...and glory to the quantum!

(part 2)

If you’re still not convinced that the wave-moose, I mean, the wave-particle duality is confusing and not even well-defined, consider this: there's not even a consensus on how much of each part makes up the quantum entity (a bit like Jesus). According to Bohr, it is strictly a wave or a particle, never both. According to de Broglie it is both (a particle guided by a wave), and according to other people (like Penrose), it is neither (the duality principle is just an illustration, not reality). And intermediate positions have also been expressed by pioneering physicists. Here, I’m going to expand on the neither-nor viewpoint, which so far only says what the object is not, and not what it is.

'It' is a quantum. The word 'quantum' is now much more commonly used as an adjective than as a noun. However, in its initial meaning it is a noun, and when we describe how this noun behaves, we are describing its… mechanics! This is the way in which I understand the term Quantum Mechanics, and it's my conjecture that this is how it was (etymologically) meant to be understood. In fact, in old papers I have seen authors talking about 'the quantum' in the same way that we nowadays talk about 'the particle'. Now, of course words have no intrinsic meaning hard-wired into them - it is us who load them with meaning. In that sense, there is no a priori reason why a 'quantum' is a better term than a 'particle' or a 'qauntum particle', but there is a pretty good a posteriori reason: the word 'particle' is already loaded with too much meaning - it inevitably evokes a picture of something round, solid, and localized in a tiny region of space, and none of these properties are necessarily properties of the quantum. 

So what is the quantum? It's hard to define it with no mathematics and in just one sentence, but let's say that it's the smallest amount of energy that can exist on its own. However, there are different types of quanta – an electron quantum, a photon quantum, etc., which we commonly refer to as different particles. A crucial difference between the former and the latter is that the quantum does not need to be localized at a particular point of space. It could be, but it doesn’t have to be. In fact, its natural state is not localized, but due to the fact that quanta interact with each other, it's hard to isolate them into this natural state. Thus, in reality - and in experiments - a quantum often appears localized, but only because there is an external force that confines it. Now, it is often said - by people I do respect as scientists, mind you - that the de-localized nature of quantum objects is completely non-intuitive: here are just two examples where I recently heard that statement. The argument is that it is non-intuitive because we never experience anything like it in our lives. But this is a poor argument: the fact that the Earth is round and rotating, that a bowling ball and a feather in vacuum fall in the exact same way, or that a body in motion will stay in motion unless an external force stops it, those are all aspects of nature that we never experience directly but can only infer from observations. Yet we never say that they are non-intuitive, in fact we never even question them and have accepted them as almost mundane.

I see a sort of a vicious circle around Quantum Mechanics: by now it's practically a cliche to say that it is non-intuitive, and this is repeated and perpetuated every time the theory is mentioned. However, there is nothing intrinsically non-intuitive in the theory, because intuition - just like the meaning of words - can evolve, and has evolved many times in the history of science. However, a prerequisite for QM to ever become intuitive is that we stop reiterating that it's not. 

In fact, I think there is a curious analogy to be found in the history of science. Insisting that objects are naturally localized into 'particles' is very similar to insisting that an object's natural state is at rest. There are forces acting all around us, so it looks as if everything eventually comes to a rest, if there's nothing pushing it. But if you remove the surrounding forces (as Newton realized and postulated), an object will continue moving indefinitely. Everyone accepts this today, although it would have sounded very non-intuitive to Aristotle and everyone else in his generation. In the same sense, we are used to objects being localized, because electrons interact with nuclei to form atoms that interact with other atoms to form molecules that interact to form, well, everything around us. What Quantum Mechanics simply tells us is, the localized nature of everything around us is not its natural state; it is due to interactions. Remove interactions, and the 'particle' spreads out and is no longer a 'particle' in the sense we would typically attach to that word.

We actually do observe this all the time: light is made out of quanta that interact weakly with other quanta - in most situations, in fact, negligibly. This is why we are used to thinking of light as waves - it is much closer to its natural (by which I mean non-interacting), de-localized state, since there are no forces confining it. When we do the double-slit experiment that I outlined in part 1 of this post, the photon quanta first propagate freely and look like waves, but then they interact strongly with the detector in the end - e.g. a photographic film in which they get absorbed - making them appear localized. This property looks particle-like, but really we always have the same object - a qauntum - but in the presence or in the absence of interactions. Eventually, the jump in intuition required to accept that fact is no bigger than the one that was needed to accept Newton's principle that an object will keep on moving if there's no interaction to stop it. 

The current intuition that stuff should ultimately be localized comes, of course, from trying to imagine everything as particles. Why this obsession? Why think of anything as a particle? Why do we say it's hard to imagine a baseball as a wave? Why don't we just imagine everything as a wave, or better: as energy! That's what Einstein tells us anyway! (E = mc²)The Earth is not a hard sphere! Neither are atoms nor nuclei nor anything. They are all a bunch of energy clustered together because it attracts itself. How do you define 'hard' anyway? How would you define the size of a particle if you want to stick to that notion? You can never actually touch it, you can only get to a given distance before the repulsion gets too strong. The Earth appears hard because, while it attracts us on the large scale, it's repulsive on the short scale (due to electrostatic repulsion between our atoms and Earth’s atoms). Most of us know all of those individual facts, yet strangely cling to the image of everything as made out of tiny matter-balls that we call particles.

