Theory of Everything
The first article explains what string theory is and why many physicists study it. The second article explains what the first superstring revolution and the second superstring revolution are in laymen’s terms. I also explain the significance of my calculation of string coupling constant. The last article derives a simple mathematical relation that is very important in string theory. fel
In these two articles, I explain the basics of the basics of chemistry as chemistry partly shares history with physics up until the early 20th century. Of course, I am not an expert on chemistry, but having taken AP chemistry, I know chemistry just enough to explain the basics of its basics.
Some basic mathematics and statistics, the first part
Vector is a very important basic mathematical concept in physics. To learn more about vectors, read the sections “Dot product” and “Linear Algebra and Quantum Mechanics.” The article on vector is the only prerequisite to “Electricity and Magnetism, the first part,” which in turn is the only prerequisite to the section “Special Relativity.”
Basic high school physics without calculus, the first part
I have written “Newton’s first law” so that those who have never learned physics can have a chance to learn about the wonderful ideas of the two great physicists, Galilei and Newton. To understand “Newton’s second law,” you have to understand what a vector is. The last article explains how one can relate the distance and the time an object falls due to gravity on Earth. It serves as a prerequisite for the articles listed in the section “Some basic ideas in General Relativity.”
Even though you may naively imagine that trigonometric functions are important only for geometry, they are actually very important and widely used in mathematics. For example, trigonometric functions are important in describing waves in physics. As quantum mechanics is about particles and waves, knowledge of trigonometric functions is essential in understanding quantum mechanics.
One of the purposes of the article “Rotation in Cartesian coordinates” is to serve as a basis to make an interesting comparison with the Lorentz transformation in my article on that topic in the section “For students learning high school physics.”
Historical Introduction to Quantum Mechanics, the first part
After realizing that some quantum mechanics is needed in “History of astronomy from the early 20th century to the early 21st century” and “Discrete area spectrum and the Hawking radiation spectrum I,” I divided “Historical Introduction to Quantum Mechanics” to two parts, and placed the first part in front of the section “Astronomy.” The first article is not about quantum mechanics, but a prerequisite to the next two articles. These three articles do not assume any knowledge of physics.
Some prerequisites for "Astronomy"
In this section, we present some concepts that are required to understand some of the articles on the history of astronomy in the section “Astronomy.” However, most of the contents in that section will be accessible without reading these articles. “Conic sections” is useful to understand Kepler’s contribution to astronomy. “Curved space” is useful to understand Einstein’s general relativity.
History of astronomy from the early 20th century to the early 21st century
In the first article on history of astronomy, I provide a brief overview of history of astronomy from ancient time to the time of Johannes Kepler and Galileo Galilei. In the other two articles on history of astronomy, I provide a brief overview of history of astronomy from the time of Sir Isaac Newton to the present. Physics was essentially applied to and received feedback from astronomy during the time period treated in these two articles. Those of you who are not really interested in astronomy may still want to read some parts of these three articles on history of astronomy, because they also treat some interesting concepts and developments in physics, such as Newton’s explanations of Kepler’s laws, Einstein’s general relativity, and Verlinde gravity, which I didn’t explain in details in “A short introduction to the history of physics and string theory as a “Theory of Everything.”” In the last article, I provide some evidences that the Earth is round, as it is alarming that the Flat Earth proponents are gaining momentum these days. They are idiots who reject scientific evidences. They don’t deserve to enjoy modern scientific innovations.
Bosons and Fermions
These five articles explain what bosons and fermions are, and how the concept of these categories of particles is applied in physics. The first article explains how supersymmetry relates bosons and fermions to one another. Supersymmetry is a very important topic in the contemporary particle physics community. First theorized in the early 1970s, many expect that it will soon be supported by experiments at the LHC in Geneva, Switzerland. The second article can be read independently of the first. The third, fourth, and fifth articles can be read independently of one another, but you may have to read the second article in order to understand them. The third article explains how the idea of color charge of quarks was introduced to reconcile the theory with Pauli's exclusion principle, and goes on to explain some interesting characteristics of the strong force. The fourth article explains the microscopic statistical properties of identical particles which may seem bizarre to macroscopic human beings. The fifth article supplies another derivation of Pauli's exclusion principle via an approach alternative to that taken in the second article.
