Source: Varsha Y S, Wikimedia Commons

Introduction

In today's post we're going to talk a little bit about Physics (finally!). As you all may know from my introductory post, I'm a physics undergrad student. But apart from being just a student, I have a knack for teaching :). So, my intention with this Physics post, and some others that will follow, is to teach you useful things; maybe not something that'll help you build a lightsaber, but certainly enough to get you curious about what (we) physicists do, and enough to not get scammed with 100% efficiency fridges, perpetual motion machines, dehydrated water...

... or an "analysis" of your quantum numbers to cure a neckache.

What a Quantum Number is and isn't

Some while after its conception around 1920, and undoubtedly at the hands of a group of remarkable pitchmen, the most successful physical theory in history became the number one metaphysical sensation of the 21st century. Ranging from things like "love particles" and "positivity tunneling", metaphysics has mutated and disfigured fundamental concepts from quantum mechanics indiscriminately. Today, in hopes of clearing some of this confusion, we'll talk about Quantum Numbers.

So let's start from the top: what the hell is a quantum number? Drumroll please...

It is a tag. That's it. Yes, you read that right. It's no more than a tag that, usually, has some integer value. It is, by no means, a mystical thing. It says absolutely nothing about you. They will not say if you're going to be in love next month. It is simply a tag that serves to identify the state of some system, whose behaviour is of quantum character. It this is not clear to you yet, do not worry. I'll be following up this post explaining a bit more what "quantum" means.

So, in essence, quantum numbers are the tags you assign to a particular set of functions that, in turn, solve a completely mathematical problem associated with the system. And that mathematical problem is, well, Schrödinger's equation:

That greek letter that looks like a trident is the so called wave function

The need for these tags is that there isn't one and only one solution to this mathematical problem. There are quite a few, with different shapes, and a great way to tell them apart and organize them is by giving them tags that characterize them completely. Makes sense, right?

Therefore, quantum numbers, as seen from the perspective of the entirely mathematical problem that is Schrödinger's equation, have no additional meaning. However, this isn't the end of the story...

Quantum numbers are intrinsically related to many physical quantities in quantum systems, and that's what make them extra valuable. You can think of them in the following way: I imagine you've seen a meme like this not too long ago:

Britney's quantum numbers

If you assigned to each day of the week a number that indicated your mood, then you would be able to say something like

"Today is monday; my tag for mondays is 1, and I'm not very productive on mondays. Tomorrow will be tuesday; my tag for tuesdays is 2, and I'm twice as productive on tuesdays compared to mondays."

That's how quantum numbers work; they determine completely the state of the system, and are also related quantitatively to physical values of interest, like momentum, angular momentum and spin.

For example, when solving Schrödinger's equation for the hydrogen atom (a pretty cumbersome calculation, by the way), one gets that the corresponding wave function is

And that the average distance between the electron and the atom's nucleus is

I certainly do not expect you to understand these mathematical expressions completely. What I want you to notice is the following: do you see those letters n, l, ml, that appear over and over? Those are quantum numbers! In the top equation, they appear as indices that characterize the wavefunction, and thus characterizing the system's state. In the bottom equation, they also appear as indices, but they even appear in the formula itself. They pop up everywhere, in all formulas and results, and because of this they are of extreme importance when studying quantum fenomena. But they're just that: numbers.

Even still, don't let yourself be fooled. Quantum numbers won't tell you if you'll be lucky with Bitcoin.

As final comments, it's very important to know that these aren't the only quantum numbers that appear when solving problems in quantum mechanics; there are a bunch of them, because they're just super convenient as tags! Furthermore, even if many of them have integer values, some of them are not and they come as the fractions of some integer; these are called half-integer values. I hope that this cleard up a little what a quantum number really is. If you want to read more about it, I've left a book down in the bibliography that presents quantum mechanics in a very elementary and tractable way, especially for the less mathematically minded. I hope it serves you well in getting to the core of the issue and learning more about the quantum character of nature.

-SA

Bibliography

Eisberg, R. M., & Resnick, R. (2009). Quantum physics of atoms, molecules, solids, nuclei, and particles. New York: Wiley.

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