The beginnings
In mathematics (or any other walk of life I assume) an iterative process is one which is repeated. If I do my process 4 times, I can say that I have done 4 iterations for example – the point is I have repeated the same process four times. We can express iterative processes as very simple expressions. Say for example my process was to take a number, then add 2 to the number with each iteration. In order to express this mathematically I would write:
x_{1} = x_{0} + 2
What I am saying here is that the next term in the series, which here is x subscript 1 is simply the previous term (x subscript 0) plus 2. Now I could write the next term as:
x_{2} = x_{1} + 2
but really you didn’t need this information – you have all the information you need from the first expression and can iterate away as many times as you like – there is no limit, your time is your own. There is of course no explicit advantage to expressing these things things as formulas, we can communicate the same information with words, there is just an efficiency benefit since as the expressions become more complex words become unwieldy, while mathematical expressions stay more compact. It also means if someone is reading this blog for example who speaks no word of English, but is a mathematician they will be able to tell you that I have used one hell of a lot of words to talk about the iterative process of adding 2 to a number, in their mother tongue.
We need to briefly discuss sets, before we proceed to the Mandelbrot. A set is a collection of objects, or in this case numbers that has some rule which defines if they should be inside or outside of the set. Generally a set is enclosed in curly brackets; as a basic example I might write {1,3,7,23,51}. There is no real logic to this, but if I give you any number one can quickly tell from inspection if an item is inside or outside the set – 1 is inside, 2 is outside. A slightly more fanciful set might look like A = {a ε ℝ : a > 4} . This looks wonderful, but reading left to right all this says is the set A is defined as the set equal to all real numbers greater than 4. Clearly we had to use more powerful notation here, since this set has an open upper bound and we could not have listed out all the elements.
Mandelbrot sets curiously involve a function no more complicated than the types taught at GCSE grade – but don’t beat yourself up if you are rusty! We consider the functions of the form
f(x) = x^{2} + c
Where c is a constant. Now this function can be altered lightly to make a series of iterations like we had before –
x_{1} = x_{0}^{2} + c
x_{2} = x_{1}^{2} + c
and so on, where I take the previous number, square it and add my constant. If I asked you to write me out 100 iterations – hopefully you would ask me for two things – the value of c and the value of x_{0}. The value of x_{0} is of great importance, so it will have a special name – the seed, for it is the seed which will drive the growth. The value of the constant is also of tremendous importance for Mandelbrot sets, although depending on the relationship with the seed in some examples it is quite trivial.
A few Mandelbrots
I want to present a few very basic sets for simple values of c and x_{0} before going on to bigger things. Say for example I take my constant to be 0, and my seed to be 1. Then I have a very boring set indeed! The first term is 1² + 0, so 1. Then the next term is the previous term squared plus 0 – uh oh, looks like I’m stuck in a loop. So by taking these values I have the set {1}. If I had of let c equal zero as before, but let my seed also take the value of zero I would of had the most boring set of all – the set {0}. Moving on to something slightly more interesting – see if it follows to you that if we let c equal 0 and we have our seed as -1, we get the set {-1,1}.
If I take less trivial values, say for example letting the seed and the constant take the value of 1 then hopefully it is clear to see this thing will blow up pretty quickly – the first term will be a nice sensible 2, then a manageable 5, followed by 26 and we are off – the next term is 677 and the one after that into the hundreds of thousands. This is because, if my seed and constant allow the squared term to dominate the growth is explosive giving a set where the terms quickly get larger. Unlike the other sets we discussed, this one is infinite in size. Interestingly when we put in different values the system tends to different behavior. If for example we hold our seed at 0, then a value of c equal to -1.6 gives us chaotic behavior however a value of c equal to -1.75 gives us a repeating pattern with a period of 3 – this is by no means intuitive. It is the changing nature of the sets relative to a seed of zero that interests us today.
Getting complex
I need you to make a conceptual leap and consider complex numbers. We won’t learn about them in full, but let me tell you a few facts about them. A complex number has a real and imaginary element to it – the real part is the bit you are comfortable with, the ordinary numbers. The imaginary element is just another number added to the real element and denoted i. It is worth knowing that i² is equal to -1, so these numbers are very hand as we can square root negative numbers.
You might think this sounds totally made up – and if you do feel that way you are correct, however what is spooky about complex numbers is that the addition of the imaginary element allows us to model and solve real world problems. So can we really say they are imaginary when they solve things which are real? Or can we say the things are real when they are solved by numbers which are imaginary? One for another day.
Rather than representing complex numbers in a line, like we would ordinary numbers we are required to introduce an additional axis – we have two number lines, at 90 degrees to each-other with one representing the complex numbers and one representing the real numbers. Because the complex numbers occupy all of this space we say the complex numbers are represented in the complex plane – this isn’t anything technical, it’s just a result of the fact that we can plot all the real numbers on a line but we need an area, or a plane for the complex numbers. Here are a few complex numbers plotted in the complex plane.
So here we have the number represented by a as 1 + 4i; 1 being the real part and 4i being the imaginary part we just introduced. Lovely.
The Mandelbrot we love
Now we have battled through all that maths we can finally produce something artistic! Now earlier we discussed a seed of 0, and we highlighted the fact that when the value of c takes different values we have different options;
The Mandelbrot set is interested in the entire complex plane – so to give you a few examples if I let c = i and my seed as zero, I have;
x_{0} = 0x_{1} = 0^{2} + i = ix_{2} = i^{2} + i = -1 + i
This set does not escape to infinity it stays locked up with fairly small values. There are many sets for which this happens – and many for which it does not. We want to be able to “see” this.
A visual representation of the Mandelbrot set is interested in considering the values within the complex plane where I can have this phenomena – if I let my seed equal zero, what values of c will give me a finite set. This is the result.
The black areas indicate the regions where this is possible, with the areas outside being values which tend to infinity.So any area within the black represents a value of c, with a seed of zero which produces a finite set. It should be clear this is a highly intricate pattern from this very zoomed out image, but what happens when we zoom in closer is truly breathtaking. Have a look at this zoomed in image.
What is most interesting is that the more we zoom in on these boundaries, they still look just as crinkly. The detail is almost beyond comprehension! The main heart of the Mandelbrot is decorated finely with bulbs and antenna all different in shape – the nature of these crinkles is one of the big problems still open in mathematics today.
There are however many interesting things we do know. For example, if we choose any number within a bulb of the Mandelbrot we will know the period the pattern tends towards. This is shown here – so for example if my c value is in the area labeled 3, then my iterations tend to the same value with each 3 goes.
The conclusions we can draw from these sets is an underlying order among chaos. We are using geometry to understand the dynamics of the function, and using the dynamical information to understand the shape of the geometry – this is a big leap forwards in human thought. Whilst this interplay is not fully understood, it has fundamentally shaped the way we look at the world. Benoit Mandelbrot once said;
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
This roughness, or chaos in the world is something Mandelbrot celebrated – and in fact on a small scale these chaotic things are similar to the overall larger things. Small clouds for example is very similar to the overall thing; allowing some logic and unity to be applied. This is the very heart of fractals and has found successful application in the fields of cosmology, medicine, engineering, genetics, art and music. The discoveries into this area are relatively new, due to the fact that they required computers for the powerful visualizations – something Mandelbrot had access to while working at IBM. There is a particularly interesting application he was able to use; through fractals he could accurately predict profits and losses made by traders over time. In 2005 he warned that traders have a tendency to act as if the market is predictable and immune to large swings; something we may well argue was vindicated a few years later. This is truly a powerful area of mathematics, which through comprehension can only deepen the human need to bring structure to chaos.
