The Most Important Idea in Theoretical Physics — “and so can you”

What we’ll talk about — Harmonic Oscillator

There is a relatively famous saying in the physics circles that the work of a theoretical physicist consists in treating the harmonic oscillator in increasingly complicated number of ways. I like this saying a lot, and even though it isn’t strictly true, of course, it does produce kind of a self reflection in physicists and will definitely get at least a chuckle from one who hasn’t heard it before.

The saying itself speaks much more to the people who work in physics related fields, be it physicists, mathematicians, or even engineers, but I think its message is actually quite deep and probably interesting even to the more general audience. Let us then analyse this saying.

Who are theoretical physicists

Theoretical physicists work in a wide range of fields, and I’m not even referring to the fact that some of them end up in finance or IT. What I mean is that even two people who would tell you that they do theoretical physics for living could be working on topics so different one from another that their only common topic of interest is football. The spectrum of topics studied in today’s theoretical physics is probably as wide as science itself. Give me a physical system that admits mathematical description and I’ll show you a topic that a theoretical physicist may be working on. This means that there are theoretical physicists studying protein folding, materials, plasmas, elementary particles, large scale structure of the Universe and anything in between.

Now you’re probably even more confused how come that the work of a theoretical physicist consists in treating the harmonic oscillator in increasingly complicated number of ways?

The central idea of this saying is surprising because it claims that there is something that all these different areas of research have in common, and that particular thing appears to be totally absurd. It is something that you’ve studied in high school (if your science curriculum was any good).

What is a harmonic oscillator

To draw a simple picture, a harmonic oscillator can be thought of as a light ball connected to a wall via spring, then laid down on an icy surface such as a hockey field, and then finally pricked lightly so that it starts oscillating about its equilibrium. Another intuitive mental picture which you can use is that of a pendulum oscillating around its minimum.

Alright, so what’s going on here? What ‘exactly’ is that harmonic oscillator? I mean, I’ve given you two completely different physical systems to have in mind as a representation of the harmonic oscillator. It is almost as if the harmonic oscillator is something so ubiquitous that it can be used to represent completely different things?!

(Oh, I see what you did there… Nice.)

So the point is that the harmonic oscillator is a mathematical concept more than it is a physical one. Using a little bit of jargon, if I may, one could say that the harmonic oscillator is a system with a property that the force acting on it depends linearly on its position. Linearly means that if I double the distance the force doubles. Now this holds in both of the examples that I’ve described above. In first case if the spring is stretched twice as much it will pull the ball back with twice as much force, and in second case if the pendulum is made to swing with twice as large angle with respect to its minimum position then the force pulling it back towards the center will be twice as large.*

Ubiquity of the harmonic oscillator

The rest of the discussion can be made math-free, but then you lose out on some very cool stuff. To try and satisfy as many people as possible I’ll split the rest of this text in two parts. First we’ll have a nice, intuitive, equation-less talk about why harmonic oscillators are cool and appear everywhere. For those not familiar with derivatives, and in particular something called the Taylor series this will probably be all they can follow, and the rest will, plainly speaking, be just some crazy talk. Now the minority which does speak analysis 101 / calculus 101 (I don’t mean fluently, I just mean you can mumble a word or two) will benefit from this discussion because it will lay the ground for all the mess that comes later. When mathematics is overwhelming it is always good to have some intuitive general idea to lean on. Alright, enough chit-chat, lets get to the meat** of it.

Imagine a swing, with little Tommy sitting on it. We’re going to turn this thing upside down later on, so don’t be cruel, imagine that swing’s seat is fixed with some metal rods instead of ropes.

Now, Tommy is sitting comfortably in his seat, so the swing is in a position where Tommy’s feet are closest to the ground. This position is called minimum, for obvious reasons, and it is also called an equilibrium, for some slightly less obvious reasons which we will talk about now. A thing is said to be in equilibrium when it is not changing in time. If the kid doesn’t do anything it will just stay in place so we call this position equilibrium. If he just tries to swing himself once then sooner or later the swing will come to a stop, back to initial position. This particular kind of equilibrium is also called a stable equilibrium. This means that even if you give little Tommy a push he will eventually be attracted back to the equilibrium. Say you give him a light push, the swing goes front to back to front a couple of times, and then eventually it stops back at the minimum.

One of the ways to avoid returning to the equilibrium is to give Tommy a slightly harder push so that you tear up the swing and poor Tommy just flies away. But we are not monsters so we will not pursue this line of thought any further.

Another way to avoid going back to the original position, only slightly less amusing, is to push the swing just enough that it reaches the upside down position, but not as much to make it go the full circle. You give it exactly that much of a push that it stops midway through the hole circle, with Tommy’s legs pointing in the direction of the sky and his head pointing in the direction of his torturer***. What we’ve done here is we’ve pushed the swing to another equilibrium. It may seem a bit counter-intuitive at first, since you don’t often see swings in parks hanging around that way (nor children, hopefully), but the reason for this is that this new equilibrium is something called unstable equilibrium. Unstable equilibrium is the complete opposite from the stable one, just as what we’re doing to this poor child is in complete opposition to any good taste. What this means is that, whereas most attempts to push the system out of the stable equilibrium will fail because after some time the system will merrily settle back to its initial position, even the slightest push will bring the system out of the unstable equilibrium.

Think about it this way, if you try to push the swing to stand upside down, in most cases you’ll either not going to push enough or you’ll overshoot and it will do a full circle. It takes a good amount of fine tuning to put the system in that position. Furthermore, once you’ve finally put it in that position, even the slightest disturbance will simply drop it down like a castle of cards.

