Essay:ADK's Occasional Guide To Quantum Mechanics
So, you want a no-nonsense, no-bullshitting, accurate and easy way into the fascinating world of quantum theory? Oh well, go on then, I suppose I can give it a shot. Here, I will try and describe the basics of quantum mechanics in a way that requires absolutely no prior knowledge, and doesn't woo people with a load of mystical half-truths about its "weird and wacky" implications. It's really not as hard as you might think, even if this document does look a little long. I'll keep the swearing to a minimum.
 First things first...
If you want to learn anything about quantum mechanics, you have to get a few things out of your head first, and then start cramming a few more things in just so that the facts have a good base for where to settle.
The first thing you really need to do, is to remember that everything said here is backed up by experiment. Even if things are "odd" and don't "make sense", they still manifest as real and genuine effects. Some of these effects even have very real implications that we can see with our own eyes, and wouldn't make sense without a quantum mechanical explanation. The following will describe some of the experiments in loose detail, but remember that there is a wealth of other experiments that not only back up these findings, but also go into it in far greater detail. Quantum mechanics is to physics and chemistry as evolution is to biology; it's not so much a theory under scrutiny as a method that underpins everything and allows us to make sense of the world because everything fits so perfectly with it. While I would never encourage anyone to blindly accept something if they don't quite get it or have any sensible questions, when learning quantum mechanics there will invariably be a point where you will have to blindly accept something until you learn a bit more. Trust me, it falls into place eventually.
Secondly, remember that quantum mechanics consists of nothing more than a series of mathematical equations, formulas and functions that help us predict the world as it appears at the atomic level. These equations can be used to spit out numbers, and these numbers agree with measurements made by finely tuned instruments - they agree with experiment. Interpretations of what these functions mean are just that; interpretations. Maybe one day we'll have some experiments that go into these interpretations, but until then treat them as nothing more than just-so stories. The important part is that what the equations say and what we observe in reality match up pretty well. All the interpretations are about trying to "make sense" of the equations, because when we try to think about what they actually mean, they seem to imply some things that simply don't mesh well with the way we think reality works. But remember, the universe is under no obligation to act in a way that makes sense to us.
That last paragraph describes what is sometimes referred to as "instrumentalism" - it's the scientific philosophy that the only thing that matters about a theory is whether it is any good at making useful predictions, and that whether it "makes sense" is secondary. This might seem obvious and almost irrelevant, but becomes important in some specific examples. For instance, objects will follow - under Newtonian mechanics, which predicts how objects move - the "principle of least action". This is sort of analogous to electrical current passing through the path of least electrical resistance. It predicts paths and motion very well, and with some mathematical transformation can be converted into many of Newton's laws of motion, which also predict paths and motion very well. The trouble is, this "principle of least action" would imply that the moving object, no matter what it is, is somehow intelligent and can predict its future path, somehow "choosing" the right path to take. This is often a problem from the point-of-view of scientists who don't think atoms can really think, yet alone have psychic powers.
Some philosophers of science (who are mostly philosophers, not practicing scientists) don't like "instrumentalism" because it's unsatisfying, and lets you hold ideas that might be absurd - such as the apparent conscious intentions of moving atoms following the principle of least action. I would agree, to a point, but I only urge people to think this way when thinking about quantum mechanics because it is simply too easy to get bogged down and mesmerised by the "zany" properties, or be fooled and tricked into thinking that interpretations a literal truth. Once you're comfortable with the physics, what the theory actually predicts, and are confident you can line up particular ideas with the experimental results they make, then you can start thinking about how to interpret them. So for this, while I'll have to give some degree of interpretation to avoid getting very maths heavy, I'll try to describe it in terms as close to what the maths say as possible, without trying to "make sense" of it.
 The atom
As part of the "no prior knowledge" part of this, I'm going to briefly describe the atom. The atom is not like a rounded sphere, and it certainly isn't like a solid round sphere. It consists of a nucleus which is absolutely tiny in physical dimensions and is orbited by electrons. The nucleus itself is composed of protons, which have positive electrical charge, and neutrons, which posses no charge. The electrons are negatively charged, and as a result are drawn towards the protons in the nucleus - as opposite charges attract. The neutrons effectively pad out the nucleus so that the positive charge of the protons don't blow the nucleus apart - as like charges repel. There are a host of other particles too that compose the nucleus, known as quarks, and there are the forces that interact between them like gluons and photons. This, however, is getting deep into the realms of particle physics and, for now, isn't really important.
The development of atomic theory is fascinating, and quantum mechanics has played a very important part in stabilising our model of the atom so that it can work in accordance with reality. Essentially, an atom obeying simple classical mechanics, wouldn't be stable. Recall that opposite charges attract; then how do the electrons not just go crashing into the nucleus? Classical mechanics posit that it should, and it was quantum theory that stopped this from happening in our models, giving us the best explanation of nearly all the properties of the atom. For the rest of this, you will need to be aware of two "particles". Despite the fact I'm going to spend a lot of time convincing you that they're actually waves and not point masses, they're still referred to as "particles". These are:
- The electron
- Carries a negative force and circles around the atom. These can be stripped from the atom and fired about, making them quite convenient for doing certain experiments. Their mass is ~1/2000 of a proton that rests in the nucleus - yet the force they emit is equal to the proton, just oppositely charged.
- The photon
- This is the "particle" that makes up light. Light isn't just visible light that we can see with our eyes. The photon also transmits x-rays, gamma rays and at the other end, microwaves and radio waves. The difference isn't magical, it's just to do with the frequency of the photon. Why a "particle" would even have a frequency is what a large part of the first section will cover.
 The fundamentals of quantum
While the above sets everything up to be horrendously complex, quantum theory rests only on a few basic principles, and I'll lay them out in more detail below. For now, these principles are:
- Particle-wave duality - that particles have both wavelike and particle-like properties.
