Blog

LIGO Laser modulation
Electronics
We present a comprehensive analysis of laser modulation techniques used in the Laser Interferometer Gravitational-Wave Observatory (LIGO) for cavity length sensing and…
Jul 17, 2025
tags LIGO,laser,gravitational-waves

Grover’s Algorithm
Quantum Computation
Grover’s search algorithm represents one of the most significant achievements in quantum computing, providing a quadratic speedup for searching unsorted databases. While…
Jul 7, 2025
tags Grover,algorithms,quantum circuit,quantum state,quantum measurement,Qubit,quantum search

Shor’s Algorithm
Quantum Computation
Shor’s algorithm represents one of the most groundbreaking achievements in quantum computing, demonstrating that quantum computers could theoretically break widely-used…
Jul 6, 2025
tags Shor,algorithms,quantum circuit,quantum state,quantum measurement,Qubit,quantum search

\(\text{Integral of the month: } \int \frac{x^\alpha dx}{x^2- 2 \beta x + 1}\)
Math
This article presents a fun exercise in complex analysis by evaluating the integral \(\int_0^\infty \frac{x^\alpha dx}{x^2 - 2\beta x + 1}\) using residue calculus. We explore…
Jul 4, 2025
tags Integral,Residue Calculus,Integral of the month,keyhole contour

Superconducting Bits
Quantum Computation
This article presents the theoretical foundations of superconducting qubits based on Josephson junctions. We derive the dynamics of Cooper pair tunneling through thin…
Jul 1, 2025
tags resonance,qubit,quantum computer

Getting Started with Qiskit
Quantum Computation
This tutorial provides a comprehensive introduction to getting started with quantum computing using Qiskit, IBM’s open-source quantum computing framework. We demonstrate the…
Jun 26, 2025
tags Quantum Computation,Qiskit,quantum circuit,quantum state,quantum measurement,Qubit

Parametric Amplifier
Math
This work presents a theoretical analysis of non-degenerate parametric amplifiers, which achieve signal amplification through time-varying reactive elements without…
Jun 21, 2025
tags parametric amplifier,nonlinear capacitance,parametric oscillator

\(\text{Integral of the month: } \int ds\,CRAZY(s)\)
Math
This article presents a detailed solution to a crazy ass integral that gained internet fame on Math Stack Exchange. The integral was later solved in full detail by Ron…
Jun 4, 2025
tags Integral,residue calculus,Integral of the month,keyhole contour

\(\text{Integral of the month: } \int \frac{ dx}{x^n+ 1}\)
Math
This article presents three distinct methods for evaluating the integral \(\int_0^\infty \frac{dx}{x^n + 1}\) using complex analysis and residue calculus. We explore a clever…
May 4, 2025
tags Integral of the month,Integral,Residue Calculus,keyhole contour

Johnson Noise
Math
This paper explores Johnson-Nyquist noise, a fundamental electronic phenomenon arising from the thermal agitation of charge carriers in conductors. We examine the…
Apr 17, 2025
tags Johnson noise,thermal noise,Nyquist noise,Johnson-Nyquist theorem

Adding two Weibull random numbers
Statistics
We analyze the probability distribution of the sum of two Weibull random variables by examining two redundant system configurations: sequential and interleaved operation. We…
Apr 12, 2025
tags Weibull,gamma distribution,reliability,redundancy,statistics

Scattering fermions and scalars
Quantum Field Theory
We present a detailed calculation of scalar-fermion scattering via Yukawa interactions. Starting from the Lagrangian with a \(\phi\bar{\psi}\psi\) coupling, we derive the…
Apr 10, 2025
tags fermion,scalar,cross-section,scattering,QFT,Feynman,Yukawa

Second Quantization
Quantum
We take a close look at the tedious steps of second quantization, which is a fundamental concept in quantum mechanics and solid state physics. The goal is the put together…
Apr 7, 2025
tags Quantum,Second Quantization,quantum field theory,quantum mechanics,quantum field,quantum state

