Tetra Quark

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Re-optimizing aLIGO RF filter

Electronics

This post analyzes the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) photodiode circuit and computes its noise performance. We apply the theoretical framework developed in previous posts to reoptimize the filter circuits and component values. The optimized circuit demonstrates substantially improved signal-to-noise ratio performance compared to the baseline design.

L4C Seismometer transfer function

Electronics

A seismometer is essentially a mass-spring system with a resonance frequency of 1 Hz, where the motion of the coil contributes to its impedance. We derive the transfer function of the system using Laplace transforms, starting from the differential equation for a damped and forced harmonic oscillator.

L4C,seismometer,LIGO

Silicon Based Quantum Computer

Quantum Computation

This article presents a detailed mathematical derivation of Kane’s silicon-based nuclear spin quantum computer, reproducing the original work with explicit calculations of the energy eigenstates and eigenvalues. We derive the single-qubit Hamiltonian and show how the hyper-fine interaction between electron and nuclear spins enables control of individual qubits through A-gates and J-gates. The two-qubit interaction is analyzed using perturbation theory, revealing how electron-electron coupling enables entangling operations between adjacent qubits. Finally, we explain how the qubit states can be read out through capacitance measurements after adiabatically changing the coupling strength.

Kane’s computer,qubit,quantum computer

Interactive Dashboard of JPL Missions

Space

This is an interactive dashboard of all Jet Propulsion Laboratory(JPL) missions.

Solenoid plungers

Electromagnetism

We derive the equation of motion for a magnetic plunger in a solenoid using magnetomotive force (MMF) and energy methods. Starting with a simple toy model, we calculate the force on a partially inserted magnetic core and show how it depends on the gap distance \(s\) and material properties. We then extend the analysis to realistic solenoid designs with return paths and derive the time-dependent motion of the plunger under constant current conditions.

coils,magnets,magnetic fields

Musings on the Weibull distribution

Reliability

This is a quick overview of the Weibull distribution with an interactive visualization. We start from simple reliability ideas and build up to the standard Weibull formulas. Along the way, you can play with sliders to see how the shape and scale parameters change the curves and what that means in practice. The goal is to make Weibull feel less mysterious and more like a practical tool.

weibull,reliability

\(\text{Integral of the month: } \int_0^1 \sqrt{x(1-x)} dx\)

Math

We evaluate the integral \(\int_0^1 \sqrt{x(1-x)} dx\) using a dog-bone contour. This exercise demonstrates how to handle branch cuts from the square root function. The dog-bone contour wraps around the branch cuts on \([0,1]\) from both sides, allowing us to relate the real integral to a contour integral. We evaluate the integral using the Beta function as well.

Integral,Residue Calculus,Integral of the month,Dog-bone contour

Mathematics of Mortgage

Math

We derive a mathematical framework for understanding mortgage payments using differential equations. Starting from the exponential growth of debt without payments, we solve the differential equation that includes constant monthly payments to obtain closed-form expressions for the loan balance over time. We derive the relationship between the initial loan amount, interest rate, loan term, and monthly payment, and provide analytical expressions for the total interest paid over the life of the mortgage. We explore limiting cases for both low and high interest rate scenarios, and demonstrate how the mathematics reveals the true cost of long-term mortgages.

Mortgage,Mathematics

Rice distribution

Statistics

In coronagraphy, atmospheric scattering creates speckle patterns that obscure faint astronomical companions. Stochastic Speckle Discrimination (SSD) exploits statistical differences between faint sources and speckle patterns to separate them. The speckle pattern follows a Rice distribution, while faint sources exhibit Poisson statistics. This document derives the fundamental equations of the Rice distribution and demonstrates its application to power spectrum analysis in coronagraphic observations.

Rice distribution,statistics

An analysis of aLIGO PD circuit

Electronics

We perform a numerical analysis of noise in the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) photodetector demodulation circuit. The analysis examines the complete circuit architecture, including the photodiode bias circuit, notch filters for harmonic suppression, and frequency selector circuits. This work provides a comprehensive understanding of noise sources and their contributions to the overall circuit performance.

LIGO,Gravitational-waves,Electronics

The Legendre Transform

Classical Mechanics

This is a deep dive into the meaning of the Legendre transform. We discuss how it appears in classical mechanics when transitioning from the Lagrangian description to the Hamiltonian description, and provide geometric interpretations using both slopes and areas.