- Quantisation of energy - that energy is confined to discrete quantities that can't simply be divided infinitely.
- The uncertainty principle - that our ability to know about the location and velocity of particles has a fundamental limit placed on it by the universe.
I'll cover these three basics in their non-bullshit forms below.
 Particle-Wave Duality
This is one of the main areas where people can get massively confused by quantum theory. Sub-atomic particles, like electrons, photons and even the nucleus of the atom itself, have the properties of both particles and waves. Indeed, the equations - particularly the de Broglie one - suggest that everything has a wavelength associated with it. It just depends on the size and speed of a particle. A golf ball will have a wavelike quality to it, just that the mass and speed of the ball means that the wavelength is far too small to be even meaningful. Even with something smaller and much faster, like a speeding bullet, still has a wavelength that is far too small to be measured or manifest in any experiment.
To illustrate just the results of this, if things act as waves we can diffract them. In the case of light, you can diffract them through a grating. Because photons in the visible spectrum have a wavelength around 400-600 nanometers, we can position the gaps in a grating close enough together to cause them to interact and diffract. The wavelength even of a small bullet traveling at the fastest speed would have a wavelength smaller than the nucleus of an atom, so it would be impossible to build anything to diffract it though. Hence, we don't see these properties at the scale that we live in and have to really zoom into the world of the atom instead.
Often, this is the mind-blowing stumbling block for people, as we see particles and waves as very, very different things. When it comes to waves and particles, it's easy for us to visualise either or, but not both. So, the way I would best advocate thinking about it is not that waves and particles are distinct things, what these sub-atomic entities like electrons and photons are actually doing is being something else entirely. It's not that they stop making sense at the atomic level, it's that the distinction between particles and waves only makes sense at our macroscopic level, where quantum waves are too small to be detected and we can build emergent waves like sound waves and water waves. We aren't quite able to visualise it but, whatever it is that these photons and electrons are, they just exhibit "particle" and "wave" properties under certain conditions. The fact that they posses these properties is important, as that is what lets us predict and observe certain things in reality. As said above, the universe is under no obligation to make sense to us, and so treating these things as something else entirely, even if it's quasi-magical and impossible to visualise - for the time being - helps smooth out that transition from our everyday world to the quantum world.
 WTF is a wave?
This is nice and all, we know that the distinction ceases to exist at the atomic level, but what do we actually mean by "particle" and "wave" properties? I'll lay it out:
- A discrete entity.
- Acting like a precise point with no size, or as a solid (albeit tiny) sphere.
- Precisely measurable location.
- Precisely measurable position.
- Position is described as a mathematical function.
- Possesses a wavelength.
- Possesses an amplitude.
Describing what a wave is in a few bullet points is actually more difficult than describing a particle, so I'll expand on it here. When we think of a wave, we think of sound waves or waves in water. You can see waves traveling through water and appearing to move, BUT, there is actually no net movement of the water molecules. As a wave travels from one end of a fish tank to the other, it's not taking any water with it - otherwise one side of the tank would quickly empty and all the water would pile up at one side. You can test this by just splashing your hand into a tank of water and seeing the waves go out from it; all those waves are moving, but the water itself isn't. Instead, a wave is a movement of the water that only appears to travel, because the movement has a fixed periodicity. As a wave travels along the length of a fish tank, the individual portions of the water are moving up and down in sync with each other but the water isn't actually moving in that direction. The physical movement along the length is effectively an illusion, similar to how a corkscrew appears to move along its length as you twist it. What is really moving along are the wavelike properties, such as the amplitude and phase, as well as the energy used to carry the wave along and cause the physical, up/down, movement of the water in the tank. In this respect, the wave does move, but the material that it is formed of stays (relatively) still.
The visible and physical movement of the wave through the water is what you might call an "emergent" phenomenon. It's something that simply emerges out of something unrelated doing something in a particular sequence. For instance, you can flash a row of blue LEDs in sequence to make it look like the flash is traveling along them, when really the lights are just flickering in sequence. In the case of waves in a fish tank, the water is moving up and down in sequence, in the corkscrew, it is rotating. This leads to a good question: waves in a fish tank are made of water molecules, sound waves in the air are made of the molecules comprising the air (nitrogen and oxygen), so what are these quantum mechanical waves made of? Is a photon just traveling along as a solid particle with its wavelength stamped on it as some diagrams seem to suggest?
This is actually a question I put to the guy who taught me QM initially, and either either didn't explain it very well (or there was a lost-in-translation between English and Russian) or it underlies one of the big stumbling blocks of quantum mechanics. The best part of a decade later, and I think I have a decent enough answer: the wave of an electron is made of the electron itself. As is a photon, or quark, or gluon, or the Higgs boson. When we draw a wave, it's usually on a graph with an X and Y axis. A sine wave, for instance, oscillates up and down on the Y axis as it moves along the X axis. When we graph the electron, the Y axis is the "density" of the electron - how much of it is there at any one time, as if instead of a fine and finite particle, it's smeared out in a particular pattern. Its motion is like the wave traveling along the fish tank, its wavelike properties are traveling along
There are a few caveats to this last point, and I'll get to them later when discussing what the wave is, how it's calculated, and how the particle-like properties really screw with this image of it. For now, though, start to think of absolutely everything as a wave made of itself. As much as we talk about "particle" physics, a lot more of quantum mechanics relies on the wavelike side of things and its certainly the less intuitive of the two things to think of.
 The experiments
I promised to loosely go over how all this fits into experimental data. For particle-wave duality there are two major experiments that need to be looked at: the double-slit experiment that "proves" that these things are waves, and the photoelectric effect that "proves" they are particles.