Adding images to Plotly
Coding
This article provides a step-by-step guide to adding images to Plotly plots. We will walk through the process of adding images to Plotly plots, including dependencies and…
Apr 5, 2025
tags Plotly,JavaScript

Installing and Using GetDP on macOS
Numerical Methods
This article provides a step-by-step guide to installing and using GetDP, a finite element software package, on macOS. We will walk through the installation process…
Apr 4, 2025
tags GetDP,finite element method,FEM,Gmsh,Electromagnetism,numerical simulation

\(2-\)D Quantum Oscillator
Quantum
This article explores the two-dimensional quantum harmonic oscillator through interactive visualizations. We consider a perturbation potential \(V(x,y)\propto xy\) and compute…
Apr 2, 2025
tags harmonic oscillator,perturbation theory,degeneracy

A refresher on statistical mechanics
Thermodynamics
This article provides a comprehensive refresher on fundamental concepts in statistical mechanics, drawing inspiration from Leonard Susskind’s lectures. Beginning with…
Mar 21, 2025
tags Statistical Mechanics,Thermodynamics,entropy,information theory,Boltzmann distribution,Lagrange multiplier,calculus of variations

Biot-Savart Law
Electromagnetism
This article presents a rigorous derivation of the Biot-Savart law from Maxwell’s equations. Starting from the fundamental observation that magnetic fields are…
Jan 3, 2025
tags Electromagnetism,Biot-Savart Law,Maxwell’s Equations,Vector Potential,Green’s Function,magnetostatics

Inductance of a Wire Pair with Neumann’s Method
Electromagnetism
This article presents a detailed derivation of the mutual inductance between two parallel wire segments using Neumann’s method. Starting from the fundamental electromagnetic…
Jan 2, 2025
tags Electromagnetism,inductance,magnetic field,magnetic flux,magnetic coupling,Neumann’s formula,self-inductance,mutual inductance,electrical circuits,transmission lines

Inductance of a Wire Pair
Electromagnetism
This article examines the mutual inductance between parallel wire segments, a fundamental configuration in electrical circuits and transmission lines. Building upon our…
Jan 1, 2025
tags Electromagnetism,inductance,magnetic field,magnetic flux,magnetic coupling,Neumann’s formula,self-inductance,mutual inductance,electrical circuits,transmission lines

Self-Inductance of a Wire
Electromagnetism
This article explores the calculation of self-inductance in a wire segment using the Biot-Savart law and energy methods. We derive expressions for both the external and…
Dec 28, 2024
tags Electromagnetism,inductance,magnetic field,magnetic flux,magnetic coupling,Neumann’s formula,self-inductance,mutual inductance,electrical circuits,transmission lines

Magnetic Dipole
Electromagnetism
This blog post provides a comprehensive treatment of magnetic dipoles, starting from first principles. We begin by deriving the vector potential for an arbitrary current…
Dec 27, 2024
tags Electromagnetism,magnetic dipole,vector potential,elliptic integrals

Red balls, Green Balls
Statistics
This work delves into a probability puzzle involving an urn with an unknown number of red and green balls. Initially, the number of red balls is randomly selected. After…
Aug 29, 2024
tags binomial,probability,bayesian,simulation

Fabry-Perot Cavity
Optics
This article explores the physics and applications of Fabry-Perot cavities, with a particular focus on their role in LIGO (Laser Interferometer Gravitational-Wave…
Aug 20, 2024
tags optics,Fabry-Perot,cavity,interferometry,LIGO,bullseye,fringes,finesse,laser,gravitational waves,precision measurement,reflection,transmission

Renewal Processes
Statistics
This article explores renewal processes and their relationship to Poisson processes in probability theory. We examine the fundamental concepts of interarrival times…
Jun 1, 2024
tags exponential distribution,poisson distribution,renewal process,poisson process,probability theory,stochastic processes,interarrival time,counting process