There's nothing that's actually solid. Matter is energy (is quanta of energy). If anything appears solid to you, it's because the energy that constitutes it is repulsive to the energy that constitutes your hand.

Maybe right now this sounds hard to grasp, and you're thinking -'what's the use, if it's abstruse?' (no rhyme intended). Could one argue that better intuition is gained in the wave-particle picture than in this 'quantum' picture? My answer to that is another question: why is then 'non-intuitive' the most common adjective assigned to Quantum Mechanics? My argument is that this is largely because of trying to describe an object by starting from the notion of a classical particle, and then outlining all the aspects in which it's not like one. Isn't it better to just define the object through its properties, complicated as they might be? Isn't that the way to break the vicious 'non-intuitive' circle?

I want to discuss in more detail said properties of the quantum, and the way it compares to the wave-particle view, as well as the difference between the quantum and the wave function. But in the interest of me being terribly late with new posts, I'll leave this for a future part 3.  

Down with duality...

(part 1)

Where I come from, we make jokes, for some absolutely unknown to me reason, with people from the Chukchi Peninsula located at 'the northeastern extremity of Asia.' One example is the following joke. A guy from the region goes on a trip to Africa, and when he comes back, the whole village gathers to hear his stories. Says he, 'I saw a giraffe!' but the people don't know what that is, so they ask. He replies, 'well, you know what a moose is, right? A giraffe is like a moose with a really long neck.' Everyone is amazed and asks what else he saw. 'I saw a zebra' - adding, due to the bewildered faces around - 'it's like a moose, but black and white.' They want to hear more, so he says, 'I also saw a crocodile.' He stops, thinks for a while how to describe this one, and says, 'well, you know what a moose is. Imagine something that has nothing in common!'

I think something similar commonly happens when we speak of quantum mechanical effects, especially when the wave-particle duality - the notion that quantum mechanical objects sometimes behave as waves and other times as particles - is invoked. The idea that stuff around us is ultimately made out of 'particles', in the sense of tiny balls of matter, is heavily ingrained in our worldview. For example, even though we know that the picture of an atom with the electrons orbiting as planets around the nucleus is wrong, it's what everyone imagines when thinking about atoms, and especially Big Bang Theory fans who see this at every cut-scene. There is this general peculiarity in the thinking about all small-scale physics, even when the thinking is done by physicists: we know that it is the wave function that rules the world on those scales, but we try to translate that world into something that's made out of particles, which just happen to have some weird properties. This inevitably and quickly results in statements of the form, 'a quanum mechanical particle is nothing like an ordinary particle in that...' We end up describing something not by defining its characteristics, but by outlining the characteristics by which it differs from something that we are very familiar with... just like the Chukchi peninsula guy in the joke. I find this equally absurd.

This is again due to the Copenhagen interpretation, and came about in the following way: Niels Bohr insisted on separating reality from the wave function; for him, reality was made out of observable objects, and the wave function was simply a tool to (probabilistically) predict how those behaved. It was important, however, to always discuss quantum physics in classically accessible terms, which is why he introduced the concept of complementarity, one manifestation of which is the wave-particle duality. In short, Bohr's view was, never mind the mathematics; that's not reality. Reality is anything that would come out in a measurement, and, in terms of that, the same quantum mechanical object sometimes has properties of a wave and sometimes has properties of a particle (in the sense that we are familiar with from classical physics).

My claim is that this has hindered the understanding of QM, rather than helped it, and that we need a revolution.

A good illustration comes from an experimental result which recently went viral in science-oriented social media. I will use this discussion of it to deconstruct the misconceptions, but I want to highlight that I am not trying to criticize neither the particular website nor the particular research group; the problem is much more fundamental, and this is just a handy, recent example. I encourage you to have a look at the article before continuing reading. It's a very sexy piece of news as far as developments in Physics are concerned, hence I'm not surprised by the attention it received. But, essentially, I have a problem with each of the first three sentences of the news article.

'Light behaves both as a particle and as a wave.' This is only within the Copenhagen interpretation, and in addition only within its very classical formulation. It is certainly what we all get taught in Physics classes, but anyone who has made one step further in thinking (or reading) about the philosophy of quantum mechanics might have questioned the statement. But, more importantly, the wave and particle behavior is essentially how some people choose to visualize quantum mechanical objects - it is not a scientific truth in any sense because it isn't even rigorously defined. There is no mathematical formulation of the principle; instead, it is used in 'hand-waving' explanations with the goal of providing some intuition. Occasionally, this works quite well, and I guess it was in particular helpful in the early days of the theory. I've just started feeling like we've outgrown it.

'Since the days of Einstein, scientists have been trying to directly observe both of these aspects of light at the same time.' They sure haven't. Or, [citation needed]. Physicists generally realize that the wave-particle duality is meant to help us visualize the behavior predicted by the wave function, but I had never heard anyone discussing capturing both aspects at the same time as an important experimental goal. Actually that 'goal' is, again, not even strictly defined. Which brings me to the next point.

'Now, scientists at EPFL have succeeded in capturing the first-ever snapshot of this dual behavior.' First of all, what does a 'snapshot' even mean, in this case? You probbly know that in order to take a photo, you always need some exposure time - the same is true here. I don't want to explain the details of the experiment - you can read the article and watch the video, I think it's fairly accessible. The bottom line is, however, that they had to average over many electrons interacting with many photons to capture the 'snapshot', so, even though everything is happening fast, is it really 'snap'?

