**Approximation of the naïve black hole degeneracy**

Approximation of the naïve black hole
degeneracy

__Comments:__

¡°Approximation
of the naïve black hole degeneracy¡± was my first research article published in
a journal. In it, I calculated the corrections to the Bekenstein-Hawking
entropy. Here, I review it in a pedagogical manner. This research of mine can
be important if mini black hole is created at LHC in Geneva, Switzerland.
Calculus, the concept of ¡°composition,¡± the concept of
¡°the Kronecker delta symbol,¡± and ¡°Discrete area spectrum and the black hole
entropy I & II¡± serve as prerequisites to understand the last article. You
may want to skip the article, if you feel it is too complicated, as it doesn¡¯t
serve as prerequisites for other articles.

**Planet¡¯s motion around the Sun **

Re-visiting angular
momentum conservation in central force and Kepler¡¯s second law

Planet¡¯s motion around the Sun

__Comments__:

These articles
explain how planets move around the Sun under its gravitational force. They
assume the familiarity with cross product as covered in the section ¡°The
products of vectors.¡± The first two are 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 **

The
Lagrangian formulation of classical mechanics

Central
force problem solution in terms of Lagrangian mechanics

__Comments__:

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 **

The Hamiltonian
formulation of classical mechanics

__Comments__:

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 19^{th} 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 **

Dirac¡¯s bra-ket notation: an exposition for science and engineering
students

A short introduction to quantum mechanics I: observables and eigenvalues

A short
introduction to quantum mechanics I addendum: revisiting double slit experiment

A short introduction to quantum mechanics
II: why is a wave function a vector?

The transpose and Hermitian conjugate

Information Conservation and Unitarity in
Quantum Mechanics

Eigenvalues and eigenvectors
of symmetric matrices and
Hermitian matrices

A short
introduction to quantum mechanics V: the expectation value of given observables

A short
introduction to quantum mechanics VI: position basis and Dirac delta
function

A
short introduction to quantum mechanics VIII: global gauge transformation

A short introduction to
quantum mechanics IX: the Unitarity of the time evolution operator

A short introduction to quantum mechanics X: position and
momentum basis and Fourier transformation

A short introduction to
quantum mechanics XII: Heisenberg¡¯s uncertainty principle

A short introduction to
quantum mechanics XIII: Harmonic oscillators

Group velocity and phase
velocity

__Comments__:

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, as well as my other articles in section ¡°Tasting
linear algebra,¡± these articles explain quantum mechanics along with all the
basics of linear algebra you need.

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 **

Transition from classical
mechanics to quantum mechanics

__Comments__:

The first article explains Dirac¡¯s ¡°derivation¡± of quantum mechanics from
classical mechanics, which he developed in the early 20^{th} century.
The Hamiltonian formulation of classical mechanics and the Poisson bracket are
19^{th} 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.

**Thermodynamics **

Kinetic theory of gases

__Comments__:

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**

The determinant of 2x2
matrices

The determinant and its
geometric interpretation

Finding eigenvalues and
eigenvectors

The determinant of
product of two matrices

Rotation and the
Lorentz transformation, orthogonal and unitary matrices

What is a
group? What is representation?

__Comments__:

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 Doppler effect, and the twin
paradox revisitied

4-vector, Lorentz transformation
and de Broglie¡¯s derivation of matter waves

Mass-energy equivalence

Compton scattering

__Comments__:

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.¡±

**Subtleties in Quantum
Mechanics**

Neutrino oscillation,
clarified

__Comments__:

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. The second article 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 ****C****rash ****C****ourse in ****C****alculus, the ****fourth**** ****p****art**** **

The Jacobian and change of
variables

Flux

Divergence and Stoke¡¯s
theorem

Why is the curl of gradient
always zero?

Polar coordinate, the area of
a circle and Gaussian integral

Spherical coordinate
system, the surface area of sphere, and the volume of ball

__Comments__

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 ****C****alculus,
the second part**** **

__Comments__

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 **

__Comments:__

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.

**Application of the
Lagrangian Formulation of Classical Mechanics **

Geodesics in the presence of
constant gravitational field

__Comments__:

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.

**Angular momentum in quantum
mechanics **

Angular momentum in
quantum mechanics

__Comments__:

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.

**Feynman
diagram **

Expectation values in quantum field theory (2)

__Comments__:

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 **

Another example of
differential equations

Coupled harmonic oscillator system

__Comments__:

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

Why was very early universe
dominated with light?

__Comments__:

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.

**Maxwell¡¯s Equations **

Revisiting Gauss¡¯ law and the
¡°derivation¡± of Coulomb¡¯s law

Gauss¡¯ law expressed
using divergence

The
Biot-Savart law

Faraday¡¯s law of induction in
Maxwell¡¯s equations

Ampere¡¯s law as corrected in
Maxwell¡¯s equations

Light as electromagnetic waves

Electric
potential and vector potential

Maxwell¡¯s equations in matter

__Comments__:

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 **

Differential forms, vector calculus, and
generalized Stokes¡¯ theorem

Maxwell¡¯s equations in differential forms

De Rham cohomology,
homology and Künneth formula

__Comments__:

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.

**Statistical Mechanics **

Planck¡¯s law of
black-body radiation

The Bose-Einstein distribution, the
Fermi-Dirac distribution and the Maxwell-Boltzmann distribution

__Comments__:

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 these 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.¡±

**Loop quantum gravity approach
to black hole**

Quantum corrections to
Hawking radiation spectrum

Maxwell-Boltzmann type
Hawking radiation

Hawking
radiation of single-partition black hole

__Comments__:

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.¡±

**Further electrodynamics **

Electrodynamics
in the Lagrangian and the Hamiltonian formulations

Energy density of electromagnetic field

Faraday's law of
induction and the Lorentz force

__Comments__:

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 **

Time-independent
perturbation theory

The ground state energy for helium
atom

__Comments__:

¡°Hydrogen
atom¡± deals with the solution of Schrödinger equation of hydrogen atom which
was first done by Schrödinger himself.