Quantum Physics III: Revenge of the Commutator

This post will be wrapping up our adventures through Hilbert space for the time being, and in grand fashion. This is a very dense post, with a lot of incredible physics. As a result it is also a bit longer than others. But if you have an emptiness inside you that is screaming for beauty, symmetry, geometry, abstraction and concreteness at the same time, put everything you're doing aside and read this one. Approx 15-20 min read.

The Fundamental Group

Topology is one of the coolest things I've ever studied and in this post, I'm going to tell you about one of the core concepts of the subject - that of the fundamental group. I'll be talking about loops and transformations a lot, but we do have a bit of ground to cover. Luckily for you, any closed subset of the reals is compact, so we can find a finite subcover of this ground, which is precisely what I've tried to do here. It ain't a party without a sick as proof though, but party animal that I am, I'm bringing the goods. And it's gonna be fun. Approx 15-25 min read.

Why Special Relativity and Quantum Mechanics do not Mix

Have you ever wondered what would happen if you took Schrodinger's Cat and squished it into an unimaginably, horrendously, pitifully, and inescapably tiny box? Well, look no further, and strap yourselves in ladies and gentlemen, because today's post answers that question and motivates the need for quantum field theory.
I'd say, 5-10 minute read probably.

[Spoiler: the cat will die, whether the box is poisoned or not.]

Noether’s Theorem

In this post you will find out about one of the coolest things in all of physics. From a mathematical standpoint it's not really that big of a deal, but from a physical perspective it absolutely kicks ass. This is a slightly long post with a lot of maths, but if you can get through it all and understand it, it will have been worth your while! Approx. 20-25 min read depending on how much maths you skip through or follow.

The Density of Q in R

I want to tell you about one of my favourite theorems. It is a theorem in the subject of real analysis, which deals with analyzing the properties of real numbers, their sequences, series, and functions. Real analysis is a very pure subject and is very fun if you have an appreciation for aesthetic and enjoy mental gymnastics. This proof is typically seen seen at the beginning of a lot of analysis classes and I think it is very pretty, so I will show you it. Approx. 5 min read. But just like all other maths, if you want to understand it, that means an extra 3 hours thinking time, if you are so inclined.

Newton’s 2nd Law and Symmetries, but Better – Rotations, and Double Pendulums

I'll show you how easy physics becomes when you possess the Euler-Lagrange equation, and you will see first hand just how fundamental it truly is.
Not only that, but I will explain to you a very interesting phenomenon that you no doubt are aware of yourself but perhaps don't fully understand, as well as show you some very nice double pendulum systems in gif form! There's a lot of maths but also some nice summaries if you don't know maths. Approx. 10 min read.