Going with this train of thought, I have two questions to that statement: have they? And is it the first-ever? The answers cannot be both 'yes'. If this experiment counts as a snapshot of light as both a wave and a particle, then so should the following one, which is so old, fundamental, and pioneering, that it is by now a classroom experiment. I'm talking about the famous double-slit, which goes like this: you shine light towards a screen with two holes in it, and record the intensity at a certain distance behind.

It has been known already for centuries that the intensity presents an 'interference pattern' of alternating minima and maxima, since the waves coming from each slit add up in either a constructive or a destructive fashion. Now, if you decrease the intensity a lot, at some point what you start recording begins to look like discrete points on your screen - we would call those photons, or 'quanta' of light. The funny thing is that if you record many of them and average out, you would still get the same interference pattern, even though you were recording them one by one, as 'particles'.

The same experiment can be done with electrons, in which case it is supposedly more striking, because, supposedly, we are used to thinking of light as a wave and of electrons as particles. But, essentially, they are both quantum mechanical objects, so no wonder they behave in the same way! Anyway, the same thing happens also with electrons, and you can see a nice experimental result of the build-up of the interference pattern. So, doesn't this experiment also produce a 'snapshot of the dual behaviour?' I see no reason not to count it as one.

In fact, in the double slit, it is quite clear what the wave and what the particle aspects are. But have a look at the 'snapshot' from the recent experiment that I discussed above:

I think an obvious thought is, 'I see the wave, but where is the particle?' That sort of confusion can for example be seen in most of the reddit comments on the article.  It's natural to not see the 'particle' if you're not a specialist, and it's difficult even for specialists, because the axes are not labelled in this photo. One of the axes represents position, but the other actually represents energy. What is demonstrated is, in fact, a wave-like interference pattern together with quantization of energy, something I explained here (the photo above looks quite similar to the guitar string vibrations, doesn't it?). In fact, if you look at the title of the actual research article (as opposed to titles in popular-science media), it says quantization, not particle. This is because the former is well-defined mathematically, while the latter is not.

The two experiments above do serve as a good illustration of the two properties that are most commonly associated with 'particles'. One, a particle is something which is well-localized in some finite region of space (double slit experiment), and two, it is something that comes in integer numbers - you can have one or two or three of 'em, but not one and a half (both experiments). But - think about it - both of those statements hold true for a moose as well! Is it an equally valid viewpoint, then, to imagine reality made out of tiny meese^*, and talk about a wave-moose duality? I leave it to you to decide.

In short, the wave-particle duality is not rigorously defined, leads to a lot of confusion, and would be equally valid if it we replaced it with a wave-moose picture... Given all that, I say, to hell with duality! We must define objects by their properties, not by the properties by which they differ from objects that we are used to! There are alternatives. Several, in fact! If you are a fan of waves, the Everett interpretation is the way to go. If you really want to keep imagining particles, Bohmian mechanics would be your thing. I will explain both of those in due time, but first I'd like to finish with the basics of QM within the standard interpretation. In the second part of this post, I will thus give an alternative, which, like freetown Christiania, stays within the boundaries of Copenhagen, but goes against some of the accepted norms.

* It seems that 'moose' is the most standard plural of the word, but wiktionary says that, 'the form meese [...] will in most cases be greeted with a snicker, and is thus generally only appropriate in humorous contexts,' which suited perfectly my intentions. 

Does this post shave itself?

In case you're wondering, the title alludes to Russell's paradox.

A catalog of all posts (newest first) follows. 

Quantum Mechanics
10.05.2015 : ...and glory to the quantum! : The mind-blowing second part of my rant against wave-particle duality.

19.04.2015 : Down with duality... : Wave-particle duality leads to a lot of confusion, and is not even something that's strictly defined! Do we really need it? Or has our understanding come to a point where we can do better?

08.03.2015: Jigsaw's cat : A discussion of the famous feline thought experiment and how it uncovers the measurement problem of the Copenhagen Interpretation (which is easily the main problem).

01.02.2015: Ladder to heaven : A description of what is meant by 'quantization', and how it arises from one very special property of the wave function. Visualized through a quantum guitar. 

30.01.2015: A trip to Copenhagen : A statement of the postulates of the Copenhagen Interpretation. An auxiliary (= boring) post with some necessary technicalities... and two jokes in the end. 

10.01.2015: To be or to not be : Outlining some common misconceptions (and pet peeves!) arising from asking questions which are meaningless in the framework of the Copenhagen (= standard) Quantum Mechanics. 

Relativity
14.02.2016: Gravity waved! (Q&A) : The most recent big result in Physics (at the moment of writing of this sentence) is the gravitational wave detection by the LIGO experiment. Tune in to this Q&A to learn more!

21.09.2015: Wibbly-wobbly timey-wimey... stuff : Some graphical illustrations of space and time in relativity. Time is indeed wibbly-wobbly, but still a progression of cause-and-effect, it seems.

30.08.2015: Some stuff Einstein actually said : A discussion of space and time, which are both relative, and events which we all agree happened, which are absolute. Motivated by Einstein's book The Meaning of Relativity.

02.04.2015: A relatively special post : Special relativity 101 - how it came about, what motivated Einstein, and what the theory is about, in a nutshell. 

Miscellaneous
18.01.2016: Let me teal you about the cyansce of color : What are the intrinsic characteristics that give objects their color? What color is a mirror, and why is the sky blue (sometimes)?