Did we go too far astray, especially with our poor swinging child’s analogies? Possibly. But they taught us a valuable lesson about what is equilibrium and in particular what is a stable equilibrium. Now I can tell you that a harmonic oscillator is essentially a mathematical description of a stable equilibrium.

Confused? Naaah, I don’t trust you. You can see that both the pendulum and the spring that I’ve talked about before act as a system in equilibrium, right? Well that’s (almost) all there is to it. Mathematically whenever you want to describe a system close to a stable equilibrium you start from a model which is essentially a harmonic oscillator. After that you can start complicating your life by adding extra stuff into your equations.

In the previous two examples the system is a harmonic oscillator if the spring is perfect in the sense that the force that it produces depends linearly on how much you stretch it. For true materials this happens only if you stretch or compress them very little. If you take a steel spring and stretch it so much that you double it in length, what you’ve got is not a spring any more, it’s just a twisted piece of metal. It’s definitely not going to provide as much force as it did initially. There are other effects to consider, too. For example the ice is not completely frictionless, so when the ball starts gliding over it it will be dragged a little by this friction. Similarly the pendulum suffers friction from moving through the air. And its rope is slightly elastic for example, and there’s some friction where it is connected to a stand, and so on.

Our harmonic oscillator ideal takes none of these things into consideration, so it has to be used wisely. It is a concept which is very powerful because it can describe any system which is close to the equilibrium, but you shouldn’t trust it too much when you’re trying to describe a system far from equilibrium. In other words, little Tommy shouldn’t trust you when you suggest to push him so that he does a full circle. He may just as well expect to be launched three meters away, landing face first on the concrete pavement. (Hey, I’m not the one who chose to make the children playground’s floor out of concrete, I’m just as much of a victim here as poor Tommy is.)

Harmonic oscillators in nature

Let us now name a few systems which are close to the equilibrium and to which, therefore, you can apply the harmonic oscillator model.

Let’s start from the large and work towards the small. I will take examples which I think may be thought provoking for you, but these obviously do not exhaust all of the possibilities (not even close!).

There’s something called baryonic acoustic oscillations. It is something very cool and very important for the formation of galaxies (and therefore for our existence too), so look it up in Encyclopedia Britannica****. These oscillations happened in the Universe billions of years ago, before galaxies were ever formed. What oscillated was all the material that ended up building the galaxies later on. The way it oscillated is that all the particles got attracted by the gravitational pull of the dark matter clumps that formed earlier. As the stuff started to fall down the potential well the pressure would start building up and it would provide a force acting in the opposite direction, trying to expel the stuff out so that it reduces the pressure in the potential well. Gravity and pressure thus provided two opposing forces making the baryons oscillate. This process which is nowadays observable with telescopes and which can be considered one of the seeds of today’s galaxies can be described as a harmonic oscillator. (In first approximation, same as the examples that we considered above.)

Next, much, much smaller, is the scale of single stars. Our Sun, same as all the other stars, burns nuclear fuel at its core, providing us with light and heat. The way it does this is by creating a state of enormous pressure and temperature near its center, but the pressure comes, same as in the case of baryonic acoustic oscillations, from gravity. Namely, gravity attracts all the particles making up the Sun one towards the other. When they all meet in a very compact space, which is the interior of the Sun, they start rubbing one against the other thereby creating a huge outward pressure. This pressure tries to expel the particles out of the Sun into the space, but again it is counteracted by the gravity which tries to brings them back down. As you can see, the particle starts oscillating, influenced by two opposing forces. Description? You’ve guessed it, the harmonic oscillator*****.

Scale down a couple more orders of magnitude and you’re entering the realm of ordinary humans. We have a load of things that behave as oscillators here. Besides mechanical oscillators such as, you might have guessed it, old clocks which used pendulums, there are also electrical oscillators. Such an example is provided by the FM radios. These are actually based on one more cool effect called resonance, which is also related to the oscillators. In the radios it is the electric current that oscillates, and so it does in the antennas, microwaves, and many other devices.

For the last example of harmonic oscillators we enter the quantum realm. Molecules, atoms and even nuclei can all be described, to some degree, as harmonic oscillators. For example when atoms come together to form a molecular bond, thereby creating a molecule, they are not connected rigidly. There is no metal rod connecting the two, in fact, the bond is electromagnetic and in this sense it is very much dynamical. Change the distribution of electromagnetic charges a little bit and the two atoms will either be attracted a little bit closer together or will be pushed a little bit farther away from each other. But once they’ve moved the electromagnetic force with which one acts on the other has changed slightly, and then repelling force gets substituted for an attracting one, or vice versa. This way the molecular bond between the atoms becomes much more similar to a spring than to a rigid rod. So if it walks like a spring and if it talks like a spring what is it?******

*If you speak derivatives, then harmonic oscillator is a thing satisfying the equation x’’=-kx, where k>0 is square of the frequency of the oscillator, x is the distance (represented by how much the spring is stretched or compressed, or by the angle with respect to minimum, in the two examples considered in the text), and primes denote the derivatives (with respect to time in both of our examples).

**Or tofu, of course, if you prefer your physics examples meat-free.

***All characters, including the knowledge thirsty scientist and the unfortunate Tommy are purely fictional. No children have been hurt nor have sustained injuries for the preparation of this text.

****Well of course I’m not serious. WTF?! Just google it or something…

*****Extra points for you if you’ve thought “in first approximation”. You can be proud of yourself and let all your facebook friends know!

******No, it is not a god damn duck! What is wrong with you?? Have I taught you nothing?? It is a harmonic oscillator! An o-sci-lla-tor! And when I just think about all that poor Tommy had to went through…