, I felt acknowledged in what stirred me deeply at the time. A thing I couldn’t really share with friends or anyone close in general, I suddenly shared with someone very clever whose name was denoted by the symbol for Einsteinium.
Thanks to the miracle of internet, we got back in touch, about twenty-five years later. I confessed I had always kept it and apologised. He had indeed wondered where it had been as he once wanted to show it to someone else. But I could keep it as he was cleaning out the attic anyway. And he was glad it had done something for me as he learnt about my current engagements in a bit of maths and physics.
I felt guilty. I still do. Someone else could have enjoyed it just as much as I have. And now I have prevented that from happening through his book. So, whoever you are, my sincerest apologies.
I hope, one day, I will be able to ignite sparks of joy for the beautiful mathematics of complex analysis to many others. I also hope you might experience at least a fraction of the amazement I felt and that the newly gained insight on the concept of ‘numbers’ might turn out to be beyond what you were able to imagine so far. So, let this be a beginning.
Number sets
We all know and love (or hate, depending) the natural numbers: the whole numbers we count things with. 1, 2, 3, etc. Some mathematicians will want to include the number 0 while others don’t. In any case, this mathematical set of numbers is called the natural numbers and is denoted by the symbol $\mathbb{N}$.
Then my father told me about the negative numbers, such as -1, -2, -3, etc. If you include the natural numbers and add to that these negative numbers, and add the number 0 to it (if you hadn’t already), then the result is an entirely new set of numbers called the integers, denoted by the symbol $\mathbb{Z}$.
To denote that the set $\mathbb{N}$ is part of the larger set $\mathbb{Z}$, people use this symbol for subset, $\subset$. They will write $\mathbb{N}\subset\mathbb{Z}$, the natural numbers are a subset of the integers.
Of course, there’s the ratio’s. The fractions. Between 1 and 2, there’s 1.5. So, in fraction-notation, that’s $\frac{3}{2}$. They’re obviously not whole numbers. They’re rational numbers because they can be represented by a ratio of integers. This number set is symbolised by $\mathbb{Q}$. We now have $$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}.$$
It is interesting to note that, therefore, by this expression of subsets of subsets, even numbers such as 9 are rational numbers. On the surface, it’s not a fraction. Below the surface, however, it can be expressed as a ratio of integers: $9=\frac{9}{1}=\frac{18}{2}=\frac{36}{4}$, for example (and infinitely more).
But wait, there’s more. Fractions such as 1.5 and 3.2 are finite. What if the decimals don’t end? What if you can’t write a particular kind of numbers as ratios, such as with the number $\pi$ or $\sqrt{2}$? These numbers are called the irrational numbers. They are all the numbers which aren’t rational. There’s no symbol for that(beginfootnote)Often, mathematicians circumvent the lack of a symbol by writing something like $\mathbb{R} \backslash \mathbb{Q}.$(endfootnote).
Instead, there’s a symbol for all the natural numbers, the integers, the rational numbers, and the irrational numbers altogether(beginfootnote)Yes, indeed, my dear fellow mathematician, you thought correctly, I am skipping transcendental numbers here (and algebraic numbers, for that matter). As all transcendental numbers are irrational numbers but not all irrational numbers are transcendental, I decided it over-complicated things in what was supposed to be an introductory text on complex enough numbers anyway.(endfootnote). They’re called the real numbers and this set is denoted by $\mathbb{R}$. This is the set we’re all used to working with. We now have $$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}.$$
The set of real numbers $\mathbb{R}$ contains all the numbers. Or does it?
The secret of del Ferro, del Fiore, Tartaglia, and Cardano
Well, you guessed it. Here they come, the complex numbers. Let’s do just a tiny bit of maths. Remember what the quadratic of a number was? And what a square root was? What is the square root of 64, in other words, $\sqrt{64}$? Yes, that’s 8. Because 8 times 8, or 8 squared, or $8^2$ equals 64.
Okay, suppose $x^2 = 64$, what is $x$ then? Well, you do exactly the same thing, you un-square $x$ by taking its square root. And you have to do the same with the number after the equal sign. So, $\sqrt{x^2} = \sqrt{64}$, in other words, $x = 8$.
Tartaglia
Maybe you remember this comes in handy when calculating the lengths of the edges of your piece of land. Suppose, the surface area of your square piece of land is 64 square kilometre (or square miles). What is the length of an edge of that land? That’s 8 kilometre (or miles).
All these calculations take place in the realm of $\mathbb{R}^+$, the positive part of all real numbers. Note that no surface area of a piece of land can be negative. In other words, a surface area of -64 square metres is nonsensical. Also, the square root of -64 has no solution. It’s not -8, because -8 times -8, or $(-8)^2$ is simply 64 again, because a negative number times a negative numbers equals a positive number as we proved in an earlier post.
Cardano
Sometime in the sixteenth century, somewhere in Italy, Scipione del Ferro, professor of the University of Bologna, solved a slightly different kind of equation. It was a so-called cubic equation. Where we basically found the solution to a quadratic equation such as $x^2 = 64$ from the top of our heads, he found solutions for a cubic equation such as $x^3 + x^2 + 6x + 3 = 0.$ Del Ferro was known for not wanting to publish any of his proofs and solutions. He kept a secret notebook and that was it.
On his death bed, however, he told his pupil Antonio Maria del Fiore the secret to solving it. Del Fiore went on to challenge Niccolò Fontana Tartaglia, a mathematician residing in Venice at the time. Tartaglia had actually solved it himself before and trusted the formula to Gerolamo Cardano, the then Milan-based polymath and genius. Tartaglia messaged the solution in the form of a poem (no less!) but didn’t entrust the proof to him.
Of course, Cardano was able to reconstruct the proof anyway. As he learnt that del Ferro had also found the solution, he then proceeded to publish it all in his Ars Magna from 1545, much to the chagrin of Tartaglia.
So, what was the secret so many large minds had been secretive about? A new type of number.
imaginary
Let’s take a simpler example. Suppose, we have the following simplistic quadratic equation: $x^2 – 4 = 0$. To solve it, we ‘move’ the 4 to the other side of the equal sign, by adding 4 to both sides: $x^2 – 4 + 4 = 0 + 4$, which simply becomes $x^2 = 4$. If you apply the square root to both sides, you get $\sqrt{x^2} = \sqrt{4}$. The solution to this equation is thus $x=2$ or $x=-2$ (because $-2\times -2 = 4$ too).
Good. Basically, the mathematicians of the sixteenth century opined that they should be able to solve a variation of this equation as well: $x^2 + 4 = 0$. Let’s bring the 4 again to the other side of the equal sign by subtracting 4 on both sides: $x^2 + 4 – 4 = 0 – 4$, which becomes $x^2 = -4$. Now, again, the question is, what is $x$?
Let’s try and apply the square root to both sides again: $\sqrt{x^2} = \sqrt{-4}$. Halt. Stop. What is the square root of -4? What is the square root of a negative number?
We have the same situation where we are to apply the square root of a negative surface area. The answer isn’t -2, because $-2\times -2 = 4$, not -4. What then?
Before del Ferro, Tartaglia, and Cardano, people would have said that there simply is no solution. Thanks to them, however, we can solve it. The answer lies in the following definition: $$i^2=-1.$$
This seemingly simple act enables us to solve $x^2=-4$. We can then write $x = 2i$ or $x = -2i$.
Let’s take our first solution, $x = 2i$. If we square this, we get $x^2 = (2i)^2$, which we can also write as $x^2 = 2^2i^2$. Now, since $i^2 = -1$, we can substitute that to get $x^2 = 2^2(-1)$, which is, of course, $x^2 = -4$. Ecco!
The same goes for the other solution, $x = -2i$. If we square this, we get $x^2 = (-2i)^2$, which we can write as $x^2 = (-2)^2i^2 = 4i^2 = 4(-1) = -4$. Ecco!
So, you may ask, what devilish entity is this $i^2=-1$? The letter $i$ stands for ‘imaginary’ and so, $i$ is a so-called imaginary number.
Now, because $i^2=-1$, you can also write(beginfootnote)Although, I actually prefer to use $i^2=-1$ over $i=\sqrt{-1}$ even though the latter has been mentioned in many school books. However, I believe it might lead to confusion. Since we have the rule that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ where $a$ and $b$ are positive real numbers, you might try to apply this rule to negative real numbers, such as when $a=b=-1$. You would then get the incorrect statement $\sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$, which is wrong as it should be equal to -1. That’s why I try to avoid using $i = \sqrt{-1}$ where I can.(endfootnote) that $i = \sqrt{-1}$. And that’s the crazy part: how can you calculate the square root of a negative number? How can you calculate the square root of a negative surface area? The answer is, you can’t. Not in the realm of the real numbers $\mathbb{R}$, that is. However, we’re not in Kansas anymore, Dorothy. We’re in a new land called the complex numbers. Bye $\mathbb{R}$, and welcome to $\mathbb{C}$.
Here are some examples of complex numbers: $2i$, $\frac{2}{3}i$, $i\sqrt{2}$, $i \pi$, $-0.25i$. What’s more, you can add these imaginary numbers to a real number such as 3, like so: $3 + 2i$ or $3 + \frac{2}{3}i$ etc. These sums are their own answer. They are complex numbers.
A complex number $z$ is of the form $z = a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$. The first real number, $a$, is called the real part of $z$. The last real number, $b$, is called the imaginary part of $z$. The set of all complex numbers is denoted by $\mathbb{C}$.
Note that every real number is a complex number but not every complex number is a real number. That is what one thing being a subset of another thing means. For instance, the real number 9 is a complex number where $b=0$. In other words, the real number 9 can be written as the complex number $9 + 0i$, which is simply 9, which thus happens to be a real number too.
But $z = 3 + 2i$ is not a real number, because it has an imaginary part which is not equal to zero. So, $z$ is now exclusively a complex number.
Complex plane
Graphically, all the real numbers of $\mathbb{R}$ can be thought of as a point on the number line.
So, where do complex numbers reside?
Owing to people such as Wallis, Wessel, Argand, Buée, Mourey, Warren, Français, Bellavitis, Gauss, and Euler[1], the idea to extend the real number line with an imaginary number line perpendicular to the real number line came to fruition. What you get is the so-called complex (geometric) plane, sometimes called the $z$-plane, Gauss plane or Argand plane.
So, a complex number such as $z = 3 + 2i$, ‘contains’ the real number $3$ along the real axis, and the imaginary part, along the imaginary axis, sits at $2i$. A complex number is therefore always represented by a point in a two-dimensional space. Note that all the numbers from all the subset of complex numbers, i.e. $\mathbb{R}$ all the way down to $\mathbb{N}$, can also be represented by a point in this same two-dimensional complex space – it’s just that they all reside on the real axis.
As you can -heh- imagine, doing calculations with complex numbers has become an exercise of geometry now! In fact, one of the most beautiful equations in mathematics (at least to my taste) pertains to trigonometry in the complex plane; it’s called Euler’s Formula.
Not so imaginary
It’s unfortunate that this number $i$ and any real number multiplication of it are called imaginary numbers. It was the renowned French philosopher and mathematician René Descartes who coined the term imaginary numbers because he considered them to be illusory. In fact, even Cardano had described them as ‘some recondite third kind of thing’[2].
It’s unfortunate because ‘imaginary’ leads to semantic ambiguity. I get it: you would never see something like $\sqrt{-1}$ in the real world. But neither would you see $\sqrt{2}$ out in the wild, for that matter. And yet, it’s the exact length of the hypotenuse of a particular right triangle, which a skilled DIY person could make while you’re waiting. To me, ‘real’ numbers such as $\pi = 3.1415926535897 \dots$ without ever ending are as real as ‘imaginary’ numbers are (and vice versa). Circles are a real thing and $\pi$ can be used to do calculations on them. Well, with imaginary numbers you can do calculations on them just as well.
Complex numbers are used in a variety of sciences. In Einstein’s relativity, which makes GPS navigation possible, you could make use of so-called imaginary time. This sounds like a concept straight from a science-fiction novel, however, imaginary time is a well-defined concept. In fact, in a previous post, we used this to derive the central set of equations in relativity, called the Lorentz transformations. See how the word ‘imaginary’ might invoke unwanted ambiguity?
To make quantum mechanics work – the most successful theory to date – complex numbers are all over the place. Without them, the computer, mobile phone, tablet, TV, VCR, even your modern fridge – they wouldn’t have worked as no engineer would have been able to produce integrated circuits. The wave function is a complex function living in a complex separable Hilbert space, taking on complex probability amplitudes, evolving according to the Schrödinger equation, which itself is a complex equation.
In mathematics, one of the better-known areas of research where complex numbers play a central role is the study of complex dynamical systems. The featured image above is a detail of the famous Mandelbrot set. It’s a special collection of complex numbers, the projection of which you see plotted colourfully in the complex plane. The study of (complex) fractals also informs all kinds of patterns in nature and growth, even weather forecasts, and climate science – they’re all informed by complex-dynamical areas of mathematical interest. Also, we’ve used them in a previous post, calculating whether a lab centrifuge with $n$ available spots can be balanced out by a $k$ number of test tubes.
A fun application of complex numbers is computer games. To calculate rotations in three-dimensional space, computer scientists make use of quaternions, which are an extension of the complex plane. A quaternion is an expression of the form $a + bi + cj + dk$, where $a,b,c,d$ are any old real numbers, and $i^2=j^2=k^2=-1$. However, this is perhaps an interesting subject for another bit of maths and physics.
[1] Cooke, R. (2005) The history of mathematics : a brief course. 2nd edn. New York, N.Y.: Wiley.
[2] Open University (2014) Essential mathematics 1. Milton Keynes: Open University.
Images
Featured image: Mandelbrot set – Step 6 of a zoom sequence by Wolfgang Beyer under CC BY-NC-SA 2.0; adapted to fit layout.
It’s perhaps not as famous as Einstein’s formula but in this day and age many people may still have heard at least once of the phrase ‘Heisenberg’s uncertainty principle’. It plays an important role in quantum mechanics. You may have heard that every time you observe or measure matter, due to the crudeness or inherent inaccuracy of the measurement device, you will inevitably disturb your own observation. This would then preclude you from gaining accurate knowledge with satisfying certainty. In fact, in general, Heisenberg’s uncertainty principle states that nothing can be certain. At the risk of sounding vague and vanilla, all of these statements are completely and utterly wrong. Let’s look at what it really says, shall we?
Figure 1. Werner Heisenberg in Göttingen in 1924.
Fourier transform pairs
Trade-offs. Who doesn’t hate them? Remember when your parents told you that you could have this but then not have that or maybe just a bit of this but then less or fewer of that? Unsurprisingly, at least three famous philosophers have written a few words on this, each in their own way lamenting on the existence of trade-offs and how to deal with them. One chose to become all rebellious about it and wrote: ‘I want it all, I want it all, and I want it now!’ (May, 1988). The other two, however, chose to be more pragmatic about it as they postulated that ‘you can’t always get what you want’ (Jagger & Richards, 1968). Obviously, they knew that, sometimes, life brings you Fourier transform pairs. The more well-known example is of course Heisenberg’s uncertainty principle.