Calculus of variations,Lagrangian,Hamiltonian,Legendre

Decoherence due to phonons

Quantum

A detailed derivation of decoherence due to phonons. We calculate the decoherence time of magnetic cluster qubits by analyzing phonon-assisted transitions out of the two-state subspace. The resulting exponential dependence on spin quantum number S provides a pathway for optimizing qubit lifetimes.

Quantum,Heisenberg

Computing joint confidence intervals

Statistics

A comprehensive guide to combining statistical information from separate experimental runs when data sets cannot be merged due to confounding factors. This post addresses the challenge of computing joint confidence intervals and p-values from multiple independent experiments.

Confidence intervals,Sampling,Statistics,Normal distribution,Central limit theorem

Fixing Ghostscript Issues with dvisvgm on macOS

Coding

This article documents a common issue when rendering TikZ diagrams to SVG on macOS using Quarto and dvisvgm. The problem occurs when Ghostscript is installed but not properly detected by dvisvgm, leading to character jumbling and rendering errors. We provide a complete solution using the –libgs option.

dvisvgm,Ghostscript,TikZ,SVG,LaTeX,macOS,Quarto,Rendering

Integral of the month: \(\int dx \frac{\sin^2 x}{x^2}\)

Math

Four different ways of evaluating this lovely integral! We explore contour integration techniques using both upper and lower semicircular contours, as well as a clever approach involving shifted contours to avoid pole splitting. A couple more approaches come from other sneaky techniques.

Integral,Residue Calculus,Integral of the month

The catenary curve with a sliding end

Math

We dive into calculus of variations to calculate the shape of a rope fixed in one end and free to slide on the other. Starting with the classic catenary problem, we derive the Euler-Lagrange equation and show how the familiar hyperbolic cosine solution emerges from energy minimization principles. The real challenge comes when one end is free to slide along a vertical post, introducing movable boundary conditions.

Calculus of variations,Lagrangian,AI/ML,Neural Networks

\(L^2\) is a Hermitian operator

Quantum

A proof that square of the angular momentum vector is a Hermitian operator.

angular momentum,operator algebra

Integrating Gaussians like a boss!

Math

This is a comprehensive deep dive into Gaussian integrals, exploring multiple techniques for their evaluation including polar coordinate transformations, complex contour integration. We derive the fundamental normalization factor and demonstrate how the central limit theorem makes Gaussian distributions ubiquitous across statistics, physics, and machine learning. This is the art of evaluating Gaussian integrals like a true mathematical boss.

Integral,Residue Calculus,Integral of the month

Forced & damped Harmonic Oscillator

Math

This article presents a comprehensive analysis of the forced and damped harmonic oscillator, covering both the mass-spring system and its electrical RLC circuit analog. We derive the complete analytical solution using Laplace transforms and explore the resonance phenomenon where the amplitude grows linearly with time. Interactive visualizations demonstrate the system’s behavior under various driving frequencies and damping conditions.

Differential equations,Laplace transforms,resonance

Dirac Particle in Magnetic Field

Quantum

We solve the Dirac equation for a particle in a magnetic field using the method of separation of variables. We find the energy levels and the wave functions for a particle in a magnetic field.

Dirac equation,magnetic field

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 control. We examine the Pound-Drever-Hall (PDH) technique, which employs radio-frequency (RF) phase modulation of laser light at frequencies ranging from 9 MHz to 118 MHz to create optical sidebands. These sidebands interact with Fabry-Perot cavities to generate error signals for cavity locking.

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 classical algorithms require O(N) operations to find a specific item among N entries, Grover’s algorithm accomplishes this task in just O(√N) operations. This article explores the mathematical foundations of the algorithm, including the oracle function, diffusion operator, and the geometric interpretation of quantum state rotations in Hilbert space. The algorithm’s elegance lies in its systematic amplification of the target state’s amplitude through repeated applications of oracle and diffusion operations, demonstrating quantum computing’s practical advantages for search problems.

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 cryptographic systems. This polynomial-time quantum algorithm can efficiently factor large integers by exploiting quantum superposition and interference to find the period of modular exponentiation.

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 the keyhole contour method to handle the branch cut created by the non-integer power \(x^\alpha\), and derive closed-form solutions for various parameter ranges. The approach demonstrates the elegance of complex integration techniques in solving seemingly difficult real integrals, making it an excellent pedagogical example for students learning residue theory and contour integration.