Most people have probably heard that nothing can get out of a black hole once it has been sucked into it. The article here explores this assertion in more detail qualitatively. For a quantitative treatment, read “By how much does time go more slowly at a lower place?” after reading the above article.
Newton’s Inverse Square Law of Gravity
In this article, I explain what I think why God chose the inverse square law for gravity.
Some basic mathematics and statistics, the second part
Cavalieri's principle and the volumes of cones and spheres
Logarithm is often used in physics and mathematics. It serves as a prerequisite for “What is entropy? From a microscopic point of view” and the articles in the section “Loop Quantum Gravity (for laymen).” “Expectation and Standard deviation” serves as a prerequisite for “A short introduction to quantum mechanics XII: Heisenberg’s uncertainty principle.” “Exponential function vs polynomial function” will play an important role in understanding my research on black hole entropy, which is reviewed in our article “Approximation of the na?ve black hole degeneracy.”
Energy and Entropy
Energy and entropy are very important concepts in physics. The second article explains what entropy is and why it “always” increases. The third article explains what entropy is from a macroscopic point of view. It can be read independently of the first one. These articles serve as prerequisites for the section “Statistical Mechanics.”
Loop Quantum Gravity
Discrete area spectrum and the black hole entropy III: Newer variables
Discrete area spectrum and the Hawking radiation spectrum IV: Single-partition black hole
These seven articles try to explain, in laymen’s terms, the research I have done on the black hole partially with Brian Kong. These articles assume that readers are familiar with the mathematical expression “log,” which is explained in my article “Logarithm.” You may skip the second and the third articles if it’s too hard since you don’t need it to understand the other articles. To understand the sixth article, you will need to know the concept of “identical particles” as treated in “Bosons, Fermions and the statistical properties of identical particles.” We will explain our research again in expert’s terms in later articles.
Complex numbers is a prerequisite to quantum mechanics. Everything in quantum mechanics is calculated in terms of complex numbers, even though at the end of these calculations one must obtain an answer in terms of the real numbers to compare with experimental results, since complex numbers are not “real” and can’t be the values of measurements. The last article explains the beautiful relationship between complex numbers and the trigonometric functions, which you may appreciate better after reading “Euler’s formula” listed in “A crash course in calculus, the second part.”
Algebra is one of the major branches of mathematics. If you want to get an undergraduate degree in mathematics at Harvard, you are required to take at least one course in analysis, at least one course in algebra, and at least one course in topology. I want to remark that algebra, a branch of mathematics, is not arithmetic, but a much more generalized version of arithmetic. Sometimes, it is called “abstract algebra” to distinguish itself from arithmetic, which many people also call algebra. “Linear algebra” which is the most basic subfield of algebra is very important in physics, but other than that, the importance of algebra in physics at undergraduate level is rather limited despite its importance in certain subfields of physics. Therefore, it is very easy to find physics professors who don’t know what is “Abelian group” which you will easily learn in the first article. Even though algebra is mostly learned by sophomore undergraduate math majors, anyone who has some mathematical sense can get the basic idea. It is not the ability to solve complicated equations or factoring out polynomials quickly but ability to think deeply that counts.
The Gauss-Bonnet theorem for triangle on a sphere
Topology is one of the major branches of mathematics. While its importance in physics at undergraduate level is rather limited, it is important in certain subfields of physics. “Algebraic Topology” is a subject usually learned by math graduate students. We will just briefly touch what alegbraic topologists deal with.
Electricity and Magnetism, the first part
These six articles provide the prerequisites for the articles explaining the relation between Einstein’s theory of special relativity and electromagnetism presented in the section “For students learning high school physics.” Nevertheless, they could be interesting on their own as those who did not take physics in high school do not usually know about the phenomena presented in these articles. “What is a vector?” listed in the section “Some very basic high school physics” is the prerequisite to “Electric charge and Coulomb force.”