 Photoelectric effect
Back when Isaac Newton did his groundbreaking work on optics, we knew that light was like a wave. This made sense and waves like it were just like waves of sound or waves in water tanks - to explain the theory of light from Newton's perspective would have been easy because we wouldn't have had to confuse it with all the particle-like aspects of light. We wouldn't have though of the photon as a discrete entity and the question of what is the wave made of would have been unncessary - at the time, it was explained by the luminiferous aether.
Until, of course, the photoelectric effect came along, the experiment which awarded Einstein his first Nobel Prize. The history behind this experiment is fairly convoluted and for that I'd defer people to the excellent Quantum by Manjit Kumar, who details Einstein and Bohr in a very autobiographical fashion and explains the science as it developed. But to cut the long story short, Einstein didn't buy into quantum theory for quite some time, and even treated most of his observation as a mere "convention" to make the mathematics work out. The photoelectric effect involved firing light at photo-reactive metals and measuring the energy of the electrons that these metal plates ejected. From this we figured out several things.
- That the energy of the individual electrons depended on the wavelength of the light, not on the intensity. Higher light intensity just ejected more electrons, not ones with more energy.
- That below a certain frequency no electrons were ejected at all.
- That electrons were ejected immediately after the light struck the metal.
While those first two are also very important for the quantisation of energy (which will come later) these three points conclusively point to the photons acting as particles. Particularly the last one - although don't ask me specifically how that was worked out. Imagine this: the wavelength of a photon is 400 nanometers, the size of an atom is thousands of times smaller, yet the atom can instantly "gobble up" a photon almost as if it was acting as a particle of hardly any size at all. It's like this wavelike quality never really existed in the first place! The other points also show that the photons are delivering a set amount of energy, in small discrete units. This might be more of a conceptual jump, but if the light was really like turning on a tap and letting water trickle out at any rate, then you should be able to eject higher energy electrons just by turning the tap a little more. Yet no, the energy being delivered was fixed and it was completely dependent on the wavelength of light. The photos that were delivering the light energy must be discrete particles.
The photoelectric effect conclusively proves that photons of light are particles - smashing Newton's optics work that said they were waves. Yet something else cropped up a little later...
 The double-slit
I mentioned earlier that if you pass a wave through a grating, you can diffract it. We can see this easily with light by passing it through a diffraction grating or prism, and we can indeed see it with other sub-atomic particles like electrons if you set the grating to be the right size. So, if these entities were particles as suggested by the photoelectric experiment above, then you'd fire them at a slit and expect that they would be detected on the far side in just a single region determined by that slit. Indeed, this is what you see. If two slits, you'd expect to see two regions where the electrons pile up on the other side... except that's precisely what you don't see.
Instead, the double-slit experiment shows a diffraction pattern appearing on the far side of these slits. It's analogous to what you see when waves of water enter a harbour, they twist, turn and interact with each other. This interaction is called interference and depends on the two waves that you're interacting with. If the waves happen to have the same amplitude at the same position, they'll add constructively, making a bigger wave, but if they're opposite they'll subtract destructively, making a smaller wave. More generally, they're both adding, and the second case is just adding a negative number. The interference pattern you see on the far side of the double slit proves that these electrons are interfering with each other just like waves.
So what? You might ask. If you're sending through electrons they'll interfere with each other, and produce a distortion and diffraction pattern. Indeed, yes, but we can slow the rate that the electrons pass through the slit so that they literally pass through one at a time. If they were particles, they would have nothing to interact with and so we should see two areas where they pile up behind the slits. We send one through, and *ping* it hits the detector on the far side, appearing as a single point on the far side. This is what you'd expect if you sent a particle through. So we send a second one, and *ping* the detector lights up in a different place (yay for experimental error!), like a particle and so far so good. But if we were to keep doing this, keeping track of the impacts on the detector a pattern begins to emerge. It's not a pattern like two slits would produce, it's an interference pattern. But there was nothing for the electrons to interfere with, as they passed through one at a time! The conclusion is that even an individual electron is acting like a wave and interfering with itself. That sounds strange, but actually there is nothing wrong with it if it truly does have a wavelike nature. A wave can interfere with itself no problem because it doesn't really end or have a discrete boundary. A wave can extend over a range, be in multiple places at once and interfere with itself.
 Duality - summary
- In quantum mechanics, what we think of as "particles" have some wavelike character and what we think of as "waves" can have particle-like character.
- The truth is, that they're not either/or, but something else that posses properties of both under different conditions.
- We don't see the wave when objects are large or slow moving - just as we don't really see the particle when they're small and fast.
- The "wave" is actually intrinsic to, i.e., made of, the particle itself.
- The double-slit experiment shows that even a single "particle" can actually interact with itself like it was a wave.
- The photo-electric effect shows that things act like particles.
 Quantisation of energy
The photoelectric effect showed that when exposed to light, metals would eject electrons with a particular energy. Because that energy was related to the frequency of light, rather than the intensity, this suggested that the energy being delivered to the electrons was coming in discrete quantities. So here is where we go over what that actually means and why it's so important.
Energy is a bit of a difficult concept to pin down quickly. With woo-meisters trying to say that energy is like a magic field that surrounds us, penetrates us, and binds the galaxy together we often can't quite figure out what it really is. Basically, it's just a function between two states that measures how much work can be done by converting from one state to another. If that generalisation seems a bit "what the fuck, dude, you said no jargon", consider that in order for us to do work - that is, get up, move around, lift things, breath and so on - we need to use energy. We can get that energy by converting food from one state to another, like glucose into carbon dioxide and water. That chemical reaction releases energy, and we use it for work. Alternatively, we can use work to convert a low energy state to a high energy state - such as using solar fuels to reverse the chemical reaction that produces carbon dioxide and water. A state, in this sense, can be quite broad as to what it can be. In the case of the chemical reaction, the two "states" are merely the configurations that the atoms are in and what is bonded to what. In an atom, the states are the "energy levels" - and we'll look at those in a moment. Moving from one state to the other is the very definition of energy.