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 insulating barriers using the Ginzburg-Landau approximation, showing how unbiased junctions behave as harmonic oscillators while biased junctions exhibit sinusoidal current-voltage relationships. The key insight is that by controlling the bias current, we can create a double-well potential that supports exactly two quantum states, forming the basis of a qubit. We demonstrate how these qubits can be manipulated through RF perturbations to achieve arbitrary rotations on the Bloch sphere, measured by barrier height modulation, and coupled through capacitive elements for multi-qubit operations.

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 complete workflow from installation and setup to implementing quantum circuits. The tutorial walks through creating a three-qubit Greenberger-Horne-Zeilinger (GHZ) entangled state using fundamental quantum gates including Hadamard and controlled-NOT operations. Using Qiskit Aer’s simulation backends, we visualize quantum states, perform measurements, and analyze statistical outcomes with confidence intervals.

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 introducing thermal noise. We derive the fundamental current-voltage relationships for parametric circuits and systematically eliminate variables to obtain the signal admittance expression. This result demonstrates how parametric coupling creates effective negative conductance proportional to pump power, enabling amplification when the negative conductance exceeds the circuit’s passive losses.

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 Gordon using sophisticated complex analysis techniques. We follow Gordon’s elegant approach, which employs multiple variable transformations, complex contour integration over a keyhole contour, and residue calculus to evaluate the integral exactly. The solution demonstrates remarkable mathematical beauty, involving the golden ratio and requiring careful analysis of an 8th-order polynomial’s factorization. The final result connects this seemingly intractable integral to simple expressions involving arctangent functions and fundamental mathematical constants.

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 pizza-slice contour approach, a traditional semicircular contour method, and an elegant keyhole contour technique involving branch cuts. Each method demonstrates different aspects of contour integration theory, making this a valuable exercise for students learning complex analysis and showcasing the versatility of residue theory in solving real integrals.

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 statistical nature of this thermal noise and derive its power spectral density using both thermodynamic equilibrium principles and the fluctuation-dissipation theorem. The analysis demonstrates how Johnson noise establishes an inherent noise floor in electronic circuits, independent of current flow.

Johnson noise,thermal noise,Nyquist noise,Johnson-Nyquist theorem

\(\text{Integral of the month: } \iint_S dt' dt f(t'-t)\)

Math

When the function depends only on the difference of the parameters, we can simplify the double integral using a clever change of variables. This technique is essential for understanding the Wiener-Khinchin theorem and appears frequently in Fourier analysis.

Integral,Wiener-Khinchin theorem,Integral of the month

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 demonstrate that both configurations yield identical probability distributions represented by a convolution integral. For the special case of exponential distributions (Weibull with shape parameter β=1), we derive a closed-form solution through a Gamma distribution. For general Weibull distributions, we propose using a generalized Gamma function with carefully selected parameters to approximate the convolution. We also provide an interactive visualization of the probability functions.

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 Feynman diagrams and their corresponding amplitudes. We evaluate these amplitudes explicitly by calculating the s-channel and u-channel contributions, and demonstrate how to square them to obtain the differential cross-section.

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 the basic formalism. We will later use this to study superfluidity and superconductivity..

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 configuration.

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, including dependencies and configuration, and provide examples of how to use GetDP to solve problems in structural mechanics and electromagnetics.

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 the first-order corrections to the energy levels due to the perturbation. We also show that the problem can be solved exactly by transforming the coordinates to the new coordinates \(\tilde{x}_1, \tilde{x}_2\) and \(\tilde{p}_1, \tilde{p}_2\) which are decoupled. The interactive plot allows readers to adjust quantum numbers \(n_x\) and \(n_y\) to visualize both the wavefunctions and probability densities in real-time.

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 probability theory and Shannon’s information-theoretic definition of entropy, we establish the mathematical foundations that bridge information theory and thermodynamics. We explore the derivation of entropy formulas using both Shannon’s axioms and combinatorial approaches with Stirling’s approximation. The article presents the zeroth, first, and second laws of thermodynamics, with particular emphasis on the relationship between entropy, energy flow, and temperature in interacting systems. Using calculus of variations and Lagrange multipliers, we demonstrate how entropy maximization principles lead to the uniform and Boltzmann distributions. Throughout, we supplement theoretical discussions with visual representations and detailed derivations to provide intuitive understanding of these abstract concepts. This refresher serves as an accessible entry point for readers seeking to revisit or develop a deeper understanding of statistical mechanics and its connections to information theory.

