The Language of Quantum Physics
An introduction: kets, wavefunctions, and quantum systems.
Not only is the Universe stranger than we think, it is stranger than we can think.
– Werner Heisenberg, Across the Frontiers
If we wanted to answer the question of what’s truly fundamental in this Universe, we’d need to investigate matter and energy on the smallest possible scales. On such scales, reality starts behaving in strange, counterintuitive ways. We can no longer describe real as being made of individual particles with well-defined properties like position and momentum. Instead, we enter the realm of the quantum: where fundamental indeterminism rules, and we need an entirely new description of how nature works. But even quantum mechanics itself has its failures here. They doomed Einstein’s greatest dream — of a complete, deterministic description of reality — right from the start. If we lived in an entirely classical, non-quantum Universe, making sense of things would be easy. As we divided matter into smaller and smaller chunks, we would never reach a limit. There would be no fundamental, indivisible building blocks of the Universe. Instead, our cosmos would be made of continuous material, where if we build a proverbial sharper knife, we’d always be able to cut something into smaller and smaller chunks.
With quantum physics, new rules are needed, and to describe them, new counterintuitive equations. The idea of an objective reality goes out the window, replaced with notions like probability distributions rather than predictable outcomes, wavefunctions rather than positions and momenta, Heisenberg uncertainty relations rather than individual properties.
Enter: Ψ
Ψ. Meet every quantum physicist’s best friend — the 23rd Greek letter. Ψ (psi, pronounced with the /ps/ in “lapse” followed by an /eye/) here, represents the wave function of a quantum system. A wave function is a mathematical description of everything we know about a particular quantum system. A quantum system is of course a “system” we happen to be studying using quantum mechanics. It could be anything. Put a few electrons in a magnetic field. That’s a system. And if you use quantum mechanics to study it, then well, that’s a quantum system.
In fact, the wave function is very, very, very closely linked to the probability distribution of a system. Probability density is a “density” function, f(x). While the probability is a specific value realized between one and zero. Well our wave-function, Ψ, tells us how likely we are to find a particular particle at different regions of space in our system. Let’s say that our quantum system is a single electron. In our system, like the one we have below, a particle can move forwards or backward since it’s confined to one dimension.
Ψ² ∝ probability distribution of finding a particle at a particular point.
It’s this function like we established, that is the important one. Specifically, this function is telling us — Ψ² is telling us — that at r=2 because Ψ² is large, we’re most likely to find our electron. Whereas at 4, Ψ² is small and so we’re least likely to find our electron. Now there are a few subtleties to this but for our purpose, all we care about is that this Ψ² function is telling us the probability distribution of finding our electron in our one–dimensional system. For the purpose of this article, however, we won’t necessarily bother ourselves with the form of the wavefunction itself. All that matters is that if we wanted to, we could find the Ψ of our system. Though it’s not the wave function itself that’s directly telling us. Rather it’s the square of the wave function that really tells us the probability density of our system. It is important to understand that Ψ² is not the probability, but the probability density.
But we don’t just write Ψ when we represent a quantum system. We actually write it like this: |Ψ⟩
A straight line to the left and an angled bracket to the right. Now, this whole “thing” is called a ket. It’s written like this for a multitude of mathematical reasons. But as far as we’re concerned today, the ket allows us to differentiate between a quantum system and a constant or an operator. A ket is the quantum mechanical symbol that encodes the state of a system. A symbol within a ket may represent any physical quantity. A physical quantity is a property of a system that can be quantified by measurement. But why is this called a ket? Why do we represent wave functions with kets? Well, because related to a ket is a quantity known as a bra: ⟨Ψ|
If you haven’t figured it out already, the bra and the ket unimaginatively stem from the word bracket.
- ⟨Ψ| the bra
- |Ψ⟩ the ket
Now a |Ψ⟩ and its corresponding ⟨Ψ| are mathematically linked. Because ⟨Ψ|Ψ⟩, the inner product between the bra and the ket, is 1.
But why’s any of this important? Superposition. Well presuming you know little to nothing about superposition, it is the ability of a quantum system to be in multiple states at the same time until it is measured. Now suppose that our quantum system is an electron. And we’re measuring its spin. For the uninitiated: the spin of an electron, simply put, is a form of angular momentum. Objects that are spinning, rotating or orbiting another body are ones with angular motion. Now with angular motion, comes angular momentum. This is slightly different from the linear momentum that you might be familiar with. The thing about spin though is that some objects have an inherent angular momentum. One that is not caused by anything external. Now with electrons, there are two conceivable possibilities: spin-up and spin-down. When measured it could be in either of the following states:
|↑⟩ = spin up
|↓⟩ = spin down
But here’s the caveat. If our electron — our quantum system — is just left alone then it is said to be in a superposition of both of these states, In other words, the electron isn’t |↑⟩ or |↓⟩, it’s |↑⟩ and |↓⟩.
All we need to know is that our quantum system can be in a superposition of states and supposedly in more than one state at one time. However, that only holds for as long as it’s left alone and nothing is interacting with it. If you somehow manage to measure its spin, you’ll find that it’s in either one of the two states.
The last two hundred words or so are what the Copenhagen interpretation says. The Copenhagen interpretation is, by far, the most widely accepted understanding of quantum mechanics. Now in the situation where our particular quantum system isn’t being measured, we can write the wave function of our electron as a superposition of both these states. The way that we can do that is to add the two states together.
Essentially what we’re doing is giving a representation of the overall state of our quantum system. But here’s the thing, let’s imagine that we go and actually measure the spin of this electron. We’ll only find it either in the spin-up state or the spin-down one. What happens is that our wave function, which was initially a superposition of both these states, collapses into just one. This phenomenon is known as the collapse of the wave function. Here’s the important thing though: we’ll never find our quantum system in a superposition of two states when we measure it.
This is where the number in front of the kets comes in. These numbers are directly linked to the probability of us finding the electron to be in a particular spin state. In other words, squaring the number that is in front of the spin-up state will give us the probability of finding our electron in that particular state. And equivalently with the spin-down state. Notice that adding their squares gives a total of 1 — which is what probability is measured relative to.
So essentially the probability of measuring our electron in the spin-up state is 0.75. While the probability of measuring it in the spin-down state remains 0.25. It’s important to understand that this is as far as we can go. Before measuring a quantum system, we cannot, with absolute certainty, predict its outcome. We can only predict the probabilities of the outcome.
There are a lot of things I haven’t particularly explored since they involve mathematical intricacies which we haven’t established (Hilbert spaces, orthogonal vectors, etc.). Nevertheless, I hope that this article has helped your understanding of quantum mechanical systems.