Finally, in case you need it:
A big thank you to all our followers, new and old. This is post 100 here at RTU.
Let’s explain what this scenario is before we branch off into the different variations and pass any judgement. The brain in a vat scenario is the scenario when an evil scientist, machine or other more powerful being than a mere human, has removed the brain from a human’s body and suspended it in a vat full of a concoction of organ-preserving elements. The neurons in the brain are then connected by wires to a ‘supercomputer which can deliver electrical impulses to the brain, ‘identical’ to those the brain would naturally receive when existing within a body which is interacting with an independent reality. At the end of the day that’s all our perceptions of the outside world are, a result of electrical signals travelling through neurons as a by-product of our experiences. Stroke a puppy, the fur of the puppy is felt on your fingers which sends signals representing the sensation of touch to the brain. Hear the puppy bark, sound waves resonate on your eardrums, which sends signals representing the sensation of sound. The idea is the supercomputer has the ability to simulate all these sensations and thus create the experience of a perfectly normal environment. In fact if so it could have even inputted in the sensory experience of all the other supposed human beings around you. This then resonates with the philosophical view of solipsism – one can only be sure of the existence of their own mind. (Wacky)
Now because this could be possible, the philosophical sceptics (people who like to question our very basic assumptions about reality) say, it is therefore impossible to tell from the perspective of the brain whether it is in a human body or in a vat. Now you may say woah wait a minute, think of all the neural simulations the controller would have to manipulate to construct a flawless reality – but unfortunately this is not the point. The sceptic’s point is that as sensations are just a product of neural signals, if even one can be simulated, theoretically they all can (with enough time). Therefore, as we cannot say with certainty we are not simply brains in vats, it is impossible to rule out being a brain in a vat. Therefore the bottom line is we cannot know whether our beliefs about reality are at all true.
There is an argument from biology that believes a brain suspended in a vat is fundamentally different to a ‘in-body’ brain and therefore we cannot say it is even possible to replicate the experiences. The ‘in-body’ brain receives input through the senses found in the body, which in turn receive their input from the external environment. However, the BIV receives stimuli from a machine. Whether these can be completely mimicked is highly debatably. This line comes down to the neuroscientists, of which one I am certainly not. However, to the best of my knowledge, those at the leading edge of the field do believe all experience can be expressed in electrical exchanges between neurons, and if that is the case the sceptic is right andit is impossible to rule out BIV scenario. I can sense many people up to this point will not be very convinced, let me try something more familiar to persuade you, something that everyone has personally experienced. Dreams.
The dream argument is the idea on frequent basis we believe we are in reality when we are not. When we dream, unless one is skilled in the art of lucid dreaming, we believe whatever we are dreaming about is in fact real in that moment. It is only when we wake up that we realise we had been deceived. Often in dreams we can also have sensory experiences, like the dreading one of falling. How do we feel this? Well it is a product of experience, we have experienced what it feels like to fall and our neurons can replicate those signals. You may think well surely we would have woken up to the actual reality by now, humans have been around for centuries. But it’s not humans we’re talking about its the experience of the individual which is in question, for we can never be sure of the experience of others and an individual of on average 30 or so human years – a mere blip in the lifetime of the universe.
All of this is all very matrix-esque, minds inside a simulated reality. Within our collective reality (if there is such a thing and we aren’t solipsists) then we are already pursuing an endeavour like this ourselves – virtual reality. As the progress stands now we are nowhere near making our attempts at virtual reality indistinguishable from the outside reality but it a task we are pursuing. Remember reading my article on the Fermi Paradox? Given the vast number of stars in the observable universe, statistically speaking there are a vast number of Earth-like planets. Our sun is relatively young and therefore other earth-like planets may host life/civilisations much more advanced than ours, due to their planetary system have been around a lot longer. Who is to say an advanced civilisation hasn’t mastered a virtual reality indistinguishable to reality already? An uncomfortable thought.
Another argument put forward for the scenario of the existence of brains alone is the weird and wacky Boltzmann brain paradox. According to thermodynamics systems tend towards a high level of entropy (a state of chaos) see ‘A descent into chaos’. Boltzmann proposed that the probability of fluctuations away from such high-entropy states are unlikely i.e. fluctuations towards order and organization. It’s rather unlikely a swarm of bees will randomly decide to form an ordered cube. Therefore, Boltzmann believes it is far more likely for the distributed matter in the universe to come together randomly to form a brain which is alone, floating around in space with a certain neural net representing memories, than it is for the existence of brains which came about through human beings. Brains as we believe them to exist arose from evolution from ancestors which required a high level of organisation in our external environment, i.e. earth, oxygen, food. ‘In an infinite universe, the number of self-aware brains that spontaneously and randomly form out of the chaos, complete with memories of a life like ours, should vastly outnumber the brains evolved from an inconceivably rare local fluctuation the size of the observable Universe.’ In a nutshell the paradox is that it much more likely that the brains that exist are Boltzmann brains, floating around freely as opposed to evolved brains in a life-welcoming environment.
[Futurama’s episode on Boltzmann brains!]
To summarise, philosophers argue that seeing it is impossible to verify that we are not just brains in vats or brains floating alone in space we cannot be sure that this is not what we are. (Take a moment to digest that philosophical point.) And, as such we cannot be sure if our perceptions of the environment are genuine representations of reality or a trick of our senses. Whether we should be so sceptical just because we cannot rule out the scenario is another question. It is undeniable that the philosopher’s point is valid but can it be truly be taken so seriously? Does it all even matter anyway if either way we still experience the pain of having to get out of bed to go to work in the morning?! I’d been keen to know what you think… provided you aren’t just a simulation within my mind of course. Drop a comment below!
There is an overarching principle that light takes the shortest path possible from A to B. From the below diagram, a simple right angled triangle, you can appreciate there are in infinite number of paths I could draw to go from A to B. I could go from A to C to B, or I could totally ignore the lines – go from A to Amsterdam, enjoy a lovely coffee by the canal and return to B after a little airport shopping. My path is still A to B. Of course what we can say is that there is only one shortest path, which is the path from A to B labelled c – the hypotenuse of the triangle, calculable with some before Christ mathematics.
To say that light takes the shortest path possible isn’t wrong – but it is the kind of thing often said by grown ups when they don’t fully understand something. Perhaps if you placed me at A and asked me to get to B as quickly as possible I would run in a straight across c to B, because I am a sentient being and it makes sense. But what if I were to be robbed of my senses? What if you were to place me in a densely overgrown forest with no clear sight of B, C or anything but shrubbery and thousands of different paths – then I would need to simply try out the different paths until I found B. So assuming light is not sentient, how does it know the shortest path from A to B? Does it send out scoutons (scout photons) to work out the path and report back to HQ before the troops ride on?
