Rohan Joshi
9 min readMar 5, 2023

In the last article, we looked at the motion of infinitely many coupled oscillators in their normal mode frequencies, and how the concept of a standing wave arises from extending the coupled oscillator to infinitely many oscillators. Indeed, we realized that discrete systems at large scales can be modeled as continuous functions, depending on both space and time. In this article, we will explore standing/compression waves, but not through the normal, cookie cutter lens of a wave on a string. We will be looking at it through the lens of energy distributions, also known as…particles.


Quasiparticles are a relatively new theory in classical mechanics that serve to analogize quantum mechanical concepts with those in classical mechanics. So, where do we start analyzing this phenomenon? Let us start with defining what a quasiparticle is.

Assume that one has an infinitely long chain of atoms, or simple harmonic oscillators, in one dimension, linked together by strings. This isn’t different at all from our coupled infinite-oscillator system studied in the previous article. The only difference here is that the system is unbounded and therefore a single vibration of even one oscillator will travel all the way to the end of the spring before bouncing back. Further, since the chain is infinitely long, that vibration will not come back — it will keep travelling down the line.

If we give a slight push to one of the ends of the chain, say to the first mass on a spring, we will be giving it some initial energy that remains constant throughout the motion of all the masses. Soon, of course, the second mass will start oscillating with the same amplitude as the first, and then the third, and then the fourth, and so on. The energy is getting transferred from particle to particle as they’re linked by the springs. Therefore, at any given moment in time, we will see that one mass/oscillator is vibrating with maximum amplitude, and then the next one is, and so on.

What does this have to do with particles? Well, a particle is a phenomenon in physics, just like a wave. So, this “compression wave” which was described in the previous paragraph is doing something. What is it doing? One major feature of particles is that they have momentum and energy, and that they can transfer those quantities to other particles. What is happening in our system? Energy is being transferred between masses on springs, and there is, indeed, a medium that is transferring that energy — the spring itself! Each compression and extension of the wave, each vibration of a mass can therefore be considered to be occurring under the influence of a “particle” of sorts that is going around the chain giving and taking energy from the blocks. We will now try to model this particle, and we shall see that it gives rise to wondrous links between classical and probabilistic mechanics — but that shall be reflected upon later in this article.

First, we shall move from the discrete world to the continuous world. In this scenario, the “particle” can be felt as a region in which the atoms/medium have maximum energy. Our goal now is to find a function that gives us the displacement y of each individual particle as a function of distance x along the oscillator string. For now, we will also assume that our system is bounded and of length l. In this scenario, we will need to set a few conditions for y(x). Our first, of course, is that at any given point x0, at which the “crest” or maximum vibrational amplitude is at at any given point, we need our function to be maximum. Second, since our “particle” is localized, we want our function to vanish as our distance gets larger and larger. Lastly, we need the integral over all space of our function to converge, because the total energy of our system remains a constant.

Is there a function with all these properties? It turns out that yes, there is! It is…the Gaussian function, or, in more colloquial terms, the normal distribution. While it shouldn’t be surprising that a continuous set of infinite data will lead to a normal distribution, it is worth reflecting on what has happened. As we moved from discrete to continuous, the vibration of particles became less of “they happen in only one place” to “all particles are affected by the vibration of one, and therefore each individual displacement can be modeled as a y(x)”. Isn’t that beautiful?

Getting back to the matter at hand, the wave function y(x) now becomes

Where A is the amplitude of the most energetic vibration taking place at any given time. Now, this expression is obviously a guess. However, we cannot underestimate the importance of just guessing in physics, because, as I have said before, when something reminds you of something else, it most likely is that something else.

Now that we have a y(x), we can find the differential maximum energy dE of a mass of the continuous oscillator dm as follows:

Here, ω is the natural frequency of each spring. If the chain of atoms is assumed to have an approximately uniform mass density λ=m/l, where m is the total mass of the system, then our expression becomes

The graph of this is below:

As you can see, the function is the normal distribution, and it shifts with time as the oscillation moves ahead. Further, it is concentrated at “peak” that also continually shifts. Thus, the total energy of our system will be given by the sum of all individual energies.

