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Newest Articles

These are some of my most recent articles. I'm putting them up top so you can see what I'm currently working on. Also, the more views an article gets in the earliest part of its life, the more the algorithm promotes it.

How to Discover Finite Fields While Bored in Class

In this series, I want to talk about my journey from messing around with cellular automata led me to a deep understanding of Abstract Algebra. I moved this part up because The Road to Quantum Mechanics takes up a lot of room.

  1. Cellular Automata
  2. Algebra and Group Theory
  3. Rings and Polynomials
  4. Fields in Abstract Algebra
  5. Finite Fields
  6. An Intro to Abstract Linear Algebra
  7. Eigenvalues and Eigenvectors
  8. Representations of Finite Fields
  9. Linear Algebra in a Finite Field (Half-Finished)
  10. The Jordan Normal Form (Not Started)
  11. Circulant Matrices (Not Started)
  12. Special Polynomials (Not Started)
  13. Modules in Abstract Algebra (Not Started)

The Road to Quantum Mechanics

It's a long road to understanding Quantum Mechanics, and there are no shortcuts. In this series, I'll show you everything you'll need to understand Quantum Mechanics. This series will also serve as a good discussion of many major topics in Classical Mechanics. I'm also putting the honorable mention Approaches to Solving Problems in Physics in this section because it's a great precursor to the rest of the series.

Intro and Newtonian Mechanics

In this first arc, we set up the goals of the series along with a general plan for the series (we repeatedly modify this plan based on what I find informative). We also set up Newtonian Mechanics, our first approach to Physics.

  1. Intro, Coordinates, and Kinematics (4,959 words)
  2. Kepler's Laws and Newtonian Mechanics (5,405 words)

Partial Differential Equations

In this second arc, we derive Poisson's Equation, Laplace's Equation, the Heat Equation, and the Wave Equation. We then discuss the two main methods of solving these PDEs that will generalize to Quantum Mechanics: eigenfunction expansions and the method of Green's Functions. We also discuss a lot of the mathematical background for these methods, which includes introducing Bra-Ket notation and projection operators.

  1. How to Use PDEs to Model Reality (4,881 words)
  2. Eigenvalue Equations and PDEs (4,684 words)
  3. The Spherical Harmonics (3,911 words)
  4. Bra-Ket Notation and Orthogonality (5,104 words)
  5. The Dirac Delta Function (5,358 words)
  6. The Wave Equation (4,984 words)

Lagrangian Mechanics

In this third arc, we work on converting Newtonian Mechanics into a form that works in arbitrary coordinate systems. We first do so by introducing Differential Geometry, then using Differential Geometry to derive Lagrangian Mechanics from first principles. From there, we talk about the Principle of Least Action and apply it to several systems. We then extend the Lagrangian to work with velocity-dependent forces like magnetism and fields.

  1. An Intro to Differential Geometry (5,177 words)
  2. Lagrangian Mechanics (5,097 words)
  3. The Principle of Least Action (5,398 words)
  4. The Classical Double-Slit Experiment (5,271 words)
  5. The Charged Particle Lagrangian (5,233 words)

Differential Forms

Up to this point, we've been introducing a lot of math piece by piece. On occasion, we've even pulled concepts and formulas out of thin air. To fill those gaps, we're going to spend the fourth arc providing the foundation for the mathematical objects we've been working with up to this point. We'll also introduce a few more that will be useful throughout this series.

  1. Vectors and Covectors (3,835 words)
  2. An Intro to Differential Forms (5,196 words)
  3. The Exterior Derivative (3,456 words)
  4. k-Chains and k-Cells (3,362 words)
  5. An Intro to Manifolds (5,227 words)
  6. The Generalized Stokes' Theorem (4,539 words)

Tensors in Classical Mechanics

Tensors show up everywhere in Classical Mechanics. In our fifth arc, we're going to rewrite everything we've done up to this point in terms of tensors. Then, we'll use tensors to describe and analyze Physical systems.

  1. An Intro to Tensors (3,870 words)
  2. The Levi-Civita Symbol and Alternating Tensors (3,188 words)
  3. The Rigid Body Lagrangian and the Inertia Tensor (5,130 words)
  4. The Multipole Expansion (Not Started)
  5. The Physics of Elasticity (Not Started)
  6. Maxwell's Equations and Differential Forms (Half-Finished)

Electromagnetism and Relativity

We've talked about Electromagnetism before in different contexts, but usually only as an example. In our sixth arc, we're going to introduce Lagrangian Field Theory so that we can describe the interactions between the electromagnetic fields and currents. As we look at the various interactions between the two, we'll notice that something is rotten in the state of Physics. Our equations for electromagnetism will predict the existence of electromagnetic waves that, curiously, always seem to move at exactly the speed of light no matter how fast you're moving. To correct this error, we'll need to introduce Einstein's Relativity.