As much as I wish I could spend the rest of this post talking about the curious adventures of the scouton, the real truth lies by considering the wavelike quantum nature of the photons – i.e. light as an electromagnetic wave. If we are to accept that light travels at a constant speed, then it makes sense that it takes the least time for light to travel directly from A to B – this is a given by virtue of the fact it is the shortest length. If light were to take other paths we would expect it to take more time to reach B. The following diagram has been taken from the popular science book QED: The strange theory of light and matter by Richard Feynman.
The first part of the diagram shows us an illustration of some paths light may take to get from one place to another, as discussed. It is not exhaustive, there are an infinite number of possibilities. The graph at the bottom left shows the various times light takes to travel those paths, which is in a U-shape with paths C,D and E the shortest. This is intuitive from looking at the diagram.
We begin by explaining the passage of light in simple terms, in a fashion similar to Feynman before we go on to explain the complexity that has allowed the simple explanation to be offered. The arrows beneath the time graph are imagined to be the whirling hand of a stopwatch, which stops when the photon lands at the source. If you want to imagine this stopwatch hand in real time you better let it whirl 30,000 times for every inch light travels! The direction of the arrow can be considered similar to the direction of travel when the light reaches the source. Now the arrows can be added together to form a path, not just here but anywhere in physics . When we add these arrows by butting the head of one by the tail of the other, it isn’t so important where each one takes us rather the overall direction we have traveled. As such when we finish adding arrows we just draw one big one from start to finish. What we see is that some arrows cancel each other out, for example B and F are almost exact opposites of each other so their sum is nothing exciting at all. Similarly A and B are quite close to cancelling, so if you net these together you haven’t really gone anywhere at all. The arrows in the middle however do not cancel – these arrows are the more horizontal ones.
Now the above illustration is of course a small snapshot; we are considering many different possible paths so we will end up with many more arrows than this although we do not need any more to understand. The core idea here is that the when we say light travels in a straight line what we are really saying is the light would probably fail to make it from the source to the detector if we were to remove the horizontal arrows. In fact the length of the arrows is representative of the probability of occurrence, which as you can see is exactly the same! So we are not saying that the other paths do not occur, they are just as a likely to occur however they cancel with other paths. The ones we are left with, the ones that actually mean something are the straight line paths which are the ones which give us that age old adage – light travels in straight lines.
We could build a really complex path using lots of funny arrows inching us closer and closer which is nothing like a straight line at all! This path would be expensive though, all those probability arrows mean that the overall path is going to be highly unlikely to occur. If we want to start worrying about things like that then for goodness sake buy a lottery ticket. Of course it is highly possible that one of the very improbable routes from A to B may be traversed by a photon at some point; indeed to should if you allow the clock to run for long enough but this is almost certain. But what’s a rouge photon among friends? A photon here or there isn’t enough to stimulate sight, but it is enough to change our belief that light travelling in straight lines is simply a general probabilistic expression of quantum electrodynamics rather than an absolute fact inherent in the properties of light.
So if you are still unclear, light is most likely to travel in a straight line path – which means over reasonable distances we can consider the straight line path. If you want to get really quantum reduce the distance to less than a stopwatch turn and watch things get weird. Another day.
Disclaimer: This is an optional end to the post. For the interested reader we elucidate the mathematical interpretation of the above explanation. This is done briefly for completeness and is unlikely to be of interest without a mathematical background.
I couldn’t leave you without putting a little more flesh on the bones – spinning arrows and adding together seemingly random paths may seem unsatisfying to the curious or mathematical mind. What Feynman is getting at in his book is path integral formulation. In classical mechanics we can consider a unique path for a particle such as a cannonball flying from A to B or a person jumping from the ledge of a tall building trying to understand quantum mechanics. In quantum mechanics however, we have to use mathematics that allows us to compute the sum over all possible trajectories. This is powerful mathematics which evaded human comprehension for some time – understandably too for it is a fairly out there idea to consider all possible paths rather than just one based on initial condition. It was Feynman who first worked out this precise description – and for this he should be remembered.
It was found that the quantum action could often be considered as a number of discrete classical actions. This redefines the way we can look at such problems and greatly enhances our toolkit for solving them. The probability assigned to a particular event is given by the modulus of the probability amplitude, which is a complex number. The probability amplitude itself is given by adding together all the paths in the configuration space we consider. We can then compute the contribution of the path by considering the time integral of the Lagrangian over a particular path (the process involves considering an exponential function, but this will suffice here). For any given process, we add up all of the paths which then gives us a wonderful elaborate anything can happen picture. We find that the amplitudes assign equal weight (modulus) but different phases (arguments) to the paths which is what allows the path which differ considerably from classical paths to cancel in a very similar way to interference. This is analogous to what we considered in the diagram above.
Storms on the Sun blow out streams of electrically charged solar particles across space and the Earth is in the path of this stream. The solar wind, as its called, distorts the Earth’s magnetic field and allows charged particles from the Sun to enter the Earth’s atmosphere. This is most easily done at the poles of the Earth where the magnetic field lines originate or end, this is why we predominantly see the lights at these polar locations where charged particles are more easily trapped.
The charged particles then collide with atoms in the Earth’s atmosphere and these atoms undergo a process called excitation. What does it mean for all these little atoms to get excited? Let’s recap the structure of an atom:
Atoms consist of a nucleus in the center where the protons and neutrons are stored and then a cloud of orbiting electrons. The orbiting electrons can live in various shells or ‘energy levels’ (which are usually given a number n, n=1 the first energy level) for a full description of this see the much more detailed post ‘Electron Configuration Demystification‘. When the charged particles from the solar winds collide with the atoms in the earth’s atmosphere the energy transferred in the collision causes the electrons in the atom to move up to higher energy levels. Also called – excitation. However, the electrons cannot stay in this state and eventually fall back down to their original energy level. When they fall down their relinquish the energy they took from the energetic particles in the solar wind and this is given off in the form of photos. Photons remember are particles of light. This is the light we see in the Aurora Borealis, but wait what about the magnificent colours?
Different gases/molecules in the Earth’s atmospheres give off different colours when they are excited. Oxygen gives of the colour red at the highest altitudes when excited atomic oxygen emits a photon in the 630nm wavelength range of visible light. At lower altitudes, emission from excited oxygen occurs at 557nm wavelengths which causes intense green colours to dominate, the most common colour of the aurora. At even lower altitudes atomic oxygen is uncommon and atomic nitrogen takes over, emitting in the blue wavelengths of the spectrum. Finally, colours such as yellow and pink are a mixture of the primary reds, blues and greens.
Often the Northern Lights, as so called, can only be seen in the North, but if a particularly violent storm happens the strength of the solar wind can be so strong that it can penetrate the Earth’s atmospheres at lower latitudes. So if you’ve managed to see the Northern Lights in England it’s very likely to be tied to recent extreme solar conditions.
As disappointing as it might be that the magnetic field of the Earth blocks out the spectacle of the Northern Lights across the majority of the globe, it really is our silent defender. Were we to have a constant stream of charged particles bombarding our planet’s atmosphere, not only may we not have an atmosphere in the first place but we would suffer the constant changing of chemical compositions of matter and the wreaking of havoc with our electronic infrastructure. So it seems the Aurora Borealis remains, as is the case with most things beautiful, rare. What a tick off the bucket list it would be to go and see their spectacular show in the polar regions of this wonderful planet.