31.05.2015: 'Energy' is a great name for a band* : What exactly is a semiconductor, is it really different from an insulator, and why in the world has it had more influence on our lives than anything in the history of humanity, ever?

14.02.2015: Astrooonooomyyyyy : Some random thoughts inspired by some astronomical images. 

A relatively special post

In the end of the 19-th century, it seemed like Physics was almost complete. Analytical mechanics was practically finalized with the work of Newton, Lagrange, Hamilton, and of course many others, and Maxwell had just written down a complete description of electromagnetism, building upon the work of Gauss, Faraday, Ampere, etc. There appeared to be a few things left to straighten out here and there, a few small outliers that didn't exactly fit the otherwise beautiful theories, but few people expected any big surprises. You won't believe what happened next!

The beginning of the twentieth century saw the development of two theories - both arising from scratching these dimples of remaining inconsistencies - that completely changed our understanding of the world. This is by now a textbook cliché, but it's also the simple truth, so I'm OK with re-iterating it. 

The first of these theories is Quantum Mechanics, about which I've already written (every word is a separate link). The second one is the theory of Relativity, which I'll (start to) explain here. A remark right off the start: there is the Special Theory of Relativity (STR) and the General Theory of Relativity (GTR); the former is much simpler and came first, which might or might not sound strange (it depends on whether you think deduction is a more natural train of thought than induction). This post is about the STR only.

The story of STR - and how Einstein came up with it - follows, in a nutshell, the same pattern that has brought some of the greatest insights in the history of science: someone refusing to take for granted something which is obvious for everybody else.

In the case of Einstein, the statement was, 'obviously, if I see you moving with velocity v₁, and I see him moving with velocity v₂, then you see him moving with velocity v₂ - v₁'. This 'obvious truth' has a slightly more complicated formulation than the ones above (it involves math!), but I'm sure that anybody would agree that it does seem pretty obvious if we only restrict our thinking to practically everything we experience in our lives, ever. But Einstein was totally, like, 'Hmmmm..."

Of course, he wasn't just toying with an alternative idea out of an abstract what-if-like curiosity. To abandon such 'truths' (postulates), which are deeply in-grained in our thinking, we usually need a very strong nudge. In the case of the STR, this came from the fact that Maxwell had derived an equation for the speed of light, and it had turned out to be a constant - without assuming any particular reference frame! In other words, you assume you're sitting on Earth, you write Maxwell's equations, and you get that you should see light moving with a velocity c. But if you assume that you're moving with velocity c with respect to the Earth, and you write the equations, you get again that light should be moving with velocity c! Initially, people just thought that this is some strange feature of electromagnetism (light is just electromagnetic waves, by the way). They thus tried to propose various extensions to the 'classical' theory, but as experiments kept proving them wrong, the add-ons themselves became more and more convoluted, even for philistine physicists who wouldn't usually care about aesthetics in equations. Einstein always cared about the aesthetics of Physics, by the way.

Maybe this is the reason why he realized there is an alternative way: instead of adding ever-more-unlikely complications, remove just one assumption, even though it seems completely and obviously true - but who knows. So instead of taking the addition of velocities as a postulate, he took a new one: 'light propagates with a constant velocity regardless of the reference frame' (obviously, you cannot have both of those true at the same time). Then, he set out to derive the new laws describing how to go from one reference frame to the other. This transformed the universality of c from a peculiarity of electromagnetism into a fundamental property of space-time. A clarification: by talking about space-time, I'm not trying to sound all sci-fi-y. Instead, this only refers to the fact that if we want to describe any event happening in the world, we have to say where and when it happens. The property of events of having a 'where' and a 'when' makes them embedded in some measure of space and some measure of time, which, combined, we call space-time. Furthermore, I should define more rigorously what a 'reference frame' is: it is the 'where' and the 'when' of the world - so, the space-time - as seen from some particular standpoint. But a standpoint need not be standing with respect to other standpoints, and in fact the relationship between moving standpoints is what the STR is all about. Simple, huh?

I'd like to highlight what I think are the most important aspects of the theory.
1. Einstein assumed just two postulates: that c is constant in all reference frames, and that the laws of Physics are the same in all reference frames (after having been properly translated). In other words, there is a fixed relationship between what I see and what you see, regardless of whether we're moving with respect to one another.
2. Given those two assumptions, he derived the way in which I can translate what I see to match exactly what you see. The theory of relativity, in its purest form, is nothing but a prescription of how to describe an observation to someone who is moving (fast!) with respect to you, in a way that would agree with the other person's observations.
3. All the crazy effects you might have heard of - like space contraction and time dilation, or the new law of addition of velocities (a modified version of the v₂ - v₁ above that complies with c - c = c) - as well as the not-so crazy but super-fundamental ones - like E = mc² - can be derived based on those two postulates (and the other standard postulates of Physics, like energy conservation, and, you know... mathematics). I always found it truly amazing how much one can achieve with so little to start with.

In the remainder of this post, I’ll focus on the effects of length contraction and time dilation. These refer to the fact that different observers might disagree on the distance between two points, or on how much time passes between two events. That is to say, they would disagree as long as they don't use STR to 'translate' their observations in a way that makes them agree. These effects can be derived in many ways; below, I present one illustration of why something fishy has to happen either space or time - or both - if light is to move with the same velocity in different reference frames. Assume that a beam of light travels between two mirrors. If we think about the distance that light covers in a round-trip, there's obviously some difference to be observed depending on whether we are moving with respect to the system (which is equivalent to the system moving with respect to us). 