  1. Lagrangian Field Theory (Half-Finished)
  2. The Electromagnetic Lagrangian (Half-Finished)
  3. Electromagnetic Waves (Not Started)
  4. Classical Electromagnetism in Materials (Not Started)
  5. Electromagnetic Optics (Not Started)
  6. Time Dilation and Length Contraction (Not Started)
  7. The Lorentz Transformation (Not Started)

Hamiltonian Mechanics

In this seventh arc, we introduce Hamiltonian Mechanics. Usually, Hamiltonian Mechanics is treated like a necessary evil to get to the Schrödinger Equation and only the bare minimum is covered. To remedy this problem, we'll cover Hamiltonian Mechanics in far more depth. We won't be wasting your time, though, as every concept introduced in this arc will show up somewhere in Quantum Mechanics or is necessary for a concept that shows up somewhere in Quantum Mechanics.

  1. Noether's Theorem (Started)
  2. Hamiltonian Mechanics (Started)
  3. The Lie Derivative (Started)
  4. An Introduction to Lie Algebras (Not Started)
  5. Symplectic Geometry (Started)
  6. Canonical Transformations (Started)
  7. Hamilton-Jacobi Equation (Not Started)
  8. Action-Angle Variables (Not Started)
  9. Liouville's Theorem and Poincaré Sections (Not Started)
  10. Canonical Perturbation Theory (Not Started)

Thermodynamics and Statistical Mechanics

In this final arc, we talk about Thermodynamics and Statistical Mechanics. We'll start by talking about Classical Thermodynamics before Statistical Mechanics, which covers concepts like heat engines, the phases of matter, chemical reactions, etc. Then, we'll introduce Statistical Mechanics and see how we can, in principle, replicate all these results.

  1. The First Law of Thermodynamics (Not Started)
  2. The Second Law of Thermodynamics (Not Started)
  3. The Maxwell-Boltzmann Distribution (Not Started)
  4. The Zeroth Law of Thermodynamics (Not Started)
  5. Phases of Matter (Not Started)
  6. Free Energy (Not Started)
  7. Statistical Mechanics (Not Started)
  8. The Einstein Solid (Not Started)

I have no idea how much this section will change or how I'm going to chop these concepts up into articles, so while the topics I want to cover won't change, the articles will. I'm probably going to add more and rearrange these topics.


In this section, we talk about topics that we should cover before we get to Quantum Mechanics, but don't really fit anywhere in the main body of the series.

  1. An Intro to Topology
  2. Chaos, Hamiltonian Mechanics, and Topology (Not Started)
  3. The Fourier and Laplace Transforms (Half Finished)
  4. An Intro to Complex Analysis (Not Started)

This list is bound to change multiple times as I'm going to rearrange the order of the topics and move topics around within the articles. For example, I might combine two articles, I might split an article, I might shave sections off certain articles and combine them into one article, etc.

Quantum Mechanics

After you've travelled The Road to Quantum Mechanics, you arrive at Quantum Mechanics. In this series, we're going to cover Quantum Mechanics from its inception to as long as I can keep writing. We're going to refer back to The Road to Quantum Mechanics throughout this series because we've already done a lot of the work.

NOTE: I'm not even close to writing these yet, so take the article titles listed below as if they're just topics that we want to cover. I will change the order of these articles, remove some, add others, and so on. For example, I will definitely spend more than one article on Quantum Electrodynamics and Quantum Chromodynamics each, but I only have it as one article for now (though those articles might be intro articles and I'll have them as their own arcs).

Old Quantum Theory

At the end of the 19th century, we started noticing some problems with our models. If we assumed certain parts of nature were continuous, we would get answers that were not only wrong, but absurd. During the early 20th century, we resolved these issues with ad hoc solutions. In this arc, we're going to talk about these problems and the first attempts of the scientific community to resolve these issues.

  1. The Ultraviolet Catastrophe (Not Started)
  2. The Photoelectric Effect (Not Started)
  3. Rutherford Scattering (Not Started)
  4. The Bohr Model (Not Started)
  5. The Bohr-Sommerfeld Model (Not Started)
  6. The Fine Structure Constant (Not Started)

Principles of Quantum Mechanics

In this section, we start to move to the modern understanding of Quantum Mechanics. During this arc, we'll develop the Principles of Quantum Mechanics and apply them to a few important systems.

  1. The Born Rule (Not Started)
  2. The Schrödinger Equation (Not Started)
  3. The Generalized Heisenberg Uncertainty Principle (Not Started)
  4. The Quantum Harmonic Oscillator (Not Started)
  5. The Hydrogen Atom (Not Started)

Approximation Techniques

Besides the systems I've listed and a few others, almost every other system in Quantum Mechanics is too difficult to solve exactly. To deal with this problem, we have to use various approximation techniques.