If you limit a particle’s range of possible positions in space $(\Delta x)$, you increase its range of possible momenta(beginfootnote)Momentum is the product of mass $m$ and velocity $v,$ so $p=mv.$ It’s a measure for the amount of motion of an object.(endfootnote) along the $x$-direction $(\Delta p_x),$ and vice versa.
This is formalised as follows:
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}.$$
Just to be absolutely clear: the delta-symbol $\Delta$ is a range of a certain quantity. Usually, a $\Delta$ is defined as the difference between two values. Suppose, you measure point $A$ of your garden fence to be $0.1$ metre away from your wall and point $B$ to be $0.7$ metre away from your wall, then the $\Delta$ of the distances, i.e. the length between points $A$ and $B,$ is $0.7-0.1=0.6$ metre.
In Heisenberg’s principle, it is stated that the product of the range of possible positions $\Delta x$ and the range of possible momenta $\Delta p_x$ is greater than or equal to some number. Mind you, it’s a tiny number. The symbol $\hbar$ stands for the Planck constant divided by $2 \pi,$ and the result gets cut in half yet again.
This means that whenever one is getting bigger, $\Delta p_x$ for instance, the other is getting smaller, which is then $\Delta x.$ And vice versa.
Click here if you’d like to do a bit of maths. It’s very easy.
Just to get an intuitive insight in this relation, suppose $\frac{\hbar}{2}=1,$ and so, suppose, $\Delta x \Delta p_x = 1.$ Furthermore, suppose $\Delta x = 0.5.$ What value does $\Delta p_x$ has to be to satisfy this equation? Exactly, $\Delta p_x$ has to be $2,$ because $0.5 \times 2 = 1,$ or else the equation is false.
Now, lets make $\Delta x$ smaller. In other words, we’re going to try to pinpoint the location with much more precision. So, let’s say, $\Delta x = 0.001.$ What value does $\Delta p_x$ has to become to satisfy this equation? You guessed right, $\Delta p_x$ has to become even larger: $\Delta p_x = 1000,$ because $0.001 \times 1000 = 1.$ If you were to reverse the situation – decreasing the size of $\Delta p_x$ – then, in turn, $\Delta x$ would have to become larger.
In reality, $\frac{\hbar}{2}$ is much smaller than 1. It is, in fact, about $5.273 \times 10^{-35} \text{J/s}.$ That’s thirty-four zeros behind the decimal point and then ending in 5273. It’s incredibly small. Don’t worry about this. We’ll get back to that later.
Hopefully, now you see the relation between $\Delta x$ and $\Delta p_x$ as put forward by Heisenberg’s formulation. They complement each other. Whenever a range of possible values becomes larger, in other words, the $\Delta$ or range of value-options is larger – its actual value becomes more uncertain, hence the use of the word ‘uncertainty’ in Heisenberg’s uncertainty principle(beginfootnote)In fact, it’s statistics. The $\Delta$-sign could just as well be a $\delta$-sign, so $\delta x \delta p_x \geq \frac{\hbar}{2},$ which signifies its statistical character more accurately. After all, the wave function is about probabilities.(endfootnote).
But why is this? While this principle plays a central role in quantum mechanics, it’s actually not fundamentally a quantum-mechanical law. This principle exists more generally in many instances in physics, and, even more generally, in mathematics.
In mathematics, the variables position and momentum are said to be a Fourier transform pair. Put in yet other mathematical jargon, position and momentum are said to be conjugate variables.
Sound
A well-known, non-quantum-mechanical example of the uncertainty principle is determining the pitch of a sound. How ‘high’ a note is, depends on the frequency.
The most familiar way we depict sound waves is a simple sine wave. It represents the simplest of sounds possible. Also, it’s the most boring of sounds possible.
The $x$-axis represents time. The $y$-axis represents the amplitude of the sound or the loudness, the intensity of it. As you can see, the sound wave repeats itself over time; the pattern is cyclic. One whole cycle is when the plot has completed going up, going down, going further down, and going up again. The time it takes to complete one cycle is designated by the symbol $T,$ called the period(beginfootnote)It is also possible to measure the time-distance between two peaks or two troughs.(endfootnote). So, this particular sound wave is said to be periodic.
Figure 3. A time-amplitude plot of a boring old sinusoidal sound wave. (Click to enlarge.)
The shorter the period – the quicker the cycles are – the higher the tone. Another way of saying, is that the higher the frequency, the higher the tone. The mathematical relationship between period $T$ and frequency $f$ is the following expression:
$$f = \frac{1}{T}.$$
If the period gets shorter, i.e. the value of $T$ becomes smaller, then the value of $f$ becomes larger, which means higher, which means a higher tone.
Seeing as the time period $T = 2 \pi$ seconds, the frequency diagram looks like a spike at $\frac{1}{2 \pi}$ Hz. In this frequency-diagram, the $x$-axis is the frequency and the $y$-axis is still the amplitude.
Figure 4. A frequency-amplitude plot of the sound wave of Figure 3. It shows the exact frequency at which that sound wave exists.
So, there are now two ways in which we can describe the sound wave: either by frequency (Figure 4) or by change over time (Figure 3).
Notice that the sound wave plotted as a function of time (Figure 3) has no beginning nor end. For all we know, that plot could just go on forever, to an infinite amount of time, in both directions. Suppose, we would ask the question at what time exactly does the sound exist? The answer is: always. There is no particular, specific time at which it exists.
In other words, we could write that $\Delta t = \infty.$
Notice, however, that the frequency plot looks very finite: just one stroke. One well-defined, finite stroke. If we were to ask the question what frequency exactly does the sound have? The answer is: there is a particular, specific, exact frequency at which it exists and it is $\frac{1}{2 \pi}$ Hz $( \approx 0.16).$
Fourier analysis
In reality, no sound is going to be infinitely long. Pluck a guitar string and it will fade out as the energy dissipates slowly. Also, at some point it started – meaning, before that, it didn’t exist. In other words, in reality, a sound wave usually exists in a finite range of time.
Let’s limit our sound wave to a range in time, so it looks more like the sound of a ‘blip’ and less like an infinite tone of boredom. Again, the $x$-axis represents time and the $y$-axis represents the amplitude.
Figure 5. A time-amplitude plot of a so-called wavelet, a short sound burst. Contrary to the sound wave in Figure 3, it’s not infinitely long. It’s now also more difficult to assess its frequency.
As you can see, the sound now exists in a more defined range of time – roughly 1.5 seconds. In other words, $\Delta t \approx 1.5$ seconds. That’s a whole lot smaller than the old $\Delta t = \infty.$
Now, we ask ourselves, what is its frequency? The difficulty now is that it’s hard to pinpoint an exact period $T$. The evolution of the plot is quite different from our infinitely long sine wave. Yes, we can identify kind of those cycles we’re looking for, however, no cycle has the same shape, so, technically, we’re dealing with multiple cycles at once. And guess what, its frequency-amplitude plot looks like this.
Figure 6. The frequency-amplitude plot of the wavelet in Figure 5. It’s far from being a specific, exact frequency. At varying degrees, it’s actually a few frequencies at the same time.
As you can see, it has become difficult to pinpoint the exact frequency of our wavelet. It exists at a variety of frequencies and amplitudes.
So, while the ‘time window’ of the sound wave has become more exact, the frequency has now become ‘less certain’.
The brilliant mathematician Joseph Fourier discovered that a wavelet such as in Figure 5 can actually be constructed by adding many infinite waves at many frequencies. Put differently, Fourier analysis shows that our wavelet is the culmination of a superposition of many waves at many frequencies.
Figure 7. The wavelet at the bottom is constructed by many infinite waves at many different frequencies superposed onto each other. This automatically means that the wavelet’s exact frequency is fundamentally harder to determine than the frequency of the sound wave in Figure 3.
Now you see why the frequency-amplitude plot has changed from a very specific value in Figure 4 to the wider set of frequencies in Figure 6. In the latter case, the wavelet ‘contains’ multiple waves at multiple frequencies, so when you Fourier transform its time-amplitude plot to its frequency-amplitude plot, the frequency has become ‘uncertain’.
The relation between time $\Delta t$ and frequency $\Delta f$ in ordinary classical physics is fundamentally complementary. No quantum mechanics needed.
The term ‘Uncertainty principle’ pertains to the general phenomenon that Fourier transforms (such as between time and frequency) entail a fundamental, mathematical trade-off between types of information carried by the two transformed variables. Heisenberg then showed that this principle also holds in quantum mechanics. And so, the uncertainty principle in quantum mechanics is called Heisenberg’s uncertainty principle.
The De Broglie relation
Time to go back to quantum mechanics. Remember that a particle’s best description is a wave function? A wave function is the mathematical expression of a particle containing all possible states it can assume once we measure it.
Instead of a time-amplitude plot, let’s represent a particle by a space-amplitude plot. To make it a little bit easier, let’s take the wave function of a particle of which the amplitude only varies along one dimension of space, $x.$
Here is a representation of a particle’s wave function along one dimension of space (along a ‘straight line’). The $x$-axis represents a position in space. The $y$-axis represents the amplitude of the wave function (which is proportional to the probability of finding the particle in that particular position $x$).
Figure 8. A representation of a wave function of a free particle. Note that this is not what it actually looks like. For one, an actual wave function exists in complex space, which we didn’t plot here. The goal is to illustrate, not to map accurately. Also note that the free particle has no specific position yet as it’s a free particle!
It was the eminent French physicist Louis de Broglie(beginfootnote)Many physicists have tried and mispronounced his last name. It should sound like ‘broy’ where the r is produced at the back of the throat, like the French r – a ‘dry’ kind of r. In this interview with him, you can hear the French presenter pronouncing his name (just after 0:16 seconds). It’s not ‘brog-ly’ nor ‘bro-ly’. Thank you.(endfootnote) who formulated the relationship between a particle’s wave function’s wavelength $\lambda$ and its momentum $p.$
$$\lambda = \frac{h}{p},$$
where $h$ is the Planck constant. Incidentally, this is the equation better known as De Broglie’s matter wave hypothesis, stating that matter, such as electrons, possess a wave-like characteristic(beginfootnote)Do note that this same equation shows that this wave-like behaviour of large bodies such as our bodies, brains, bowling balls, tennis balls, and animals is completely and utterly negligible as we will demonstrate at the end of this post.(endfootnote). This won him the Nobel Prize, no less.
If we rewrite this to solve for $p,$ we get
$$p = \frac{h}{\lambda}.$$
So, clearly, a wave’s momentum is determined by its wavelength. The smaller the wavelength, the greater the momentum. What is the wavelength? It’s the length between two peaks (or two troughs). The higher the frequency, the smaller the wavelength. Now have a look at Figure 8 again. As you can see, the infinite wave of a free particle has a well-defined wavelength. The logical conclusion is that the momentum is also well-defined. Nevertheless, Figure 8 also shows that the particle’s position is not defined at all!
Let’s turn this on its head and limit the range of possible positions of our particle. No longer is it a free particle. It is now confined within a finite range of locations.
Figure 9. Our former free particle’s position is now restrained between $x = 0$ and $x= \pi.$ In other words, $\Delta x$ is now limited to $\pi$ wide. There is no well-defined wavelength as the wave function has different values in different places. It’s there, but not as well-defined as in the wave function in Figure 6.
What we’ve done in Figure 9 is making $\Delta x$ smaller than it was in Figure 8 (where it was infinitely large). In fact, $\Delta x = \pi$ wide. By the same Fourier transform mechanism as with the time-frequency pair, the complimentary sister of position space $\Delta x$, namely momentum space $\Delta p_x$, will now become less certain.
To construct a limited wave function such as the one in Figure 9, Fourier analysis shows that you need – again – a bunch of waves at different frequencies in superposition (added on top of each other).
Figure 10. A Fourier deconstruction of the wave function in Figure 9. Many waves, many frequencies. Hence, the momentum is less well-defined.
So, when it comes to quanta, Heisenberg’s uncertainty principle states that there’s a fundamental trade-off between information on position and momentum(beginfootnote)Another pair is energy and time. This is interesting in the context of Hawking radiation. We’ll get to that, don’t worry.(endfootnote). This is due to the fact that they are a Fourier transform pair or conjugate variables.
This also means that if you constrain a particle to a minuscule $\Delta x,$ its wave function will start to contain momenta $\Delta p_x$ all over the place. It will occupy many more velocity possibilities, including the much faster velocities. If you were to subsequently perform a measurement, the probability of finding it moving at higher speeds is now much larger!
Scale and effect
At the scale of the big bad world, we never see this effect. If you would confine a bowling ball in a limited space, you will not see its momentum increase dramatically. It won’t suddenly start bouncing up and down. Conversely, if you swoop the bowling ball with considerable momentum, it won’t suddenly start appearing everywhere and nowhere at the same time: its position is still quite clear. You won’t suddenly quantum tunnel through the pins or be rolling on all bowling lanes of the neighbouring players at the same time. If it doesn’t hit a single pin, then that’s not because it’s suddenly in a state of superposition with regard to its possible locations of existence. You’re just not that good.
You won’t notice any of these quantum effects in your everyday-scaled objects. Only when you’re dealing with particles. Or atoms. However, as soon as the mass increases, it all changes. Why? Partly because Planck’s constant is so darn small(beginfootnote)And because the number of interactions between atoms increase exponentially, causing any quantum effect to disappear due to decoherence.(endfootnote). It’s just $5.273 \times 10^{-35} \text{ J/s},$ remember? That’s small.
All this knowledge does allow for some fun calculations. For instance, if you were to confine a bowling ball with a mass of $7.2$ kg (16 lb) inside a box where $\Delta x = 22$ cm (8.66 inches), by Heisenberg’s uncertainty principle, the ball’s speed will be $3.283 \times 10^{-35} \text{ m/s}.$ That means that after $965.9$ billion years it might have moved a distance equal to the diameter of a proton. That amount of time is seventy times the age of our current universe. Granted, quantum-mechanical effects aren’t zero, but as you can see (or rather, as one can calculate), on our everyday scale, these effects are quite meaningless.
Sometimes, weird films such as What the #$*! Do We (K)now!? and What the Bleep!?: Down the Rabbit Hole will want to make you believe such quantum things can happen anyway. They will mention Heisenberg’s uncertainty principle like it is a magical law allowing us to do whatever. I hope that this post has shown that Heisenberg’s uncertainty principle is not about that. Nor does the uncertainty principle itself have its roots in quantum mechanics. It’s basically wave mechanics, the classical stuff, which all first-year undergraduates in physics have to learn in their first or second semester.
A few months ago, I stumbled across a video showing an Australian senator’s question to the head of the Commonwealth Scientific and Industrial Research Organisation, an Australian federal government agency responsible for scientific research. Clearly, the senator had – shall we say ‘read something about Heisenberg’s uncertainty principle’. During a senate hearing for a legislative committee, the senator questioned if research done in climate change should be taken with precaution as Heisenberg’s uncertainty principle stands in the way of accurate measurements(beginfootnote)He basically sought a ‘scientific’ way to put climate science in doubt – which, apparently, he is not a proponent of. I do not claim to know anything about Australian politics, or even at great depth about climate science, however, when a legislator starts talking quantum physics – well, I do know stuff about that.(endfootnote).