So, energy doesn't make sense as an absolute concept, it's always a comparison between one state and another. In quantum theory, energy is no different. We convert electrons from one state to another, releasing or consuming energy in the process. While we're on this model of how energy works, though, I want to make a very brief detour to say that this is why zero-point energy cannot create a perpetual motion machine. Zero-point energy is the energy of the lowest possible energy state something can be in; it still has "energy" because it still vibrates and moves around, but simply cannot go lower. As it can't go to a lower state, energy cannot be extracted from it for work. At all. As I said, energy doesn't really make much sense as an absolute concept, so even if - as some legitimate science postulates - there is "infinite energy" in a zero-point state, the energy that can be extracted from it for work is, well the hint is in the name, zero.
Normally, we might think of energy as coming in any amount we like. A dimmer switch on a light bulb, for instance, can give us practically any intensity of light we like between off and full-power. If we spin the dial 10 degrees, it gives us X amount more light. If we were careful and turned it only 1 degree, we'd get X/10 more light. If we were really careful with our movements we might be able to spin it a tenth of a degree and get X/100 more light produced by the light bulb. In principle this sort of halfing could go on forever. Or in practice we might not always be able to turn that dial a precise 10 degrees - we might turn it up by 10.57468... degrees, and then turn it down again by 10.38275... and then back up by another 10.12595... because there's no restriction as to where this switch can lie. The point is, we might think that energy should work like that, but at this quantum level it really doesn't. Energy is actually coming in discrete steps, like a dimmer switch that clicks between set levels, 1, 2, 3 and so on rather than an infinite number of positions.
Like so much of quantum theory, this seems counter-intuitive when we think about it. It's like describing a car that can only travel at 10, 20, 30, 40 (and so on) miles-per-hour and nothing in between. It's a car where pressing down on the accelerator doesn't cause the car to speed up gradually, but causes it to jump immediately across those different speeds.
 Energy levels
In the photoelectric effect, electrons are being confined to one particular energy level until the light they're exposed to is just enough to cause them to break free of the metal and fire out towards a detector. Energy levels are sometimes difficult to grasp because when they're drawn on graphs it's easy to forget that the Y axis is energy, not exactly its height above something. Getting around energy level diagrams is essential to understanding some of the applications of quantum theory, but for now just figuring out what an energy level is is enough.
These levels are the energy ranges that our particles are confined to. Like the car that can only go at certain speeds, but if this is too weird an analogy, think of it like being analogous to the gear shifts. You can have 1st, 2nd, 3rd gear and so on, but you can't get a 1.5th gear. And converting from one energy level to another releases (or consumes) energy we can use for work.
For electrons, these levels are the orbitals around an atom that they occupy and move between. There are other energy levels. For instance, the rates that molecules rotate and vibrate are found in discrete energy levels, but these are a bit more difficult to conceptualise. As something operating under quantum mechanics can't rest in between two levels, giving it too little energy does nothing, as does giving it too much. So as you expose an atom to different wavelengths of light, rather than different intensities, some of those wavelengths will impart just the right amount of energy to be absorbed and cause the electrons to move from one level to another. This is the basis behind spectroscopy, the method used to probe chemicals and molecular structure, and specifically in this case the UV and visible regions of the spectrum, and so is the reason things are coloured (other ranges interact with different attributes).
 Energy in the atom
The fact that particles operating under quantum physics can only occupy certain energy levels is the reason that negatively charged electrons don't go crashing into the positively charged nucleus. They can't do that simply because they're not orbiting like planets orbiting the sun; they're there as standing waves that obey certain rules. We can easily visualise a 1 dimensional wave, just like a sine wave traveling along an X axis, and electrons around the atom simply 3 dimensional equivalents. This is the Bohr model of the atom and there are multiple energy levels based on how waves can fit around the spherical atom.
The electrons have wavelike properties, so they must obey certain rules about how waves work in three dimensions. In this case, it's spherical harmonics. There are only a limited number of ways that waves can "fit" in a sphere, and so the electrons are constrained the particular energies associated with each of the waves that fit.
 The experiments
Spectroscopy is the study of how energy (like light) interacts with matter. It can tell us an awful lot about what we're looking at, but most importantly it confirms the quantisation of energy and the fact that energy is confined to only certain levels. For this we basically scan through various wavelengths of light and see what gets absorb and what doesn't, and then plot this on a graph. Reading spectra is basically what 95% of analytical chemistry is, and also is the method that will let us find out if any exoplanets have water or oxygen in their atmosphere - yes, it can actually be that sensitive if we're careful and thorough.
The image below is a spectrum I acquired quite recently of potassium permangenate, which is a strong purple colour. Of course, with purple being a mixture of blue and red (see the line of purples if you're really keen), the spectrum shows a strong absorption of light in the green and yellow area between 500 and 600 nanometer wavelengths. This leaves the red and blue light to pass right through because those frequencies don't carry the right energy to excite any of the electrons from one level to another, the blue is too energetic and the red isn't energetic enough. You can also see individual peaks corresponding to different energy levels of electrons being probed by the different wavelengths. To prove that it's not some special piece of magic that depends on big and fancy machines, the spectrum on the right was made using simple pieces of kit you can build yourself. An LED emitting white light, a photodiode to detect the voltage and a diffraction grating to split the light up. The entire thing is mounted on two pieces of wood attached together with tape.
While spectra are nice things to have, you don't need even makeshift kit and diffraction gratings to show it. Without quantisation of energy, our eyes wouldn't pick up colour at all. This is where I get to diverge from QM for a moment, and indulge in another fond love; colour theory.