Most generally, in the first case, light travels some distance 2L in some time t, and in the second case - a distance 2L' in time t'. Our intuition would strongly suggest that t' should be the same as t, and that L' should not be the same as L. But if you think about it, this couldn't possibly be the case! Light has to travel with the same speed in both systems, and speed is just distance over time. So 2L/t has to equal 2L'/t', which implies that either we're wrong about t = t', or we're wrong about L' not equal to L, or we're wrong about both... In that particular case, we're wrong about t = t', but this is a detail. The big picture implied by STR is that both distances and time periods vary from one observer to the other. These are in fact two manifestations of the same, more general effect: space-time itself is different for different observers. 

Now, the illustration above gives a good intuition of why something fishy must happen when we think of light propagation, but the interesting twist is that, since something fishy actually happens to space and time itself, all physical processes pass on different time-scales in different reference frames. This, by extension, also includes chemical and biological processes, and, ultimately, what we perceive as the passing of time in any meaningful definition (also, aging). This has been experimentally verified in various experiments. One of the seminal ones involves the decay of particles called 'muons' (they’re kinda like electrons, but heavier) created by cosmic rays in the top layers of the Earth’s atmosphere.

We can create muons here on Earth, in a lab, where we can measure their lifetime (how long they 'live' before decaying into other, more stable particles), which turns out to be quite short. We also get to detect the muons that come from the cosmic rays, which 'rain down' on us with an enormous speed (close to that of light). The funny thing is, though, that if we multiply their speed and their expected lifetime, we should get the maximum distance that the muons could travel before decaying, and... it turns out that the cosmic ones shouldn't be able to get to us at all! The distance they can travel is smaller than the width of the atmosphere, if we don't take the STR effects into account. Thus, we shouldn't be able to observe the cosmic muons here on Earth - yet we do. Fundamentally, these muons are identical to the ones we make in a lab; the difference comes only from the motion with a 'relativistic' velocity.

This is a great experiment to demonstrate why time dilation and length contraction are two different manifestations of the same effect. The fact that the muons do reach the Earth can be explained in two different ways, depending on whether you want to stay in a frame in which the Earth stands still (which you are probably used to^{[citation needed]}), or in one where the muon stays still (remember that the Physics has to be the same in those two frames, namely, in both of them the muons have to reach the surface). Let's first have a look at the less intuitive frame: the muon is standing still, minding its own business, and the Earth is moving up towards it with a speed close to c.

The life-time of the muon in this reference frame is exactly the one that we measure for muons created on Earth; however, since the planet, together with its atmosphere, is moving towards the particle, the atmosphere appears shorter than its length as measured on Earth. And so, the particle reaches the surface – or actually, in this case, the surface reaches the particle!

We can also look at the case where the Earth is standing still, minding its own business, and the muon is free-falling.

In this case, the length of the atmosphere is the one we are used to, but the lifetime of the muon is dilated, since it is moving fast with respect to us – so it has more time to traverse the atmosphere and reach the surface. And so, as expected, the laws of Physics are preserved.

Many people, when they first learn about the relativity of length and time intervals, are tempted to ask questions like, 'What happens exactly? Does the atmosphere actually get thinner? Does time actually pass more slowly?' The problem with these questions is that the whole point of the theory of relativity is that there is no 'actually' - hence the name of the theory! There is no 'absolute' series of events in the Universe - they are always described relatively to a given reference frame. In some reference frames, some lengths appear shorter; in others, time seems to pass by more slowly; but in the end, everybody agrees that the muons reach the surface of the planet, one way or another. 

P.S. There is some polemic about Einstein's role in the whole matter, and how fair it is for him to get all the credit, which he often does. Here's my opinion: it's true that a lot had already been done - the constant speed of light is within Maxwell's equations, and Lorentz had derived the famous transformations that also imply time dilation and length contraction, but it really was Einstein who realized that these two effects are not some strange, minor aspect of electromagnetism, but instead carry a fundamental implication about the very nature of the world. This is as far as STR is concerned; regarding the GTR, Einstein deserves even more credit. So, no, I don't think he's overrated.

Jigsaw's cat

Quantum mechanics written in terms of the wave function and the Schroedinger equation is quite neat. It is only when we include the 'measurement' component that things usually become messy, and people start getting lost in 'paradoxes'. But measurements are a part of the postulates of the Copenhagen interpretation, thus they play a central, unavoidable role in the theory. The conceptual difficulties that this poses are collectively labeled the 'measurement problem', which represents - I think - the main reason for people to look for interpretations beyond Copenhagen. I'll try to outline the problem here. 

To do that, let's use the good ol' Schroedinger cat. The setup is the following: we take a box and put inside a cat and a vial of poisonous gas. We also put a quantum particle which has a probability of ½ to decay within one hour, as well as a detector and some mechanism that breaks the vial as soon as a decay is detected. 

Thus, quantum mechanics seems to say that, after one hour, the particle is in a superposition state of being decayed and not decayed, and the cat is therefore in a superposition state of being dead and alive. It is at this point important to note that when Schroedinger proposed this famous thought experiment, he did not give it as an illustration of quantum mechanics. Instead, he was demonstrating how there is something wrong with the way we think of it, because of the nonsense zombie-cat result. He was pointing out a problem, not giving a solution. This is very important and from now on you should never let anyone get away with saying that Schroedinger's cat is simultaneously dead and alive, as if that's some strange paradox of QM that one has to accept as true. Point out their mistake! Be that geeky 'technically, you're wrong' guy or gal! Together, we can start a revolution! And make the world a better place! Yes we can!