I suspect this discussion pertained to a study where a satellite uses infrared radiation to perform surface and/or atmospheric remote sensing. He continued to state that as infrared light has lower frequencies than visible light, it’s ‘very difficult’ to understand the properties of infrared radiation based on Heisenberg’s uncertainty principle.
Many things were going on (wrong) in this one short bit of speaking time of the senator, as is usual when someone hasn’t caught up on quantum physics as much. Which is understandable, but no less gnawing to watch (the link opens a new tab and leads to a short video on Twitter).
In any case, I genuinely hope that this article contributed at least a sliver of knowledge to educate the electorate of the world, so we can all vote as informed and responsible as possible for the right persons for the right jobs, besides one’s preferred socioeconomic idealism.
If you should take one thing from this post, it’s that Heisenberg’s uncertainty principle is not about anything spiritual nor does it have anything to do with scientific measurement mistakes: it’s good, old wave mechanics and Fourier analysis taught to undergrads in their first year at university. It works and it works well. It does not lead to science not being able to know things about the universe. In fact, it increased our knowledge of it. In fact, no modern information device would have worked without it. After all, you’re reading this with an electronic device which exists thanks to Fourier, Heisenberg, and De Broglie, among others. All that with a bit of more maths and more physics at the same time.
Photo Werner Heisenberg by Friedrich Hund, a German physicist who took this photo in Heisenberg’s place of residence, Göttingen, in 1924. It was uploaded to Wikimedia Commons under CC BY 3.0 by Friedrich Hund’s son, Gerhard Hund, a German mathematician, computer scientist, journalist, and chess player. We have used a colour-corrected version by Martin Geisler.
When the state of a subatomic particle cannot be described by a wave function without taking the state of another subatomic particle into account, we speak of quantum entanglement. It’s the special case where both particles can only be described by one and the same wave function. No longer are they separate entities nor do they have separate wave functions. The astonishing consequence is that performing a measurement on one particle has an immediate effect on the measurement of the other particle, no matter how far apart they are from each other. In this article, the second part of our mini-series on quantum entanglement, we will discuss the EPR paradox which Einstein and colleagues put forward. After that, we will discuss Bell’s Theorem which allowed physicists to test Einstein’s proposal. Was Einstein correct?
A representation of an electron’s spin – do note that this is not what an electron actually looks like nor is it what its spin looks like. The quantum world is simply too strange to depict accurately using ‘classical’ notions as done here. Here we drew a vague ball-like thing which seemingly spins around, which it isn’t and it doesn’t. But it’s the best we’ve got. Although, the best we’ve got is actually something else: a mathematical expression, the wave function.
Quick summary
Firstly, let me give a quick summary of the previous post:
1. we used the property of spin as a way of distinguishing between the two entangled electrons;
2. the orientation of an electron’s spin is expressed as spin up (anticlockwise) or spin down (clockwise) along the axis of measurement;
3. you can arbitrarily choose along which axis you want to measure its spin, in three dimensions;
4. no matter which axis you choose, the result is always going to be a spin up or spin down (there is no spin-a-bit-to-the-right, for instance);
5. we are able to entangle particles in such a way that they will either always yield opposite spin or they always yield identical spin; once prepared this way, they will never deviate from this correlation when measured;
6. we used the opposite-spin entanglement in our example and we will do so again here;
7. quantum mechanics states that before measurement neither electrons have a specific spin: the wave function contains all possible measurement outcomes, in this case pertaining to both spin up and spin down (which can be characterised as having no definite spin yet)(beginfootnote)Analogously, the double-slit experiment showed that before measurement, particles don’t have a specific location yet.(endfootnote);
8. as soon as you measure one electron’s spin along a certain axis, the other electron’s spin immediately snaps to the opposite orientation along that same axis, regardless of spatial distance between the two entangled particles(beginfootnote)Or, if their entanglement were prepared in such a way that they always have identical spin, the other electron would then immediately snap to the identical spin orientation along the same axis of measurement.(endfootnote).
EPR paradox
Even though Einstein understood quantum mechanics like few others, and while accepting these predictions and results, he didn’t quite like the non-local implications brought forth by quantum entanglement. He didn’t like point 8 of the previous section. There seems to be zero time delay between influencing a particle in Amsterdam (through measuring its spin) and influencing its entangled particle in Boston. It violates a pivotal consequence of Einstein’s theory of special relativity: no signal or piece of information – anything within this universe, really – can exceed the speed light(beginfootnote)In a vacuum.(endfootnote) or else causality would not exist. In other words, if information or signals were able to travel faster than light, an effect could occur before its cause had taken place. To put it mildly, this doesn’t seem to be the universe you and I are living in.
So, Einstein, Podolsky, and Rosen (EPR) hypothesised that something else, something secretive was going on in nature – well out of sight for theoretical and experimental physicists. Quantum mechanics as it was known then had to be incomplete. Obviously, they acknowledged its successes, but when it came to quantum entanglement, they asserted something was missing in the theory of describing nature through wave functions.
To solve for the seemingly faster-than-light signal, they proposed that what really was going on was that the particles have always been in a specific state. When the electron pair were separated from each other, they have always had either spin up or spin down from the start from the moment of their creation.
Suppose, a pair of gloves were made. Like all pairs of gloves, they always were each other’s opposite with respect to ‘handedness’(beginfootnote)‘Handedness’ in this context is a form of the more generalised term chirality.(endfootnote). One has always been left-handed, the other has always been right-handed. And if the first one happened to be right-handed, then the other was left-handed. (Or else you’re holding a glove from another pair.)
Suppose, the machine which had made the pair put each glove in a separate box. We can’t see which glove went in which box until we open the box. The boxes were sent to Amsterdam and Boston. The experimental physicists then open the box in Amsterdam: it’s the right-handed one! And so, we now instantly know, the one in Boston is left-handed. No magic, no non-locality, no lightspeed-breaking shenanigans.
This is what Einstein and friends said was happening in the case of electrons. An electron pair always had specific spins to start with. It’s only in Amsterdam and Boston that we ‘open the box’ aka measure their spin. It’s only logical now that as soon as you know which spin the Amsterdam electron has, you immediately know which spin the Boston electron has.
So, said Einstein, non-locality is an illusion. It’s all just normal local laws of nature and a bit of logical thinking. For one, spin orientation is merely hidden from us and not principally uncertain. Secondly, there’s no spooky action at a distance[1], as he famously described it(beginfootnote)In German, he wrote ‘spukhafte Fernwirkung'[1].(endfootnote).
In everyday parlance, physicists call this a local version of the ‘hidden variables’ theory. ‘Hidden variables’ pertain to the stuff that we can’t see yet (such as spin orientation or other variables influencing this) because our quantum mechanical description (the wave function) is incomplete, however, they are there, they do exist – they do not not exist yet, according to the hidden variables theory.
Bell’s inequalities
Unfortunately, Albert Einstein passed away in 1955. And Niels Bohr, the other great physicist with whom he used to debate the fundamental nature of quantum mechanics passed away in 1962. In both cases too soon for them to be able to read John Stuart Bell’s 1964 paper called ‘On the Einstein Podolsky Rosen Paradox'[2]. Bell realised that Einstein’s proposal was in principle testable. It yielded a clear prediction, called Bell’s inequality.
At this point, we must note that over the years, more than one Bell’s inequalities have been put forward by physicists(beginfootnote)Besides his original inequality, there’s the much-used CHSH-inequality, for instance.(endfootnote). To explain Bell’s inequality, we will apply a version of David Mermin’s original version as mentioned in his fantastic Boojums All the Way Through: Communicating Science in a Prosaic Age[3].
Recall from point 3 before that we can measure an electron’s spin orientation along any axis. We’re going to be measuring along three axes. These axes will be at an angle of 120° relative to each other.
The first axis will be the spin orientation along the vertical axis, which we will denote with the following symbols for spin up and spin down:
$$\uparrow \downarrow$$
The spin orientations up and down will also be measured along this second axis:
$$\nwarrow \searrow$$
And the spin orientations along the third axis will be denoted by:
$$\nearrow \swarrow$$
So, imagine two entangled electrons being separated in space from each other. The usual quantum-mechanical description of each electron is that they are in a superposition of spins up and spins down for all three axes.
Except, Einstein says, no, no, not really: hidden behind the ‘veil of superposition’ they are in fact already in definite, specific spin orientations for each of the three axes. We just don’t yet know which until we measure them!
He says, the electron in Amsterdam may already be in the specific spin states as follows:
$$\left( \uparrow \searrow \swarrow \right)_A$$
So, along axis 1 it’s spin up, along axis 2 it’s spin down, and along axis 3 it’s also spin down.
Einstein continues and says that the entangled electron in Boston has to already be in the opposite states:
$$\left( \downarrow \nwarrow \nearrow \right)_B$$
And so, Einstein concludes, as soon as you actually perform a measurement in Amsterdam along the first axis, of course, you get the opposite spin in Boston. Only logical!
Bell’s insight was that if you would work out this entire argument for all possible combinations, you could actually get a prediction of a ratio of outcomes. Here’s how that goes.
First of all, if you measure along axis 1 in Amsterdam, that doesn’t mean you have to measure along that same axis in Boston. You could just choose to measure along axis 3. So, with the two examples above, your results would simply be that in Amsterdam you get spin up and in Boston you also get spin up:
$$\left( \uparrow \right)_A \text{ and } \left( \nearrow \right)_B$$
Bell then argued, if you would count the number of times you would get the combinations up-up, down-down, and of course up-down and down-up like this, you should get ratios of these combinations which should match experiment. If, however, these ratios don’t appear in the experiments, then Einstein’s hypothesis is incorrect. In that case, something entirely different is going on. The electrons were not already in a specific state, which in turn means that the non-local measurement effect in quantum entanglement does exist!
Bell’s theorem
So, let’s put them all together. Let’s first take our example above:
If you measure along axis 1 in Amsterdam and along axis 1 in Boston you get spin up, spin down. If you measure along axis 1 in Amsterdam and along 2 in Boston, you get spin up, spin up. And so on, and so forth! We’ve put it in a little table:
Here you can see all the possible combinations of measurement outcomes along the three possible axes of the electrons in Amsterdam (A) and Boston (B). We used U for spin up and D for spin down.
Bell then says that if Einstein was correct, and the states of the spin orientations along these three axes were already there, then these are the expected outcomes.
Let’s focus on the number of UD or DU combinations, in other words, let’s focus on the number of times we find the opposite spin orientations, irrespective of the axes along which they are measured. We’ve marked them yellow.
Exactly five out nine times you will find the opposite spin directions.
Let’s check for other spin combinations. Suppose, the electron in Amsterdam is secretly in the following spin states, $\left( \downarrow \nwarrow \swarrow \right)_A$, and the electron in Boston is then the opposite, $\left( \uparrow \searrow \nearrow \right)_B$. If we count again the number of times the measurement outcome of opposite spins, we get, again, five out of nine.
Okay, I think you can imagine where this is going. We’re not going to go by all the tables, but I do want to do one more, just for fun. Suppose, the one in Amsterdam is all spin down, $\left( \downarrow \searrow \swarrow \right)_A$, and, obviously, the Boston one is its opposite, $\left( \uparrow \nwarrow \nearrow \right)_B$. In that case, we would get opposite spins in nine out of nine times.
And so, this particular Bell inequality states that the probability (P) of finding opposite spins along all three axes is at least $\frac{5}{9}$ or 55% (and at most 1 or 100%). In other words, $P(\text{opposite}) \geq \frac{5}{9}$. If this inequality were violated by experiment, the underlying theory will have been proven to be incorrect.
Experimental outcomes
Over the past thirty years, many experiments were carried out to test multiple versions of Bell’s inequality. Usually, these tests involved photons rather than electrons and pertained to measurement of polarisation rather than spin.
Freedman and Clauser did the first Bell test. They used a version of the so-called CH74 inequality[4].
The most well-known test was performed by Alain Aspect and colleagues. As Bell had originally suggested, they were able to have the two measurement devices randomly select the method of measurement before the entangled photons had arrived[5].
In all tests, all versions of Bell’s inequalities were violated. Instead, the statistical outcome was congruent with the predictions of quantum mechanics. The conclusion has to be that Einstein’s local hidden variable theory was incorrect. There is nothing local about measuring entangled particles.
In our particular inequality, the result was that the occurrence of opposite spins turned out to be exactly 50%, not 55%.
Conclusions
Let’s summarise what we have established over the course of the last two posts, including this one.
In quantum mechanics, particles which have not been measured yet don’t have a definite, specific state. Instead, they are best described by a wave function which incorporates all the possible future states it can snap into once measured.
When a particle can only be described in tandem with another particle, i.e. both particles can only be described by one and the same wave function, they are maximally quantum entangled(beginfootnote)In practice, in the real world, particles aren’t maximally entangled like the way we can prepare them in the laboratory. The world is too messy for those ‘pure states of entanglement’ to exist for any significant amount of time. There are simply too many particles around to not interact with any other particle. Every particle will invariable interact with thousands of trillions of other particles and so any previous entanglement will quickly decohere into either a very weak version of the original entanglement or simply to zero entanglement. Every interaction represents a measurement. Since our brains are too large and consist of thousands of trillions of particles, they will never be in a pure state of superposition nor entanglement. Not to mention our much larger body, which will never be in any sort of quantum state. It is statistically so unlikely that you’d have to become as old as $(10^{100})^{100}$ times the age of our current universe to witness such an event. And that number was a metaphorical one. It’s much larger.(endfootnote).
If their entanglement entails their spins will always correlate in a certain way – be it identical spins or opposite spins – a measurement on one particle, causing it to snap into one of the possible, specific, definite states, has immediate effect on the state of the other particle: it instantly snaps out of its wave function haze into a correlating, specific, definite state.
Einstein didn’t like this as this would imply some kind of information was somehow transported beyond the speed of light from one particle to the other.
He postulated that particles have always been in a specific, definite state to begin with. The only reason we don’t know which is because we haven’t measured it yet. There is no ‘snapping out of the haze’ going on.
John Bell showed that Einstein’s hypothesis can be tested. If you would perform many, many measurements of many, many maximally entangled particles, eventually, the occurrences of the variety of correlated states should show up in a certain ratio, an inequality, as it happens.
Experiments showed they do not. Instead, the ratio is exactly according to the predictions of quantum mechanics.
This demonstrated that particles indeed snap out of their haze upon measurement and not that particles had always been in a hidden but definite state.
And if that is true, then non-locality has to be true – there is no other way the other particle snaps into the correct, correlated state.
Nobody knows how this happens. Certain non-local but still hidden-variables hypotheses have been proposed. One of the more famous versions is called the ER=EPR conjecture by Juan Maldacena and Leonard Susskind. Perhaps we’ll dive into that later on.
Einstein’s aversion to this ‘particles have no definite state until measured upon’ made him utter his famous complaint, ‘God does not play dice’.
Unfortunately, he was wrong here on two occasions. God(beginfootnote)We are using the word ‘God’ in a purely metaphorical way. This does not pertain to any specific religious entity as revered by many in a variety of societies in human culture.(endfootnote) does play dice. Moreover, He throws them where we can’t see them. Even God seems to be bound by Heisenberg’s Uncertainty Principle. But that’s a subject for another bit of maths and physics.
