These articles are written with diverse types of audiences in mind, including ordinary high school students, laymen, students learning high school physics, students learning calculus, mathematically oriented high school students, science and engineering students, and physics majors. However, I tried to write these articles as self-contained as possible so that those articles targeted for higher level readers can be understood if lower level readers read their prerequisites. Also, I have tried to present these articles in a suggested order of reading. Nevertheless, you may want to skip some articles and jump ahead to more interesting ones, unless the articles you skip are prerequisites to those. Of course, if you are a high level reader, you may not want to read the articles that examine topics with which you are already familiar. In any case, prerequisites are stated in the comments for the grouped articles, and you can check out the flowchart of these articles here.
One of the biggest goals of these articles is that you understand the loop quantum gravity research that I have done with Brian Kong. As quantum mechanics and general relativity are prerequisites to loop quantum gravity, the next biggest goal of these articles is that you understand these two grand subjects. (Even though I present articles on quantum mechanics and general relativity to help you understand loop quantum gravity, I don¡¯t confine myself to the strictly necessary prerequisites, as some topics are interesting on their own.) Besides these two subjects, loop quantum gravity is intricately related to thermodynamics and statistical mechanics, which I also present in these articles. Besides giving an opportunity to understand cutting-edge research, reading my articles will give you sufficient preparation to learn more advanced material on your own by reading standard quantum mechanics textbooks, general relativity textbooks, and statistical mechanics textbooks.
Quantum mechanics, which physics majors usually learn in their junior year, requires the knowledge of calculus and linear algebra, which science and engineering students usually learn in the first and second semesters of their freshman year. Besides this mathematical background, quantum mechanics requires some basic physics knowledge, usually taught in a high school or freshman college physics course. However, if you want to understand how quantum mechanics can describe every phenomenon classical physics can describe, you have to learn classical mechanics, which physics majors usually learn in their sophomore year. Therefore, before teaching you quantum mechanics, I will first present classical mechanics to serve as a prerequisite for the deep connection between classical physics and quantum mechanics.
The main thing that sophomore physics majors learn in their classical mechanics courses is how to solve key equations from classical mechanics for various situations. However, this is something that I keep to minimum, and instead I focus more on the basic formalisms (i.e. the derivation and motivation behind the key equations) of classical mechanics. Only then do I present basic quantum mechanics based on basic physics, even though this doesn¡¯t strictly require sophomore classical mechanics. Here, I completely amalgamate a course on linear algebra with this section. By this way, this section will give readers more motivation for studying linear algebra. They will understand how certain concepts in linear algebra are useful in applications. Science and engineering students usually learn linear algebra in the second semester of their freshman year, but in the U.S. I was surprised to find out that some American high school students learn it in prep schools. After presenting basic quantum mechanics and basic linear algebra, I will present the connection between classical physics and quantum mechanics, as well as more advanced topics in linear algebra and quantum mechanics.
General relativity, which physics majors usually learn in the first year of graduate school, requires the knowledge of calculus, special relativity, and classical mechanics. Basic special relativity is usually taught in a freshman physics course, but advanced special relativity, which is required to understand general relativity, is taught in a classical mechanics course.
Special relativity is very beautiful, and, depending on the level, it can be learned either with or without ¡°electricity and magnetism.¡± To truly appreciate its beauty, I will teach you special relativity with electricity and magnetism.
Science and engineering students usually learn electricity and magnetism in the second semester of their freshman year, and physics majors learn these concepts more deeply in their sophomore year. Using no difficult math, I will first treat electricity and magnetism in three parts, and in the first part I will provide prerequisites for special relativity. Then, in ¡°Maxwell¡¯s equations,¡± I will re-visit the three parts using multivariable calculus, culminating in the derivation of Maxwell¡¯s equations. The main thing sophomore physics majors learn in their electricity and magnetism course is how to solve Maxwell¡¯s equations. Again, this is something I keep to minimum, and instead I focus more on the basic formalisms.
Statistical mechanics is a huge subject that junior physics majors usually learn after learning the first semester of quantum mechanics. However, we only examine the basics of the basics of statistical mechanics, the minimum prerequisites for loop quantum gravity.
A freshman physics course for science and engineering students covers a little bit of classical mechanics, a little bit of thermodynamics, a little bit of electricity and magnetism, a little bit of special relativity, a little bit of optics, and a little bit of modern physics (i.e. quantum mechanics and some miscellaneous topics.) In particular, it doesn¡¯t assume any knowledge of high school physics. I cover the classical mechanics part in ¡°Basic high school physics without calculus¡± and in ¡°Mathematical Introduction to Physics.¡± The electricity and magnetism part is included in my three part treatment of the subject and in ¡°Maxwell¡¯s equations,¡± which I mentioned earlier. However, I completely omit the circuit problems, which are crucial for electrical engineering subjects. I cover the thermodynamics part in ¡°Entropy¡± and in ¡°Thermodynamics.¡± I cover the optics part in ¡°Wave.¡± The modern physics part is covered in ¡°Historical Introduction to Quantum Mechanics.¡±
Physics is written in the mathematical language called ¡°calculus.¡± Most South Korean high school students and advanced American high school students learn ¡°single variable calculus.¡± Then they learn ¡°multi-variable calculus¡± for a semester right after they enter college, if they intend to major in science or engineering. We will examine both single variable calculus and multi-variable calculus.
After writing out most of these articles, I decided to add articles on high school mathematics to make this webpage as self-contained and accessible as possible. After all, even though most South Korean adults know quite a bit of math, many American adults don¡¯t.
At the end of every article, I included a summary, which you should remember. For some articles I included exercises for the readers to solve. Some of them are intended for the readers to check whether they have grasped the key concepts in the article. Others are intended to give the readers the opportunities to derive the key concepts on their own, which are deliberately left out of the main article, as people remember better when they derive things on their own than when they just watch someone else derive them. I have tried to make such exercises as easy as necessary, so that if the readers understand the key concepts, they are expected to solve such exercises correctly. Other exercises are intended to fill-in the gaps between the reasoning. Of course, I could have presented the solutions in the main text, instead of assigning exercises, but I wanted to give the readers some chances to think before checking the answers. I will upload the solutions later if I get a chance. As I give out hints for most exercises, make sure that you take some time trying to solve the exercises by yourself.
Normally, you would get to understand loop quantum gravity only after getting an undergraduate degree in physics and studying for a couple of years as a graduate student in physics. This would be about six years after you enter college. Nevertheless, as I try to tend to avoid topics that are rather not essential to understanding this big subject, it may take much less time if you learn it through my articles. This is an ambitious plan, but I dare try! Go ahead, and enjoy my articles!
The articles here explain high school mathematics, also known as ¡°middle school mathematics¡± in South Korea. They also cover the Cauchy-Schwarz inequality, which is usually left out in a standard high school curriculum, but is actually simple and interesting to learn.
The articles here can benefit everyone. Most require no specific knowledge, although some assume familiarity with high school mathematics. They can be understood both by laymen and by students learning high school physics.
The articles here are primarily targeted for mathematically oriented high school students and for students learning calculus. Science and engineering students can also benefit from some of these articles, especially the ones on quantum mechanics and on special relativity. Physics majors can also benefit from some of these articles, especially the ones on differential forms and on the Feynman diagram.
The articles here are suitable for advanced undergraduate physics majors and physics graduate students.