Uhm, anyway. To better understand the measurement problem, let's take a small modification of the thought experiment: let's put a scientist in the box instead of the cat. Let's also put the particle and the vial of poison inside another box. And let's say the scientist inside the first box is to open the second box one hour from the start of the experiment

Ok, let's say the box will open on its own one hour from the start of the experiment.

And let's say there's another scientist outside the big box.

who's creepily happy about the whole situation

because it's actually a cat controlling a humanoid robot!

It seems that in this situation it is extremely difficult to say at which point of time a measurement occurs and the particle's wave function 'collapses'. Say that the detector measures a decay already at the fifth minute. Has the wave function collapsed already? As far as both scientists are concerned, no it's not, because they haven't 'measured' it yet – they don't know whether the decay has happened. On the other hand, some sort of measurement did take place: the apparatus interacted with the system, right? So it must have also broken the vial of poison. Then, when 55 minutes more pass and the box opens, the second scientist is dead. Was that a 'measurement'? Is the wave-function collapsed now? As far as the cat-scientist is concerned, no it's not, because he hasn't 'measured' it: the big box is still closed. Or, let's have mercy and remove the vial of poison. When the small box opens, the second scientist can simply read out what the detector says

Has the wave function of the particle collapsed then? The detector has now been read out, so maybe yes, but again that's only the case from the point of view of the scientist in the box. As far as the cat outside is concerned, the system is still in a superposition state, and if he didn't know about us removing the vial of poison, the second scientist is himself in a superposition state of being dead and alive (Schroedinger's original nonsense result). Can we say that the wave function has collapsed because a conscious being has read out a detector? That's obviously a very dangerous path to take. What is consciousness anyway? Is a cat a conscious being? What if it's smart enough to build a controllable humanoid robot? 

Bottom line, the measurement postulates present many difficulties on the philosophical level. When does the measurement happen? Does it happen instantaneously? This is a huge problem in cases in which it would imply that an action at one point of space has an instantaneous effect at a different, distant point: that could break the idea that there is a cause-effect relationship in nature. Anyway, how do we even split the world into 'quantum' and 'classical' – isn't everything inherently quantum? That we are able to do such a split is pre-supposed in the Copenhagen postulates, but no rigorous way to do that is given. What if we want to study the whole universe as a quantum system?

In case that hasn't convinced you yet that there is a problem, I'll now return to the original problem as posed by Schrodinger. There is actually more than one way to 'resolve' it within the Copenhagen interpretation. Here I suggested one: do not ascribe any classical properties to the cat before it is observed - it's not a cat, it's a wave function in a superposition state. But it is also possible to think of the cat as a nice, ordinary, classical kitty, which has some probability to be dead, but is not in a superposition state (because that wouldn't make any sense, or would it?) In other words, since there is no rigorous definition of how to split the world into quantum and classical, it's up to us to decide to which world the kitty belongs. What's the difference between the two representations of the situation? I hope this makes it clear:

Different people might prefer one or the other version. I think Schroedinger had the first version in mind when he proposed the problem. In the second version, it's fun to think of the cat as a classical measurement apparatus which is constantly measuring the wave function of the quantum particle... by dying or not dying! In that sense, the wave function cannot be in a superposition state, it's either the wave function of a decayed particle or of an un-decayed one, because at every moment the cat interacts with it and 'measures' which is the case, causing an immediate collapse.

The problem is not with the versions themselves as much as with the fact that they are both allowed. The problem is, in other words, with the supposition - inherent to the Copenhagen postulates - that we can divide the universe into a quantum part, which is described by quantum mechanics, and a classical part, in which we live. Cats, too - or do they? What if instead of a cat we had a bacteria? Should we include it in the quantum system or keep thinking of it classically? What if we had a molecule that would dissolve if exposed to the gas? Is the molecule 'quantum' enough for us to have to include it in the wave function and not in the classical world? And if yes, where do we draw the line between a molecule and a cat... and is this line different for every different setup?

At this point, people who really want to stick to the Copenhagen interpretation will inevitably start talking about decoherence, which is a complex topic, but which, to me personally, always leaves a tang of not really solving the problem. In other words, I think that everybody who is sufficiently bothered by the problem to feel the need to dig deeper has to inevitably leave Copenhagen in search of further answers. And, of course, I'm planning to get to that one day!

Astrooonooomyyyyy

The title (and the multiplicity of vowels therein) is inspired by a hugely unknown Metallica song from the Garage Inc. album called... well, 'Astronomy' - which is, like everything on the album, a cover. The original song is by Blue Öyster Cult. Do check it out (both versions are pretty cool)!

So, to illustrate that I won't be talking only about quantum mechanics here, let me share some random thoughts about some astronomical images. Like, maan, nebulas are pretty:

This one is a vary famous one, the Orion nebula. Credit for the image goes to the Hubble Space Telescope, which has in fact done an amazing job: the actual resolution of the image is 18,000 by 18,000 pixels, and a fun zoomable version is available here. Until recently I had never thought about how big nebulas actually are, but I know that they are formed after a star burns out and explodes into a supernova, so I thought that gave me a pretty good idea. They turned out to be slightly bigger than that idea, but not too much. Well, although I cannot say that they are huge by astronomical standards (insert a 'your mom' joke here), they certainly are pretty huge for something created by a single star. Let's stop for a second and admire another one. 