[1] Einstein, A., Podolsky, B. and Rosen, N. (1935) “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,” Physical Review, 47(10), pp. 777–780. doi: 10.1103/PhysRev.47.777.
[2] Bell, J. S. (1964) “On the Einstein Podolsky Rosen Paradox,” Physics Physique Fizika, 1(3), pp. 195–200. doi: 10.1103/PhysicsPhysiqueFizika.1.195.
[3] Mermin, N. D. (1990) Boojums all the way through : communicating science in a prosaic age. Cambridge England: Cambridge University Press.
[4] Fry, E. S. and Thompson, R. C. (1976) “Experimental Test of Local Hidden-Variable Theories,” Physical Review Letters, 37(8), pp. 465–468. doi: 10.1103/PhysRevLett.37.465.
[5] Aspect, A., Dalibard, J. and Roger Gérard (1982) “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Physical Review Letters, 49(25), pp. 1804–1807. doi: 10.1103/PhysRevLett.49.1804.
Featured image: Portrait of theoretical physicist John Bell at CERN, June 1982 (CERN, CC BY 4.0)
To this day, quantum entanglement and its effects are phenomena which still leave physicists scratching their heads when trying to get a deeper understanding of what is actually happening. This series on quantum entanglement is going to be a two-parter. In this post, we will discuss what is meant by locality and non-locality and what quantum entanglement is. The term quantum entanglement has been used in many instances of popular culture pertaining to spirituality, healing, and a flurry of new age approaches to human consciousness. This is not the kind of ‘quantum entanglement’ we will discuss here. We will purely look at the physics of it, its original and proper meaning. We will study the state of a two-particle system. In the next post, we will discuss what Einstein and his friends proposed, what Bell wrote, and whether Einstein was right. And then there are also exciting caveats which we will explore.
The basics
Let’s go over the basics one more time. ‘Particles’ aren’t particles in the classical sense at all – they’re absolutely not like tiny balls or pellets. They are best described by the wave function, a mathematical expression containing all possible states the particle can be in. This pertains to its energy levels, its positions or a number of other properties it can have.
As long as no measurements have been performed on it, the particle has no definite state or states. It displays wave-like behaviour like being caught in a haze of all possible states. However, as soon as you measure it, the particle will snap out of its haze and it will appear to be a particle, an actual particle in the classical sense, with a definite state.
Note that ‘the state of an electron’ can refer to a particle with no definite set of states when no measurement was performed. The state of an electron is then best described by the wave function, which contains all possible definite states upon measurement.
Hereafter, ‘wave function’ and ‘state’ are used interchangeably.
Isaac Newton knew he had a problem when he formulated his theory of gravity. While it beautifully described the extent to which two masses exert gravitational forces upon each other, his theory didn’t explain how they did that. He didn’t like the conclusion that the gravitational influence between Earth and the Moon seemed to spookily operate at a distance through the vacuum. He wrote it was ‘so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it’. He famously stated to leave this unsolved mystery to ‘the Consideration of my readers’[1].
In other words, Newton wasn’t big on non-locality. And yet, his own theory did entail an invisible force operating over vast distances through the vacuum. Moreover, it seemed to be an instantaneous effect: if the Sun were to suddenly disappear, then Earth would be flung off its trajectory immediately. Of course, today, we know that nothing can travel faster than light, so the gravitational changes of the Sun would take about eight minutes to ‘reach’ Earth.
The following years, physical phenomena such as magnetism and electricity proved, in fact, to be very local indeed. It became clear there is always an indirect way through which one object is able to influence another object at a distance. What is meant with locality? Here’s the mechanism: an object interacts with its immediate environment, a field embedded within the three-dimensional space we live in, i.e. the electromagnetic field, which then passes on that ripple of disturbance onto the other object. In terms of ‘fields’, one could say that at one particular location the field’s value is changed by some object. That value change then changes the values of the field in the direct vicinity, which then change the values in their vicinity, and so on. It’s a bit like ‘the wave’ done by thousands of sports fans in a stadium. Or like falling dominoes. Every change is ever local and the propagation of that change through space is limited to the speed of light.
Tumbling telephone boxes are definitely a ‘local phenomenon’. The sculpture Out of Order by David Mach is situated in Kingston upon Thames (UK). Photo by 272447.
Many years later, Einstein replaced Newton’s theory with his own theory of gravity, General Relativity (GR). It showed that Newton’s intuition was correct. Gravity couldn’t be non-local and Einstein showed it isn’t. In GR, space and time itself are the stretchy substance through which gravitational disturbances propagate at the speed of light towards the other object. When a mass curves or disturbs spacetime around it, that curvature or disturbance then ripples through the universe, on its way to influence other objects. In fact, on 11 February 2016, a large collaboration of incredibly talented scientists physically measured these gravitational ripples in spacetime as predicted by Einstein in 1916. It won three key figures the Nobel Prize.
And so, it seems there is no spooky influence at a distance in physics. Even still to this day, in modern quantum physics, our best understanding and most successful theory is that quantum fields pervade our universe, forming the mediums through which forces are propagated, limited by the speed of light.
Non-locality entails a change in one patch of space instantaneously influencing another patch of space irrespective of their distance. Locality entails the propagation of change through space by influencing only neighbouring patches of space at a maximum of the speed of light.
Spin
Electrons have several properties. One of the more obvious is (negative) charge. The Stern-Gerlach experiments showed that they possess another property which was given the name spin angular momentum or simply spin for short, for lack of a better term as electrons aren’t exactly like spinning balls.
Nevertheless, as it stands, electrons have an intrinsic spin, which cannot in any sensible way be described like a classical-mechanical rotation. Like with any object in three-dimensional space, you can measure its spin along any angle within 360 degrees in three dimensions. With respect to whichever axis you choose, they can only ever spin clockwise or anticlockwise(beginfootnote)Yes, this does sound like there is an actual rotation around an axis in the classical sense. And maybe, in some deep sense, there is after all, however, this deserves a post of its own, so suffice to say for now, our language is simply too limited to avoid using classical terms for quantum mechanical phenomena, misleadingly.(endfootnote). The latter is called spin up and the former spin down, according to the right-hand rule.
If electrons were like tiny, fluffy balls such as displayed here, you could picture their spin as an anticlockwise or clockwise rotation about the axis of measurement. Using the right-hand rule, we can designate this spin-up or spin-down. Of course, in three dimensions, any axis of measurement at any angle can be chosen with respect to which it will be found spinning. Disclaimer: this classical-mechanical illustration does not portray actual electrons nor actual quantum mechanical spins. But it’s perhaps useful as a simile.(Illustration by KJ Runia)
Symbols
As we take our readers seriously, we’ll take this opportunity to introduce a few mathematical symbols which will prove to come in handy at later stages of this series.
Let’s use the symbol $\lvert A \rangle$ to denote the state of the electron in Amsterdam with respect to its spin. As long as we haven’t performed any measurements on the electron, it has no definite state. However, upon measurement, its spin with respect to the vertical axis of measurement is ever either spin up or down. Let’s write these two possible measurement outcomes as $\lvert\uparrow\rangle_A$ or $\lvert\downarrow\rangle_A$.
Likewise, if the state of an electron in Boston $\lvert B \rangle$ is spin up or spin down, we write $\lvert\uparrow\rangle_B$ or $\lvert\downarrow\rangle_B$.
Assuming the state of the electron in Amsterdam hasn’t been measured yet, we can express this (with respect to spin) as a combination of both spin states:
$$\lvert A \rangle = \alpha \lvert\uparrow\rangle_A + \beta \lvert\downarrow\rangle_A .$$
This is why physicists often poetically say that the unmeasured particle is in a state of both spins at the same time while it’s more accurate to say it has no definite state. Mathematically, its state is an amalgam of all possible, linearly superposed (added together), algebraic solutions to the Schrödinger equation, hence, it’s said to be in quantum superposition.
What’s that $\alpha$ and $\beta$, you ask? Well, they’re numbers of probability we need to find in order to complete our expression. The Born rule states that if we square the (modulus of the) wave function (the state), we will get the probability (density) of either possible outcome after measurement. Now, experiments have shown that either outcome, spin up or spin down, $\lvert\uparrow\rangle_A$ or $\lvert\downarrow\rangle_A$, appears in 50% of the total number of measurements. In other words, the probability of measuring either spin state is exactly $\frac{1}{2}$. So, if we put $\alpha=\beta=\frac{1}{\sqrt{2}}$, then $\lvert\alpha\rvert^2 = \lvert\beta\rvert^2 = \frac{1}{2}$. After all, $(\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$, which is exactly what we want. So, the state (wave function) of our Amsterdam electron with respect to spin can be represented by
What you need to take from this is the following: the state of an electron before measurement is the sum of all possible states (multiplied by a probability factor, in this case $\frac{1}{\sqrt{2}}$).
In the case of spin as measured along the vertical axis, the state of the electron is the sum of two possible states, spin up $\lvert \uparrow \rangle$ or spin down $\lvert \downarrow \rangle$.
Note that there are other possibilities: we could measure the spin along a horizontal axis. We could represent this with spin left $\lvert \leftarrow \rangle$ or spin right $\lvert \rightarrow \rangle$. Or we could measure the spin at angles of +120 or -120 degrees from the vertical axis, which we might represent as $\lvert \nwarrow \rangle$ and $\lvert \searrow \rangle$ or $\lvert \nearrow \rangle$ and $\lvert \swarrow \rangle$. We will get to that in the discussion of Bell’s Theorem in the next post.
Quantum entanglement
So, what is quantum entanglement? Recall that the most complete description of a particle is the wave function. This has always been about a free, single particle, not interacting with anything. In the case of quantum entanglement, however, this doesn’t fly anymore.
When the state of a particle can no longer be described without a description of the state of another particle, those two particles are said to be quantum entangled. No longer can we describe either particle by one wave function each. They can only be described as a two-particle system by one and the same wave function.
This has an astonishing consequence. Suppose our two electrons become entangled in such a way that they always have opposite spins(beginfootnote)Producing spin-entangled electrons is difficult but clever experimental physicists have their ways.(endfootnote). So, if one has ‘spin up’, $\lvert \uparrow \rangle$, the other always has ‘spin down’, $\lvert \downarrow \rangle$, or vice versa(beginfootnote)It’s also possible to have them correlate such that they have identical spin, but for our example, let’s not.(endfootnote). So, we now have one system with two particles who always have opposite spins, which means that the total spin of our system is 0, zero. Let’s denote the total spin of our system with $\lvert S \rangle$.
Before our experiment takes place, they are both separated. One is staying in a laboratory in Amsterdam. The other is transported to Boston. Since no measurement has taken place on either particle, they are in a superposition according to the one wave function. They haven’t an exact location (although one is very likely to be somewhere in Amsterdam at the moment of measurement and, likewise, the other in Boston), their energy levels are all over the place, and their spin isn’t either spin up or spin down along this or that axis.
We can represent this whole situation with respect to spins as follows:
When you’re looking carefully at the expression above, you can see that the state of the total spin $\lvert S \rangle$ of our two-particle system is a combination of two situations: the electron in Amsterdam is spin up and so the electron in Boston is spin down or the electron in Amsterdam is spin down and the electron in Boston is spin up. They need to be subtracted from each other because the total spin equals 0, remember? Hence, the minus sign. Lastly, both states are multiplied by the fraction $\frac{1}{\sqrt{2}}$ because both states have a 50% chance of occurring (which you get if you square the whole thing).
And so, what does this mean? As soon as you perform measurements on the one in Amsterdam, and you find it has spin up, the other electron in Boston immediately has spin down along that particular axis upon measurement, even though the probability before measurement was still 50%! How does the electron in Boston ‘know’ what the measurement result in Amsterdam was? En how does it know this so fast? Faster than the speed of light! Besides this, turns out, you’ll always get a definite spin from the other particle opposite to the one you measured first. As soon as the measurement in Amsterdam took place, the measurement outcome in Boston being the opposite result is always 100% all of a sudden! (Or the other way around.) There are never any exceptions!
In other words, as soon as you do the measurement, the mathematical description changes from
And here’s the astonishing part: this will always work this way, no matter how great the physical distance between the two particles. Locality out the window. Welcome back, non-locality.
Einstein accepted this prediction in quantum mechanics as being correct. However, he didn’t like it. How did the other particle instantly ‘know’ which spin to exhibit when Einstein’s fantastically successful theories of relativity relied on the universal law that nothing can exceed the speed of light? He accepted the theory but he concluded it wasn’t complete. There had to be some sort of hidden mechanism which they had overlooked.
We will discuss Einstein’s attempt at saving the principle of locality and the universal speed limit in the next post. As well as John Bell’s and Alain Aspect’s subsequent work. For now, the question of whether Einstein was right, we will ‘leave up to the Consideration of our readers.’
[1] Newton, I. (1756) Four Letters from Sir Isaac Newton to Doctor Bentley: Containing Some Arguments in Proof of a Deity [Online]. Available here. (Accessed: 14 May 2020)
When micro- or molecular biologists do research on viruses, bacteria, fungi, human or animal cells, one of the many instruments they will use is a laboratory centrifuge. This equipment allows them to separate substances contained within a test tube. This way scientists are able to obtain, for instance, purified enveloped viruses, such as the novel coronavirus, SARS-CoV-2. Or they can isolate nucleic acids, such as DNA.
Often, the rotor of the machine rotates at incredible speeds. It is vital that the test tubes have been placed in a perfectly balanced way. If not, the machine might break down and potentially dangerous glass shards and substances might be flinging about(beginfootnote)Although sensors may be installed to prevent the machine from operating in case of force imbalance. See also the Final remarks down below.(endfootnote).
Fortunately, there is a nifty way to calculate whether you can – in principle – place a certain number of test tubes in an evenly balanced way. To crack the code, we will use my favourite type of number: the prime numbers. Fun fact: this funky little trick wasn’t proven until fairly recently in 2010.
Before we begin, we assume that the mass of each test tube, including their contents, is equal. Also, I would like to remark that, of course, we could do this the physics way, using angular velocity and torque and all that, but in this case, we’re going to be all mathy about it, or specifically, in a way, number-theoretical.
Suppose, the machine can hold eight test tubes. Eight holes are positioned in a circle on the rotor bit of the machine.
If we have just one test tube, there’s no way we can make it balanced. That much is clear. If we have two test tubes, however, no problem. They can be balanced easily. Just put one on either side precisely opposite each other. Three test tubes? Hm. I don’t see how. Whatever arrangement we try, it’s always going to be asymmetrical. What if you have four test tubes? Well, this is easy enough. Make it symmetric, like a square.
Okay, so what about five test tubes? Well, that’s just the same as when we had the inverse of this, with three test tubes! That couldn’t be done, so, this can’t be done either.
Six? Yeah, of course, we can do that. It’s just the same as having two test tubes, it’s just the inverse! Three on one side and three on the other side. Now you have two open spots on either side. Perfectly symmetrical, just like the inverse situation, where you had two test tubes and six open spots.
Seven? No. You will have guessed it by now. Having seven test tubes is exactly the same as having just one test tube in a rotor with eight spots.