This section groups together eight articles on special relativity, in a suggested order of reading. (You may also want to read pages 177-209 of “The Evolution of Physics.”) While the third and fifth articles require the basic knowledge on electricity and magnetism, as covered in the section “Electricity and Magnetism, the first part,” others can be read without these prerequisites. The third and fifth articles show that Einstein's theory of special relativity is consistent with electromagnetism. Of course, this should be the case since light is an electromagnetic wave, and Einstein's theory of special relativity is based on the fact that the speed of light in vacuum is always constant. Nevertheless, it is exciting to check these consistencies between electromagnetism and special relativity in easy ways, since understanding them from the original construction of special relativity requires advanced knowledge. Moreover, it is always exciting to see that you arrive at the same conclusion even from very different perspectives.
The third article makes use of the relationship between electric current and magnetic force. Later, in the fifth article, this relationship is partly derived by “using” the special theory of relativity, but the relevant facts are stated in the third article and can be taken for granted for the purpose of understanding the point of the third article. If you don’t wish to take it for granted, you may want to read the articles in the section “Electricity and Magnetism, the first part,” which deal with the interplay between electric fields and magnetic fields.
Unlike other articles, the last three articles related to Lorentz transformation are mathematical to some extent. Even though you only need to know the concept of square root to comprehend them, the derivation of Lorentz transformation might be complicated. “Lorentz transformation and Rotation, a comparison” assumes prior knowledge on "Rotation in Cartesian coordinates" covered in the section "Trigonometric functions."
Paradoxes in Special Relativity
Theory of relativity is quite confusing and unintuitive. At first glance, it seems wrong. Therefore, many people could come up with a lot of paradoxes that seemed to show that theory of relativity was wrong. However, it isn’t, since these paradoxes can be resolved. These articles deal with such paradoxes.
Electricity and Magnetism, the second part
Magnetic monopole and Gauss’s law
The prerequisites to “Electric field” are “What is a vector” and “Electric charge and Coulomb’s law.” The prerequisites to “Gauss’s law” are “Electric field” and “The inverse square law and the 3-dimensional world.” These articles do not use any complicated math, but we will formulate Gauss’s law mathematically in our later article “Revisiting Gauss’ law and the “derivation” of Coulomb’s law” listed in the section “Maxwell’s equations.”
Basic high school physics without calculus, the second part
Hooke’s law and harmonic oscillator
There are two aims for this section. The first one is completing the course on very basic high school physics that began in an earlier section. The second one is presenting the elastic collision problem in 2-dimension, which plays an important role in our later article “Compton scattering.”
Electricity and Magnetism, the third part
Faraday’s law of induction
Faraday experimentally discovered his law of induction in the 19th century. The first article explains Faraday’s law while the second article explains the origin of Faraday’s law; Faraday’s law nicely fits with the picture presented in that article. These articles serve as prerequisites for “Faraday’s law of induction in Maxwell’s equations,” in which we present the mathematical equations for the material described qualitatively in the above articles.
Modified Newtonian Dynamics
In “History of astronomy from the early 20th century to the early 21st century,” I explained how Modified Newtonian Dynamics tried to explain galaxy rotation curves. Here, I provide the details.
Dimensional Analysis, the first part
Dimensional analysis is widely used in advanced physics. Although it may not be apparent at first glance, dimensional analysis is essential to understanding nature. There is a sequel to these articles in the section “Dimensional Analysis, the second part.’’
These two articles explain what Kobayashi and Maskawa won the Nobel Prizes for. These articles could be a little bit difficult to understand for beginners.
This article explains at a laymen level what is the neutrino oscillation for which Nobel Prize in Physics was awarded in 2015. We will revisit it fully in our later article “Neutrino oscillation, clarified” as neutrino oscillation seems so bizarre and cannot be understood without quantum mechanics.
Young’s interference experiment
Interference from thin films
Familiarity with trigonometric functions is the prerequisite to the above articles. They serve as prerequisites to “Bohr model,” “De Broglie’s matter waves,” and “Schr?dinger equation” listed in “Historical introduction to quantum mechanics.”
A Crash Course in Calculus, the first part
Calculus is an essential tool for physics. It’s a very basic area of mathematics which you cannot avoid with if you truly want to understand physics. Calculus was first discovered by Newton to apply it for physics, even though it is widely believed that Leibniz discovered it independently, despite the fact that Newton accused Leibniz of plagiarism. Calculus is composed of two parts: differentiation and integration. Interestingly, integration is anti-differentiation. If I make an analogy, integration is to differentiation as subtraction is to addition and division to multiplication. In the first part of the crash course, I explain differentiation, which is also called “taking the derivative.”