That's the Crab nebula. Credit goes again to Hubble, and again there is a zoomable version. Pretty amazing, huh? Let's see how big it is. Let's start from the standard comparison, Earth, Sun, distance between the two.

On the left side, the dot representing the Earth is to scale compared to the ball that is the Sun; on the right side, the dot representing the Sun is to scale with the Earth-Sun distance, but the dot representing the Earth is obviously not. The distance between the Earth and the Sun is a unit of length commonly used in astronomy, and as such, it's rather appropriately named... 'astronomical unit'. 1AU is about 150 million kilometers. And here is how this distance compares to the rest of our Solar System (and a bit beyond! Have you heard that Voyager 1 not long ago became the first mam-made object to leave our star system?)

We're however still far from the required length-scale to describe the size of a typical nebula, i.e. they are much larger than the entire Solar System! I cannot come up with anything familiar whose size is in between the Solar System and the nebula size I am trying to get to. Also, I seem to have an unexpected problem in writing this post in that I want to put a 'your mom' joke in almost every paragraph. Resisting the temptation, let me define the size of the Solar System (twice the distance from the sun to the heliopause) as a unit of length, for which I'll use the (unfortunate) abbreviation SS. For people more familiar with standard internet units of measure, 1SS is approximately 179.51744 × 10¹² bananas, although it depends where you buy them from. That's 179.5 trillion, for the people unfortunate enough to not understand scientific notation. 

Ok, so having defined that, we can define one Deca-SS (10 times SS), and one Hecto-SS (you guessed it!).

The Hecto-SS is a unit that's finally big enough to be visible on the scale of the Crab nebula.

More precisely, the nebula is about three thousand times bigger than our entire Solar System! So, yeah, pretty big, huh? Or maybe you're disappointed, cause you expected it to be bigger? (zing!) Anyway, with astronomical distances it's really hard to know what to expect, but now you do, in case you ever want to be a millionaire (and by incredible chance you get asked just that). 

Now, what got me thinking about nebulas in the first place was not their size. Instead, I was wondering if the above images are just pretty pictures, or if they are a representation of reality. This is because scientific images which are taken using some apparatus, like a telescope or a microscope, are sometimes in 'fake color', which means that the apparatus does some coloring following some rules. There is usually a very good reason to do that, and the rules usually encode some information in the colors. So what about the nebulas, could we ever hope to see them in all their majesty (i.e. as in the images above) with the unaided eye? Not exactly. First of all, to see the nebulas as big as in the images, we would definitely need some aid: either a telescope, or, if we have somehow gotten close enough to need no magnification - a filter, because the light will be too bright. But in addition to that, for nebulas, Hubble does use a color-coding: namely, it assigns a particular color to a particular chemical element. For example, in the image of the Crab nebula, blue represents neutral oxygen, green indicates singly-ionized sulfur, and red indicates doubly-ionized oxygen. 

So, then, are those just pretty images, or are they also an accurate portrayal of reality? I would say that they are the latter. What is reality anyway - we definitely cannot restrict the definition to whatever we can see with our eyes. And in a way, in the Hubble images reality is portrayed even better, because, while the nebulas will no doubt still look stunning unmodified, this color-coding allows us to differentiate the constituent elements even better. In other words, as long as the image conveys scientifically accurate information, to me it is indeed a representation of reality, regardless of what part of it our eyes could see if unmodified. Take as another example this stunning image of the Milky Way

This is reality right there, it's something that is always just above our heads, ready to be gazed at, but we can never hope to see it like that (unless we teach our brains to take long exposure shots...) But still, this awesomeness exists above us and it is only our sensory limitations that prevent us from appreciating it. Fortunately, however, we are getting better and better at using tools to capture reality, and represent it in an accurate way that our crippled senses could actually appreciate. Great stuff. 

But this is not always the case: there is another type of 'fake' images which sometimes appear especially in relation to astronomy, and that brings me to the original inspiration for this post. This image of a recently discovered exoplanet became mildly viral in the last weeks

Now, the important thing to realize here is that everything we know about Kepler-186f is inferred through observations of the star it orbits; we have no way of capturing any image of the planet itself, let alone one as detailed as the one in the picture. What's nevertheless pretty cool is that just by observing the star, we can infer with quite a lot of certainty that there has to be a planet orbiting it, and, furthermore, we can estimate the size and orbital radius of the planet. Based on this, we know that around a certain star there orbits a planet that's slightly larger and slightly colder than the Earth, but that's literally everything we know with certainty. We don't even know its mass, so we can only guess what it could be made out of, and we certainly have absolutely no idea if there's water or not. So the planet on the left is just someone's imagination of what a slightly larger, slightly colder Earth could look like - one of the infinite number of possibilities. Unlike the Hubble images, there is nothing scientifically accurate in the details of this picture. Images of this type are called 'artist's conception'. Don't get me wrong, I'm not calling the image fake or anything, and I'm not against images like this, but I do think that if they belong to that type, that should be made very, very clear, because a lot of people could be fooled. 

To conclude, since I mentioned the Milky Way earlier, we all know and love our home galaxy, right? 