And eight, well, of course, we can do eight. It’s also the exact same as having no test tubes at all. So, yes, that’s balanced.
Do you see a pattern here? You might. Notice how the number of occupied spots and empty spots always complement each other.
Prime factorization
Just for clarity’s sake, I’m going to call whole numbers integers since that’s what they’re called in mathematics.
So, I’m assuming we all know what a prime number is: an integer greater than 1 which cannot be formed by multiplying two smaller integers. In high school or even in primary school, you may have been taught that prime numbers are numbers which can only be divided by 1 or by itself (not including 1). So, prime numbers are 2, 3, 5, 7, 11, 13, 17 and so on.
Prime factorization is writing down any non-prime integer as a multiplication of two or more prime numbers. The fundamental theorem of arithmetic states that any integer is either itself a prime number or can be written as a product of prime numbers. This is one of the reasons why they’re my favourite. Primes are the building blocks of any integer.
So, for instance, we take the number 15. This number can be written as $ 15 = 3 \times 5 $. Or take 279. We can write $ 279 = 3 \times 3 \times 31 = 3^2 \times 31 $. Let’s take 16. This number can be written down as $ 16 = 2 \times 2 \times 2 \times 2= 2^4 $.
As you can see, prime factorization is pulling apart a non-prime number into a product of prime numbers. We call the latter prime factors.
So, that’s what that is. One of the many applications of prime factorization is finding the greatest common divisor between two integers, for example. Or encrypting (and decrypting) secret files and messages. Here, we’re going to use it for calculating whether test tubes can be arranged in a balanced way.
The trick
Suppose, your machine has $n$ spots available. Suppose, $k$ is the number of test tubes. The number of empty spots is $n-k$. Here’s the trick.
Determine the prime factors of $n$. If (and only if) $k$ can be written as a sum of these prime factors and the number of empty spots $n-k$ can be written as a sum of these prime factors, you can in principle balance the rotor.
The mathematics
It’s too technical to discuss at length the proof given by Gary Sivek in his 2010 paper (or here). However, the gist for the more mathematically inclined is available by clicking ‘expand’. You may skip this paragraph if this is (understandably) still too technical.
Expand
Striving to obtain an $n$-th cyclotomic polynomial (or prime polynomial), we obtain a series of complex numbers $z^n$ which satisfy $z^n = 1$, all being $n$-th roots of unity where $n$ is the number of total spots on the centrifuge. We then map the test tubes onto the roots of unity in a non-overlapping way. As is well-known, the values of $z \in \mathbb{C}$ are given by $e^{\frac{2\pi i}{n} k}$, where $1 \leqslant k \leqslant n$.
So, now we have $k$ roots of unity among the $n$-th roots of unity representing the occupied spots in the centrifuge.
Sivek proved, using Leung’s and Lam’s Theorem, that if (and only if) the sum of the $n$-powered $k$ roots of unity and the sum of the $n$-powered $n-k$ roots ‘vanish’, i.e. are equal to zero (using good-old de Moivre’s formula, if you remember from your very first semester at uni), as long as $n \geqslant 2$ and $1 \leqslant k l\eqslant n-1 $, then balancing is a fact (where $k=0$ and $k=n$ were regarded to be trivial cases for obvious reasons).
As you can see, no classical mechanics required.
An example with eight roots of unity in the complex plane
Obvious examples
Suppose, we take our centrifuge which was capable of handling 8 test tubes. We have 6 test tubes. First thing we do is calculate which prime factors the number 8 has. We know this, it’s all 2s. So, the only prime factor of 8 is 2. We can write the number of test tubes, 6, as a sum of this prime factor 2: $6 = 2 + 2 + 2$. The number of empty spots, that’s $8-6 = 2$, is the prime factor itself! So, yes, if you have 6 test tubes, you can balance the machine.
Let’s take 7 test tubes. Can this be written as a sum of the prime factors of 8? No, it can’t. Well, that’s it then. We cannot arrange the test tubes in such a way that it’ll be balanced out.
A counter-intuitive example
Suppose, our centrifuge is capable of handling 12 test tubes in total. We only have 7 test tubes. Hm. Surely, we can imagine 6 test tubes working, but can we make a balanced arrangement with 7 test tubes?
Let’s first do some prime factorization with 12. So, $ 12 = 2 times 2 times 3 = 2^2 times 3 $. In other words, the prime factors of 12 are 2 and 3.
Now, can we write 7 as a sum of these prime factors? Yes, we can: $7 = 2 + 2 + 3$. Okay, so far, so good. Can we write the number of empty spots as a sum of these prime factors? Well, $12-7 = 5$. And yes, we can also write 5 as a sum of 2s and 3s: $5 = 2 + 3$.
So, yes, we can balance 7 test tubes in a rotor with 12 spots! It’s likely this outcome wasn’t immediately apparent to you. If you were to see or draw a depiction and a working out of the arrangement yourself, however, I think it’ll become clear how this would work. Bonus points if you can draw a balanced configuration for 5 test tubes. Because you should know by now, you can.
Bonus trick
The beauty of it all is that all of the above does give us another quick way to assess whether we can balance the centrifuge. I’m going to be honest with you: it may be the easiest. If you can express the number of test tubes as the sum of two numbers of which you already know you can balance the rotor, then you can balance the rotor. Heh.
Final remarks
In real life, most machines have sensors to prevent force imbalances from taking over. The rotors have markings so that users won’t have to think about where to place the test tubes. Besides, in a university lab, you would simply make sure you prepare the number of samples which make a balancing act trivial. Moreover, many rotors contain three compartments containing sets of test tubes. This makes adjusting for mass variability much easier. And some machines, in hospital labs, for instance, have fully automated robots doing the heavy lifting.
Therefore, the reason for why this type of mathematics is done, isn’t so much for the applicability as it is for the joy of exploring deep connections such as between prime numbers and complex geometry, if you will. It’s first and foremost a fun and fruitful exercise of human exploration of the lands of number theory, algebraic geometry, and finite fields, on the continent that is pure mathematics.
Over three hundred years ago, grumpy old men with 17th-century wigs or 18th-century black-ribboned man ponytails were divided into two camps. They were squabbling over what type of phenomenon light is. ‘Light is waves’, said Huygens, Hooke, Euler and friends. ‘No no, light is particles’, said Newton, Laplace and colleagues. Fast forward to 1990 and even my physics teacher in high school confesses he still wasn’t sure about the correct answer.
His confusion is understandable. Even though he could have known the correct way of thinking about it, the reason for these murky waters can be traced back to the now famous set of double-slit experiments.
So, join me in tumbling through the slits of science, into the mad world of quantum physics, where one thing was proven be in two places at once. Or was it?
(not really meant as podcast since referrals are made to figures in the article)
The set-up
Suppose you had a shotgun capable of spraying a cloud of numerous tiny lead pellets in one shot. If you’d aim it at a screen containing two thin slits, so only some might get through, what shooting pattern should you expect to appear on a screen behind it?
Figure 1. A screen with two slits
I’m quite confident your answer will correlate strongly with the situation as depicted in Figure 2.
Figure 2. The screen with the double slits and a screen behind it with the typical impact pattern of pellets or particles
This is exactly what you would expect if the things you’re using to shoot with are tiny pellets or tiny particles. No surprise here.
Now imagine, we’d slowly submerge the screen with the two slits half-way into a pond. Water waves are slowly rolling towards the first screen as depicted in Figure 3.
Figure 3. Both screens are now partly submerged in water. Water waves are approaching the first screen.
What would these waves look like after they’ve gone through the slits? When seen from above, it would look like Figure 4.
Figure 4. As the waves go through the two slits, they transform into two circularly spreading waves like two stones in a pond.
The two slits transform the waves into two circularly spreading waves. Like two stones thrown into a pond. You can see how the waves will intersect with each other. You might expect some interaction to occur at these crossroads and you would be right.
In fact, let’s have a look at a real pond. In the GIF of Figure 5, you can clearly see how these two circular waves interfere with each other. If two crests meet, they amplify each other’s amplitude, whereas two troughs meeting, they amplify each other’s trough-ness (also amplitude but in the other direction). And where a crest meets a trough, they cancel each other out!
Figure 5. Two circular waves in an actual pond.
Now have a look at the animation of Figure 6 and observe especially what the second screen receives: patches where the waves hit the screen are white and patches where there are no waves at all are black.
Figure 6. The white areas are where the (amplified) waves hit the second screen, the black areas are the parts where no waves are present due to mutual cancellation
So, now we know what happens if waves would be thrown at the two slits. Contrary to what you see when you would shoot pellets towards the screen, you would see what’s depicted in Figure 7.
Figure 7. The double-slit set-up with the typical pattern on the second screen when waves have gone through
So, now we have two options. If whatever we’re shooting at the slits is particles, we get what’s on the left in Figure 8. If we’re aiming waves at the slits, we get what is on the right in Figure 8.
Figure 8. If particles went through the slits, you get to see the pattern on the left. If it’s waves, you get the pattern on the right.
Young’s interference experiment
Thomas Young was a polymath and physician. In the 1790s, he wrote a thesis on the physical and mathematical properties of sound. In 1800, he presented the Royal Society, the UK’s national academy of sciences, his theory that light is a wave too. He was met with great skepticism as the likes of Newton and Laplace were proponents of the light-is-particles theory.
Young then showed how they were wrong. A notable fact is that he didn’t actually use two slits. He had a bundle of sunlight pass through a pinhole so as to obtain a very tiny bundle of sunlight. He then placed a ‘slip of card’ in front of the pinhole, essentially splitting the small bundle in two even smaller bundles which then interfere with each other. The resulting light pattern would have looked like the one shown in Figure 9.
Figure 9. The pattern which Young produced by splitting sunlight
If light were particles, you would have seen an entirely different pattern. This result, however, completely corresponds to the wave theory of light. Young concluded therefore that light is indeed a wave phenomenon. He called this the most important of his achievements.
This marked the beginning of the acceptance of the wave theory of light (yay for Huygens and friends) and a departure from the particle theory of light (nay for Newton and fr… well, colleagues, at least).
Or particles after all?
Figure 10. Individual electrons
Of course, Max Planck, Albert Einstein, and a few other colleagues would later show that light is particles after all. In a previous post, The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time, we discussed Einstein’s finding which won him the Nobel Prize.
In short, Max Planck and Albert Einstein showed that certain behaviour of light could only be explained if it consisted of small packets of energy, quanta as they were labelled.
But apart from that, experimenters found another peculiarity. In the 1960s, electrons were generally expected to behave like particles – like pellets or ball bearings. So, instead of light, they fired one electron at a time towards a splitter and have a screen behind that capture the electron. What they initially saw was to be expected. A few (11) loose dots on the screen as shown in Figure 10. However, as the individual electrons kept being fired, one after the other, an astonishing pattern started to emerge – the kind you would expect to see in the case of interfering waves! Wait, what?
Are they waves after all? But they were individual electrons! How?
Needless to say, experimenters did the same thing with individual photons, the quanta of light Max Planck and Einstein were talking about. Extremely low-intensity light was produced up to the point where single photons were shot at the screen. The same result. They seem to behave like particles at first but then this wave pattern emerges.
Two places at once?
Theorists then theorised that the only explanation was that a single electron and a single photon somehow went through the two slits at the same time, enabling some kind of self-interference so that this wave pattern would emerge while also preserving a particle pattern at the same time.
To test this theory, people put particle detectors at the two slits in order to see if the single electron or the single photon indeed flew through both slits at the same time.
The result was again astonishing: the wave pattern disappeared and what they got was instead the pattern you’d expect to see if the particles were actual particles – the pattern was like the pattern in Figure 2. At the same time, they never detected the particle at both detectors. They were ever only detected by one detector – as if they were particles.
As soon as they removed the detectors, however, the wave pattern emerged again.
And even if they placed just one detector at one slit, the wave pattern disappeared again and the particle pattern showed up.
It was as if the electron and the photon knew when they were being watched and then decided to behave differently.
This is called the measurement problem. In the next post, we will discuss this at greater depth.
People now started talking about the wave-particle duality of elementary particles. Are particles truly particles or waves? They’re both, people now said. Sometimes they’re waves, sometimes they’re particles.
Fields
Of course, nowadays, the reigning theoretical paradigm is quantum field theory – mathematical field descriptions to capture the behaviour of ‘particles’ such as the electron, the photon, and a whole zoo of elementary constituents of our reality. The most successful quantum field theory to date is called the Standard Model of particle physics. In a previous post, Why, exactly, do glass and liquids refract light?, we dive a little bit into quantum field theory.
In short, the question of whether light is particles or waves has been answered: it’s fields. The same goes for electrons. And all the other elementary ‘particles’. It’s all fields.
As long as no interaction with the outside world such as detectors take place, a photon or electron are part of the wave functions of their respective electromagnetic and electron fields, governed by the Schrödinger equation. They are very much like waves. However, as soon as they interact with something, such as a detector, what is detected is a particle, merely a slice of a photon’s or electron’s entire wave function.
I promise we will unpack these two last paragraphs in a later post. We expounded on that a little bit already in This is not an atom.
But, please, tell me now, are they in two places?
No, not even technically. Linguistically then? Also, no. That statement is likely the result of mixing-up or lack for a better way of providing both metaphorical and physical descriptions of what is going on – it also reveals the still-present, outdated notion of what photons and electrons were supposed to be. If electrons were in two places at once, you still imagine them being small, little pellets, two copies of which fly through both slits, somehow interfering with each other. And that’s just not so.
The correct expression is that electrons and photons and the likes don’t have a definite location: their existence is simply spread out in space according to the wave function, the time-evolution of which in turn obeys the Schrödinger equation. Again, do give This is not an atom a read where this is explained in more detail.
Pretty mind-bending stuff, right? Good. Welcome to the club. Great minds before you have had to take their time to wrap their heads around the double-slit experiment. Now you’re one of them.
Sunlight diffraction pattern by Aleksandr Berdnikov under CC BY-SA 4.0
Single electron build-up series. Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000 (e). By Belsazar under CC BY-SA 3.0.
This Conjecture is probably one of the easiest to understand which hasn’t yet been proven in the history of mathematics. The beauty of this one is that a student in the last forms of primary school might very well be able to do the calculations while, thus far, not even the greatest mathematical minds have been able to prove if and why the Collatz Conjecture is true or not. The great, late Hungarian mathematician Paul Erdős has been quoted as saying: ‘Mathematics may not be ready for such problems.’[1]
The rules
There is some controversy over whether the prolific German mathematician Lothar Collatz was actually the first to come up with the idea in 1937, two years after his receiving his doctorate. It is also known as the ‘3n + 1 problem’, the Ulam conjecture, Kakutani’s Problem, the Thwaites Conjecture, Hasse’s Algorithm or the Syracuse Problem. If you were under the impression most of these refer to other, actual people then you are correct.
As the rules of the Conjecture are so simple, it is likely many people have had the same idea independently of one another.
Here are the rules:
Take any positive, whole number – a positive integer, as it’s called.
Do either of the following:
if the number is even, divide by 2;
if the number is odd, multiply by 3, add 1.
Take the result and do either of the following:
if the result is 1, stop;
if the result is not 1, go back and do step 2 again but this time using the result to do either of the two operations, and so on.