Statistical Mechanics,Thermodynamics,entropy,information theory,Boltzmann distribution,Lagrange multiplier,calculus of variations

Stirling approximation for factorial

Math

Stirling’s approximation for the factorial function.

Thermodynamics,factorial,Math

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 divergence-free, we develop the vector potential formulation and use Green’s function methods to solve the resulting differential equation. The derivation demonstrates how the familiar Biot-Savart law emerges naturally from these first principles, providing both the general form for continuous current distributions and the specialized case for line currents. This treatment emphasizes the deep connection between the absence of magnetic monopoles and the mathematical structure of magnetic fields.

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 energy expression involving the vector potential and current density, we evaluate the mutual inductance through direct integration. The analysis assumes thin wire approximation and provides the final result in terms of the wire length and separation distance. This approach offers an alternative perspective to the more commonly used flux-based calculations, while arriving at the same well-known logarithmic dependence on the geometric parameters.

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 previous analysis of single-wire self-inductance, we derive the magnetic coupling between current-carrying conductors using the Biot-Savart law. We address the mathematical challenges of finite-length conductors and present a complete solution that includes both the self-inductance of each wire and their mutual coupling.

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 internal contributions to the self-inductance. The external component is calculated by integrating the magnetic flux over a finite region, addressing the inherent challenges of infinite wire assumptions. The internal contribution is determined through energy considerations of the magnetic field within the wire.

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 distribution and apply it to the specific case of a circular current loop. The exact solution is expressed in terms of complete elliptic integrals, and we provide explicit forms for the magnetic field in both spherical and cylindrical coordinates. We then develop the magnetic dipole approximation, showing how it emerges naturally as the leading term in a multipole expansion. Finally, we extend our analysis to continuous distributions of magnetic moments, introducing the concept of bound currents and demonstrating how they provide an elegant framework for describing magnetized materials. Throughout, we emphasize the mathematical techniques and physical insights that connect these various aspects of magnetic dipole physics.

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 observing a red ball on the first draw, the likelihood of drawing another red ball is calculated using Bayesian probability, which adjusts the probability distribution over possible urn configurations. The puzzle is further explored by modifying the setup, introducing a scenario where the number of red balls is determined by repeated coin flips, resulting in a binomial distribution. Theoretical results suggest a 2/3 probability of drawing a red ball again in the original setup, and a 1/2 probability when using the binomial model. These outcomes are validated through simulations, highlighting the nuanced effects of initial conditions on probability outcomes and underscoring the importance of rigorous reasoning in probabilistic analysis.

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 Observatory). We derive the fundamental equations governing cavity transmission and reflection, introduce the concept of finesse, and analyze different coupling regimes. The formation of interference patterns and bullseye fringes is explained through rigorous mathematical treatment. Building upon previous posts about LIGO’s electronics, this article complements the series by delving into the optical principles that make precision gravitational wave detection possible.

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, counting processes, and their distributions, with particular emphasis on exponential and gamma distributions. The discussion includes rigorous derivations of key probability distributions, including the distribution of arrival times and the number of events. Special attention is given to clearing up common misconceptions about Poisson processes and their relationship with exponential distributions. The article provides both intuitive explanations and mathematical proofs, making it accessible to readers with basic probability theory knowledge while maintaining mathematical rigor.

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 distributed random variables can be derived iteratively using convolution integrals. Starting with the case of \(n=2\), we explicitly calculate the probability density function and cumulative distribution function, then extend the result to \(n=3\) and generalize to arbitrary \(n\). Second, we present an elegant alternative derivation using Laplace transforms, leveraging their properties to convert convolutions into multiplications in \(s\)-space. Both methods arrive at the same result, showing that the sum of \(n\) independent exponentially distributed random variables is Gamma distributed.

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 exponential distribution is uniquely characterized by its memoryless property through two distinct approaches. The first proof utilizes the survival function and its homomorphism properties, extending from natural numbers to rational and real domains. The second, more physics-oriented derivation employs differential equations to arrive at the same conclusion. We conclude by connecting these results to the hazard function formalism, providing an intuitive framework for understanding why exponential distributions are fundamental to describing microscopic systems such as radioactive decay.

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 at in this post.