Pretty compact, isn’t it? The l vanishes, giving us our answer independent of the length of the system. Now, if the average stiffness of our material is given to be K=ω2m, then the energy can be expressed as

Therefore, this is the total amount of amount of energy carried by this “particle”, which we shall now call a “quasiparticle”. Why a “quasi” particle? Because it isn’t, of course, really a particle. It just shares certain properties. Indeed, the vibration of lattice atoms due to some external force, such as sound, can be considered as occurring due to the transfer of quasiparticles, or a “phonon” in this case, between individual atoms, giving and taking away energy from them as it moves.


Now that we have been introduced to the concept of a quasiparticle, let us now look at the phenomenon of standing waves from a new perspective. You may have learnt of them in the form of waves on a string. I shall try to explain them from the perspective of coupled oscillators and quasiparticle energies.

So, let us look at our setup. Once again, we have a line of continuous oscillators, but this time bounded on both sides and therefore having a fixed length l. Such an oscillator system, of course, can have infinitely many normal modes and resonant frequencies. Let us look back to what we uncovered in the last article — that in our first mode of vibration, our continuous y(x) has one extreme, that in the second it has two, and so on. If our system is forcibly oscillated by a driving force at some resonant frequency ω, then our associated eigenmodes are such that some part of the system will move in one direction with some amplitude, and the other parts in other directions, oscillating back and forth across space. So, we shall make another guess for y(x) in this scenario. If A is the maximum possible amplitude of an oscillator in the system (dictated by its input energy, of course), then

Here, n corresponds to the nth mode of oscillation. We can check that this works. Why? Because if we plug in n=1, our graph is half a wavelength long, from x=0 to l. For n=2, the graph has two wavelengths, and so on.

So, what is the significance of this? It tells us that, if our system is “trapped”, then my system can only oscillate in very specific modes. Because it is continuous, it turns out that that only an integer number of half-wavelengths at very specific frequencies can exist in this system. This is unlike a discrete system, in which arbitrary frequencies can be used to induce motion. Further, we can think of each eigenmode, again, as a quasiparticle, or a phonon, trapped inside a finite box. For different external vibrations, it can have different vales of energy. How? Let us do the same analysis as before and see where we end up.

Now, y is obviously not just a function of x. It is also a function of time t. Therefore, we can write the following:

Here, ωn is the frequency corresponding to the nth mode of oscillation. We can simplify the above as

Here, k=nπ/l is the wavenumber of the oscillator. The above equation looks eerily like the equation for two separate waves moving in opposite directions superimposed on each other

– the prerequisite for a standing wave to exist. We only have one last bit to prove. We know that ωn/k represents the speed v of a wave in the material. But since we know that k is quantized in multiples of π/l, ωn has to be quantized as well for the wave speed to be the same for all modes of vibration. Therefore, we have come to another wonderful conclusion — that the eigenfrequencies ωn are also quantized as ωn, or integer multiples of the lowest, or natural frequency. We can thus infer that, given quantized resonant frequencies ωn , one can create phonons in matter with distinct energy states depending on the frequencies. But what is this total energy? Let is find out.

Again, we consider the energy differential dE at any given time.

Plugging in what we know,

Now, integrating over the domain [0, l], we get

Pretty neat, isn’t it? What’s even better is that the above equation tells us that even the energy of the oscillator/quasiparticle is itself quantized!

So, what is the relation to quantum mechanics here? I think that a misconception must be cleared up about something being quantum. It is true that quantum mechanics generally plays a much larger role in the behavior of subatomic/fundamental particles that macroscopic phenomena like sound waves. However, this is because one major tenet of modern particle physics is the existence of fields and the emergence of particles from those fields. The same thing is happening here — a phonon is arising from the continuous field of atoms all around it — though discrete, at large enough scales a system such as this can definitely be considered to be continuous and uniform. And, as we have just seen, trapped oscillating systems can exist only in integer normal mode frequencies — something known as quantization. Quantum mechanics isn’t just about electrons or whatever — it is about any, any system, in which energy can exist in only fixed states.

Thus, what we have just studied is very, very analogous to the quantum harmonic oscillator. However, there are still a few more things we can play around with to fully complete our study standing waves. Maybe that shall be the next article…