The Collatz Conjecture goes as follows: no matter which positive integer you start from, irrespective of the number of steps, you will always get 1 as final outcome.
(Note that if you would continue to do the steps with 1, you would simply cycle back to 1 in just three steps until the end of times. I mean, that’s just boring and silly. So, stop at 1.)
Example
Let’s try it out. Let’s start with 10. 10 is even; divided by 2 equals 5. 5 is uneven; multiplied by 3 plus 1 equals 16. 16 is even; divided by 2 equals 8. 8 is even; divided by 2 equals 4. 4 is even; divided by 2 equals 2. 2 is even; divided by 2 equals 1. We’re there!
This took us 6 steps. You can try a few numbers yourself if you’d like. Well, that is, have your computer, mobile phone or tablet do the boring work, which you can do here.
Visualisation
As you probably saw via the link above, we can also visualise the steps produced by the Collatz algorithm. We will quickly show a few alternatives before moving on to the most famous one, the Edmund Harriss visualisation (for which we wrote a JavaScript applet, yay!).
Suppose, we would plot the progression of the results of our example. We started with 10. As you can see, the values fluctuate a bit before descending to 1:
That’s a whole lot of numbers before the algorithm leads to 1. Hundred and fourteen steps, to be precise. This is what it looks like:
As you can see, the whole plot fluctuates quite a bit. It’s a bit of a mess, really. There’s no distinct pattern other than it eventually converging to 1. As the values may become quite large very quickly, let’s plot the same graph in a semi-log grid. The y-axis is logarithmic, the x-axis remains linear.
Just to humour ourselves, let’s reverse the step order, so that the plot is mirrored and converges to the value 1 in the bottom-left corner:
As we’re not sure how many steps it might take before a sequence of numbers ends with 1, let’s also change the x-axis to a log scale. We get this:
With this one, we can probably visualise a whole bunch of sequences! Lastly, let’s now plot a series of sequences! That is, firstly, we let our JavaScript applet calculate the sequence starting at 10000. Then we let it calculate the sequence starting at 9999. And so on, downwards, until it reaches 4. And then have it all plotted, all at once! To prevent it from becoming too dense, as several sequences will overlap each other, we add a little transparency to each plot. If the same ‘path’ has been taken, that path will appear ‘darker’. This is the result:
This almost becomes some form of art. If you would frame this plot, without the titles, axes, and scales – just the plot – you could have a piece of geometrical art bearing the title ‘The Collatz Conjecture’ or something like that. I might do that, actually.
With this applet you can generate your own ‘art’ like the one above.
The Edmund Harriss visualisation
Edmund Harriss, a mathematician working at the University of Arkansas, came up with a beautiful visualisation of progressions of Collatz sequences, which was featured on the mathematical YouTube channel Numberphile. We highly recommend following their channel.
A screenshot of Numberphile’s video showing a partly hand-made version of Edmund Harriss’s visualisation
The rules of visualisation are simple. While iterating through the Collatz rules, the algorithm goes as follows. If the current step is twice the value of the next step, rotate a fixed amount clockwise, otherwise rotate half of that fixed amount anticlockwise (and, again, stop at value 1). The result is a bundle of threads which looks like some sort of organic entity – messy and seemingly randomised within certain constraints, just like nature.
We wrote a JavaScript applet to try and produce an approximation of his visualisation. We used it to produce the featured image at the top of this page. You can try it here yourself.
Having visualised in several ways the kind of disorderly fashion in which the Collatz sequences progress, thus far, it may not seem surprising it has proven to be hard to crack the code. Well, the underlying mathematical code that is, not the JavaScript code.
[1] Guy, Richard K. (2004). “E17: Permutation Sequences”. Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 336–7. ISBN 0-387-20860-7. Zbl 1058.11001.
Today, many people know that all things around us – the chair we sit on, the screen we look at – are a composition of all sorts of different molecules and that they are in turn composed of all sorts of different atoms. Indeed, ancient Greek philosophers such as Democritus hypothesized matter consists of tiny, physically indivisible entities, which they then named atoms.
However, we also know that the Greeks weren’t entirely correct: the atom itself is composed of electrons, protons, and neutrons. We also know that the latter two are composed of even smaller things – quarks and gluons.
What not many people know, however, is that this classical picture:
is absolutely not what an atom is!
Old ideas
If you hung out in the wrong street corners, you might have been told that electrons whizz around the nucleus like tiny planets around the Sun or tiny moons around a planet. If that were the case then you have been lied to.
If you hung out in yet other unsavoury street corners, you might have been told that quantum mechanics is something magical, spiritual, and the doorway to a deeper understanding of love, consciousness, and healing. Telepathy, even. Again, you have been lied to.
Admittedly, the famous physicist Richard Feynman is often quoted as saying that nobody understands quantum mechanics. In a specific way, that’s true. Particles don’t behave like everyday objects and that is a strange fact. Furthermore, the mathematical descriptions of particles tell us what they do but not what they are. We know all the equations but we don’t know what they mean – as opposed to knowing the meaning of the words ‘microscopically tiny ball’.
However, this doesn’t mean we shouldn’t make an effort to making particles predictable, useful, and less mysterious and esoteric. It doesn’t mean we can’t harness the power of a good theory of quantum behaviour.
In fact, that’s exactly what we’ve been doing rather successfully since quantum mechanics took shape in the 1920s. Hence, the existence of your mobile phone, computers, cameras, and self-checkout in the supermarket.
So, let’s slice off the fat and cut to the chase.
Classical mechanics versus quantum mechanics
In high school, we were taught Newtonian mechanics. We were told that the world is reigned by Newton’s laws, the most powerful of them being the second: force equals mass times acceleration,
$F = ma.$
We were taught that when an object is moving, it moves according to Newton’s second law. The beauty of his mechanics was that physicists and engineers were now able to predict the future (and retrodict its past) of a sliding block, for instance, based on just a few known initial conditions.
Inclined plane problems are the staple in physics class for senior high school students
More generally speaking, and in physics jargon, Newtonian mechanics is capable of describing the state of a system over time with mathematically infinite precision based on a sufficient set of initial conditions. We call this a deterministic theory as it’s possible to determine past and future of the state of a system. Thanks to this property even space vessels such as the Apollo Lunar Module and Mars Rover Curiosity were able to arrive successfully at their extraterrestrial destinations.
In quantum mechanics, we study the behaviour of subatomic stuff, such as electrons and quarks. After many twists and turns throughout history, it turns out we can’t actually determine the past and future of, say, an electron – not as we could for blocks and balls in Newtonian mechanics. Why not? Because it’s simply not a tiny block or ball. It’s not even a particle in that sense (assuming a particle is like a tiny ball)! It’s a wave function, a mathematical expression describing all the possible states of a ‘particle’. Quite a different beast.
Possible states? Yes, in quantum mechanics, things aren’t so deterministic. Turns out that to describe the state of an electron, for example, Newton’s second law doesn’t apply. It’s fundamentally impossible to predict or retrodict where an electron will be at any given time, for instance. Or how fast it’s moving at a particular point in time. The best we can do is calculate the probability it’ll be here or there or whizzing at this or that velocity. In other words, the state of an electron can only be described in terms of probabilities.
It also turns out that the probabilities of this set of possible states may change over time. Luckily, like in Newtonian mechanics, there’s an equation for that. In quantum mechanics, the analogue of Newton’s second law is called the Schrödinger equation. It tells you how a wave function, i.e., the set of all possible states of a particle, changes over time. In its most compact form(beginfootnote)Although, technically, using Newton’s notation instead of Leibniz’s, an even more compact form is $i \hbar \dot{\Psi} = \hat{H} \Psi.$(endfootnote) it goes like this:
No need to understand all the symbols but here you can see that also in quantum mechanics there’s a beautiful equation at its centre, and it’s this one(beginfootnote)There are other ways to calculate the time-evolution of the wave function, of course, such as in Heisenberg’s matrix mechanics, Feynman’s path integral formulation, and Dirac’s formulation for matrix mechanics and the Schrödinger equation combined. However, this one is invariably taught at undergraduate level.(endfootnote). It tells you the evolution of $\Psi$, the symbol for the wave function.
Deterministic theories, such as Newtonian mechanics, are called ‘classical’ as opposed to quantum mechanics, dealing with probabilistic wave functions(beginfootnote)Note that the Schrödinger equation is deterministic. It’s the wave function itself that yields probabilities, or, if you’re a stickler for accuracy like me, it’s the wave function’s norm squared that yields probability densities (by integrating the norm squared over a volume, area or distance).(endfootnote).
Wave functions
As we’ve learnt in the previous section, a particle is not a particle. Granted, we still talk about a ‘particle’ but that’s only for lack of a better term. It’s an artefact of humankind’s limited understanding of the Universe back in the day. The idea of a particle simply fits among the things we already know. We can picture a little ball because we grew up playing with little balls. Or marbles, or whatever. Admittedly, sometimes particles do look like particles, which we’ll discuss in the last section.
Nevertheless, experiments from the early 20th century proved that tiny balls were definitely the wrong idea. Therefore, nowadays, our best descriptions of ‘particles’ are indeed wave functions, mathematical expressions. The fundamental question is whether the wave function is the particle or just a mathematical representation of it. This question hasn’t been answered yet, however, the personal opinion of the author of this post is that after about a century of the highly successful theory of quantum mechanics, it’s maybe time to start regarding the wave function as the thing that is a ‘particle’.
Just to illustrate the difference between a particle and wave, have a look at this point-like particle. The horizontal axis is the x-coordinate in space and the vertical axis is the y-coordinate in space.
Now tell me, where in space is the particle located? You probably got the answer straight away. It’s at coordinate (2,3). Good.
Now, have a look at this (two-dimensional) wave.
So, tell me, where in space is the wave located? You may find it harder to pinpoint the wave to a specific set of coordinates. That’s because it’s in several places at once. It doesn’t have a specific position. In physics speak, this is called a superposition.
This is also the case for an electron (or any other ‘particle’). It’s in a superposition, and not just in terms of its position: it’s also in a superposition in terms of its energy, momentum, and a few other properties. In other words, it’s in all places at once at several energy levels at once, whizzing at all kinds of velocities at once.
Note, however, that the wave function is absolutely not the same as a simple sine wave in normal space which was merely displayed here for reasons of clarity(beginfootnote)And, indeed, those who read a previous post on light refraction in glass know that ‘particles’ are oscillations in their respective three-dimensional quantum fields in quantum field theory. The wave function plays a central role in this highly successful theory. Secondly, the wave function is a so-called complex function and therefore exists in so-called complex space $\mathbb{C}$, not in the ‘regular’ number space $\mathbb{R}$, we all grew up with. Lastly, and at the cutting edge of our scientific knowledge, there is actually only one wave function, the wave function of the Universe. Every field and particle in it are mere parts of that wave function, which can be thought of as small, individual wave functions to keep it manageable.(endfootnote).
Maybe now you can appreciate how revolutionary quantum mechanics truly is compared to the simple mechanics of the blocks and tiny balls of everyday life.
Picture of an atom
So, we’ve arrived at the correct picture of an atom. We already learnt that an electron isn’t a point-like particle, it’s a wave function. What do wave functions look like then? Well, they are cloud-like but not clouds, smeared out in space, yet both size- and location-less. They don’t have a specific position, they don’t have a specific momentum, they don’t have a specific energy value. They are in a superposition of all these possible states. The probabilities of these states may oscillate over time as dictated by the Schrödinger equation.
That doesn’t help much, does it?
Well, fear not. Dillon Berger, a PhD student of Theoretical Particle Physics at UC Irvine, made a beautiful animation of a cross-section of a hydrogen atom using the Schrödinger equation. The contours represent the wave function of the electron. The colours denote the probability of the electron being in that particular state. Note, there is only one electron in a hydrogen atom. So, yes. It’s almost everywhere at the same time, while also oscillating over time. (This animation has time slowed down by a thousand trillion. The nucleus, a proton, is too small, so it’s invisible.)
Time Evolution of Hydrogen in Superposition of Energy Eigenstates (3,2,0 & 4,1,0 states) •The probability density of the electron's position oscillates in time when in a superposition of states of definite energy pic.twitter.com/Yx8oFPaKb5
That whole tiny balls or planets revolving around the nucleus analogy? Flush it out of your system. For good.
Unless we’re looking
Now, hold on, you might say. Why is it then that professional physicists still talk of particles? And what did you mean, when, earlier, you said they’re size-less? How then do you explain the fact that in scientific tables we saw in high school, actual, physical sizes of particles are listed? And how do you explain this classical picture of the readout of a cloud chamber demonstrating the existence of a subatomic particle? That trajectory certainly looks like it was created by a point-like particle and not at all a wave.
Source: Anderson, Carl D. (1933). “The Positive Electron”. Physical Review 43 (6): 491–494
You’re quite right to doubt the whole story about wave functions in the face of these empirical findings. You’ve also arrived at a mystery that is at the heart of quantum mechanics, worthy of a Nobel Prize, which, of course, has a name: the measurement problem.
Turns out, particles are indeed wave functions but only if we’re not looking. As soon as we do measurements, trying to gauge their position, for example, we won’t find them at all places at once, like a wave. Instead, we will find them at one particular location – just as we would expect from an actual particle!
This is what the famed double-slit experiment demonstrated. In another post, we discussed this.
This is why, to this day, you might have heard of the ‘wave-particle duality’ of the subatomic world. The wave function is the most complete description of a particle. As soon as we do measurements, we see only a sliver of its original wave function, a mere shard of the set of all possible states.
Therefore, Dillon Berger’s animation shows a hydrogen atom when left completely alone. This is its fundamental state of being: its electron being a wave function in superposition, oscillating over time according to the Schrödinger equation. And as soon as it interacts with its environment, only a metaphorical slice of its full existence will show (a slice corresponding to the disguise of an actual particle).
How this happens or what actually happens when we do measurements is still up for debate. We will most certainly dive deeper into a variety of views on how to tackle this phenomenon in another post. Expect a post on the Copenhagen interpretation of quantum mechanics, the Many-Worlds interpretation, and others soon.
Now you have a better mental picture of an atom, at least. Probably.
While it’s one of the most well-known and well-trodden proofs among proofs, the irrationality of $\sqrt{2}$ shouldn’t be lacking on a blog about mathematics and physics. So, here it goes.
What is irrationality?
For those who aren’t too familiar with mathematical jargon, let’s first discuss what it means to be irrational. Obviously, we’re not talking about the psychological attribute but the mathematical one.
You might remember primary school when you had to learn about fractions such as
A practical application of a fraction is when you were reading a recipe for a delicious dish with a certain ratio of water and rice, which, even if you might not be aware of it all the time, can be written as a fraction, representing the ratio between water and rice. In fact, a fraction is a ratio.
For instance, in order to cook the perfect, fluffy rice without the need to pour off excess water when the rice is cooked, the ratio is that for 1 cup of rice, you add 1.5 cups of water(beginfootnote)Rinse the rice thoroughly to remove the starch and dust for a nice fluffy texture. Add water by 1.5 times the used volume of rice. Add salt if you must. Bring the water to the boil as quickly as possible. Bring down the heat but keep the water bubbling softly. Give it one good stir. Put the lid on and don’t remove it for eight minutes. Don’t look inside; the water needs to stay in the pan. After eight minutes, shut off the heat and let it rest for another eight minutes. Still, don’t look. Keep the lid on the whole time. That’s it.(endfootnote) So, the fraction is $\frac{1}{1.5}$.