Principle Quantum Number
The principle quantum number is as it sounds – the important one which if you have taken any basic Chemistry qualification you will know about. The principal quantum number, denoted n, is essentially the average distance that an electron will find itself from the nucleus. It is often referred to as the number of electron shells or orbitals – this isn’t wrong, since when an atom is in its ground state (none of the electrons are excited) the highest principal quantum number will be the number of orbitals that atom has although be warned “orbitals” is misleading. The further from the nucleus of the atom you go, the higher the energy level of the shell. Much like objects will naturally flow to the point of lowest gravitational potential energy electrons will naturally move to the orbital of the lowest energy – as such it should be no surprise that the ground state of the a hydrogen atom, which has one electron has a single electron in the first orbital.
The final question that often arises is why don’t all electrons sit in orbital 1, if this is what electrons naturally want to do? Much the same reason you might leave a crowded underground train and wait for the next one. There are more scientific reasons that this – but consider the fact that as you retreat from the nucleus the size of the orbital increases. Electrons have the same electric charge (along with a host of other clashing qualities), which much like me and other humans, means they refuse to be within a certain proximity. It is not so much that I can’t be that close to other humans, it is more that beyond a certain closeness other arrangements look more attractive. It’s the same deal here – the electron may yearn for the lowest energy arrangement possible, but if it’s overcrowded then the next best will suffice.
Diagrams in any scientific or mathematical discipline are essential – they allow the human mind to model a situation and to rationalise it into something we can understand. However it is important that once we are finished with a diagram or a model, we do a reality check to ensure we haven’t in some way lost sight of the physical situation. The above picture of the atom is what we would consider a Niels Bohr model – very useful for understanding what we have just discussed, but there are two important and fundamental differences with reality – firstly the nucleus in that image is massive. The nucleus is dense and tightly packed – indeed if we were to increase the size of a hydrogen nucleus to the size of a basket ball, the average position of its lone electron would be approximately a two mile walk. Secondly, the orbitals are shown as concentric circles – this isn’t correct in any sense, for two reasons. The first is that electrons don’t orbit the nucleus – orbiting anything requires acceleration, which in turn requires energy. If the electron were to orbit the nucleus they would gradually loose energy and crash into the nucleus meaning life as we know it is destroyed. The second reason, is the approximate shape of an average map of the positions of electrons is only spherical in certain circumstances – that leads us nicely on.
Azimuthal Quantum Number
Isn’t science just the best? Azimuthal. Whilst this number may sound like a number bestowed upon atoms by Zeus. I assure you it is much cooler (sort of). This number is also known as angular quantum number and helps us understand the shapes of each shell. The plural on shape was no error – each shell is broken down into subshells. So for all the electrons in a particular quantum number n, there are a number of different subshells which are helpfully labelled s,p,d,f,g,h,i*… by chemists, or from 1 upwards by normal people. *(This used to stand for sharp, principal, fundamental and diffuse until it was realised this didn’t mean much and it was abandoned)
The natural first curiosity is how do I know how many subshells I will have? For all discussions in this post we assume the atom is in ground state – all of the considerations still exist when the atom is excited, but the electrons will not follow the logical ordering of the lowest possible energy configuration. Let me show you the maximum number of electrons in some subshells and try to spot the pattern
Okay so it isn’t all that simple – but basically for any principal quantum number we can have a maximum of 2n² electrons. Then within a specific subshell;
4ℓ+2
where ℓ represents the azimuthal quantum number – which works providing you label the s subshell as 0, then p as one and so forth.
This may all sound very interesting (or it may not), but why do we care? Well ℓ determines the shape of the orbitals, and when we are trying to visualize an atom it is very hard to do so without knowing the shape. Angular momentum is a function of the inertia a body has along with it’s angular velocity, so it shouldn’t come as too much of a surprise that the angular momentum, or the angular quantum number ℓ dictates what shape the shell should have. If you want to understand why a shell has that shape you are looking at some hardcore mathematics, for which this is not the place. The diagram below shows the various, rather beautiful shapes of orbitals.
Unfortunately the ordering does not obey beautiful logic whereby all of the subshells attached to the principal quantum number 1 are filled first, before the shells attached to n=2 and so on. The s subshell is closer to the nucleus that the p and so on, however there is overlap, which means the outer subshells attached to certain principal quantum numbers are actually further from the nucleus than the inner subshells of higher principal quantum numbers. This means the inner subshells attached to the higher principal quantum number are a lower energy configuration and should therefore be filled first. The following diagram shows you the order in which things should be filled.
Finally let us do our reality check – we are building a clearer picture of what the electron looks like, however the thing which remains potentially at odds with reality is the “shape of subshells”. When we say the shape of a subshell you must not think of this like some kind of plane which the electron is constrained to move in or you will not be looking at reality. Instead think that the electron can go to many different places, however some are more likely that others. Therefore whilst it will go to all places it can go in time, it will visit the most likely ones more often. Now if we could attach some sort of ink to the electron, which left a mark in every place it had been then the final cloud of marker traces would look something like the orbital shapes. That isn’t to say I can tell you where the electron is however – I can tell you where it is likely to be, but due to probability and the uncertainty principle pinpointing that slippery little tyke isn’t possible beyond certain bounds. If you want to get really behind this thing you need to consider the fact that electrons are actually behave like standing waves as they occupy the orbitals – complicated but interesting.
The magnetic quantum number
The final two quantum numbers are shorter to explain so stay with it. The orbital pictures drawn above have three different configurations on the p row, and four on the d row. Whilst it is logical for a human to just have one orientation which is “upright” that is a relative term and nature does not care for human logic. The magnetic quantum number, m_{ℓ}, must obey the following inequality where ℓ is the azimuthal number as described above.
−ℓ ≤ m_{ℓ} ≤ ℓ
The magnetic quantum number determines which of our orientations the subshells should have – so in the above picture on the p row you can see -1, 0 and +1. Origins are human constructions, but it is interesting and curious that when we do peg down our origin we are constrained for the p subshell to have three orientations, for the d subshell four and so on. It should be no surprise that a spherical orbital therefore only has one possible projection – a feature of the quantum world or human measurement?
The spin quantum number
The spin quantum number is a description of the way in which a particular electron is spinning in an electric field. For an electron m_{s} will be ±½, which you can loosely think of as clockwise and anticlockwise. In physics these two options are called spin up and spin down. Under the Pauli exclusion principle (not for this post) electrons in a particular suborbital cannot have the same quantum numbers – so that means we can only ever have 2 electrons in a suborbital. The spin number is most restrictive! If you are wondering how this works with everything I have just said, look at the p shell in the above diagram. As we discussed you would expect this to hold a maximum of (4 x 1) + 2 = 6 electrons- if you look you will see there are two dumbbell parts to the orbitals plus one you can’t see in the center – it is these which are the suborbitals, so when we say each suborbital can only hold two electrons we mean these – one with spin +1/2 one with spin -1/2. Similarly you can see four suborbitals in the d subshell plus one in the center, which gives the 5 suborbitals for the maximum (4 x 2) + 2 = 10 electrons.
Not that for the magnetic numbers and the spin numbers the values are “random” – there is no set way to know specifically which of these configurations a lone electron is going to choose and there are not universally fixed values – the above are naming conventions which must hold when we perform calculations. I can of course say if I have an electron with spin up, if another joins the party it will have spin down, but as hinted previously these are indeed features of human measurement.