Colour as we see it is effectively an illusion created by the brain to show us a very simple spectrum of visible light. What it really is, is just the particular frequency that a photon is at, but that's not a particularly quick or easy thing to work with. A scanning UV/Vis spectrometer will take hundreds, if not thousands of data points, my home-made spectrum above uses 35 points, most undergraduates get about 7 points. The human eye, however, does the same thing - sort of - but with only 3 data points.
The three types of cone cells in the retina are stimulated by different frequency ranges as shown on the diagram. Light coming into the eye will therefore stimulate these three cells in different proportions depending on its frequency, and our brain converts this into what we see as colour and intensity. If the light is mostly stimulating the cell sensitive to the longer wavelengths, we perceive it as red, if it stimulates the cell sensitive to the shorter wavelengths we perceive it as blue. Firing these off in different combinations and different intensities produce colour in our heads. When TV screens and computer monitors produce colour, they do so by producing "red", "green" and "blue" light separately, just so our eyes can pick it up and convert it into another colour.
All of colour theory, how colours mix, primary, secondary, tertiary colours, RGB values, the colour wheel, hue and saturation, the line of purples, it all steps from the fact that our eyes pick up three different frequency ranges. Chemical differences inside the cell cause this to be the case, and if energy levels weren't quantised by quantum mechanics, this would simply not be possible!
 Quantisation - summary
- Energy is transfered in individual packets.
- Energy can only take discrete values, of certain levels.
- While the exact energy of these levels can change, they're still quantised and you can find a stable position between two levels.
- You can get something to transfer between one energy state to another by applying the right amount of energy. Too much or too little will cause nothing to happen, hence spectroscopy.
 The uncertainty principle
The Heisenberg Uncertainty Principle, named for Werner Heisenberg not for Walt's moniker in Breaking Bad, puts a universal limit on how precise we can be when describing a particular particle. Specifically, it states that the margin of error in a particle's momentum combined with the margin of error in its position is greater than a particular constant. This constant is very low, of course, so we're not too worried about macroscopic objects like golf balls or speeding cars. Other properties can also be compared like this and are included in a more general "uncertainty principle" but position and momentum are the most commonly cited and easiest to cover.
On account of the "no prior knowledge" promise, I feel I should clarify the following. Speed is the distance something can cover in a particular time, 10 metres-per-second, for instance. Velocity is a vector, which is speed combined with a direction, 10 metres-per-second heading in that direction over there. Momentum is the product (multiplication) of mass and velocity. This is sometimes overlooked in certain descriptions of the uncertainty principle that seem to treat it as if it was just position and speed, but is a useful thing to bear in mind.
Often the principle is phrased along the lines of "if you know an exact position, you cannot know momentum at all, and if you know the exact momentum you cannot know the position at all" - however, I personally think that's a slight misrepresentation as it implies that knowing such things exactly is even possible.
Finally, just to get it out of the way, I would like to quote what Wikipedia decided to put in big bold letters on its page on the uncertainty principle. Upon hearing of the uncertainty principle for the first time, people will always try to figure out ways around it:
“”The uncertainty principle states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.
 Fun side effects
There are a couple of side-effects of the uncertainty principle that manifest in reality. Again, this should underscore the point about it being a fundamental property of the universe, rather than due to us having limited precision due to our equipment.
- Quantum tunneling
Quantum tunneling is the effect where a particle can apparently jump trough near impossible barriers - at least ones that would be impossible under classical mechanics. In fact, to make the barrier impossible, it must require infinite energy to cross. In a scanning tunneling microscopy (STM), which is the tool responsible for all the most close-up images of atoms its possible to get, it's quantum tunneling that allows the electrical signal to jump from the surface being looked at to the tip of the detector. It allows the ammonia molecule to vibrate in such a way that it spontaneously inverts at a far lower energy than you'd expect. It allows for nuclear fusion in the sun. It's a direct result of all this wave-like nature and the uncertainty principle.
- Fast electrons
As the uncertainty principle states that a better precision in position produces a lower precision in velocity, if you start confining a particle to a smaller and smaller space, then its speed has the potential to grow larger and larger. In the thought experiment version, if we were to confine a single particle to a box and then shrink the box, the particle's energy would steadily increase. So much so, that it could break out of the box quite happily, aka quantum tunneling. In the real world, we can see this in atoms. Because the larger and heavier atoms have a stronger positive charge in their nucleus (due to more protons packed in there) the electrons are drawn in tighter. Roentgenium, for instance, is the 111th element and is expected to have a smaller radius than copper, despite having 82 more electrons in there. The result is electrons traveling faster deep in the cores of these heavier atoms. The electrons in these heavier atoms push close to the speed of light and is where special relativity clashes headlong into quantum theory, which doesn't actually include any special relativity in it at all!
- Hearing the uncertainty principle
Because quantum theory is all about waves, there are a lot of analogies to sounds and music. In this fascinating ScienceBlogs post the analogies gets even more explicit; we can actually hear the uncertainty principle in action. The shorter the sound, the less precisely we can figure out what its frequency is. A sine wave that travels along infinitely doesn't really deviate has a very precise and clear wavelength. There can be no doubt that it's 540.55448... Hz for instance, but the shorter it gets the less obvious it is what the exact frequency of it is because, basically, we don't have much of a sample to analyse and get the frequency exactly. Multiple (infinite) sine waves of different frequencies could fit through the shorter pulse equally well, and the shorter the pulse the wider the range of frequencies that can do this.