But this is not the Milky Way! That's another galaxy (NGC 6744), and we just think that the Milky Way should look something like this. It took me an embarrassingly long time to realize that we cannot take an 'outside' picture of the Milky Way like the one above, cause we are hopelessly trapped on the inside of it. To put it simply, we don't have a giant, inter-galactic selfie stick.

Ladder to heaven

I hope I've made it clear how within (my interpretation of) the Copenhagen interpretation one doesn't really need the notion of a 'particle'... Cause now I will go back to using the word 'particle' occasionally, in a very general sense. There are several reasons to do this. There certainly is a semantic one: quantum mechanics describes things like atoms and electrons and photons and so on, and we are used to calling those things by the generic name 'particles'. There's also the physical one, namely the fact that in observations the wave function sometimes shows particle-like properties, like a very well-defined position in space. But I think the semantic reason is stronger - I really don't want to go on writing 'a quantum object' instead of 'a particle' whenever I want to refer to what the wave function describes. However, I do urge you, whenever you see the word 'particle' and are dealing with QM, do not imagine a hard, solid ball shooting around through space: the wave function obviously has little to do with that. On the contrary, it is in many ways like a standard wave - like on the surface of a pond, for example - but has one additional property that ultimately leads to the whole particle/wave fiasco. Let me try to explain it, which in a way will help me put the 'quantum' into quantum mechanics. But before doing that, let's take a look at a guitar string.

The tone that a guitar string makes when pinned between two points is determined by, among other things, its length. More precisely, the length determines the wavelength of the possible vibrations of the string, which is inversely proportional to the frequency of vibration, which is what our ears register. Different wavelengths mean different frequencies mean different tones. 

The ‘fundamental’ tone of the string is the one with the longest wavelength, and this is what mostly comes out when we pluck the string, although the other tones are generally also present, and are called 'overtones'.

By pressing on various frets, we make the fundamental wavelength shorter or longer, and produce various tones. So here is how the beginning of Stairway to Heaven goes, if played on one string (well, for the fifth row, which represents two notes played simultaneously, you'll need two identical strings).

That's actually not at all relevant to my point but I thought it's cool enough to show. My point goes along the lines that any vibration of the string has an associated energy to it. For a given wavelength, the energy is also related to the amplitude of the vibration: the farther the string goes up and down, the more energy is present in its motion. Consequently, more energy is transferred to the air in the form of sound waves, and we hear the tone louder. So there is a continuous relationship between amplitude and energy:

One of the simplest QM systems to study is called the 'particle in a box', and is very similar to the guitar string. Assume the wave function can only be non-zero between two points in space, but in between those points it is free. This practically corresponds to 'trapping' a quantum particle inside a 'box'. What are the possible solutions to the Schroedinger equation? (i.e., what happens?) Well, it turns out that the allowed 'vibrations' of the wave function look exactly the same as the fundamental tone and the overtones of the guitar string.

But in the case of QM we don't call these 'tones': instead, we directly refer to them by their associated energies, which I've labelled in the figure. So far, the wave function looks like an ordinary wave (which is practically the reason why it's called like that in the first place), but here comes the fundamental difference: for every 'tone' of this 'quantum guitar', the amplitude is fixed! For example, there is exactly one energy E₁ which is associated to the fundamental tone, and we cannot modify the amplitude of the vibration.

Here's why. The wave function defines a probability density, which stems from the fact that results of measurements can be (probabilistically) predicted by it. Thus, if we call the wave function Ψ(x), then |Ψ(x)|² gives the probability of observing a particle at a given position, x. Now, probabilities must always have the following property: when you add them all up for all possible outcomes, the sum must be equal to one (i.e. 100%). This is practically the definition of 'all possible outcomes': if you try to measure the particle at any possible place, you are 100% sure to find it somewhere. This means that the wave function follows a pretty standard wave-like equation, similar to the ones we find in electromagnetism or fluid dynamics, but on top of that there's an extra condition – its absolute value squared has to 'integrate' to one.

This is what fixes the amplitude. In other words, then, one cannot modify the energy of the tone by 'plucking the string harder'. In fact, if we want to 'pluck' the wave function, that is, to add energy to it, we can only do that by adding exactly the right amount so that we excite a higher-energy overtone. If instead we try to give it an amount of energy which is 'not good', nothing will happen: the wave function cannot 'accept' that, because it cannot use it for anything!

The fact that only discrete energies (physicists call them 'energy levels') are allowed is what is called 'energy quantization'. The word 'quantum' basically means a 'packet', or more precisely some quantity which cannot be divided into smaller quantities. The fact that energy comes in 'quanta' is one of the very fundamental properties of QM (hence the name!) - and is always the case for a finite quantum mechanical system (and one can argue that infinite ones are unphysical). By the way, the energy levels of the system, E₁, E₂, E₃, etc. are sometimes called a 'ladder', which is the origin of the admittedly lame-ish pun in the title of this post.

So there you have it. If you've followed this far, you can brag around that you know what 'quantum' means. It has always struck me as quite funny that many people know a little about quantum mechanics, but very few non-physicists have any idea what the word 'quantum' actually refers to! Of course, I'm not claiming that I have given here a strict, exhaustive definition of the term - not at all, I just gave a simple illustration. But that's practically where the word came from in the first place: people saw that certain systems accept only some very well defined packets of energy, and called them quanta, and... that's it!