Newton’s laws of motions are three laws that together form the basis of classical mechanics. Newton’s insight was so advanced for his day (17th century) and it was he who taught us for the first time how to understand the spatial behaviour of objects and the relationship between forces and motion. Physics may have advanced leaps and bounds since Newton’s day but the three laws remain solid pillars of the classical world.
The First Law:
The first law states that if the sum of all the forces acting on an object is zero, then the velocity of the object is constant. [Note constant, not necessarily zero.] i.e if no force acts upon an object it continues to do whatever it was doing before, if it was at rest it continues to sit still, if it was moving it continues to move without changing its velocity. This is known as enacting uniform motion. Myth bust: let’s just clarify what velocity is -velocity is a quantity that incorporates the speed and direction of a moving object. The speed of an object (say 5 miles/hour) is the same regardless if the object is moving in a straight line or a circle. However if the object is moving at 5 miles/hour but in a circle it is changing direction and hence its velocity is not constant. Now let’s get some notation involved with this first law:
To represent force we use the symbol F. Now this sign: ∑ means summation and in front of the F would mean a sum of all the forces in question. So this first expression below means the sum of all the forces on the object add to 0. Now the position of an object is represented in physics by the symbol x. The rate at which an object changes its position over time is its velocity, represented by the symbol v (commonly thought of as speed but we know the difference now). How do we represent this rate of change over time? Well there are two common ways: either by writing d(insert quantity of interest)/dt (where t is time), the d represents the ‘change in’ so this effectively reads ‘change in quantity of interest over change in time’. Or it is given by writing the quantity of interest with a little dot over the top.
So back to our first law. If there are no forces acting on the object the velocity of the object is constant, or in other words, the rate of change of the velocity is zero. [It does not change, it stays the same/constant!]
There we have it, the first law in all its simple glory.
The Second Law:
The second law states that the rate of change of momentum of an object is proportional to the force applied. Let’s recap what momentum is, momentum is a quantity which combines the mass and velocity of an object. A momentum of a tiny person running very fast is roughly the same as a fat person walking slow. You can get an idea of how different objects would rank with their momentum by thinking of which you would least like to be hit by! You definitely wouldn’t like to be hit by a fast moving car because not only is it heavy (large mass) but its velocity is very high. Momentum is represented by the symbol p and is equal to the mass of an object times its velocity. (p = mv) Now to recap the initial statement, you would need to apply a greater force in order to produce a greater change in the momentum of an object – this is intuitive, if it were the other way around our world would seem very strange indeed.
So as used before, d(quantity of interest)/dt indicates the rate of change of that quantity and in this case we are talking about momentum – so we’ll stick in a p or an mv, whichever you prefer. This expression is equal to our Force F. Therefore what this law is saying is that you need a force to act on upon an object in order to change its momentum.
Final step: how does the product mass x velocity (mv) change with time? Well Newton’s laws are valid only for constant mass systems so the m can sort of be ignored when it comes to the rate of change. We take the m out of the d(mv)/dt bracket so we now have md(v)/dt as seen below. What, however, is the rate of change of velocity? Acceleration! Acceleration is the rate of change of velocity of an object how much it speeds up or speeds down (deceleration). The symbol for acceleration is a. Therefore dv/dt can be replaced with simply a.
So there we have Newton’s second law in all its glory. In a nutshell: a net force applied to an object of mass m, produces an acceleration a.
If we know the force on and mass of the object we can work out the acceleration. If we know the mass of the object and acceleration it is experiences we can work out the force. And finally if we know the force we’re applying on an object and can measure how much it is accelerating we can deduce its mass! Hurrah!
(Why are some letters typed in bold you may ask? The letters typed in bold represent vector quantities in mathematics – a quantity that has direction as well as magnitude)
The Third Law:
The third law states that for every force that exists there is an equal and opposite force pushing back. If an object say you, exert a force a second object (you push a door) then the second object exerts a force back on you with equal magnitude (a.k.a strength) yet in the opposite direction. If we call the two objects A and B then the notation for this force is simply: F_{A} = −F_{B }
In this set up force exerted by object A is referred to as the ‘action’ and the force exerted back by object B the ‘reaction’. This is why sometimes Newton’s third law is referred to as the action-reaction law. This law gives as the insight into forces to understand they are all fundamentally interactions between different objects or bodies. There is no such thing as a force without the presence of its equal and opposite partner. This third law is essentially what allows us to get anywhere. Think about it – when you walk you push back against the floor and the floors pushes you forward. When you swim you push the water backward with your arms and the water simultaneously pushes you forward. The tires of a car push against the road and the road pushes back on the tires, driving the car forward.
Has anybody seen the new sci-fi film Passengers? I do love a good sci-film (especially one set in space) the plot line was a little lack-lustre and quite frankly disturbing in its premise at parts but this isn’t a film review blog so i’ll stop. Anyway there’s a scene near the end of the film where one the main characters is floating in space but being pulled towards the fuel-buring backside of the spaceship. He has a heavy object in his hands and cleverly remembers Newton’s third law to get him out of the sticky situation. By throwing this object towards the fire he himself is propelled backwards in the opposite direction due to the reaction force and manages to return to the safety of the ship. It pays off knowing your basic physics, you never know what situation you might find yourself in…
Anyway that’s all for today – I hope this post was educational without being too taxing! Any questions, post them below!
My last post simple harmonic motion, was the first in a new technical series – posts which will appear infrequently but contain the mathematical rigour required to get behind the details of a particular subject. As promised, we now return today to a more discursive post to consider one of the big ticket quantum effects – quantum entanglement. Towards the end of the post, as we cover Bell’s theorem the reader will be required to think through some basic probability – this can be done by a beginner with no mathematical background.
The basic principles
“Spooky action at a distance.” – Albert Einstein
You may recall from previous posts that describing a quantum particle’s state is characterised by the wave function- a partial differential equation describing the wave-like properties of a quantum particle (we keep saying quantum particle to indicate that such complexities are not required when considering a classical particle). Imagine the wave function as a machine, the type where you put things in and get something else out. A classic machine.
Into this machine we can throw different information (for example the degrees of freedom within a quantum system), then (ignoring the inner workings) out of the machine comes a crisp mathematical expression offering a complete probabilistic description of the particles options. Everything a particle may do, expressed in a probabilistic expression. We have been considering all the information we need to feed the machine about a particle to arrive at this description of its universe, but what happens if some of the information we need is in fact the information about another particle? There you have it, after less than 300 words we have quantum entanglement.
Let’s enrich our understanding with an example. Quantum particles posses a quality known as spin. It isn’t quite the same as the dictionary definition of the word – that is the Physicists favourite game, to alter the meaning of words – but in this instance it isn’t too far away either. For those who are interested, quantum spin is a form of angular momentum carried by particles; which for the really astute is what omega represents in a classical system in my post on simple harmonic motion. For the purposes of our illustration let the spin take one of two values; up or down (these expressions used in particle physics). The spin can be one of these two values here, the choice is binary. A photon; the quantum of light, is indeed a particle which may possess one of these two spin values. If you take a laser, and aim it correctly at a certain type of crystal you can create pairs of photons which become entangled – that is to say the qualities one possess has a direct consequence on the other. It is observed that if we measure the spin one of the photons as up, the other will certainly be down.
