Joseph Mellor
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If you like my work, the easiest way to support me is to read and share my articles. Doing so helps out a lot and allows me to dedicate more of my time to writing these articles. If you really like my work, you can support me by donating to my kofi account. If you have trouble accessing any of the articles, let me know and I'll fix it.
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 Do Well in Your STEM Classes
An Intro to Differential Forms
MLK's Top Five Quotes
Vectors and Covectors
How to Be Taken Seriously as a Scientist
An Intro to Abstract Linear Algebra
An Intro to Topology
Elon Musk to Remove 75% of His Organs
The Charged Particle Lagrangian
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.
Algebra and Group Theory
Rings and Polynomials
Fields in Abstract Algebra
Finite Fields
Eigenvalues and Eigenvectors (Half-Finished)
Linear Algebra in a Finite Field (Half-Finished)
The Conclusion (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.
Intro, Coordinates, and Kinematics
Kepler's Laws and Newtonian Mechanics
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.
How to Use PDEs to Model Reality
Eigenvalue Equations and PDEs
The Spherical Harmonics
Bra-Ket Notation and Orthogonality
The Dirac Delta Function
The Wave Equation
Lagrangian Mechanics and Electromagnetism
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. In studying these concepts, we eventually derive Special Relativity. We end by talking about Noether's Theorem.
An Intro to Differential Geometry
The Principle of Least Action
The Double-Slit Experiment
The Generalized Stokes' Theorem (Half-Finished)
Lagrangian Field Theory (Half-Finished)
The Electromagnetic Lagrangian (Half-Finished)
Electromagnetic Optics (Not Started)
An Intro to Tensors (Half-Finished)
Electromagnetism and Special Relativity (Not Started)
Special Relativity (Not Started)
Noether's Theorem (Not Started)
In this fourth arc, we then talk about 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.
An Intro to Manifolds (Not Started)
Hamiltonian Mechanics (Started)
Canonical Transforms (Started)
Hamilton-Jacobi Equation (Not Started)
Action-Angle Variables (Not Started)
Liouville's Theorem and Poincaré Sections (Not Started)
An Introduction to Lie Algebras (Not Started)
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.
The First Law of Thermodynamics (Not Started)
The Second Law of Thermodynamics (Not Started)
The Maxwell-Boltzmann Distribution (Not Started)
The Zeroth Law of Thermodynamics (Not Started)
Phases of Matter (Not Started)
Free Energy (Not Started)
Statistical Mechanics (Not Started)
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.
Chaos, Hamiltonian Mechanics, and Topology (Not Started)
The Fourier and Laplace Transforms (Half Finished)
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.
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.
The Ultraviolet Catastrophe (Not Started)
The Photoelectric Effect (Not Started)
The Bohr Model (Not Started)
The Bohr-Sommerfeld Model (Not Started)
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.
The Born Rule (Not Started)
The Schrödinger Equation (Not Started)
The Generalized Heisenberg Uncertainty Principle (Not Started)
The Quantum Harmonic Oscillator (Not Started)
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.
The Variational Principle (Not Started)
The WKB Approximation (Not Started)
Nondegenerate Time-Independent Perturbation Theory (Not Started)
Degenerate Time-Independent Perturbation Theory (Not Started)
The Helium Atom (Not Started)
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.
The Potential Step (Not Started)
Alpha Decay (Not Started)
Time-Dependent Perturbation Theory (Not Started)
Fermi's Golden Rule (Not Started)
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.
The Klein-Gordon Equation (Not Started)
The Dirac Equation (Not Started)
The Dirac Hydrogen Atom (Not Started)
The Path-Integral Formulation (Not Started)
The Path-Integral Hydrogen Atom (Not Started)
Gauge Theory (Not Started)
Quantum Electrodynamics (Not Started)
Feynman Diagrams (Not Started)
Non-Abelian Gauge Theory (Not Started)
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.
Crystals (Not Started)
Reciprocal Lattices (Not Started)
Brillouin Zones (Not Started)
Phonons (Not Started)
The Debye Model (Not Started)
The Fermi-Dirac Distribution (Not Started)
Kronig-Penney Model (Not Started)
Semiconductor Crystals (Not Started)
Fermi Surfaces (Not Started)
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.
The Road to Quantum Mechanics: 2022 Review
I Started Using Blender
I Am Once Again Splitting Articles
Three Articles Coming Soon
I Made a Full Video
My Plans as of May 2022
I'm Streaming the Writing Process for My Aricles
Goofs and Gags
Here's where I try to prove I'm goofy so she'll let me hit.
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.
What Would a Sustainable Society Look Like?
A Layman's Intro to Material Science
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.
Let's Derive the Power Rule from Scratch!
What is a Limit, Really?
Uniform and Pointwise Convergence
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.
Stop Saying Entropy is Disorder
Creationism and Entropy
Let's Derive the Ideal Gas Law from Scratch!
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.
Beyond Calculus: The Math Classes You Didn't Take
Approaches to Proving Statements in Mathematics
When I eventually write more computer science articles, I'll probably give them their own section. Until then, I'm leaving them here.
How to Talk About Math and Science
This Week Had Some Truly Awful Science Takes
Switch Statements Won't Fix Yandere Simulator
The Memory Hierarchy
TUACM
Basic Data Visualization in Python
Making Sense of C
Representing Integers in Binary
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.
Photo by Felix Mittermeier from Pexels