Approximation of the naïve black hole degeneracy

Composition

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 doesnt serve as prerequisites for other articles.

 

Planets motion around the Sun

Planets motion around the Sun

Keplers first and third laws revisited

 

Comments:

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

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 Fermats 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

The Poisson bracket

 

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 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

Matrix inverses

Eigenvalues and eigenvectors

Diracs 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?

A short introduction to quantum mechanics III: the equivalence between Heisenbergs matrix method and Schrödingers differential equation

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 IV: inner product in Hilbert space and the orthogonality of eigenvectors of 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 VII: the Hermiticity of the position operator and the momentum operator

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 XI: comparison with de Broglies matter waves and the time-dependent Schrödinger equation

A short introduction to quantum mechanics XII: Heisenbergs 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

Ehrenfest theorem

 

Comments:

The first article explains Diracs 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 Diracs derivation and nails down the fact that quantum mechanics reduces to classical mechanics in the macroscopic limit.

 

Thermodynamics

Boyle-Charles law

Kinetic theory of gases

Specific heats of gases

Entropy as a state function

Boltzmann factor

Adiabatic process

 

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 Levi-Civita symbol

The determinant of 2x2 matrices

The determinant

The determinant and its geometric interpretation

The trace

Finding eigenvalues and eigenvectors

The cross product revisited

The determinant of product of two matrices

The similarity transformation

The diagonalization

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

Relativistic momentum

Relativistic energy

4-vector, Lorentz transformation and de Broglies 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 Einsteins 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

EPR paradox

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 Crash Course in Calculus, the fourth part

The Jacobian and change of variables

Flux

Continuity equation

Divergence and Stokes theorem

Curl and Greens 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 Maxwells equations. Curl and Greens 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

Stirlings formula

Lagrange multipliers

Non-Euclidean geometry

 

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

Coriolis force

Coriolis force, revisiting

 

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

Noethers theorem

Geodesics in the presence of constant gravitational field

 

Comments:

Most physics students first learn Noethers 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

Pauli matrices and spinor

Angular momentum addition

 

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)

What is a Feynman diagram?

 

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

Forced harmonic oscillator

Another example of differential equations

Coupled harmonic oscillator system

Travelling spring waves

The speed of sound

 

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 hadnt seen his solution on a blackboard.

 

The Very Basic Cosmology

The expanding universe

Pressure of photon gas

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 Boyles 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.

 

Maxwells Equations

Revisiting Gauss law and the derivation of Coulombs law

Application of Gauss law

Gauss law expressed using divergence

Electric potential revisited

Amperes law

The Biot-Savart law

Faradays law of induction in Maxwells equations

Amperes law as corrected in Maxwells equations

Light as electromagnetic waves

Electric potential and vector potential

Maxwells equations in matter

Capacitor

 

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. Maxwells equations are very beautiful, and many agree that Maxwell was as great physicist as Newton or Einstein. Moreover, you have to know Maxwells 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

Maxwells 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 Maxwells 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

The definition of temperature

Boltzmann factor

Density of states

Plancks 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 Hawkings 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

Poissons equation

Energy density of electromagnetic field

Electric dipole

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 Poissons equation article is to serve prerequisites to understand how Einsteins theory of general relativity can be reduced to Newtonian gravity as explained in A relatively short introduction to general relativity. Gausss law expressed using divergence is a prerequisite to the Poissons equation article. Energy density of electromagnetic field shows that electromagnetic fields carries energy and calculates their value.

 

Further quantum mechanics

The Aharonov-Bohm effect

Hydrogen atom

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.