There is more to the above – intuitively it might seem as though it makes sense, if two photons are fired off near to each other that they have some sort of influence on each other but the effects of quantum entanglement are not interested in how close they are to one another. The same phenomena will be measured even when the particles are separated by great distances; this is what elevates quantum mechanics to being spooky action at a distance.
There are a couple of things worth illuminating at this point. Firstly, we are not suggesting that the particles are sending messages to each other, while we may not be able to fully explain how quantum entanglement works we are reasonably sure there are no secret messages. Secondly, whilst we are saying that the particles will have opposite spin we are not saying we can accurately predict the spin of both of them; we can just say A if we already know B. Finally, whilst we may strive to understand to understand how nature works it is unwise, particularly at the present, to attempt to understand why nature works in that way. Indeed a question of why nature does something may in itself have no meaning, but let us climb out of the philosophical rabbit hole and progress.
Is it actually real?
In order to progress in the field of Physics we need to abandon common sense – whilst it can take you so far it is riddled with human bias. Although this is a most excellent practice, it is prudent to ensure that as you abandon common sense you seek evidence wherever possible to confirm the nonsense.
In this spirit, I offer you an overview of Bell’s inequality, which is the single most compelling piece of evidence supporting quantum entanglement. It also gives a more rounded understanding of what we are dealing with. Locality is the idea that an event in one location cannot instantaneously cause some effect in another location, without having something travel from A to B (for example light or sound). This is common sense – if a gun is fired in Australia, a person won’t instantly die in London (at least not because of the Australian gunshot). Einstien tried to show the world that quantum mechanics must be flawed since it breaks locality, which is clearly at odds with the universe. Bell killed locality, and showed the universe was at odds with us.
To begin, assume locality is true. Mathematicians often do this – it is called proof by contradiction where we assume the contra is true and then reduce it to absurdity. We say that locality is true – i.e. quantum entanglement as we described it cannot exist because the very heart of it is causing an effect instantly somewhere else. The most common way to test anything in particle physics is with detectors; there are all sorts of different detectors (a photomultiplier being a personal favourite), but truly for comprehension just let the detector be a device which measures information about a particle. We want to do an experiment where we measure information about locality – so I can assume you will be comfortable in the assertion that it is useless to have only one detector. We need two detectors, which we will place in locations such that we can take measurements at exactly the same time and there is no chance one particle will influence the other. Under our assumption that locality is preserved, I should see no dependence in the results from the two detectors.
In the experiment electrons are used – they are fairly easy to come by and manipulate in laboratory settings. In order to get things going, we place both the detectors with the same set up, separated by a good distance and fire electrons at them and observe the result – in this case we get the same reading on both detectors (don’t worry about what the reading actually is, it only distracts from comprehension of the key points). From this we conclude that they must be coming out of the source the same – we can fire millions and millions of electrons and the result is always the same and agreement between the two detectors so locality lives on.
Now we tinker with our experiment a little and move the set up around. Our detectors can only give the value 1 or 0; there is no choice in-between. As we mess around with the detectors we can gather the following information;
Now with last point, if we were to randomly set up our detector each time, and repeat the experiment again and again and again how many times would you expect both detectors to have the same result? That is a 1 on both or a 0 on both? You need to think this through, it’s not that intuitive I am afraid.
The probability part:
Take detector A and detector B, both of which can be in three possible positions, call them – 1, 2 or 3 for obvious reasons. If we write the positions of the two detectors as (A,B) the possible configurations are (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2) and (3,3). Now based on the last bullet point “we get a 1 in 2/3 of the detector arrangements and a 0 in the other 1/3” -let’s say if the detectors are in position 1 or 2 I get the measurement 1, and if they are in 3 I get the measurement 0. How often will the readers have the same value? The answer is 5/9 – four from the arrangements where I get the measurement 1 on both readers [configurations (1,1) (1,2) (2,2) (2,1)] and one from the arrangement I get the measurement 0 on both readers, (3,3).
In this experiment the orientation of the detectors, as in the set up that gives the reading of 1 or 0 is totally random and changing. We don’t actually know what the instructions are being sent from the the source, so we can’t say it will be 5/9. But what we do know by bullet point 1 is that the electrons being fired have the same instructions. The instruction and the set up might always deliver the same result, as per bullet point 2 or, or we might get the set up we have just described where we said 5/9 should be in agreement. Actually it does not matter because what we can construct is a lower bound on the probability – this is what Bell’s inequality actually is –
Probability of agreement between detectors > 5/9
What we have done up to now is outline an experiment and make some very broad brush assumptions. The reason we have had seemingly confusing random arrangements of detectors with 0s and 1s is so that we don’t get lost in a long discussion about the different states of quantum particles and how or why we get certain readings. When you are trying to understand something, it makes no sense to layer on top another 4 or 5 things – let the brain digest a layer at a time. It also means I have something else to talk to you about in future weeks! So to this point we have devised an experiment with totally bullet proof logic which says the absolute minimum number of times we will see agreement between the two detectors is 5/9 – so 5 out of 9. Of course doing this experiment 9 times may well yield the incorrect result; probability can only make certainly and laws when it is done enough times to make it totally implausible that we are seeing a chance anomaly. If I did my experiment 9 billion times, and saw something that was not close to 5 billion in terms of the agreement between the detector we would know something is wrong.
The result of the experiment when performed, however, is that the probability is less than 5/9 – it is actually around 1/2 and you can do it as many times as you like. Our theory has failed us! Many different laboratories have done the experiment, the machinery has been tinkered with and the electrons have been fired again and again. Millions and millions of times and the only conclusion is that the upper bound on probability is broken. There are two candidates here – either we have made an assumption which has turned out to be wrong, or the preservation of locality is wrong. I appreciate my detector set up looked arbitrary, but you will either have to trust me that these arguments correspond to an acceptable simplification of real world phenomena (or indeed read into the detail). Many have tried and there has never been any suitable candidate to suggest something is wrong with the assumptions. To cut to the chase, some of the brightest minds in the world have been hoping to come up with something easier to understand to explain the phenomena and have not managed. Locality cannot, and is not preserved in the quantum world.
The result embodies the weird world of particle physics; not only are we impacting the properties of electrons by measuring it the electrons are impacting each other without even being in the same place. What can our little human minds even trust? I love it.
Next time I will be doing a brief run through of the quantum numbers, which I hope will be interesting in its own right and serve as a useful reference article to have on the site.
A black hole can be described in its full glory by three quantities alone: mass, angular momentum (how fast it spins) and electrical charge. The fact that any other information about what formed the black hole is redundant is known as the ‘No Hair Theorem.’ Hawking went on to show that the size and shape of a spinning black hole would depend on those three quantities alone. At this time some further laws of black hole mechanics were also settled, collectively known as black hole thermodynamics: the mass of a black hole is related to its energy, the area of a black hole is related to its entropy and the surface gravity of a black hole is related to its temperature.
With that out the way, onto the good stuff… Hawking radiation is Hawking’s theory that a Black Hole emits energy in the form of radiation until it exhausts its energy supply. Remember due to Einstein’s infamous equation E = mc²; energy and mass are proportional therefore a loss in energy is equivalent to a loss in mass. So as the Black Hole spews out energy it looses mass, gradually becoming smaller and smaller until it effectively ‘pops’. This is known as ‘Black Hole Evaporation’. But wait, wasn’t it the case that nothing could escape a black hole?! How is this energy suddenly being spewed out and where does it come from?! What is this mysterious Hawking radiation made of? To this we must turn back to quantum mechanics, antiparticles and uncertainty.