  1. The Variational Principle (Not Started)
  2. The WKB Approximation (Not Started)
  3. Nondegenerate Time-Independent Perturbation Theory (Not Started)
  4. Degenerate Time-Independent Perturbation Theory (Not Started)
  5. The Helium Atom (Not Started)
  6. Electron Orbitals and the Periodic Table (Not Started)

Quantum Dynamics

At this point, we know how to find things like energy levels and eigenstates, but we want to know how systems change over time. Currently, this section is a little dry, so let me know if there are other tricks I can bring into the system.

  1. The Potential Step (Not Started)
  2. Alpha Decay (Not Started)
  3. Time-Dependent Perturbation Theory (Not Started)
  4. Fermi's Golden Rule (Not Started)

Quantum Field Theory

At this point, it should become clear that this list is definitely incomplete and will be revised in the future. For example, I've made no mention of the Weak Force at all. I'm definitely going to add more to this section and possibly split it up. Alternatively, I might keep it as is and give stuff like Quantum Electrodynamics and Quantum Chromodynamics its own section.

  1. The Klein-Gordon Equation (Not Started)
  2. The Dirac Equation (Not Started)
  3. The Dirac Hydrogen Atom (Not Started)
  4. The Path-Integral Formulation (Not Started)
  5. The Path-Integral Hydrogen Atom (Not Started)
  6. Gauge Theory (Not Started)
  7. Quantum Electrodynamics (Not Started)
  8. Feynman Diagrams (Not Started)
  9. Non-Abelian Gauge Theory (Not Started)
  10. Quantum Chromodynamics (Not Started)

Solid State Mechanics

In this arc, I want to cover the study of solid materials. Our goal is to somehow get from a microscopic description of a material to its macroscopic physical properties.

  1. Crystals (Not Started)
  2. Reciprocal Lattices (Not Started)
  3. Brillouin Zones (Not Started)
  4. Phonons (Not Started)
  5. The Debye Model (Not Started)
  6. The Fermi-Dirac Distribution (Not Started)
  7. Kronig-Penney Model (Not Started)
  8. Semiconductor Crystals (Not Started)
  9. Fermi Surfaces (Not Started)
  10. The Physics of a Diode (Not Started)

Density Functional Theory

I don't know what exactly I'm going to add here, but I definitely know I want to cover it. I'll figure it out when I get here in a year or two.


General updates to my current situation and future plans. Most recent updates are at the top.

  1. I Can Now Make Vector Fields in Blender
  2. I'm Making Some Art Tonight
  3. I'm Going to Publish Some Programming Articles
  4. The Road to Quantum Mechanics: 2022 Review
  5. I Started Using Blender
  6. I Am Once Again Splitting Articles
  7. Three Articles Coming Soon
  8. I Made a Full Video
  9. My Plans as of May 2022
  10. I'm Streaming the Writing Process for My Aricles

Big Bois

These are articles that I put a lot of effort into, but are unfortunately too long, meaning they get rejected by the algorithm. To help them survive, I've decided to give them their own section.

Real Analysis and Calculus

I originally wrote an article on how to derive the Power Rule in Calculus from scratch, but the title was inaccurate because I relied on the Moore-Osgood Theorem, which is from Real Analysis, the field that underpins modern Calculus. Someone pointed this fact out, so I decided to write an article proving the Moore-Osgood Theorem. That article became three articles. If I decide to write more, I'll put them here.

  1. Let's Derive the Power Rule from Scratch!
  2. What is a Limit, Really?
  3. Uniform and Pointwise Convergence
  4. When Can You Switch Limits in Calculus?


Someone once decided to call entropy disorder, and the scientific discourse has never recovered. I have taken the crusade to rid the world of this nonsense.

I'm also putting this video in this section because it's relevant.


I have a few articles about math in general that don't fit anywhere else, but these are some of my biggest articles so I want to bring attention to them.

Computer Science

Despite my earliest articles being CS, I haven't written too many CS articles. Since Medium is filled with a lot of people interested in CS content and I want to write CS content, I've decided to start writing CS content.


In this series, I'm going to talk about how to go from setting up python to using it to do cool things. The first six articles will focus on setting up python to do some basic data analysis, but I plan on doing a lot of things with python.

  1. Getting Started with Python in 2023
  2. The Core Objects of Python
  3. Control Flow in Python
  4. Basic File Handling and Parsing in Python
  5. Importing Modules in Python
  6. How to Write a Program in Python

Making Sense of C

The best way to understand how a computer thinks is to learn C. In this series, I'll go into all the details of C and why C made the design decisions it made.


These articles don't really fit into a category, so I'm putting them here.



I have a YouTube channel where I upload videos on topics related to physics, math, and coding. Right now, I'm focusing on the series with some occasional streaming, so it's a little bare-bones at the moment.

Request an Article

If you would like me to write an article for you, email me and we'll discuss rates. $30 per 500 words baseline, and the price will go up if I have to make my own pictures (including pictures of mathematical equations), write code, or do a significant amount of research.

Do not offer to pay me in exposure.