A Crash Course in Calculus, the second part
In the second part, I explain integration and other topics which are usually covered in the second semester of calculus.
Applications of Calculus, the first part
We already covered Snell’s law in the section “Wave,” but we provide an alternative derivation using calculus in the second article. To appreciate Fourier transformations, you must know linear algebra well, but it is simple enough to understand without it. I included the article on Fourier transformations here because it is not usually covered in freshman mathematics while its construction is interesting. For more motivations behind Fourier transformations, please read “What is Fourier series?” listed in the section “Trigonometric functions.” Actually, the whole book “Who is Fourier?” treats this topic, along with all its prerequisites such as trigonometric functions, complex numbers, calculus, and vector. I highly recommend the book. “Expectation values in quantum field theory (1)” prepares you to understand Feynman diagram. The last two articles explain some statistics.
The products of vectors
There are two different ways to multiply two three-dimensional vectors: “Dot product” and “cross product.” These articles assume the familiarity with vectors as treated in “What is a vector?” Unlike cross product, dot product can be defined in any dimensions, and plays a very important role in physics, math and engineering.
A Crash Course in Calculus, the third part
In the third and fourth parts, I cover the content of the third semester of calculus, which most science and engineering college students who have taken a year of calculus in high school take in the first semester of their freshman year. The first four articles are important in understanding quantum mechanics.
Some basic ideas in General Relativity
These articles survey two ideas Einstein discovered in 1907 and found very useful for his final discovery of the theory of general relativity in 1915. Both articles assume very basic high school physics knowledge as covered in “The free fall and Newton’s second law.” The second article assumes the knowledge of the first article, familiarity with the constancy of the speed of light, in addition to very basic high school physics knowledge. For an alternative approach to the second article, please read “Why time goes more slowly at a lower place and what a black hole is” listed in the section “Black hole.”
Mathematical Introduction to Physics
I wrote the first four articles for those who haven't taken high school physics to read before going on to my articles on quantum mechanics. The articles in listed in this section assume familiarity with calculus at the level of the section “A Crash Course in Calculus, the third part.” For those who have read the articles in our earlier sections “Some very basic high school physics” the first three articles, “Earth’s Gravity near its surface” and “Centripetal force revisited with calculus” will be a review. Nevertheless, it is good to study the same material for the second time, from a slightly different approach using calculus. In addition to some physics, “Kinetic energy and Potential energy in three dimensions, Line Integrals and Gradient” explains some basic concepts in multivariable calculus using physics as an example. The last two articles are about waves and require the section “Wave” as prerequisites.
The main purpose of these articles is to introduce the concept of the magnetic dipole which will turn out to be crucial in our later article “Stern-Gerlach experiment.” Nevertheless, if you are impatient and want to jump into “Stern-Gerlach experiment” as soon as possible, you can go ahead and read it without reading these four articles as I explain there the concept of magnetic dipole needed for “Stern-Gerlach experiment” in a light manner. Nevertheless, if you want a firm basis, you would need to read “Magnetic dipole.”
Historical Introduction to Quantum Mechanics, the second part
I wrote these articles to complement my articles on quantum mechanics listed in the section “Linear Algebra and Quantum Mechanics,” because I worry that those articles may be too mathematical and allow readers to lose the physical picture. However, those articles can be read more or less independently as well without knowing these articles much. The first two articles do not assume any knowledge of physics. “Bohr model” assumes knowledge of high school physics, such as centripetal force and Coulomb force. “De Broglie’s matter waves” should help to understand my article, “A short introduction to quantum mechanics X: comparison with de Broglie’s matter waves and time-dependent Schr?dinger equation.” “Schr?dinger equation,” “Tunneling,” and “Infinite potential well” assume knowledge of differential equations.
College students or even math majors don’t usually learn enumerative geometry, but I included the second article to give readers a sense of what kind of mathematics is used in string theory. The first article helps understanding the second article. The third article is a prerequisite for “Kepler’s laws, revisited.’’
Dimensional Analysis, the second part
Building on the ideas in our earlier article “Natural units” we explain the important concepts.