A wise man called Werner Heisenberg in 1927 postulated Heisenberg’s uncertainty principle. The uncertainty principle asserts a fundamental limit to the precision with which a certain pair of physical properties of a particle can be known. There are two of these expressions – one for position (x) and momentum (p) and another for energy (E) and time (t).
The first says a particles position and momentum cannot be known simultaneously beyond a certain precision – for a fixed position there is a unknown range the momentum can assume. This range is shown by the triangle symbol (greek letter delta), which effectively means ‘change in’. Likewise, over a fixed time there is an unknown range the energy can assume. These effects come into play on very small length scales which is what the h represents in the equation. When a mass is very large, the uncertainties become very small and classical physics is applicable. As such these phenomena are only noticeable in the quantum world. So in quantum physics, in a moment there can be a change in the amount of energy at a point in space. This is known as a quantum fluctuation. Energy can temporarily appear in what was, previously empty space (empty space is often referred to as a vacuum).
You may be thinking, hang on everything I know is going out the window now – what happened to energy always being conserved?! But i’m afraid you’ll have trust me for now when I say the wacky world of the Quantum realm allows this with peculiar behaviour of quantum superpositions of particle states when at the lowest-energy levels (i.e. the vacuum state/empty space). Suffice it to say the violation is allowed and I might ask Joe, the resident master on Quantum behaviour here at RTU, to do a post on it in the future. Anyhow, this energy can manifest itself in the form of energetic particles, and in order to conserve quantum properties like charge, spin etc these energetic particles are produced as particle-antiparticle pairs. The pairs are known as ‘virtual particles’ because they often pop out of existence as soon as they created.
Strange things happen when these fluctuations occur near the horizon of the Black Hole. [For a recap on what the horizon is in detail see Black Holes #1, but in brief: the event horizon is the distance from the centre of the black hole from which nothing can escape due to the immense gravitational tug of the black hole]. If a fluctuation creates a particle, anti-particle pair just outside the black hole event horizon, the gravitational energy of the black hole can ‘boost’ these particles into becoming real particles. Then the funny thing happens… Hawking predicts that one of the particles is sucked into the black hole, past the horizon, while the other particle escapes.
Because nobody can see past the horizon, it appears as though matter/radiation/energy (all the same thing at the end of the day) is being spewed out of the black hole. Instead of the being void of all life, the horizon is actually hub of activity, with exchanges like this going on all around the perimeter. Through spewing out the particles energy is given off and the black hole is not black after all, in fact it glows with this radiation emission.
Now here’s the final jump, seeing as the energy and mass have been shown to be equivalent from Einstein’s E=mc², and the creation of the particle pair came from the gravitational energy of the black hole itself, if one of the particles escapes to the outside world, the black hole will loose energy or – loose mass. But you might think, what about the particle that was swallowed, surely this adds to its mass to counteract the loss. Things get a little fiddly here, we’re nearly at the end, bear with me if you can…
Since the particle that is emitted has a positive energy, the particle that get absorbed by the black hole must have a negative energy, relative to the outside universe – in order to preserve the laws of thermodynamics (conversation of energy). We must not confuse the particle and antiparticle as being the positive energy and negative energy particle. Popular accounts of Hawking radiation explain the process as it being the antiparticle that is always absorbed, this is not the case. It is a 50/50 chance that either it will be the particle or the antiparticle that is the unlucky partner that gets sucked in but in order to preserve the law of thermodynamics the absorbed particle is assigned negative energy.
Ultimately, the particle that gets spewed out, is a product of the energy fluctuation of the black hole and as such we now have a particle, bobbing freely around the universe, that we never would have had before without the black hole. In a sentence: the black hole suffers a net energy loss due to this phenomena, therefore a net mass loss as it radiates away particles, meaning it gets smaller and shrinks. Shrinks and shrinks until it eventually disappears all together. The complete description of the dissolution requires a model of quantum gravity but it is hoped that tiny black holes might be experimentally re-created in extreme conditions at CERN in the future! So there we have it, black holes… not so black or devoid of life after all, according to Hawking. In fact, fantastical beasts which continually glow and over time shrink as they leak away Hawking radiation until they cease to exist all together. A pretty cool life for the elderly years of a massive star if Hawking is correct.
Particles have various quantum numbers that characterise their nature. The three that shall concern us today will be charge, baryon number and lepton number. Time for some terminology and book-keeping. Baryons are quark-based particles and are susceptible to the strong force, protons and neutrons are therefore baryons because they are made of three quarks. Baryons, funnily enough, have a baryon number of +1. Other particles which are not susceptible to the strong force and are not quark based are known as leptons, things like electrons and neutrons. These leptons have a lepton number of +1. Bit silly doesn’t it seem giving baryons and leptons a corresponding number of +1, what else would they have? Well that’s the whole point of the anti-particles, they have the number with an opposite sign! So an anti-proton has a baryon number of -1, an anti-neutron has a baryon number of -1 and an anti-electron, properly known as a positron, has a lepton number of -1. Finally any lepton has a baryon number of 0 and visa versa. Then of course, last but not least, we have charge. If a proton has a positive charge of +1 the anti-proton has a negative charge of -1. All of this was theorised and the experimental results at particle accelerator sites went on to confirm the existence of such entities matching the descriptions.
Collisions between particles and antiparticles lead to annihilations and give off energy (in the form of photons) proportional to the total mass of the particles in accordance with Einstein’s equation E=mc^2. Antiparticles are also created in regions of the universe where high-energy particle collisions take place, such as high-energy cosmic rays colliding with the Earth’s atmosphere. This antimatter can then be detected in the traces of its products after it has annihilated with matter – for example gamma rays which are themselves photons of the highest observed range of photon energy.
Now looking around us at everyday objects, the Earth, planets in the solar system, stars… why is it that matter seems to vastly outweigh antimatter? Since, all known processes for creating massive particles create both in equal quantities. The asymmetry of matter and antimatter in the universe is another great mystery but we should be very thankful for it. If an equal amount had existed shortly after the Big Bang all matter would have annihilated with its antimatter counterparts leaving nothing but energy behind. In fact it is estimated that in the early universe there was only one extra matter particle for every billion-matter antimatter pairs! Whatever hypothetical process that took place in the early universe that caused the matter-antimatter asymmetry is termed, by cosmologists, Baryogenesis (confusingly not solely to do with baryons). The difference in the abundance of matter and antimatter in the early universe left behind the matter remnants we see today, which allowed the formation of the universe as we know it… thank you Baryogenesis.
Anti-matter is a classic go-to weapon of science fiction stories due to its annihilating power and high energy potential. Recent examples include Angels & Demons where Professor Langdon tries to save Vatican City from an antimatter bomb and Star Trek’s starship the Enterprise which uses the energy released from matter-antimatter annihilation as propulsion to achieve faster-than-light travel. Though the substance anti-matter exists in the real world the feasibility of harnessing its powers is best left to the realm of sci-fi. Think about, in order to harness its power and manipulate it as a weapon, we would have to, as a first step, be able to store it. By what means could we achieve this?! Any container we put it in, it would annihilate with the matter that constitutes the walls of the container itself! Antimatter is a tricksy beast, storing it would be like trying to grasp a fistful of air.
