Casual Physics Enjoyer

Physics is a two-part story — and part two is where it starts to get weird.

In my second year of university, I sat down with anticipation for my first winter term lecture. It was entitled ‘quantum mechanics’. My impression of quantum mechanics was built from sci-fi actors on TV who used the word to sound intelligent. To be brutally honest, I didn’t even know what kind of objects quantum mechanics tried to model. I didn’t know what kind of math I would need to tackle.

The lecture series went on for around 2 months — terms in Cambridge are squeezed into eight weeks. The pace of my lectures was so fast and the math so intensive that I barely got any time to reflect on the meaning and philosophy of what I learned in quantum mechanics. At the time, my main concern was to get up to speed with the linear algebra, calculus and probability required to do quantum mechanics. Only later had I started to understand what I was doing.

I wrote this article to outline the biggest takeaways from that course, both in physics and philosophy. The main concept I learned was that a particle and a wave are essentially one and the same. From this idea, many wacky effects can be both proved and observed.

A change in paradigm

As an undergrad, I learned that the history of physics is sliced into roughly two parts — the classical and the quantum. ‘Classical’ physics broadly refers to the work of Newton and Maxwell. It consists almost entirely of Newton’s laws of motion in different forms, paired with Maxwell’s classical laws of electromagnetism. Classical physics structures the universe as a reliable, clockwork combination of particles and waves. Each particle, and its interactions with other particles, behave predictably — provided you know the initial ‘scene’ they are set in. Our philosopher colleagues might even call this paradigm a ‘deterministic’ universe. In addition, waves and particles are viewed as separate things.

Quantum mechanics, on the other hand, are philosophically different. It models small objects like electrons and protons. It is a theory that models nature with the language of chance and uses probability distributions to give us predictions about the physical properties of particles. Electrons and other small objects are no longer single points in space. Rather, they are viewed as probability clouds that only have an interpretable meaning when you repeatedly observe them over and over. This uncertainty is physically evident. Heisenberg’s uncertainty principle gives us a lower bound to determine the position and momentum accurately.

In using this language, we discover that the split between waves and particles is not as clear cut as once thought. Particles can be treated as waves, which explains phenomena like quantum tunnelling. Conversely, the waves that Maxwell used in his theory of light can be modelled as particles. Photons are waves that ‘shift’ discretely between different fixed frequencies, holding only limited satchels of energy. All in all, we have blurred lines between what is a particle and what is a wave. When I was a physics undergraduate, most if not all of the strange effects came from the fact that objects can be both at the same time.

The Consequences of treating Particles as Waves

The classical view

In classical physics, we view objects as particles, each modelled as single points that take the least path in some potential. A potential is a backdrop of which the particle behaves, and it is usually modelled to be unchanging throughout time. In physics, the potential is a function that assigns some number at each of the coordinates — a landscape, if you will. In the first part of my series on geometry in physics, I wrote about this in a previous article about the path of least action.

For example, suppose I have a bowl. If I wanted to model the motion of a particle sliding from the rim of the bowl into the centre, then the potential, in this case, would be a fixed geometrical shape resembling that of an upside-down hemisphere. In the same vein, suppose I had a wall. The potential in this case, in two dimensions, would just look like the cross-section of a wall.

Classical mechanics is then concerned with solving for the motion of a particle given some potential. None of the results is that surprising. Many classical results are intuitive and easily visualisable. Even if you were a five-year-old, you could probably predict most of the results that classical physics merely adds a level of quantification to.

If you have ever tried running into a concrete wall, and you might find that you can't run through it to get to the room on the other side. If you had a ball trapped in the middle of a bowl — it’s not gonna be able to leave the bowl if it is undisturbed.

The quantum view — quantum tunnelling and potential hopping

These are all obvious and intuitive conclusions you get from classical mechanics, and they come from the view that particles are solid objects with a well-defined position and velocity. However, in quantum mechanics, things are fundamentally different. We model the position — and other quantities like momentum — with a mathematical object called a wave function. The wave function is an abstract function that spits out a probability depending on where we look at the wave function.

This means that if, say, an electron has a particular wave function that is ‘large’ in a given area, then if I observe the electron repeatedly in a series of experiments, I am likely to see the electron in that area more often.

When we model the position of a particle as a wave function instead of a solid position in space and time, things get weird. In quantum mechanics, we no longer solve for the position of a particle as a function in time but rather for something a bit more abstract. Given a backdrop, say a bowl like in the previous example, we get an equation to solve. This is the famous Schrodinger equation. Once we solve it, it gives us a wave function from which we get a probability of a particle at every point in space.

If the potential barrier is small enough, one of the exercises you can do is to find the wave function you get when you solve the Schrodinger equation, where the potential is modelled as a shallow wall. Amazingly, the probability that the particle can be observed on both sides of the wall, meaning that it has a chance of tunnelling through what is a solid barrier.

You read this right — there is indeed a chance that you might pass straight through a wall if you run at it. I suggest you try it and see what happens. It is amazingly counterintuitive that we find a non-zero probability of a particle being observed on the other side of a potential barrier. Note that this is distinct from the classical case. Particles cannot on their own jump over a wall — for this to happen, we would need the energy of the particle to be larger than it originally had.

In the same fashion, you can prove with similar mathematics that if you had stuck a particle in the well, you’d find a non zero chance that it lies outside the well. This kind of tunnelling phenomenon is indeed found in nature, and we have harnessed it to engineer materials that make life better for us. It is pivotal in the theory of superconductivity, and in 2016 scientists even showed that water molecules themselves exhibit quantum tunnelling.

The Consequences of treating Waves as Particles

We’ve talked about particles behaving as waves — but what about the other way round? In my opinion, this is where the duality gets most interesting, and I will reserve a detailed discussion for another article.

In short, however, the quantisation of waves is the theory of only allowing waves to store discrete values of energy. In classical mechanics, as you go up a ramp, you can slowly increase your gravitational energy smoothly and continuously as you go higher and higher. However, this smooth transition of energy is not permitted in quantum mechanics due to the fact that Schrodinger’s equation only permits discrete numbers of solutions.

It is this change that explains many previously unexplained phenomena. For example, it explains why hydrogen can only emit certain, discrete energies when it is excited with energy. It explains why the angular momentum of particles only has ‘discrete’ values, much like putting a bicycle into gear.

Wrap up

I hope that this article has given you a rough primer on some strange results in modern physics. Stay tuned for more! (P.S — Don’t actually run into that wall).