This is wrapped up in Fourier transforms. A Fourier transform (FT) is just a mathematical function converts a wave from the time domain - the way the wave looks over time (think of the waveform images of the music you see on SoundCloud, for instance) - and the frequency domain, which shows the range of frequencies inside that wave. In a very handwaving way, imagine a piece of music played on a piano, the time domain would be like the sheet music with the notes as they're played in time, and the frequency domain (produced by the Fourier transform) would be like a graph of how many times each piano key was pressed. So if you played a simple C Major chord, you'd hear a single sound, but the FT would resolve this three specific spikes corresponding to C, E and G, the triad making up C Major. BUT, when you do this on actual waves as opposed to a nicely quantised piano, there is a slight problem. The shorter the duration of your sample, the wider the peaks the Fourier transform will produce - because there is a wider range of frequencies that fit the chord. A C Major chord played and allowed do decay slowly over a few seconds would produce three clear and sharp spikes corresponding to C, E and G. But a C Major chord played quickly and sharply, lasting a fraction of a second would see the peaks as broad and slightly confusing as to what the notes were. While only just perceptible to human hearing because you need millisecond long sounds to spread out these peaks over more than one note (read the blog above, listen to the sound files that go with it), the same effect is readily apparent and is a complete bitch in applications of FT such as NMR spectroscopy.
 Other important factors
 Building a wave(function)
I probably skipped a bit ahead by mentioning "spherical harmonics" above, but there are a few simple steps to bring anyone up to speed on what that means. Firstly, a wave needs to have certain properties for it to work in the real world as well as mathematically. It must:
- Be smooth: no kinks or sudden changes in direction. The function of the wave should flow nicely.
- Be continuous: no sudden breaks just for it to pick up at another point elsewhere on the graph. Everything needs to be connected.
- Not loop around: For each value of X on our graph, the wave should have only one value for Y. This makes sense because something can't have two density or probability values at the same position.
With this in mind, we can start looking at how to build a wavefunction in a series of idealised thought experiments that can also be used to map onto reality.
We call this a "particle-in-a-box" thought experiment because we imagine that we put a particle - it doesn't matter what - into a confined box and close it tight. Because we're now no longer interacting with this particle and we can't see it, we don't really know where it is, but we can say with what probability it's in certain positions. First, let's imagine that the particle is confined to one dimension, a line. Also, for the purposes of this, it's confined to only to a certain length of that 1-dimensional line and can't possibly leak out of it, because it's bounded by barriers that require infinite energy to cross (it needs to be infinite due to quantum tunneling, which is a whole other complication that we simply have to ignore for now). This is analogous to an electron in a piece of wire, for example, but it also applies to the conjugated p-orbitals of akenes (which is a fancy way of saying we can see this exact effect in some molecules).
Given the caveats above about what makes a good wave, there are only certain waves that fit in this line. We can't have a cosine wave because it suddenly stops at the edges of our line where the barriers are. That wouldn't make it smooth and continuous. We could, however, have a sine-like wave that loops up, reaches maximum height in the center and dives back down again to zero at the edges. This satisfies all the conditions described above. We only have half a wavelength of this sine wave in our "box", but this is the easiest to obtain - and thus the lowest energy. If we give the particle an extra kick of energy, it actually produces a new wave, something where a full wavelength fits inside the box. The wave goes up, down to zero in the middle, and down again before ending at zero at the far side. We can keep generating new waves that satisfy the conditions above by adding another half length of a sine wave to it. In fact, bollocks to describing just it just watch this, which shows how standing waves work using a simple rope. You can see each of the different levels appear one by one as more energy gets put into it.
We can extend this concept to what's modeled as a "particle on a ring". In this case we have the same set up as before, except instead of infinitely high barriers keeping our particle (and wave) locked on a line, the 1 dimension loops around and joins up with itself. In this case, our first and simplest wave is different because as it loops round it must still be smooth and continuous - if you put only half a wavelength onto the ring, where they meet up would form a discontinuity and that simply wouldn't do. Our first wave on a ring is actually just a constant, the second is one full wavelength where it goes up and down once around the ring, and the third wave is where it first two wavelengths around the ring. This is useful because when it comes to benzene, a ring-shaped molecule that has electrons running around it just like the particle on a ring idea, this is the sort of shape the electron waves really do take. The lowest level has the wave evenly distributed around the entire ring, the second goes up and down once across the ring, and the third level goes up and down twice. There are ones higher than that, but the electrons don't (normally) occupy them. This is what makes benzene so remarkably stable, and why it's often drawn with just a circle in the middle rather than lines connecting between them.
Finally, once you know the basic rules for how to form a wave given certain constraints, you can start thinking in more dimensions. The spherical harmonics around an atom are exactly the same as the particle on a line idea and the particle on a ring idea except moved into three dimensions. It makes more sense if we think of it not as three dimensions of X, Y and Z, but three dimensions in spherical co-ordinates, which are like latitude, longitude and height on the Earth's surface. Converting between the two is just a (relatively) simple mathematical transformation - the key is that you need three pieces of data to describe the position of something in three dimensions. You can take X, Y and Z, or you can take the three spherical co-ordinates.
The wave is then constrained around this sphere exactly as in the particle-in-a-box experiment described above.
 Waves and reality
Key to understanding how quantum theory actually lets us make predictions, is understanding how these waves related to reality. Waves are just mathematical functions, plotted on graphs with an X axis and a Y axis. More generally, they're functions that map a range of independent variables to a dependent variable - so we can have three dimensions of space that are analogous to our "X" axis, so what is analogous to our "Y" axis? From all of the crap above it's clear that the waves are intrinsic to particles and relate to where they are and how they're moving. Let's consider a very very simple wave for a moment. A sine wave, represented by the equation:
And it looks like this:
Now, I'm not saying this is a particularly realistic wave, it's just an example for now, but one thing should be obvious - the Y axis, the value of Ψ, goes positive and negative. And this is a bit of a problem. Does this mean that the particle has some sort of negative and positive properties? Yes, and no. Yes, in the sense that whether the wave is positive or negative is important to how they interact - see the constructive and destructive interference in the double-slit experiment - but no in the sense that the properties aren't particularly physical. An electron doesn't switch to between being positively and negatively charged, for instance. Neither does it mean that the wave has some kind of negative presence there.
