Learning Update 2023

Sometimes being interested in many different things can be challenging. I constantly start diving into new rabbit holes of totally unrelated things. Even though this is great fun it sometimes feels like I get lost in too many topics. Often I don't document anything about what I learn and there are no real goals so even though I am constantly learning it doesn't feel like it. Thinking about this I believe there are two main parts to changing that:

• documenting learnings to visualize progress
• defining focus and goals to align efforts

In the end, these are just tools to convince my mind that what I am learning is useful and not just entertainment. Realizing doing this has one more component: consistency. I need to actually work with these things and consistently update them for them to be useful. So first up let's just write down what I learned in the last half year.

The current rabbit hole: Physics

Note that what follows is a very basic and probably wrong explanation of some physics I learned throughout the last year. The main purpose is to have a map of my learning journey. If you want to learn similar topics please don't put any weight on my explanations and rather read the sources directly.

I just want to build an electronic circuit

Last year I wanted to do some hands-on electronics and therefore started playing with the amazing Arduino platform. It's amazing how they achieved building a product and community that teaches you the foundations of electrical engineering in such a friendly and accessible way. Just reading their introductory book while building some basic circuits was a real pleasure.

While doing this I came to understand that I don't know much about what electricity is. At first, I accepted the rules presented in the book but when I came across the concept of voltage dividers it dawned on me that my mental model of how electrical circuits work can't be right. I wanted to understand this on a more fundamental level before continuing to accept the rules and formulas of electrical circuits.

So I started the Khan Academy course on electrical engineering. It gave me a mental model of how electrons move through a circuit. A wire is full of atoms that have electrons that can be pulled to their neighboring atoms if these neighboring atoms have "space" for them and there is a force pulling them away. For example, a battery can pull electrons on one side and therefore trigger a chain reaction throughout the wire. That creates a current of electrons flowing through the cable. This in turn creates a field that can power components. Although later this mental model was refined in the discussions of this veritasium video.

How do molecules bond?

This leads me to question how atoms and molecules work. How do they bond? Where do these bonding forces come from? What is the trajectory of an electron?

This led me further down the rabbit hole to this explanation of the electronic structure of atoms and molecules. Turns out that atoms bond together in specific ways because the electron configurations of these atoms produce charge distributions. Combining these charge distributions of multiple atoms produces certain combined configurations of minimal energy. So in the end most of the school chemistry seems to be a very useful simplification or abstraction of these density maps and their minimal energy configurations. Chemistry then describes how to manipulate these configurations by adding energy or introducing other atoms into the mix to end up in other minimal energy configurations.

I love these visualizations of the charge density maps of atomic bonds. So I thought, couldn't we visualize these things in 3D and make them interactive? Let me put two atoms next to each other and see what their charge density maps look like. That would be so cool. So I asked my brother about it and he mentioned that it's sadly not that easy. He touched on that we would need to solve Schrödinger equation for multiple particles. That sounds tricky. Especially because I had no idea what the Schrödinger equation is. So there goes my next rabbit hole.

How do electrons move and get affected by fields?

So from there, I wanted to understand the basics of quantum physics so I could have a chance of understanding what this Schrödinger equation is all about. Of course, I started watching some youtube videos first. The standard entry point seemed to be explanations of the double slit experiment and then of the Stern Gerlach experiment. It was confusing for me what this idea of spin meant. I wanted to get a more founded idea of these principles so I started reading The Theoretical Minimum of Quantum Mechanics. This is a great introduction to the mathematical world of quantum mechanics. Turns out that these quantum phenomena can be nicely described using the tools of linear algebra. You can think of the state of a quantum system as a vector in a high-dimensional space. In the example of the position of an electron that would mean the vector is a vector of probabilities. Each element of the vector describes the probability of the electron being in a specific position in the universe. This is a pretty large vector though as there are quite a few positions in the universe. To be exact it's an infinite dimensional vector (because there are an infinite amount of possible positions of the electron in the universe). That is kind of hard to imagine. The main interesting question in quantum mechanics is how this vector evolves over time. Because if you know the current state vector of an electron and how it evolves over time you can predict where it will be in the future.

So some very smart people came up with mathematical descriptions of how these vectors evolve in our universe. To simplify the explanations and analysis of these explanations we usually only look at a tiny piece of this state vector. The smallest useful part of this state is the so-called spin. From what I understand this is a property of an electron that describes how it interacts with a magnetic field. We can imagine it similar to the polarity of a magnet as that also affects how the magnet interacts with a magnetic field based on its orientation in the field. The difference though is how this polarity works. It's not the same as with a classical magnet.

My understanding is that the way this property behaves in experiments is best described using a two-dimensional complex vector. Confusingly the components of this vector are called spin up and spin down (because that corresponds to how electrons with these spins behave in a magnetic field). Spin up and spin down can be represented using the following vectors: $$ s_{\text{up}} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \medspace s_{\text{down}} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$ The spin state of the electron can be described as any linear combination $$s = x_u s_{\text{up}} + x_d s_{\text{down}}$$ of these two as long as its length is 1. Note that $x_u$ and $x_d$ are complex numbers. Interestingly when measuring the spin state of an electron the measured state can only be exactly $s_{up}$ or $s_{down}$ and nothing in between (hence the state is said to be quantized). The probability that you measure ether corresponds to the components of the vector before the measurement. After the measurement, the probability collapses to the measured outcome. So if you measure the same spin again directly after you will always see the same outcome.

That's pretty cool. Seems like probability is baked into our universe.

So how does the probability change over time when we are not measuring? That's what the Schrödinger equation explains. It defines how probabilities in the quantum state evolve over time. We can use this for the evolution of the spin state just as we can use it for the evolution of position. We just need to figure out how to map what we want to measure to this idea of the quantum state.

So the weird thing is why doesn't the probability evolve according to the Schrödinger equation once we measure it? Why does the probability function collapse to whatever we just measured? Why doesn't it evolve the same?

There seem to be multiple explanations for that. This video about the many worlds interpretation seemed elegant to me. I started reading Sean Carolls book Something Deeply Hidden which gives an overview of the different interpretations with a strong bias on the many worlds interpretation. This different view on the topic gave me a better intuition of what is going on. I don't think many worlds is the only viable interpretation but I feel it's a useful tool for understanding quantum physics.

So that's where I am at right now. I feel like I start to understand some basic ideas of quantum physics and I appreciate the complexity of it while being amazed at how we can use these theories to build incredible technology. The precision of what we can describe with mathematical formulations seems crazy to me. It's amazing that humanity created a language that can express such unintuitive ideas in so much detail.

This appreciation for the expressiveness of mathematics leads me to the fact that I want to understand more of that. I learned mathematics with a view of it being given. The deeper I get into physics the more mathematics itself seems more like a language created to express complex ideas in a concise way. So I would like to learn more about how that evolved and understand it on a deeper level. That should also be useful for a deeper understanding of physics.

Does this format make sense?

It was really fun writing this down and I think it helps me keep track of the different levels of the rabbit hole. When coming back out of that hole this should be able to guide me on where to go next. It also helps to understand why I jumped into that hole in the first place.