Of course, it’s conventional to write a fraction using whole numbers (integers) only, so, $\frac{1}{1.5} = \frac{2}{3}$. Just multiply the numerator and the denominator by two. In other words, for 2 cups of rice, add 3 cups of water.
If we use our calculator, we get $\frac{2}{3} = 0.666\dots$ There is no end to this number, but the number can be perfectly written down as a ratio: $\frac{2}{3}$.
Of course, $\frac{2}{3}$ is the same as $\frac{4}{6}$, or $\frac{10}{15}$, or $\frac{200}{300}$, since, and this is crucial, all the other fractions (ratios) are simply multiples of our original fraction: they can all be simplified to their ‘simplest’ form, $\frac{2}{3}$. A slightly more technical way of saying this is that the fraction $\frac{2}{3}$ is the form in the lowest terms of the fraction $\frac{200}{300}$. It’s very important to remember this.
Now we’ve arrived at what irrational numbers are.
Premise 1. A number is irrational when it cannot be written as a ratio in lowest terms.
Two of the more well-known examples of irrational numbers are $\pi$ and $\sqrt{2}$. If we use our calculator, we can see how there seems to be no numerical repetition in them. This is a quality that irrational numbers possess.
Babylonian tablet clay tablet showing the root of 2 (credits below)
Proof by contradiction
So, how do we prove that $\sqrt{2}$ is irrational, i.e. it cannot be written as a ratio? We do this by contradiction: if the opposite of a statement is demonstrably false (and there are really only two options), then the statement itself must be true. In a previous article, we used the same strategy to prove that ‘minus minus is plus’.
We will do that here, too.
Even and odd
Premise 2
We will also use the fact that some number multiplied by 2 equals an even number. Take any number, odd or even, multiply that by 2, and you will get an even number. This isn’t rocket science, really, as a characteristic of an even number is that it’s divisible by 2. In our proof, we will use the symbol $k$ for ‘some number, any number, odd or even’ being multiplied by 2.
Premise 3
If you take the square of an odd number, the result is always odd. If you take the square of an even number, the result is always even. Conversely, if you take the root of an odd number, the result is always odd. The same idea goes for even numbers. Check in your head to see if that’s true (it is). We will provide a proof for that in another post.
Okay, ready? Let’s go.
Proof that the square root of 2 is irrational
Anti-Premise 1. Suppose, by contradiction, that $\sqrt{2}$ can be written as some ratio in lowest terms: some (whole) number $a$ divided by some other (whole) number $b$ in lowest terms.
Now, we see that $a^2$ is an even number as $b^2$ – whatever that number is – is multiplied by 2. It also means that $a$ cannot be an odd number – it’s even. Remember premise 3?
Conclusion 1: $a$ cannot be odd – it’s even.
We can then also state that $a = 2k$, where $k$ is some number, any number, odd or even. If we substitute that into equation (4), we get
This means that irrespective of what number $k^2$ is, because it’s multiplied by 2, the result is an even number. In other words, $b^2$ is an even number, which also means that $b$ is an even number.
Conclusion 2. $b$ cannot be an odd number – it’s also an even number.
Looking at conclusions 1 and 2, we arrive at
Conclusion 3: $\frac{a}{b}$ is not a ratio in the lowest whole numbers as $a$ and $b$ can still be divided by 2.
This is a contradiction. The fraction $\frac{a}{b}$ cannot both be the lowest fraction and not be the lowest fraction. Conclusion 3 contradicts Anti-Premise 1. Therefore, there is no fraction $\frac{a}{b}$ in lowest terms that exists that can be equal to $\sqrt{2}$. Hence, the original statement Premise 1 is true.
Credentials of the Babylonian tablet clay tablet showing the root of 2: Photograph by Bill Casselman under CC BY-SA 3.0, and the Yale Babylonian Collection as the original holder of the tablet. A black and white rendition of Casselman’s own photograph of the Yale Babylonian Collection‘s Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras’ Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888…(30 x square root of 2).
Probably the most famous equation on this planet is $E = mc^2$. Energy equals mass times the speed of light squared(beginfootnote)Usually, in mathematics, we leave out the multiplication sign ($\times$).(endfootnote). Usually, the formula is associated with Albert Einstein. This relationship between, energy, mass, and the speed of light, this equation, has a name: the mass-energy equivalence. Perhaps you’ve read or heard people explain that ‘Einstein taught us’, that mass is a form of energy, mass is frozen energy, mass can be converted into energy, and that, in the end, all matter is essentially pure energy.
The equation is, however, definitely not about all of that. At least, not in this Universe. Here, we will discuss the actual meaning of it. We think it’s time for disposing of some of the unnecessary obscurantism accompanying many popular explanations. We think the involved mathematicians and physicists of yore deserve better.
Albert Einstein (right) with Dutch physicist Paul Ehrenfest (left) and Ehrenfest’s son in Ehrenfest’s home in Leiden, The Netherlands.
Standing on the shoulders of giants
Einstein wasn’t the first to write down this very relationship between mass and energy. There were many others before him who, one way or the other, explored the connection between mass, energy, and velocity. However, most hypothesised that a mechanical mass increase was exclusively due to interactions with electromagnetic fields. They called it electromagnetic self-energy of some kind, giving rise to a form of electromagnetic mass (Miller, 1981; Okun, 1989). Einstein then showed that there was no need for such a concept and was the first to derive the relation correctly.
However, over the course of a few years, he published several derivations of the equation, none of which were literally written down as $E = mc^2$. You might not recognise them if you saw them. Furthermore, the versions he did write down, weren’t universally true.
Lastly, the famous equation isn’t the complete version. Usually, people only know the snazzy edition fitting on baseball caps. The full equation is valid in a more universal way and sometimes referred to by contemporary physicists as the ‘correct version’(beginfootnote)cf. https://youtu.be/mkiCPMjpysc(endfootnote). However, this one wasn’t first formulated by Einstein but by Paul Dirac (Eisberg & Resnick, 1974; Miller, 1981). We will get back to that.
Among others, these incredible minds have all derived and used some version of the famous equation before Einstein.
Energy is a mathematical idea
We should remind ourselves that energy isn’t any ‘thing’. As we mentioned in a previous article, it isn’t some invisible, immaterial, fluid-like ‘essence’ which everything is made of, within and behind the façade of the tangible world. Fork, no.
It has always been just a number, an accounting tool, an important and practical, mathematical concept, first proposed by the 17th century, German scholar Gottfried Wilhelm von Leibniz. How is it a mathematical concept? It’s the number you get when you multiply an object’s mass with its velocity squared(beginfootnote)Which was later calibrated to 1/2 times the mass times velocity squared.(endfootnote). It’s a pragmatic way of keeping the books on these two things.
Suppose, we have a billiard table with three billiard balls. We ignore any sort of friction. Imagine this isolated system of balls changing internally: the balls constantly collide and bounce off of the edge of the table. He assumed that what doesn’t change, is their mass. What does change, are their velocities. Leibniz then noticed that if you multiply for each ball its mass with its velocity squared, and summed all three products, that sum remained constant, irrespective of how the balls were bouncing in which direction, and how fast, at each point in time (until you changed something to the system by introducing a whack by a cue stick, for example).
This was the early formulation of what we now fanciful call the law of conservation of energy(beginfootnote)He called this product quantity vis viva, so, not even energy yet. Perhaps he was being poetic. Furthermore, Leibniz also had some fierce competition: his rival Newton had come up with a different quantity, a different way of keeping the books. He stated that the sum of the products of mass and velocity — just velocity, not velocity squared — remained constant. This is the law of conservation of momentum. Later, it was understood that the two laws were complimentary, not contradictory.(endfootnote), which, by the way, is true in a small enough patch of the Universe, such as Earth or the solar system, but in the context of the entire observable Universe, for instance, energy is not conserved. So, indeed, the law of conservation of energy is fundamental enough for us, Earthlings, and our physics experiments, however, contrary to what people usually think, in the grander scheme of the Universe, it’s not(beginfootnote)Courtesy of the genius Emmy Noether (Noether’s theorem) and Albert Einstein (general relativity).(endfootnote).
For the purpose of this post, however, this is irrelevant. What is important to note, is that since then, with their propensity to invent intricate systems of categorisation, humans have made distinctions between various forms of energy. Think of potential energy (something’s high up and can fall down or wound up and unwind rapidly), thermal energy (something’s hot), chemical energy (something’s ‘charged’), and kinetic energy (something moves). All of these, including their mother-concept ‘energy’, are human constructs. Thinking of any of these nouns as referring to physically separate, physical, tangible things is, ironically, a category mistake.
Sometimes, people misinterpreted Albert Einstein. In the old days, again, being the talented labellers that they are, humans split up the term ‘mass’ into rest mass and relativistic mass. Rest mass is the mass when the object is at rest. Relativistic mass is the mass when the object is in motion. If then the object would start to move faster and faster, then this particular mass would become larger and larger, because, you know, that’s what he said.
Well, no. He wrote in his third 1905 paper (1905a, p. 920), Zur Elektrodynamik bewegter Körper, an equation for the kinetic energy of an electron, which went as follows:
While it may not look like it, you could say that this was his first expression for the relationship between (kinetic) energy, mass, and speed of light squared(beginfootnote)Incidentally, Max Abraham had published Walter Kaufman’s work just before Einstein, showing the same equation for kinetic energy. Einstein probably wasn’t aware of this (Miller, 1981).(endfootnote). It does require a little translation, but it’s easy. Ignore the part in the middle, focus on the letter $W$, and the part behind the last equals sign. You should know that in modern notation, kinetic energy $W=E_k$, rest mass $mu=m_0$, and speed of light $V=c$. So, what it says is:
So, this is slowly starting to resemble our familiar $E=mc^2$. It doesn’t state, however that mass increases. It only says that when the speed of the object $v$ approaches the speed of light $c$, the result of this whole equation is infinity – infinite energy.
As the standing interpretation is that mass equals energy (mass-energy equivalence), people nevertheless concluded that if the energy of a moving object becomes infinite at the speed of light, that an object’s mass becomes infinite, or relativistic mass, to be precise (in the minds of the old folk).
This, however, is something of the past – well, technically. As soon as 1940, the great Lev Landau and Evgeny Lifshitz ignored the distinction between rest mass and relativistic mass in their book The Classical Theory of Fields. The legendary John A. Wheeler and Edwin F. Taylor also brought an up-to-date Spacetime Physics to the reading table. Unfortunately, many textbooks today still mention archaic notions, terms, and notation.
Contemporary professional physicists don’t speak of relativistic mass anymore. In special relativity, Einstein showed that observations and measurements depend on one’s frame of reference. By definition, there are at least a couple of things that do not depend on the motion of observers. Besides the spacetime interval, the laws of physics, and the speed of light, this turns out to be rest mass.
An object’s rest mass is invariant, i.e. it doesn’t vary or change, regardless of the motion of the observer relative to the object (Taylor & Wheeler, 1992, p. 211). And, as you can see in Einstein’s equation for the kinetic energy, only rest mass is used. No mentioning of relativistic mass whatsoever. In fact, Einstein himself wrote (as cited in Okun, 1989, p. 32):
‘It is not good to introduce the concept of mass $M = m/(\sqrt{1-v^2/c^2})$ of a body for which no clear definition can be given. It is better to introduce no other mass concept than the ‘rest mass’ $m$. Instead of introducing $M$ it is better to mention the expression for momentum and energy of a body in motion.’
Of course, $M$ is what humans would later call ‘relativistic mass’, which means that they did it anyway, against Einstein’s wishes.
Today, however – well, at least since 1940 – professional physicists speak only of mass. We tossed out relativistic mass as, with Einstein, it’s ‘not good’. Furthermore, the adjective ‘rest’ in rest mass is redundant. Mass is about an object at rest. If it’s in motion, we speak of the product of some proportion of mass and velocity: either momentum or energy. An object’s mass does not grow by its motion.
Resistance is crucial
So, what is mass then? Well, just to be clear, mass isn’t weight: the same amount of mass has different weight on different planets. We were taught this in high school. A spring scale measures weight, not mass. A balance with calibrated counter-weights – *cough* masses – is your best option during interplanetary travels. These masses should have been calibrated to the definition of a kilogram according to the International Bureau of Weights and Measures.
Mass isn’t matter either. Mass and matter are different categories. The first is a property, the second is a ‘thing’. Elementary ‘particles’, things(beginfootnote)We use quotations marks in ‘particles’ because, while it’s easier to use that term, we acknowledge it’s actually quantum field oscillators we should be talking about or, even prettier, wave functions.(endfootnote), such as the electron, have a certain amount of mass. An electron gains its mass through interacting with the Higgs field, the existence of which was proved in 2012 at CERN. And the amount of elementary particles does correlate with the amount of mass. However, the mass of an object isn’t defined by merely the amount of elementary particles: it’s also the motion of gluons inside protons, the motion of electrons, atoms, molecules – the kinetic, thermal, and chemical energy contained within the (resting) object.
Einstein wrote this in his fourth paper of 1905, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (1905b, p. 641):
If a body releases the energy L in the form of radiation, its mass decreases by $L/V^2$,
where, in modern notation, $L=E$, and $V=c$. Also, the energy he’s talking about is not the kinetic energy (of a body in motion) but the internal energy (of a body at rest), such as thermal energy. And yes, this does mean that mass increases or decreases depending on the object’s internal energy.
Two structurally identical balls of steel have different mass if one ball is hotter (more mass) than the other due to their diverging thermal energy content. Two structurally identical mobile phones have different mass if one is charged (more mass) and the other is out of juice (electrochemical energy). Note, we are talking about the whole object being at rest in its reference frame.
As soon as either object starts radiating light or heat, they lose mass. The fraction $E/c^2$, however, is very small because the speed of light is very high. And so, the extra mass gained or lost is so small that this may be a reason for people confusing mass with the amount of matter. It’s almost the same. It’s the amount of matter plus something more, its internal energy.
Mass could be best described by the resistance to acceleration – a ratio between the force needed to accelerate it to the extent it’s accelerating. Mass is inertial mass, an object’s inertia (as Einstein put it in the title of his 1905a paper).
The complete equation
If you’re not into equations any longer, skip this section. If you want to know the real equation, let’s go.
Einstein published several derivations. One of the more familiar was the following for the total energy:
If you just to happen to be fluent in algebra, then you could see how we obtain $E=mc^2$. If not, not to worry. If an object is at rest, is has no speed, so $v=0$. If you would fill in that number in the equation, the denominator of the big fraction becomes the value 1. And anything divided by 1 equals exactly that same anything. So, that means that what you get is $E=m_0c^2$.
Of course, since, nowadays, there is only one mass, which is $m$, since ‘rest’ is redundant, we should really leave out the subscript 0. This also means that the famous equation $E=mc^2$ is only applicable if the object isn’t moving in our (inertial) reference frame. Moreover, it’s not applicable to phenomena without mass either, such as a photon. Hence,
$E=mc^2$ is not universally true. Only in an inertial reference frame, where the object isn’t moving, and only in the case of ‘particles’ with mass, does this equation hold, so, this equation isn’t valid for photons and gluons.