Fun facts to round off, antimatter is the most expensive stuff on the planet, weighing in at a whopping $62.5 trillion a gram! However as obvious from the storage problem, a gram has not been produced. The most that has been created and stored at a time is about a billion antiprotons, but that’s only a meek one millionth of a billionth of a gram. If you fancy getting up close with some antimatter however, the answer is most likely sitting in your kitchen. Go grab a banana! Bananas contain potassium-40 which as it decays, gives off a positron (an “anti-electron”) at a rate of about one every 75 minutes. Too bad it collides with the nearest electron pretty much instantly…
I make a conscious effort to avoid arbitrary targets – it seems nonsensical to center your appreciation of a loved one on the 14th of February, to delay making a positive change because the Gregorian calendar has not reset or to be more charitable once a year due to a festival from a religion you are not a member of. That said, there is a relative amount of calm in the weeks surrounding the holidays that gives the opportune moment for self-evaluation. With regard to Rationalising the Universe I am pleased with the progress we have made and continue to be astonished with the number of writers from all disciplines who engage with the site. I offer no great change in direction for a site which is not broken – but rather propose some refinements.
This site was set up for a number of reasons – but the overarching aim was to bask in the glory of science for nothing but pleasure. Over the last few months I have written increasingly about the forefronts of theoretical and particle physics – something of great interest to us all, but as caveat-ed I have more questions than answers in these areas. Whilst this is a noble pursuit, I fear I am being permanently seduced while avoiding the foundations of mankind’s knowledge; popularizing popular science is actually fairly easy, it is popularizing the bread and butter which requires a certain zeal. I propose over the coming year, very occasionally we cover an essential area in any undergraduate physicists toolkit. This will, I hope, be (slightly) more interesting than the average textbook. This is not to say musings on the forefront of theoretical physics are suspended – they will remain the main event, interspersed with essential physics to ensure we do battle properly equipped. This may be a little painful I am afraid.
Thank you for your continued interest and support in the site, I hope you choose to stay reading Rationalising the Universe through 2017
– Joe
Simple harmonic motion
What an arrogant title – unfortunately the name is too embedded in science for us to be able to make any changes to it, so whilst you may not find this topic simple, we will call it simple harmonic motion, or SHM for brevity. I often like to start with the dictionary definition of a word or phrase before uncovering the detail, which for SHM we have;
noun
Forces like this can be modeled very neatly using mathematics, since we have a force which depends only upon the position of the particle. We will consider the case of a spring, with one end attached to a fixed end and the other free to move with a particle attached to the end of it. The set up can be seen in the diagram below.
All I intend to do today is to derive from first(ish) principles an equation which will allow us to model the motion of the particle. Note that the above diagram is showing phases of time – the only motion we have is in the vertical direction. This is called the j-direction which in the below formulas, for now you will see denoted j with a hat (^) on it. This is not really any different to saying upwards, but what it actually means is a unit vector in the vertical direction which we have defined. Generally when you try to model anything which has classical motion you first need to consider the forces. There are two forces here – the weight of the particle and the spring force.
Dealing with the weight first, the weight of any object is the mass of the object multiplied by the acceleration due to gravity. It is this second part which changes depending on the gravitational field you find yourself in – hence why you can say you weigh less on the moon than on Earth. If we have expressed the positive j-direction as vertically downwards, then the weight of the particle can be expressed as follows;
Now we move onto the spring force – the spring force obeys a law known as Hooke’s law. Often when you see formulas written down they seem very difficult to construct – actually they are just the result of people who mess around with springs and know how to draw graphs. From experiment, it turns out the force a spring exerts is proportional to the deformation of the spring (which may be compressed or stretched) by a constant known as the stiffness of the spring (k). The force is directed towards the center of the spring – which leaves us with an initial dilemma for our model; should we take the force as being in the positive or negative j-direction, since it can be both? Spoiler – it makes no difference. So we can set up our spring force as follows, where x is the position of the particle.
Note that the term l with a subscript 0 is the natural length of the spring. That was a little dry but now we are in business, we have forces giving rise to motion of a particle which can only mean Newton. We want to express the motion of the particle on the spring as F=ma, however there is an issue in that a introduces a further variable. However let us define the x-direction to be vertical (I know this isn’t a Cartesian set up, but it’s my model so deal with it). Now the position of the particle at any moment in time can be modeled as some function of t, say x(t). Since velocity is the rate of change of position, and acceleration is the rate of change of velocity I may denote the acceleration as the second derivative of x, or in applied mathematics notation x with two dots above it. A derivative is simply the rate of change – so I am taking the rate of change twice of x. This means I can rewrite our famous F=ma as;
The next logical step from here is to replace F with the forces we have define above. All of a sudden I am going to drop my unit vector j – this is okay. What I have done is reached a point where I have realized I only have j-components so resolved in this direction (see note 1), so I can just consider these forces with the unit vector implied.
Now using the above I consider only the last two terms, which I expand out and collect all x-terms on one side with all constant terms on the other side. This gives;
This is a differential equation in x – an equation that links the derivatives of x. All other terms here are constants, so really once we have set up a real system these will just be numbers. If you are really mathematically minded, or just like jargon, what I have constructed is a second order constant coefficient in-homogeneous differential equation. The solution to a differential equation is a function which satisfies the equation – i.e some kind of mathematical expression that when we substitute it into the above, the equality is satisfied. I have illuminated more of the logic in the notes for those who are interested, but the first stage is to solve the equation as if the right hand side were to equal zero (which is called the complimentary homogeneous function). To do this I have defined a new constant omega such that;
Now an equation which will satisfy my differential equation if the right hand were to be zero is as follows (note 2);
Here B and C are arbitrary constants. Next we need to deal with the right hand side – since the right had side is not actually zero we need to add a term to the above solution to make it work. In fact what we need to add is the equilibrium position of the particle, multiplied by the stiffness of the spring. Please see note 3 for further discussion – but if you think about what is happening; a particle oscillating around an equilibrium position this hopefully will roughly”feel right”. This gives us the following final general solution.
And there we have our solution – we can define the position of a particle, oscillating in harmonic motion from a spring with the above equation. We would need to know the stiffness of our spring, the mass of the particle and the natural length of the spring in order to quantify omega, k and x-eq. Following on from this, we would need an initial condition in order to remove the arbitrary constants B and C – this would look something like initially the particle is released from rest at position x = 0.1m for example. This will be specific to the model we look at.
Why do I care?
Being able to model the motion of particles under forces is central to all of Physics; and it is entirely implausible to think that one might go on to model complex quantum mechanical systems if simple classical skills have not been gained. That is not to say the above skills will be directly used (they generally will not), but the mindset of the Physicist (and the Mathematician) is to be able to take a real world situation, express as much of this information mathematically and produce a model which is a close match to real world results. This is a most wonderful thing. This article goes into a little more depth around the applications of harmonic motion.
Finally – my apologies for this dense technical post. Please ask anything you have not understood, I can give confident answers on this topic; my next few posts will be light and much less technical.
Notes: