Black hole entropy in loop quantum gravity picture
The content of the second article was a part of my first research article published in a journal. ¡°Composition¡± which was also included in this research article is included here as a prerequisite for the second article. ¡°Discrete area spectrum and the black hole entropy I & II¡± and ¡°Exponential function versus polynomial function¡± are other prerequisites.
Planet¡¯s motion around the Sun
These articles explain how planets move around the Sun under its gravitational force. The first article is prerequisites for ¡°Hydrogen atom,¡± while the last one is prerequisites for relativistic predictions for the orbit of mercury, as explained in ¡°A relatively short introduction to general relativity.¡±
Lagrangian Formulation of Classical Mechanics
Most physics students learn Lagrangian mechanics in their sophomore year. You can learn the fundamental ideas of Lagrangian mechanics only knowing a little Newtonian mechanics and multivariable calculus as taught in freshman year. I have included the above articles here so that you can understand ¡°A short introduction to general relativity,¡± listed in the section ¡°For physics majors,¡± and ¡°the Hamiltonian formulation of classical mechanics¡¯¡¯ without obstacles. Also, it would be helpful to read our earlier article ¡°Fermat¡¯s principle and the consistency of physics,¡± before reading this article. For an alternative approach please see Chapter 2 of ¡°The Feynman lectures on physics, volume 2¡± available at http://www.feynmanlectures.caltech.edu/II_19.html. The second article shows that conservation of angular momentum and centripetal force can be easily derived using Lagrangian mechanics.
Hamiltonian Formulation of Classical Mechanics
Most physics students learn Hamiltonian mechanics in their sophomore year right after learning Lagrangian mechanics. The Hamiltonian formulation of classical mechanics and the Poisson bracket are 19th century inventions, but they play an important role in ¡°deriving¡± quantum mechanics from classical mechanics as you will see in the section ¡°Transition from Classical Mechanics to Quantum Mechanics.¡±
Linear Algebra and Quantum Mechanics
The basic concepts of complex numbers, calculus and linear algebra are prerequisites to quantum mechanics. Assuming the knowledge on dot product, complex numbers and calculus as covered in my previous articles, these articles explain quantum mechanics along with all the basics of linear algebra you need.
Some high school students study a little bit of matrices, but I don¡¯t think that they can appreciate the real value of matrices until they learn linear algebra, usually in the second semester of their college freshman year. ¡°Matrices and Linear Algebra¡± is written so that readers can appreciate the real value of matrices.
If you are already familiar with some key concepts in linear algebra, such as eigenvectors and eigenvalues, I advise you to read immediately ¡°A short introduction to quantum mechanics I.¡± It will show you what quantum mechanics is, without complicated details. You need not read the other quantum mechanics articles if you feel they are too much, but if you want to know a little more than is explained in the first article, you may want to read ¡°quantum mechanics II¡± and ¡°quantum mechanics III,¡± though you can skip the former and still understand the latter. If you find "quantum mechanics II" confusing, don't worry and skip to the next articles. It should make more sense once you come back to it after reading "quantum mechanics VI."
Transition from Classical Mechanics to Quantum Mechanics
The first article explains Dirac¡¯s ¡°derivation¡± of quantum mechanics from classical mechanics, which he developed in the early 20th century. The Hamiltonian formulation of classical mechanics and the Poisson bracket are 19th century inventions. The last article takes a step from Dirac¡¯s derivation and nails down the fact that quantum mechanics reduces to classical mechanics in the macroscopic limit.
Kinetic theory of gases
The first three articles should be familiar to anyone who learned physics in high school. Besides the knowledge on the first three articles and ¡°What is entropy? From a macroscopic point of view¡± listed in the section ¡°Entropy,¡± the third article assumes the knowledge of basic calculus. Besides the first two articles, the fourth article assumes the knowledge of ¡°Differential equations¡± as listed in ¡°A crash course in calculus, the third part.¡±
Further Linear Algebra
These articles proceed beyond the minimum knowledge of linear algebra needed in quantum mechanics. The Kronecker delta symbol and the Levi-Civita symbol are frequently used in general relativity. The concept of the determinant is important in understanding Jacobian, which we will cover in the section ¡°A Crash Course in Calculus, the fourth part.¡± The article on the rotation and the Lorentz transformation is important in understanding 4-vector, which we will cover in the section ¡°Theory of Special Relativity and 4-vector.¡± The last article deals with the concept of the group which plays a very important role in math and physics. Math students learn about group in details in courses titled ¡°abstract algebra¡± for two semesters in their sophomore year.
Theory of Special Relativity and 4-vector
The first article is essential in understanding our article ¡°Expanding universe¡± in section ¡°The very basic cosmology.¡± The second and the third articles assume knowledge of calculus and special relativity as covered in section ¡°Special Relativity" The fourth article assumes knowledge of Lorentz transformations, as covered in ¡°Rotation and the Lorentz transformation, orthogonal and unitary matrices.¡± I strongly recommend you to read this article, since it shows the mathematical beauty of special relativity. ¡°Mass-energy equivalence¡± shows the derivation of Einstein¡¯s famous equation E is equal to m c squared. ¡°Compton scattering¡± assumes familiarity with momentum conservation and energy conservation during collision process as covered in ¡°Elastic collision in 2-dimensions.¡±
Application of the Lagrangian Formulation of Classical Mechanics
Most physics students first learn Noether¡¯s theorem in a quantum field theory class, but there is an easier analogue in classical mechanics. The first article deals with this. The second article, which assumes the knowledge of Lagrangian mechanics, time dilation, and the article ¡±By how much does time go more slowly at a lower place?¡± listed in ¡°Some basic ideas in General Relativity,¡± shows how general relativity can be reduced to Newtonian gravity in a certain limit.
Subtleties in Quantum Mechanics
Einstein was well-known for his criticism against Copenhagen interpretation of quantum mechanics, and came up with a paradox with his colleagues Podolsky and Rosen to refute it. The first article deals with the paradox. It only assumes ¡°A short introduction to quantum mechanics I: observables and eigenvalues¡± as prerequisites. ¡°Neutrino oscillation, clarified¡¯¡¯ deals with theoretical background for neutrino oscillation which was already explained at a laymen level in our earlier article ¡°Neutrino oscillation.¡± Along with that article, basic quantum mechanics and ¡°Relativistic energy¡± in the section ¡°Theory of Special Relativity and 4-vector¡± are the prerequisites.
A Crash Course in Calculus, the fourth part
Gradient, curl, divergence in curvlinear coordinates
As in the third part, in the fourth part, I cover the content of the third semester of calculus. To understand the article on the Jacobian, you need to understand what a determinant is, as listed in the section ¡°Further Linear Algebra.¡± For our purposes, the Jacobian is important in understanding general relativity. Other articles are necessary to understand differential forms and Maxwell¡¯s equations. Curl and Green¡¯s theorem assumes familiarity with line integrals as explained in ¡°Kinetic energy and Potential energy in three dimensions, Line integrals and Gradient¡± listed in the section ¡°Mathematical Introduction to Physics.¡± Even though line integrals and gradient can be defined without the aid of physics, I decided to introduce them using physics examples. This is the reason why the article introducing them is listed in the physics section. ¡°Spherical coordinate system¡± is essential in understanding our later article ¡°Hydrogen atom.¡±
Applications of Calculus, the second part
The first two articles are written to serve as prerequisites for ¡°The Bose-Einstein distribution, the Fermi-Dirac distribution and the Maxwell distribution.¡±
Evidence of the rotation of the Earth
Coriolis force is present when an object is moving relative to a system which is rotating around certain axis. A good example of such a system is the Earth. The first article explains Coriolis force qualitatively and a famous experiment that showed that the Earth is rotating. In the second article we revisit Coriolis force mathematically.
Angular momentum in quantum mechanics
Apart from its own importance in quantum mechanics, I wrote the first three articles because it is essential to understand how the area spectrum is derived in loop quantum gravity, even though strictly speaking, it is not angular momentum per se but merely the mathematical structure that is needed.
To understand the first article, you have to read ¡°Expectation values in quantum field theory (1),¡± listed in the section ¡°Applications of calculus.¡± To understand the second article, you have to read the first article. To really understand quantum field theory and Feynman diagrams, you must know quantum mechanics and special relativity, but you can learn their basic methods without knowledge of physics.
More differential equations
Forced harmonic oscillator is usually treated in the freshman physics course albeit less mathematically than treated in our article. In any case, the concept of ¡°resonance¡± explained there is very important and has wide applications in engineering. The second article treats the differential equation problem solved by former Korean child prodigy Ung-yong Kim in 1967 when he was four years old appearing on a Japanese TV show. I would not have been able to solve this problem myself if I hadn¡¯t seen his solution on a blackboard.
The Very Basic Cosmology
The expanding universe
To understand cosmology, you need to know general relativity, but the very basics can be understood without it. ¡°The expanding universe¡± explains why astronomers believe that our universe is expanding. ¡°Photon gas pressure¡± deals with the photon version of Boyle¡¯s law. ¡°Why was early universe filled with light¡± builds on from ¡°Photon gas pressure¡± and shows that very early universe was dominated with light rather than matter even though our current universe is dominated with matter.
The Biot-Savart law
The prerequisites for these articles are ¡°Electricity and Magnetism, the first, second and third parts,¡± ¡°A Crash Course in Calculus, the fourth part,¡± and ¡°Kinetic energy and Potential energy in three dimensions, Line Integrals and Gradient.¡± Maxwell¡¯s equations are very beautiful, and many agree that Maxwell was as great physicist as Newton or Einstein. Moreover, you have to know Maxwell¡¯s equations to understand Yang-Mills theory, a prerequisite for loop quantum gravity. The last two articles are not necessary for this purpose.
Differential forms and their applications
The articles about differential forms are based on what I learned in honors multivariable calculus and linear algebra class. I became convinced that there is a simple theory of everything when I learned how to write Maxwell¡¯s equations using differential forms, because the equations look very simple when expressed in this way. The article on de Rham cohomology relates topology with calculus.
Black-body radiation plays an important role in Hawking¡¯s black hole thermodynamics, which in turn plays an important role in loop quantum gravity. This is the reason why I have written-up the first five articles. They assume the prior knowledge on entropy as explained in ¡°Entropy¡± and basic calculus. Boltzmann factor as listed in ¡°Thermodynamics¡± would be also helpful but not essential. ¡°Density of states,¡± which is necessary to understand black body radiation, assumes the knowledge of ¡°Infinite potential well¡± listed in ¡°Historical introduction to quantum mechanics.¡± The result of the last article whose prerequisite is the section ¡°Applications of Calculus, the second part¡± was crucial to my research on Hawking radiation which we will review in our later article ¡°Quantum corrections to Hawking radiation spectrum.¡± Currently ¡°Bose-Einstein condensate¡± finds no direct application in particle physics or quantum gravity, but as it is important its own, I included it here.
Loop quantum gravity approach to black hole
Hawking radiation of single-partition black hole
These three articles explain my own work on loop quantum gravity approach to black holes. The prerequisites are ¡°Discrete area spectrum and the Hawking radiation spectrum II¡± and ¡°The Bose-Einstein distribution, the Fermi-Dirac distribution and the Maxwell-distribution.¡±
In the presence of electromagnetic field, an electrically charged object receives Lorentz force. This is a Newtonian picture; the Lorentz force gives acceleration for the object. However, as the Lagrangian and the Hamiltonian formulation are equivalent to the Newtonian one, we should have the Lagrangian and the Hamiltonian for the object from which we can derive the Lorentz force. This is the object of the second article. Also, it serves as the prerequisite to ¡°What is a gauge theory?¡± even though it is not absolutely essential if one takes granted the key equations in the above article. On the other hand, the main purpose of the Poisson¡¯s equation article is to serve prerequisites to understand how Einstein¡¯s theory of general relativity can be reduced to Newtonian gravity as explained in ¡°A relatively short introduction to general relativity.¡± ¡°Gauss¡¯s law expressed using divergence¡± is a prerequisite to the Poisson¡¯s equation article. ¡°Energy density of electromagnetic field¡± shows that electromagnetic fields carries energy and calculates their value.
Further quantum mechanics
¡°Hydrogen atom¡± deals with the solution of Schrödinger equation of hydrogen atom which was first done by Schrödinger himself.