If you’ve read or heard anything about quantum mechanics, you’ve heard the phrase “particle-wave duality.” The common wisdom is that this means that particles sometimes behave like waves and sometimes behave like particles. And although this is right, it’s a bit misleading. The truth is:

Everything is always a wave. It’s just that waves can be made to behave like particles.

To see what I mean, let’s actually show how one can make a set of waves behave like a particle. Specifically, let’s show how a set of light waves can be made to behave like a photon, a light particle.

Light Goes Round

Just to be concrete, let’s talk about light that bounces between a special configuration of mirrors. It looks something like this:

The idea is to configure three mirrors (light blue) so that the light (yellow) bounces around in a loop so that it ends up in the same place that it started from. In optics, this is called a ring cavity. It’s often used to make a type of laser called a ring laser.

We can represent the path of the light (which lives in two-dimensions) as a position along a single line. All we have to do is demand that the lefthand side of the line be the same point as the righthand side of the line so that it wraps around to where it started, as shown below. This is called a periodic representation.

So, given our periodic plot of the path of the light in the cavity, what does a light wave look like? Since the wave has to wrap around to where it started, the wave on the left side of the line must look the same as it does on the right… In other words, it has to become itself after it travels around the cavity, as shown below:

One important consequence of this periodicity is that the waves in the cavity can’t be whatever they want. Only certain waves fit. In the figure above, if the wave stopped a little earlier on the right (say at the peak, the highest point), as shown below, then the right and left sides of the wave wouldn’t be the same. This would obviously be a problem.

 

Now, given a bunch of light waves that fit in our cavity, we want to combine them in such a way that we get a particle travelling around the ring in the cavity. An important idea that we’ll need is the principle of superposition. 

The Superposition Principle

I’ve previously discussed wave interference and the superposition principle, so if you remember my previous discussions, feel free to skip to the next section.

Imagine waves as wiggles on a very stretchy string. If I try and push up on the string (make a wiggle that goes up) and you try and push down on the string (make a wiggle that goes down) at the same time, neither of us ends up moving the string as much as we intended. This is called destructive interference. Similarly, if I push up on the string at the same time that you push up on the string, we’ll probably stretch it quite a lot. This is called constructive interference. The process of overlaying one wave over another is called superposition.

Light waves work the same way. If we make two waves overlap

What’s in a Particle

Okay, so let’s build our particle! As a first step, let’s put the largest wave that can possibly fit into our cavity, shown below. Note how the height of the wave on the left is always the same as the height of the wave on the right.

Now, we add more, higher frequency, waves to the cavity. (By higher-frequency, I mean the waves have more wiggles in them.) If we take a wave that has exactly twice the number of wiggles as our first wave and add the two waves together, we get something like this:

Interesting! Now we get a big wiggle and a small wiggle. Perhaps that big wiggle will become a particle?

(Note that I added a wave of a specific height to the first wave. There’s a lot of math involved in knowing precisely what height to add. I’m going to completely ignore that detail for now.)

After we add five waves to the cavity and sum up the wiggle heights, we get:

Now there’s one wiggle that’s clearly much bigger than the others. If we add enough waves to the cavity, we can make that wiggle all we can see:

Wow! That doesn’t even look like a wave! What is that? That, my friends, is a particle. The point where the peak is highest represents the average position of the particle and the width of the peak represents the quantum uncertainty in the position of the particle.

(Actually, the little wiggles around the peak are still there. They’re just too small to see in my plot. In principle, you can get rid of them entirely by adding up infinitely many waves.)

Enter Heisenberg

What if we wanted to make the peak narrower, thus making it possible to measure the particle’s position more precisely? Well, you might have noticed that our central peak got narrower as we added more waves to our ring cavity. This means that we would need to add more, increasingly wiggly, waves to the cavity to make the peak narrower.

How do we interpret that? We know that the number of wiggles in a wave determines both its energy and its momentum, meaning that the particle is not only made up of many waves, but has many different energies.

It has an average energy, of course, and an average momentum. But if we measure the particle, we might not measure that energy or that momentum. Instead, we’ll measure each energy some fraction of the time. The percentages look something like this:

This is a visible manifestation of the Heisenberg uncertainty principle. If we want to know the particle’s position better, we need to add waves with different energies to it, meaning that it has more energies and we know the energy (and thus the momentum) less well.

Particles Can Act Like Waves

So I’ve just described how we can make a bunch of waves like a particle. But, of course, a bunch of particles can wiggle to make a wave. This is, after all, what’s going on when you wiggle a string, since that string is made up of particles. So you might ask… can you make a wave out of particles, add a bunch of such waves together, and get a new particle?

Amazingly, you can! If you add up a bunch of sound waves travelling through a material (which is made of particles which are made up of waves), you can get a particle called a phonon!

So now we’re ready to state the principle of particle-wave duality one last time:

Everything is a wave. But particles can be constructed out of waves… and waves can be constructed out of particles.

Disclaimer: It’s Often Useful to Think in Terms of Particles

I just told you everything is a wave. But that doesn’t mean physicists always think in terms of waves. Often it’s more useful to think in terms of particles. For example, the width of the peak I constructed above can be very narrow on human terms and it can be very difficult to notice or measure uncertainty in the particle’s position. This is why we didn’t discover quantum mechanics until the early twentieth century.

But even at the quantum level, it’s often valuable to think in terms of particles. Richard Feynman’s diagrams, and Werner Heisenberg’s formulation of quantum mechanics, for example, treat particles as particles in some sense. This isn’t inconsistent, however, because one can prove that Feynman and Heisenberg’s formulations imply the wave nature of, well, Nature.

Alright folks, that’s all I have for now.

Play With it Yourself

I generated all of my animations in Python. If you’d like to play with them yourself, you can find my code here:

https://github.com/Yurlungur/make-a-particle

Related Reading

I’ve written about quantum mechanics before. If you enjoyed this post, you might also enjoy the following posts:

• In this article, I introduce particle-wave duality by describing the experiments that convinced physicists that particles have to be waves.
• In this article, I describe how Niels Bohr used particle-wave duality to unravel the mysteries of the atom.
• In this article, I explain how we should interpret particles-as-waves.
• In this article, I use particle-wave duality to explain quantum tunneling.
• Everything I just described is based on something called Fourier Analysis. I discuss Fourier analysis in a very similar context in this article.