Dirac delta,functional

\(\text{Integral of the month: } \int dr \cos r^2\)

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 fundamental wave phenomenon that limits the resolution of optical instruments like telescopes. We first derive the results in the large \(r\) limit using the residue theorem. Then we will see how to evaluate the integrals numerically.

Fresnel,Fraunhofer,Integral of the month,optics,diffraction

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 Maxwell’s equations, we derive the wave equation in free space and develop the mathematical framework using the Huygens-Fresnel principle. We examine how light waves interfere through apertures, leading to characteristic diffraction patterns. The analysis includes interactive visualizations that demonstrate how parameters like aperture size, wavelength, and distance affect the resulting intensity distributions. Special attention is given to Fresnel integrals, which are essential for calculating these diffraction patterns.

Fresnel,Fraunhofer

Coil calculations

Electromagnetism

This article provides a comprehensive analysis of electromagnetic coil calculations, from basic principles to advanced mathematical treatments. We begin with simplified models to build intuition about magnetic fields in solenoids, then progress to more sophisticated analyses using elliptic integrals. The work covers magnetic field calculations both on and off axis, vector potential derivations, and practical considerations like resistance calculations and wire selection. Special attention is given to the effects of coil geometry, number of turns, and material properties on the magnetic field distribution and electrical characteristics. The analysis includes both analytical solutions and practical engineering considerations, making it valuable for both theoretical understanding and practical applications.

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 process of decomposing the Laplace equation into its radial and angular components, leading to solutions involving spherical harmonics. The discussion includes a detailed analysis of the radial dependence, angular components, and their relationship through Sturm-Liouville theory. This mathematical treatment is fundamental to various physics applications, including quantum mechanics, electromagnetism, and gravitational field theory.

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 conformal transformations: first, a Möbius transformation maps the cylindrical boundaries to the real axis, followed by a logarithmic transformation that converts the problem into a simple Cartesian geometry. This double transformation approach elegantly reduces a seemingly complex boundary value problem to a straightforward solution of Laplace’s equation. The method showcases the power of conformal mapping techniques in solving two-dimensional electrostatic problems with nontrivial geometries.

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 approach through a simple electrostatic setup, making the method accessible and practical for common applications..

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 Cauchy-Riemann equations, demonstrating how these conditions lead to the Laplace equation. We explore the geometric interpretation of conformal mappings and their crucial property of preserving angles between intersecting curves. The article then shows how conformal maps can transform harmonic functions while preserving their harmonic properties, making them particularly useful for solving problems in electrostatics and fluid dynamics. We conclude with a practical example, using conformal mapping to solve for the electric field of an infinite line charge, demonstrating how these mathematical tools can simplify complex physical problems.

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 a crucial tool for solving differential equations, particularly in systems exhibiting cylindrical symmetry.

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 test strips were used, saving both time and money.

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 superconducting materials. We begin by examining how topological defects emerge from both global and local symmetry breaking, using examples from magnetic materials and superconductors. The mathematical framework of spontaneously broken U(1) gauge symmetry is presented, leading to the formation of vortices and flux tubes. We derive the Bogomol’nyi equations that describe these configurations and discuss their physical properties, including mass, central charge, and BPS saturation. The article provides insights into how these fundamental concepts connect quantum field theory with condensed matter physics.

Gauge Theory,Superconductivity,Symmetry

Replacing break pads of a Prius

DIY

Replacing the front wheel break pads of a 2011 Prius car.

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 density function of the absolute difference, \(S=|R_1-R_2|\). We employ various methods for simulating random numbers that adhere to the derived distribution, including inverse transform sampling and geometric approaches. The results are validated through graphical comparisons of the theoretical and simulated distributions.

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 frustrated by the common advice that small compressors won’t work - especially when I didn’t want to buy and store a huge tank. Through trial and error, I discovered a simple hack involving the back-flow valve gasket that lets my 3-gallon pancake compressor effectively clear all zones. This personal guide shares the technique that has kept my pipes burst-free through countless freezing winters, proving that with a little ingenuity and patience, DIY winterization is entirely achievable.

DIY,sprinkler,PSI

240V outlet for EV charger

DIY

Building 240V outlet for EV charging. This guide covers the complete installation process from electrical panel wiring to outlet installation, including safety considerations and code compliance. Learn how to install a 50A circuit breaker, run 6-gauge wire, and connect a NEMA 14-50 receptacle for efficient Level 2 EV charging.