This following equation, however, does hold for massless as well as massive particles, and while Einstein laid the groundwork, the genius Paul Dirac was to write this down for the first time in 1928 (Eisberg & Resnick, 1974; Miller, 1981), albeit in a slightly more technical fashion than presented here. The following equation handles all objects, including light:
The letter $p$ is the momentum. Suppose, we want to calculate the energy of a photon. Since the photon has no mass ($m=0$), this equation becomes $E = pc$, which is, indeed, the correct relation between energy and a photon. If you would try to use $E = mc^2$ to calculate the energy of a photon, you would get a silly answer.
So, $E = mc^2$ isn’t even a universal equation because it doesn’t fly for massless ‘particles’: photons and gluons. The equation first written down by Paul Dirac does, however. And it still fits on a T-shirt.
which makes it possibly even snazzier as it shows a beautiful Pythagorean relationship triangle.
Paul Dirac
The meaning of E = mc²
All well and good, but, technicalities aside, what does it mean?
What it means is that energy is mass, proportioned by a factor of $c^2$.
What it also means is that an object’s mass is a measure of its total intrinsic energy (potential, thermal, chemical, electrical, even kinetic, if parts inside the object have motion) proportioned by a factor of $1/c^2$.
$E = mc^2$ should actually be written $E_0 = mc^2$ as it’s about the energy of an object at rest and the subscript 0 usually denotes something at rest.
However, it isn’t the full equation.
What it doesn’t mean is that energy is matter, and, conversely, it doesn’t also mean that matter is energy. Mass isn’t matter. This is a category mistake.
It also doesn’t mean that mass can be converted into energy or vice versa. For one, mass cannot be converted as it isn’t a ‘thing’. Secondly, energy isn’t a ‘thing’ either. Mass is a property, a measurable property. Energy is also a property, a calculable property, which can be done by measuring mass.
Imagine an object had the following properties: size, colour, hardness, and energy. Suppose, the equation would have said $E =$ hardness $\times c^2$. Perhaps it’s more clear now that this doesn’t mean that hardness gets converted into energy. What it means, is that you have a mathematical way of calculating one measure in terms of the other measure. The only thing that’s being converted here, is a number, a quantity.
Matter is a different beast. It’s a clump of things: ‘particles’. An object is a clump of matter and matter interactions. As CERN show on a daily basis, matter in motion can be converted into a thousand other things in motion. If people insist on talking about things getting converted, then they could talk about converting particle A with motion $a$ and particle B with motion $b$ into particles C, D, E, F, G, $\dots$ with motions $c,d,e,f,g,\dots$
So, next time someone thinks they should explain to you that $E = mc^2$ means that mass can be converted into energy or that no object can gain the speed of light because its mass would become infinite, you can just reply with, ‘Nah, mate, mass is an invariant property of an object, calculable through the complete equation, you know, $E^2 = (pc)^2 + (mc^2)^2$. Although you would have to solve for $m$ and merely use the pseudo-Euclidean norm for momentum, not the whole four-vector, but that shouldn’t be a problem – it makes it easier.’ Then pause, and add, ‘In Minkowski space, obviously.’(beginfootnote)Minkowsi space is like Euclidean space but in four dimensions. This might be a good time to add the footnote that there is an even more fundamental equation, which is Einstein’s field equation of general relativity (Carroll, 2014), but that’s something to discuss at a later point in time.(endfootnote)
Also, pure energy = pure nonsense. If Leibniz were somehow able to hear this, he would cackle and turn over in his grave – if he could muster the energy for it. To be fair, physicists use the word energy all the time, all over the place. It’s short, sweet, and simple to use on a daily basis, which is fine, just as long as we’re all in agreement about what we mean.
Energy isn’t fundamental to motion, it’s motion(beginfootnote)Of quantum fields as described by wavefunctions(endfootnote) and interactions giving rise to the construct of energy.
However, if you do bump into a floating blob of pure energy down the long narrow hall upstairs of your rich aunt’s mansion, then, well, yes, that would most certainly be something strange.
References
Carroll, S. (2014) Spacetime and Geometry: Pearson New International Edition : an Introduction to General Relativity. 1st. Pearson.
Einstein, A. (1905a) ‘Zur Elektrodynamik bewegter Körper’, Annalen der Physik, 322(10), pp. 891-921.
Einstein, A. (1905b) ‘Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?’, Annalen der Physik, 323(13), pp. 639-641.
Eisberg, R. M. and Resnick, R. (1974) Quantum physics of atoms, molecules, solids, nuclei, and particles. New York: Wiley.
Miller, A. I. (1981) Albert Einstein’s special theory of relativity : emergence (1905) and early interpretation (1905-1911). Reading, Mass ;: Addison-Wesley.
Okun, L. B. (1989) ‘The Concept of Mass’, Physics Today, 42(6), pp. 31-36.
Taylor, E. F. and Wheeler, J. A. (1992) Spacetime physics : introduction to special relativity. 2nd ed. edn. New York: W.H. Freeman.
Featured image: NASA’s Solar Dynamics Observatory captured this image of an X2.0-class solar flare bursting off the lower right side of the sun on Oct. 27, 2014. The image shows a blend of extreme ultraviolet light with wavelengths of 131 and 171 Angstroms. Credit: NASA/SDO. Retrieved 30 Aug 2019, from https://www.nasa.gov/content/goddard/sun-release-x20-class-flare-on-oct-27-2014
Oliver Heaviside (1850-1925) – Science and Society Museum/ Universal Images Group. Oliver Heaviside, English physicist, c 1900.. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_541915/1/102_541915/cite
Hendrik Lorentz (1853-1928) – Science and Society Museum/ Universal Images Group. Hendrik Antoon Lorentz, Dutch physicist, c 1920.. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_523268/1/102_523268/cite
Henri Poincaré (1954-1912) – akg-images / Universal Images Group. Henri Poincare / Photo c. 1890. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 3 Sep 2019, from https://quest.eb.com/search/109_171035/1/109_171035/cite
Joseph J. Thomson (1856-1940) – Science and Society Museum/ Universal Images Group. Sir Joseph J. Thomson, English physicist, late 19th century/early 20th century.. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_547694/1/102_547694/cite. Cropped by @kjrunia.
George Frederick Charles Searl FRS(1864-1954) – Royal Society. As printed in Thomson, G. (1955) ‘George Frederick Charles Searle. 1864-1954’, Biographical Memoirs of Fellows of the Royal Society,1, p. 247. Cropped by @kjrunia.
Wilhelm Wien (1864-1928) – NATIONAL LIBRARY OF CONGRESS / SCIENCE PHOTO LIBRARY / Universal Images Group. Wilhelm Wien, German physicist. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/132_1255736/1/132_1255736/cite
Max Abraham (1875-1922) – Niedersächsische Staats- und Universitätsbibliothek, Göttingen. Max Abraham around 1905. Public domain. Slightly cropped by @kjrunia.
In 1905, Albert Einstein published an article on moving bodies and electrodynamics. He noticed that Newton’s mechanics of moving bodies weren’t compatible with Maxwell’s equations of electromagnetism. In this article, he reconciled the two by modifying the first. These ideas and mathematical derivations became what we now know as Einstein’s special (theory of) relativity. In this post, we will describe some of its important bits. We start with two fundamental propositions. For the geeks, we will end with answering why special relativity is special.
Einstein’s postulates
His first postulate is basically that, whether or not you’re standing on a moving object, such as a ship, a train, or in your car, the same laws of physics apply. If you’re standing on a platform at the train station and you throw a ball in the air, the laws of physics ensure your ball comes down again. If you’re standing in a moving train, that ball still comes down again because the same laws of physics apply. The fact that you’re in motion doesn’t change anything to the rules of the Universe.
Einstein’s second postulate is basically that, by extension of the first one, the speed of light is the same for everyone. He recognised that Maxwell and colleagues correctly describe light as an electromagnetic disturbance propagating according to the electromagnetic laws of physics. He also recognised that the velocity of the light source plays no role in Maxwell’s equations. And so, if the first postulate is correct, then irrespective of the velocity of an observer relative to the light source, light travels at the same speed $c$ ($c$ is about 300 000 000 m/s) and nothing can go faster.
Whether you’re on the train or standing on the platform, the laws of physics are the same. The propagation of light is a law of electrodynamics not involving the velocity of its source. Hence, its velocity is the same on the train as on the platform, irrespective of its source or the train’s velocity.
Intuition works mostly, just not really
Suppose, you’re standing on a train station’s platform. A train passes at a speed of 30 metre per second. From inside the train, our friend throws a tennis ball out the window but in the direction of where the train is headed, at 2 metre per second, right at you. You catch it. At what speed does the ball hit your hand?
Intuitively, you might say, that’s 30 + 2 = 32 metre per second. This way of calculating is very useful, most of the time. Instinctively, you would simply add the train’s speed and the throwing speed together. This is how Isaac Newton(beginfootnote)And Galileo Galilei before him as this is the so-called Galilean transformation.(endfootnote) would want you to do it. And, mostly, he’s not wrong. Except, well, he is kinda.
Let’s wonder what would happen if our friend didn’t throw a ball, but, instead, switched on a pocket torch. The train still passes at 30 m/s. Light, however, flies out the torch at a speed of about 300 000 000 m/s. You lift up your hand. It ‘catches’ the light. At what speed does it hit your hand?
You might say, that’s 30 + 300 000 000 = 300 000 030 m/s. But no. That’s wrong. Remember Einstein’s second postulate? Irrespective of the motion of the observer, light always travels at 300 000 000 m/s and nothing goes faster, full stop. So, by our simple addition, we would have invented a way for light to go faster than light! We would be Nobel Prize winners, surely. Except, it doesn’t, and we’re not.
Something’s gotta give
So, if the speed of light is the same to our friend, on the moving train, as it is to us, standing on the platform (do read this again and realise how bonkers this is), then how the Helheim does the Universe achieve this? After he did some relatively simple mathematics – which a student in secondary education can do – Einstein realised that something was up with metres and seconds. He proved that what is a metre to us isn’t a metre to our friend and vice versa. Furthermore, what is a second to us isn’t a second to our friend either.
Basically, since we, on the platform, measure light to be going at 300 000 000 metre per second, and so does our friend on the train, well, that means that our understanding of what metres and seconds are is wrong.
The point
Here’s what’s happening. If two ‘things’, people, ‘objects’, or reference frames as physicists call them, move with respect to each other, weird things happen to space and time. Yes, the actual space and time. They are weird. We thought they were just there. Static. Always and everywhere the same. Two unchanging entities. Well, they’re not.
In the Dutch town of Leiden, the project Leiden Wall Formulas have scattered physics equations throughout the town centre. This is the Lorentz contraction, aptly placed alongside a train track: it describes how space (in this case, a one-dimensional length) is contracted, according to Einstein’s special relativity. Click here for Google Street View.
Suppose, we are on the train this time. We are in motion relative to our friend on the platform. Our friend then observes that space along our direction of motion becomes smaller, it contracts. They will actually measure our train to be shorter as compared to when it was standing still relative to our friend. They will also observe that our time is being stretched, i.e. our clock slows down. If one second goes by on their clock, they see only 0.9999999995… seconds have past on ours(beginfootnote)This number is a metaphor. The real time difference is too small for my calculator to show.(endfootnote).
And to us, being on the train, it’s our friend who is travelling (backwards!) relative to us, in fact, the whole world is travelling relative to us, backwards. So, indeed, we measure the world to be shorter in the opposite direction of our motion as well as their time being slowed down.
Twin paradox
Now, you might say, hold on: if both of us see each other’s clock slow down, then surely, there is no difference between our clocks. However, when we ride back to our friend, stop at the platform, and compare our clock to our friend’s clock, we do see that our clock is behind. This is the so-called twin paradox. If both can state the same thing, how do their clocks still differ in the end, causing one half of the twin (on the train) to be younger than the half who stayed behind (on the platform)?
This is due to the switching of reference frames: we were on a moving frame (the train), switched to a moving frame in the opposite direction (returning to our friend), and, lastly, switched to the platform’s frame, standing next to our friend, to compare clocks, while our friend never switched – he stayed on the platform(beginfootnote)Contrary to many popular and even introductory physics texts, this has less to do with acceleration, even though this does plays a role – without it, in the real world, one wouldn’t be able to switch reference frames. However, mathematically, acceleration isn’t necessary for solving the so-called twin paradox, switching frames is.(endfootnote). So, the situation isn’t symmetric.
Length contraction and time dilation. They’re not illusory, they’re real. Many experiments showed that space contracts and time dilates for things in motion as observed by things with a different motion.
To summarise in a slightly more mathematical way – Newton taught us that one platform metre equals one train metre and that the same is true for seconds:
1 platform metre = 1 train metre, 1 platform second = 1 train second.
Einstein, however, taught us that:
1 platform metre $\equiv \dfrac{1\text{ train metre}}{\gamma}$, 1 platform second $\equiv \dfrac{1\text{ train second}}{\gamma}$.
This $\gamma$ (Greek letter gamma) is crucial. It’s a factor necessary to make sure that the speed of light doesn’t get any faster than the speed of light, even if it’s on-board a moving train. This factor is called the Lorentz factor(beginfootnote)Named after Hendrik Lorentz. The expression for his Lorentz factor is as follows: \[ \gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}, \] where $v$ is the speed of the other relative to us and $c$ is the speed of light.(endfootnote).
We never notice these things though. Usually, our speeds are way to slow for the effects of special relativity to be noticeable. Except for things that do go fast. Without Einstein’s special relativity, GPS satellites wouldn’t work(beginfootnote)Of course, we also need Einstein’s General Relativity for that, but that’s for another time.(endfootnote). Research institutes such as CERN and Fermilab also need to take special relativity in account for the high-energy, fast-flying particles.
So, our intuition (Newton) is mostly just fine though not precisely right.
What’s so special about special relativity
This is a somewhat technical question, requiring a somewhat technical answer, our apologies. Contrary to what is being said in many popular science texts, special relativity isn’t really about constant speeds as opposed to general relativity dealing with acceleration. In fact, special relativity is able to deal with acceleration. It’s just that it works fine as long as we’re assuming things exist in so-called Euclidean space, adhering to Euclidean geometry(beginfootnote)Named after Euclid.(endfootnote). However, after ample deliberation, Einstein concluded through his general relativity that we’re not living in Euclidean-geometric reality at all. We’re living in a Riemannian manifold(beginfootnote)Named after Bernhard Riemann.(endfootnote). In short, Euclidean space is a special case of the more general Riemannian manifold(beginfootnote)If we’re being precise, Hermann Minkowski developed a modified version of Euclidean space which we now call Minkowski space, while Riemannian manifold should actually be named pseudo-Riemannian manifold but many physicists simply call this Riemannian manifold anyway, perhaps because they’re not mathematicians.(endfootnote). Hence, we have special relativity as opposed to general relativity.
Featured image by StockSnap from Pixabay, modified by @kjrunia (added equations).
