Black hole entropy in loop quantum gravity picture
Composition
Domagala-Lewandowski-Meissner formula reprodcues the Bekenstein-Hawking entropy
Comments
Planet’s motion around the Sun
Planet’s motion around the Sun
Kepler’s first and third laws revisited
Comments
Lagrangian Formulation of Classical Mechanics
The Lagrangian formulation of classical mechanics
Central force problem solution in terms of Lagrangian mechanics
Comments
Linear Algebra and Quantum Mechanics
The mathematical definition of vector
Matrices and Linear Algebra (an exposition for secondary school students and laymen)
Einstein summation convention
The identity matrix
The Kronecker delta symbol
Row reduction and echelon form
Linear independence, linear dependence and basis
Change of basis
Matrix inverses
Eigenvalues and eigenvectors
Dirac’s bra-ket notation: an exposition for science and engineering students
Revisiting Fourier transformations
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 Heisenberg’s matrix method and Schrödinger’s 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 Broglie’s matter waves and the time-dependent Schrödinger equation
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
Thermodynamics
Boyle-Charles law
Kinetic theory of gases
Specific heats of gases
Entropy changes, an example
Entropy as a state function
Boltzmann factor
Adiabatic process
Mean free path
Comments
Hamiltonian Formulation of Classical Mechanics
The Hamiltonian formulation of classical mechanics
The Poisson bracket
Comments
Transition from Classical Mechanics to Quantum Mechanics
Transition from classical mechanics to quantum mechanics
Ehrenfest theorem
Comments
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
Dual space
The tensor
The cross product revisited
The determinant of product of two matrices
The similarity transformation
The diagonalization
Simultaneous diagonalization of two commuting matrices
Rotation and the Lorentz transformation, orthogonal and unitary matrice
What is representation?
Even permutation, odd permutation, and det A^T=det A
Dimension of orthogonal group
Comments
Theory of Special Relativity and 4-vector
The relativistic Doppler effect, and the twin paradox revisited
Relativistic momentum
Relativistic energy
4-vector, Lorentz transformation and de Broglie’s derivation of matter waves
Mass-energy equivalence
Compton scattering
Totally inelastic relativistic collision
Ultra-relativistic particle, an International Physics Olympiad problem
Comments
Application of the Lagrangian Formulation of Classical Mechanics
Noether’s theorem
Geodesics in the presence of constant gravitational field
Comments
Subtleties in Quantum Mechanics
EPR paradox
Symmetry and conservation law in quantum mechanics
Parity in quantum mechanics
Charge conjugation
Strangeness
The neutral kaon system
Neutrino oscillation, clarified
Comments
A Crash Course in Calculus, the fourth part
The Jacobian and change of variables
Flux
Continuity equation
Divergence and Stoke’s theorem
Curl and Green’s theorem
Why is the curl of gradient always zero?
Polar coordinate, the area of a circle and Gaussian integral
Cylindrical coordinate system
Spherical coordinate system, the surface area of sphere, and the volume of ball
Gradient, divergence, curl in orthogonal curvilinear coordinates
Comments
Applications of Calculus, the second part
Stirling’s formula
Lagrange multipliers
Non-Euclidean geometry
Solid angle
Surface area of n-sphere and volume of n-ball
Comments
Evidence of the rotation of the Earth
Coriolis force
Coriolis force, revisiting
Comments
Angular momentum in quantum mechanics
Angular momentum in quantum mechanics
Pauli matrices and spinor
Angular momentum addition
Spherical harmonics
Comments
Feynman diagram
Expectation values in quantum field theory (2)
What is a Feynman diagram?
Comments
More differential equations
Forced harmonic oscillator
Another example of differential equations
Coupled harmonic oscillator system
Travelling spring waves
The speed of sound
Transverse waves and longitudinal waves
Comments
Maxwell’s Equations
Revisiting Gauss’ law and the “derivation” of Coulomb’s law
Application of Gauss’ law
Gauss’ law expressed using divergence
Ampere’s law
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
Capacitor
Comments
Differential forms and their applications
Differential forms, vector calculus, and generalized Stokes’ theorem
Maxwell’s equations in differential forms
Electromagnetic duality
De Rham cohomology
Homology
The duality between de Rham cohomology and homology
Comments
Statistical Mechanics
The definition of temperature
Boltzmann factor
Density of states
Planck’s law of black-body radiation
The Bose-Einstein distribution, the Fermi-Dirac distribution and the Maxwell-Boltzmann distribution
Bose Einstein condensate
Fermi energy
Pressure of photon gas
Comments
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
Complex analysis
Cauchy-Riemann equations
Cauchy’s integral formula and Cauchy’s residue theorem
Application of residue theorem
Proof of fundamental theorem of algebra
Comments
Further electrodynamics
Electrodynamics in the Lagrangian and the Hamiltonian formulations
Poisson’s equation
Energy density of electromagnetic field
Poynting vector
Faraday's law of induction and the Lorentz force
Comments
Further quantum mechanics
The Aharonov-Bohm effect
Hydrogen atom
Why are string theories only consistent in certain dimensions?
Time-independent perturbation theory
The ground state energy for helium atom
Comments
Further statistical mechanics
The Helmholtz free energy, the enthalpy, and the Gibbs free energy
The partition function and the calculation of various thermodynamic quantities
The Gibbs paradox
The Gibbs factor
The Gibbs free energy revisited
The chemical equilibrium
Comments
Review Articles
Introduction
High School Mathematics
For Laymen
For Mathematically Oriented
High School Students
For Physics Students
The Most Recommended Articles
한국어
Korean
For Korean Laymen
For Korean Students
About Me
Facebook
Quora
Popular Essays
Math and Physics
Moving Beyond the Korean System:
On Learning To Love Mathematics and Physics
Surprises in math, the reason why I study physics,
and a recommendation for "The Number Devil"
Is math and science homework mechanical drudgery?
The mathematical beauty of physics:
simplicity, consistency, and unity
English as a foreign language versus English as a
mother tongue: an analogy for the language of
mathematics and the language of physics
Physics solutions for mathematical problems
Are memorizing formulas well and good arithmetic
skills essential to become a good theoretical
physicist or a good mathematician?
Math
The imagination in mathematics:
"Desargues's theorem"
Why Math Education Matters
Mathematical maturity in
solving mathematics problems
The square root 2 as
irrational number
The prisoner's dilemma
Math History
The superiority of Hindu-Arabic numeral system
Fermat's Last Theorem
The power of negative number
Nightingale, the statistician
The calculation of pi, the first part
The calculation of pi, the second part
Why are there as many even
natural numbers as natural numbers?
How to turn a complicated multiplication into a simple addition
Pythagoras and the musical scale
Physics
Studying physics as resolving paradoxes
The similarity between painting and theoretical physics
Fermat’s principle and the consistency of physics
Did Einstein really prove that Newton was wrong?
The cosmological principle, and my view on philosophy
The case for string theory
Why are theories of crackpots, the self-claimed physicists
who lack proper physics education, wrong and useless?
About This Page
Found out the reasons for my physics website
Other Resources
Recommended Physics Books
Other Physics Resources
My Harvard Application Essay
My MIT Application Essay
Ein Monat wie im Traum (in German)
Feedback
Review Articles
Introduction
High School Mathematics
For Laymen
For Mathematically Oriented
High School Students
For Physics Students
The Most Recommended Articles
한국어
Korean
For Korean Laymen
For Korean Students