DIY,240V,Transformers

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 exterior myself - even though it felt way out of my league. Over two grueling weeks, working early mornings and evenings, I discovered that the biggest obstacles weren’t the paint or prep work, but the logistics of reaching 21-foot heights safely. My solutions included telescoping poles, specialized edgers, and my secret weapon: climbing glasses that flip your view 90 degrees to prevent neck strain. It was excruciating at times, but I pulled it off completely solo. This is my story of tackling the biggest DIY project I’d ever attempted.

painting,home projects,DIY

Rendering PGF plots inside Quarto

Coding

Rendering pgf plots directly in Quarto.

Latex,pgf,RMD,Quarto,R Studio

Math mode in SVG

Coding

A quick tip on rendering math mode in SVG.

Latex,pgf,RMD,R Studio,SVG

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 \(\mathcal{L}=\sqrt{q^2+\dot{q}^2}-\frac{1}{2}q^2\), we demonstrate how transforming from the Lagrangian to Hamiltonian formalism enables more tractable analysis of the equations of motion. We first develop the theoretical foundations through variational calculus and the Euler-Lagrange equation, examining how the principle of least action leads to the fundamental equations governing classical systems.

Hamiltonian,Lagrangian,canonical,Poisson brackets,gauge invariance

Real coils are not purely imaginary

Electronics

Real coils deviate from idealized models due to finite wire resistance, skin depth effects, and parasitic capacitance. We analyze manufacturer data to extract key parameters and develop predictive models for practical coil behavior. This analysis is essential for accurate circuit optimization, particularly in applications such as LIGO RF filter design.

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 to the SBGW. We discuss the accessibility of the total background due to kinks as well as cusps to current and planned gravitational wave detectors, as well as to the big bang nucleosynthesis (BBN), the cosmic microwave background (CMB), and pulsar timing constraints. As in the case of cusps, we find that current data from interferometric gravitational wave detectors, such as LIGO, are sensitive to areas of parameter space of cosmic string models complementary to those accessible to pulsar, BBN, and CMB bounds.

LIGO,gravitational-waves,NANOGrav,SBGW

An SNR analysis of aLIGO circuit

Electronics

We develop a simplified model of the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) photodiode circuit to analyze signal-to-noise ratio (SNR) and noise characteristics. The model accounts for photodiode shot noise, thermal noise, and operational amplifier (OPAMP) noise contributions in a simplified circuit with frequency selectors and notch filters.

LIGO,Gravitational-waves,Electronics

Coherent states of Quantum Harmonic Oscillator

Quantum

A derivation of the coherent states of quantum harmonic oscillator.

Quantum,Heisenberg

RLC filters

Electronics

Analysis of RLC filters.

LIGO,coils,magnets,LTspice,Python

Quantum Harmonic Oscillator

Quantum

A detailed derivation of quantum harmonic oscillator solution using both analytical and algebraic methods. The post explores the brute force approach of solving the Schrödinger equation directly, leading to Hermite polynomials and quantized energy levels. We also demonstrate the elegant ladder operator method that provides a more intuitive understanding of the quantum system.

Quantum,Heisenberg

A crash course on JFETs

Electronics

JFETs are unipolar field-effect transistors that offer very low current noise and high input impedance, making them ideal for low-noise amplifier applications. This article provides a comprehensive derivation of the JFET drain current equation, starting from the p-n junction physics and depletion region analysis to show how the gate voltage controls the channel conductivity. We derive the quadratic relationship between drain current and gate voltage.

Noise analysis of OPAMPs

Electronics

OPAMPs have inherent voltage and current noise that must be accounted for in circuit design. We analyze noise in both non-inverting and inverting amplifier configurations by projecting noise sources into the input terminals and solving for the output noise voltage. The analysis reveals that noise currents flowing into high impedances create large equivalent input noise, which is particularly problematic for OPAMPs using BJTs in the input stages.

Wiener Khinchin Theorem

Math

A rigorous proof of the Wiener-Khinchin theorem, which establishes the fundamental relationship between the autocorrelation function and the spectral power density of a random process. This theorem is central to signal processing and statistical analysis, providing the mathematical foundation for understanding how random signals distribute their energy across different frequencies. The proof demonstrates the elegant connection between time-domain and frequency-domain representations of stochastic processes.

statistics,random,Energy flow,spectral power density,Fourier analysis,signal processing

Lagrange Multipliers

Math

We explore Lagrange multipliers as a powerful technique for solving constrained optimization problems, providing both a geometric interpretation and analytical framework. The method relies on the observation that at the optimal point, the gradient of the objective function is parallel to the gradient of the constraint function, allowing us to combine both conditions into a single optimization problem. We illustrate the technique through several examples including finding extrema on a unit circle, maximizing Shannon entropy, deriving the Boltzmann distribution, and solving the catenary problem.