To get to something a little more real, we have to square the wavefunction. This Ψ2 value is actually proportional to things we can really view. So the equation:
Looks more like:
And we have no negative areas. It's this squared function that we actually observe as the probability of finding a particle at this point. This produces some very interesting effects. For instance, in some areas the probability of finding a particle is zero, these are called nodes. That is, the particle can never be found at that point. At all. Ever. So it must mean that it can never pass through the node, right? Yet, it has no problem with being found either side of a node, and has a certain probability of being found at any particular point as described by the Ψ2 function. Just remember, this is quantum mechanics; what we are looking at are not "particles" in the sense that we normally think of them, with fixed position and limited behaviour.
So if the Ψ2 function is what describes the probability of the particle, why bother with Ψ in the first place and just include the power of two as part of the function? Simply put, if a wave just has a zero value - i.e., Ψ = 0 at a particular point - it would neither constructively or destructively interfere with another wave. To get destructive interference, the waves must be of opposite sign at the same point, one needs to be positive and the other negative. We do get such interference observed in quantum mechanics, as the double slit experiment shows, so while only Ψ2 manifests itself, Ψ still does show some very real and observable effects.
To show the difference in action, the image on the left is the graph of a wavefunction of a 4s electron, Ψ, and the one on the right is the Ψ2 equivalent converted to its real radial distribution, which is the probability of finding the 4s electron at that distance from the atomic nucleus. Note that the "real" version doesn't have any negative areas, because you can't get a negative absolute probability - and certainly it doesn't change the physical properties of the electron when it moves between.
Superposition is yet another strange concept to explain, and it's where the whole idea of "interpretations" really come into their own but, as I've said, the interpretations are nothing semi-philosophical conjecture at this moment in time. It arises from the wave-like nature of the quantum world. Put most simply, if two waves could possibly describe how a particle is moving, and we have no idea of telling which one is the "real" one (because we haven't looked at it, yet), then a superposition occurs and effectively both waves are used to describe the quantum mechanical system. As far as an instrumentalist idea of quantum mechanics is concerned, this is just a mathematical way of representing our uncertainty. In the double slit experiment, the interference pattern produced is a direct result of superposition. When we look at the wave, one of them will appear conclusively, and the superposition is destroyed.
A simple analogy would be tossing a coin. While it's in the air, it is constantly rotating and flipping and spinning - it really isn't either heads or tails yet. But when it hits a table and settles down, you can see it clearly as heads or tails. The good part of this analogy is that the tumbling coin is very much in a different state to the settled one, as often people might be tempted to think of the coin as "being both heads and tails at the same time", but neither of those make much sense of the coin hasn't landed yet! The bad part of the analogy is that it suggests a superposition might just be a quick switch between two states, or would have a dramatically different physical appearance to a non-superpositonal state. Like with particle-wave duality, the important thing is to know that a superpositional state often has the properties of both potential states, but it isn't literally both states simultaneously. I also want to go through another example of it in action.
If we consider the nucleus of the hydrogen atom (I use this example because it's research I'm directly involved in) it has a particular magnetic moment which causes it to act like a compass needle. So just as compass needles will point north due to the Earth's magnetic field, the magnetic moment of the spin will also align with any applied magnetic field. But because this is a quantum system - and because the proton is so much smaller than the average compass needle and easy to move about - it can align against the magnetic field. It is as if our miniature atomic compass needle had only a slight statistical chance of pointing north rather than south. This gives us two states that the nucleus (considered on its own) can be in at any time. We can label them however we want, aligned or counter-aligned, north or south; the technical jargon is α or β but often handwaved as "up" or "down".
Anyway, this becomes important in creating a superposition when we have more than one atom involved. As we're talking about hydrogen, we have two atoms involved; H2, which is two hydrogen atoms chemically joined together. If we could tell the difference between the two atoms there are four possible combinations we can have:
- Up and up.
- Down and down.
- Up and down.
- Down and up.
You can demonstrate this easily enough by pointing your thumbs in those respective directions, and as you can tell the difference between your right and left hand (hopefully) you should be able to note a difference between numbers 3 and 4. In the H2 molecule, though, we can't actually know which atom is which. Just as we can't conveniently tattoo and label an individual electron, we can't tattoo a hydrogen atom and any label we try to give it is entirely arbitrary. We can't say that one atom is on the left and the other is on the right because we can spin it around 180 degrees then they'd be reversed, but still completely indistinguishable. The technical term for what is going on here is permutational symmetry, which basically governs how a system should act when we have this sort of symmetry. The end result is that those last two combinations can't actually be distinguished in an H2 molecule, permutational symmetry says that those functions just aren't workable, and so a superposition between the two occurs. In the superposition state, the two nuclei can be said to be both up and down at the same time - but really, they're in a state that's just represented mathematically like so:
- Up and down PLUS down and up.
- Up and down MINUS down and up.
These are called linear combinations, because the two possible states are combined and with just a simple linear function, namely addition and subtraction. It works because linear combinations like this are also perfectly acceptable solutions to the Schrödinger equation, and so form valid functions that can operate in quantum mechanics. At this point, it is entirely mathematical. The actual equation and wavefunction that describes the "up and down" combination is being added or subtracted from the one that describes the "down and up" combination. The actual equations justifying this are written here if anyone is interested, the extra fluff is to do with integration and normalisation that results from combining multiple functions together but the underlying principle of simple addition and subtraction is the same.