Musings on the Gamma Distribution
Statistics
In this post we demonstrate the derivation of the Gamma distribution through two different approaches. First, we show how the sum of \(n\) independent exponentially…
May 1, 2024
tags gamma distribution,exponential distribution,poisson process,convolution,laplace transform,random variables,stochastic processes,statistics

Musings on the Exponential Distribution
Statistics
This article presents a rigorous mathematical proof demonstrating why exponential distributions naturally emerge in memoryless physical processes. We show that the…
Apr 1, 2024
tags exponential distribution,poisson distribution

Dirac delta with a Function inside
Math
Dirac delta function appears frequently in physics. In certain cases, it takes a function as an argument. Such cases require care, and that is what we will take a quick look…
Mar 1, 2024
tags Dirac delta,functional

Diffraction
Optics
This post explores the physics and mathematics of diffraction, a fundamental wave phenomenon that limits the resolution of optical instruments like telescopes. Starting from…
Feb 1, 2024
tags Fresnel,Fraunhofer

\(\text{Integral of the month: } \int dr \cos r^2\)
Math
Fresnel integrals are a pair of integrals that are used to calculate the diffraction patterns of light waves. This post explores the mathematics of Fresnel integrals, a…
Feb 1, 2024
tags Fresnel,Fraunhofer,Integral of the month,optics,diffraction

Coil calculations
Electromagnetism
This article provides a comprehensive analysis of electromagnetic coil calculations, from basic principles to advanced mathematical treatments. We begin with simplified…
Jan 15, 2024
tags magnets,coil

Separation of variables in spherical coordinates
Math
This article presents a comprehensive derivation of the separation of variables technique applied to partial differential equations in spherical coordinates. We examine the…
Jan 1, 2024
tags Laplace,spherical harmonics,separation of variables,Sturm-Liouville theory

Potentials of a split cylindrical shell
Math
We solve for the electric potential of a cylindrical conductor split into two semicylindrical shells held at different voltages. The solution employs a sequence of two…
Dec 20, 2023
tags Conformal maps,Symmetry,Laplace,Mobius

Electric potential of angled plates
Math
This post demonstrates how conformal mapping techniques can be used to solve the Laplace equation, a fundamental tool in electrostatics. We introduce the conceptual…
Dec 12, 2023
tags Conformal maps,Symmetry,Laplace

An introduction to conformal maps
Math
This blog introduces conformal maps, which are complex functions that locally preserve angles between curves. We begin with the fundamentals of complex derivatives and the…
Dec 6, 2023
tags Conformal maps,Symmetry,Laplace,Complex Calculus,Cauchy-Riemann

Radial Green’s Function in Cylindrical Coordinates
Math
This document discusses the Radial Green’s Function in cylindrical coordinates, a fundamental concept in mathematical physics and engineering. The Green’s function serves as…
Dec 4, 2023
tags Green’s function,Laplace,cylindrical coordinates,electrostatics

Not All Heroes Wear Capes!
DIY
This is how the expiration lock on a medical device used for home blood coagulation testing can be bypassed. By resetting the device’s date, expired but still functional…
Nov 19, 2023
tags DIY,Medical,Hack

Abrikosov-Nielsen-Olesen flux tubes
Superconductivity
This article explores the physics of Abrikosov-Nielsen-Olesen (ANO) flux tubes, which are topological defects arising from spontaneously broken local symmetries in…
Nov 18, 2023
tags Gauge Theory,Superconductivity,Symmetry

Difference of two linearly distributed random numbers
Statistics
We explore the statistics of the absolute difference between two random variables, each of which is linearly distributed within a specified range. We derive the probability…
Nov 16, 2023
tags Random Variables,statistics

Winterizing the sprinkler system
DIY
After 10 years of winterizing my own sprinkler system, I’ve learned that you don’t need a massive compressor to get the job done right. Like many homeowners, I was…
Oct 20, 2023
tags DIY,sprinkler,PSI