Lagrange multipliers,optimization,gradient

Pandemic, home-schooling, and dragons: 2021 Edition

Short Stories

An update on my daughter’s reading journey during the second year of the pandemic, featuring her impressive collection of 324 books read in 2021. This follow-up post showcases her continued love for dragon-themed books and her exploration into new genres like the Wizard of Oz series. I share insights about her reading habits, the mix of purchased and library books, and provide a complete searchable list of all the books she read that year.

story,How to Train Your Dragon,books,homeschooling

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 energy propagation in transmission lines, with a particular focus on understanding how energy and signals travel from a source to a load. While the video suggests that energy flows through the electromagnetic fields surrounding the conductors via the Poynting vector, we provide a detailed mathematical treatment showing how the traditional circuit theory approach remains valid and complete for understanding energy transfer. We examine both ideal and lossy transmission lines, analyzing how voltage and current waves propagate when subjected to different load conditions including open circuits, short circuits, and reactive loads. Through mathematical analysis of wave equations and reflection coefficients, we demonstrate that the Poynting vector is not necessary for understanding energy transfer in circuits, and that the traditional circuit theory approach is sufficient for describing the energy flow in both DC and AC circuits. We also build an interactive plot to visualize voltage and current waves propagating in a transmission line.

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 typically have white background. I don’t like the look of such images, see an example below. So, I ended up writing a quick Python code to change the background color, which can be copied below or can be cloned from my repository.

Erasure Coded Data Durability

Reliability

We calculate the erasure coded data durability in terms of the device failure rates and the recovery speed. We put more emphasis on RAID5&6.

weibull,reliability,redundancy,raid

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 Cartesian and polar coordinates, explaining the underlying concepts and their significance in complex analysis.

Complex Calculus,Residue Calculus,Cauchy-Riemann

Yas (Grief)

Short Stories

This is in Turkish. We translated a post from GSnow@reddit on grief to Turkish to share with a family member who lost a loved one.

\(\text{Integral of the month: } \oint_{c} \frac{ |dz|}{|z- \beta|^2}\)

Math

Going around a circle! This post explores the mathematical beauty of circular integration and its applications in complex analysis. We’ll discover how the geometry of circles reveals elegant solutions to challenging integral problems.

Integral,Residue Calculus,Integral of the month

Dealing with huge and tiny numbers

Math

We sometimes have to deal with very large and very small numbers in the same equation. This post looks into a probability calculation for an event which has very low success rate but repeats many times.

Probability,Math

Integral of the month: \(\int dx \frac{\sin x}{x}\)

Math

Three different ways of evaluating this lovely integral! We explore complex contour integration using the upper semicircular contour to avoid the singularity at the origin. The parametric Laplace transform method introduces a parameter and manipulates it to simplify the integral evaluation. Finally, we demonstrate the direct Laplace transform approach, showing how to create the necessary 1/x term in the integrand.

Integral,Residue Calculus,Integral of the month

HDD average seek time

Math

The average seek distance of a random hard drive access is 1/3rd of the maximum seek distance, or is it? We re-examine this rule of thumb and find that it is not exactly true. We also provide the full statistics of the seek distance and discuss how to compute the seek time.

Math,Statistics

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 visualizations, we demonstrate how these fundamental circuits process digital signals, showing the relationship between the time constant (RC), duty cycle, and the resulting output voltage. Readers can experiment with different parameters to understand how the filter smooths out PWM signals into analog voltages, making this complex topic both accessible and practical. This understanding is essential for anyone working with digital-to-analog conversion, motor control, or signal processing applications.

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 transmits them wirelessly to the wheelchair’s controller, allowing remote operation. The build requires minimal soldering and uses off-the-shelf components, making it accessible for DIY electronics projects. The final result is a compact, battery-powered remote control that can operate the wheelchair from a distance.