This should also hopefully demonstrate what I mean about "making sense" of quantum mechanics. It's easy to say that the superposition of states here consists of an atom being both up and down "at the same time", but it's actually two completely new and unique states described by two new and unique mathematical functions. You can actually separate that PLUS state from the MINUS state because they have different properties - for symmetry reasons, molecules in the MINUS state can occupy a lower lying rotational energy level, making it more thermodynamically stable. So it's possible to produce a sample of H2 where all the molecules have their magnetic properties in the "up and down MINUS down and up" state. If it's then chemically reacted in a way that allows for the atoms to be differentiated (breaking the symmetry that imposed this superposition on us in the first place) you get an equal distribution of the two "up and down" and "down and up" states back. It's as if these two states were always there from the start, but if that was so then we wouldn't be able to make a distinction between the two linear combination states and wouldn't be able to separate them out like this! Clearly, the superposition is very real, and more than just a mathematical way of representing mere uncertainty.
To recap, a superposition is where we cannot distinguish two (or more) states, and so we represent it mathematically by combining those two states together. When we examine the system, the superposition goes away and we get just the one state back.
 Why Schroedinger's Cat is bollocks
While superposition is the general term for whenever we can use more than one wavefunction to describe a particle, entanglement is when that specifically includes a second particle as part of it. First of all, let's revisit the magnetic states described above and consider what would happen without this entanglement.
If we assume the interpretation that a superpositional state is that our quantum compass needles are "both up and down at the same time" then when we break the symmetry and observe them, we'll have a 50:50 chance (roughly) of seeing it pointing up or seeing it pointing down. If we had two atoms in an H2 molecule, then each one would have a 50:50 chance of pointing up or down and when observed we'd get our four combinations back:
- Up and up
- Down and down
- Up and down
- Down and up
Each combination would appear with 25% (roughly) probability. However, this doesn't take into account entanglement and, indeed, isn't what we see! Our superpositional state doesn't include just the one atom in complete isolation, it has become entangled with the other - all of this is describing them both at the same time. When we talk about the wavefunction describing a quantum mechanical system, we're talking about describing all of it. So this "up and down" and "up and up" type description is describing both atoms at the same time, not each one separately. This is what entanglement really is; when particles cease to be separate entities and can be described together. In most cases, not only can they be described together, they must be described together. This has a few important ramifications, most specifically that whatever you do to one particle must directly affect the other that is entangled with it. They're being described by the same wavefunction, so cannot be considered separately and independently.
So, we start with two hydrogen atoms, and the magnetic field of their nuclei are in a superpositional state and entangled with each other. If we examine them by breaking the symmetry and find one to be pointing up, the other will be down. If the one we look at is down, the other will be up. No exceptions. There's no way that one of these states can produce an "up-up" or "down-down" combination when we break that superposition. There are two justifications for this. Firstly, the wavefunction of the superposition only contains information about the "up-down" and "down-up" combinations, it simply can't magically generate a "down-down" combination from that as that property just wasn't there. Secondly, it's simple conservation of momentum and energy. The magnetic moment of these atomic nuclei is a real property, and to switch it from one state to the other requires energy. Flipping from a "down-up" to an "up-up" state would require energy to be put into the system in order to balance it all out. The "down-up" combination and state has zero overall magnetic moment - as the up and down cancel each other out - and this is true of our superpositional state too.
Entanglement, therefore, is effectively nature's way of maintaining and conserving the total energy of the universe, which is its most basic and fundamental law. Allowing a random, spontaneous and, most importantly, energy free change from zero overall magnetic moment (down-up, up-down) to an overall magnetic moment (up-up, down-down) would be a heinous violation of the very fabric of reality!
The interesting part is that this happens, as best we can tell, instantly. You can separate entangled particles as much as you like, and it will still work. You can entangle them right next to each other and transport them to the opposite sides of the universe, and still, instantly, one will know what the state of the other was determined to be and will change accordingly. But, if this happens instantly is it violating special relativity and the ultimate cosmic speed limit that is the speed of light?
 Information transfer
The short answer to that question is no. For the longer answer, we need to clarify relativity: Einstein didn't say nothing can travel faster than light (indeed, breaking it is fairly trivial), he said that energy and information cannot travel faster than light. This is the important part. Entanglement doesn't transfer any information, at all, faster than the speed of light.
To justify this simply, consider what you would need to do to send a message via these entangled states. You could treat up and down as 1 and 0 in a computer binary code, for example. So, presume I wanted to send a message instantly across the galaxy. I take my entangled particles, and separate them and fly one (sub-light speed, unfortunately) to its respective destination. Now, whatever happens to my particle, the opposite will instantly happen to the other. The first thing I would need to do, is check whether the particle was a 1 or 0 so whether I need to flip it or not to encode the message. So I check... and it's a 0. But I want it to be a 1... yet that act of checking it, just looking at it has destroyed the superposition. I can now flip my particle between 1 and 0 all I like, without the entanglement the particle at the other side of the galaxy is going to feel and do nothing. And worse, the other side can't check to see if and when I've sent the message without destroying the entanglement too!
This isn't just a practical limitation, it's also about information theory. The shortest statement I can make about information, is that it is a measure of how surprised you are by a new revelation. If I already know something is a 1, telling me it's a 1 doesn't give me any new information. So if we look at an entangled particle, then we have a 50:50 chance of it giving a 1 or a 0. The second particle we look at after that will also have a 50:50 chance, and the one after that, and the one after that. We can't get a message across if it's just going to be random noise like that. For information to be transferred, we need to bring some order to this - which we do by encoding a message so that there is some structure, and we can "surprise" people by giving them a pattern that can be decoded and has meaning.
People might say that entanglement "seems to" transfer information faster than light because one particle knows the state of the other almost instantly. But there's a very precise definition of information that we need to use, and when we consider that, there's no "seems to" about it. Entanglement does not transfer information faster than light.
And no, there isn't some clever way around it.