Painting exterior of my house
DIY
When I saw the paint peeling off my house after 10 years, I knew it needed attention. As a DIY enthusiast, I couldn’t resist the challenge of painting 3200 square feet of…
Sep 3, 2023
tags painting,home projects,DIY

Canonical Transformations
Classical Mechanics
This paper explores canonical transformations in classical mechanics as powerful tools for simplifying complex dynamical problems. Beginning with a non-standard Lagrangian…
Aug 1, 2023
tags Hamiltonian,Lagrangian,canonical,Poisson brackets,gauge invariance

Gravitational-Wave Stochastic Background from Kinks and Cusps on Cosmic Strings
Gravity
We compute the contribution of kinks on cosmic string loops to stochastic background of gravitational waves (SBGW). We find that kinks contribute at the same order as cusps…
Jul 1, 2023
tags LIGO,gravitational-waves,NANOGrav,SBGW

Curl your Poynting vector
Electromagnetism
This blog post is a quantitative analysis of the concepts discussed in Veritasium’s “The Big Misconception About Electricity” video, and explores the dynamics of electrical…
Dec 30, 2021
tags Electromagnetism,circuit theory,transmission lines,poynting vector

Recoloring pixels with Python
Coding
I try to match the background color of my plots with the background color of the blog body. However, I sometimes need to borrow images from external sources, and they…
Sep 30, 2021
tags Python,PIL

Cauchy-Riemann conditions
Math
This post derives the Cauchy-Riemann conditions, which are fundamental for understanding when a complex function is differentiable. We present the derivation in both…
Sep 1, 2021
tags Complex Calculus,Residue Calculus,Cauchy-Riemann

The math of low pass filtered PWM
Electronics
This article explores the mathematical principles behind low-pass RC filters, with a focus on their response to Pulse Width Modulation (PWM) signals. Through interactive…
Mar 17, 2021
tags PWM,Electronics,RC circuit,low pass filter

Hacking into a wheelchair controller
Electronics
This project shows how to build a wireless joystick controller for a wheelchair using Arduino and nRF24L01 radio modules. The system intercepts the joystick signals and…
Mar 15, 2021
tags Arduino,nrf24l01,joystick,RF transceiver,DIY,hack

Thomas-Reiche-Kuhn sum rules
Quantum
The Thomas-Reiche-Kuhn sum rule represents one of the most elegant and fundamental results in quantum mechanics, connecting classical and quantum descriptions of atomic…
Oct 26, 2020
tags operator algebra,perturbation theory,Thomas-Reiche-Kuhn,sum rules,quantum,atomic physics

Electron Hydrogen scattering
Quantum
We use the Born approximation to compute the differential cross section for the elastic scattering of a fast electron by a hydrogen atom in the ground state. We will treat…
Oct 6, 2020
tags scattering,Born aproximation,cross-section

Simultaneous Eigenstates of angular momentum operators
Quantum
We show that if there is a simultaneous eigenstate of two components of the angular momentu, then this state has zero eigenvalues for all components.
Sep 23, 2020
tags Angular Momentum,Commutation relations

Interactive Timeline of JPL Missions
Space
This interactive visualization presents a comprehensive timeline of Jet Propulsion Laboratory (JPL) missions throughout history. The timeline features an intuitive interface…
Mar 17, 2020
tags JPL,NASA,space,timeline

Eigenvectors of \(\hat{n}\cdot\vec{\sigma}\)
Quantum
In this blog post, we explore an alternative method for finding the eigenvectors of the operator \(\hat{n}\cdot\vec{\sigma}\), where \(\vec{\sigma}\) represents the Pauli…
Feb 6, 2020
tags quantum,scattering,tunneling,wave function,Pauli matrices

Quantum scattering in one dimension
Quantum
In this blog post, we explore quantum scattering in one-dimensional systems, focusing on the case of a rectangular potential barrier. We start by examining the Schrödinger…
Jan 24, 2020
tags quantum,scattering,tunneling,wave function
No matching items