Arduino,nrf24l01,joystick,RF transceiver,DIY,hack

There is no such thing as farmer’s wife

Random Rants

A passionate critique of gender bias in children’s literature, specifically examining the problematic phrase ‘farmer’s wife’ and its implications for young readers. This post explores how language shapes perceptions and calls for more inclusive representation in kids’ books, backed by research from Florida State University on gender representation in children’s literature.

Pandemic, home-schooling, and dragons

Short Stories

A personal reflection on homeschooling during the pandemic and the joy of reading, featuring my daughter’s journey through the ‘How to Train Your Dragon’ series and other beloved children’s books. This post shares practical advice for parents navigating the challenges of teaching their children to read, especially in a non-native language. You can also find a list of great books to get your kids to read.

story,How to Train Your Dragon,books,homeschooling

Time dependent spin-spin coupling

Quantum

We analyze the quantum dynamics of two spin-1/2 particles subject to a time-dependent spin-spin coupling interaction. The problem is solved using both exact analytical methods and time-dependent perturbation theory approaches. The exact solution employs Clebsch-Gordan coefficients and eigenbasis decomposition to derive transition probabilities between spin states.

operator algebra,perturbation theory,angular momentum

Computing hydrogen ground state wavefunction on a sunny day

Quantum

A practical approach to computing the hydrogen ground state wavefunction using minimal information. This post demonstrates how to derive the wavefunction from basic principles of normalization and the Schrödinger equation, then applies this knowledge to calculate transition probabilities in beta decay processes. The analysis covers the transition from tritium to helium-3, showing how quantum mechanics governs nuclear decay processes and atomic state transitions.

hydrogen,transitions,quantum mechanics,wavefunction,beta decay

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 oscillators. This sum rule states that the total oscillator strength for all possible transitions from a given atomic state equals the number of electrons in the atom.

operator algebra,perturbation theory,Thomas-Reiche-Kuhn,sum rules,quantum,atomic physics

Filter Matching

Statistics

A mathematical analysis of pattern matching in digit sequences, specifically examining how many numbers contain a given pattern like ‘01’. This post develops a combinatorial approach to count pattern occurrences in n-digit numbers, accounting for over-counting through inclusion-exclusion principles. The analysis reveals that as the number of digits increases, most numbers will contain any given pattern, with connections to the mathematical properties of irrational numbers like π.

Combinatorics,Pattern Matching,Statistics,Probability

One-dimensional attractive potential of any shape supports a bound state

Quantum

We use the variational principle to prove that one-dimensional attractive potential of any shape supports a bound state.

variational principle,bound states

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 the hydrogen atom as a fixed target with a time-independent charge distribution.

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.

Angular Momentum,Commutation relations

Hydrogen Atom in Magnetic Field

Quantum

A detailed analysis of a hydrogen atom placed in a uniform magnetic field, focusing on the addition of angular momentum and spin interactions. This post demonstrates how to find eigenvalues and eigenstates for both zero and non-zero magnetic field cases, showing the intricate quantum mechanical behavior of coupled spin systems and the formation of entangled states through angular momentum coupling.

Angular Momentum,Commutation relations,Spin,Quantum Mechanics,Magnetic Field

Nuclear Magnetic Resonance Based Quantum computer

Quantum Computation

This article presents the theoretical foundations of Nuclear Magnetic Resonance (NMR) quantum computers, one of the earliest experimental implementations of quantum computation. We derive the complete Hamiltonian for spin-1/2 nuclei in magnetic fields and demonstrate how radiofrequency pulses can implement arbitrary single-qubit rotations through the rotating wave approximation.

resonance,qubit,quantum computer

A romantic walk under rain and partial derivatives

Short Stories

A personal story about discovering the beauty of partial derivatives through observing rain patterns during long commutes to high school. This post explores how mathematical concepts like gradients and partial derivatives can elegantly describe the changing intensity of rain as you move through space and time, culminating in a romantic walk where these mathematical insights were shared with a loved one.

gradient,partial derivative,three.js,rain,love

A Mountain on top of a mountain

Short Stories

A short story celebrating a daughter’s 6th birthday, inspired by Richard Feynman.

Feynman,story,How to Train Your Dragon

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 that allows users to explore missions by their operational status, mission type, and destination. Users can filter and analyze JPL’s diverse portfolio of space exploration initiatives, from planetary missions to Earth observation satellites. This interactive tool serves as both an educational resource and a historical record of JPL’s contributions to space exploration and scientific discovery.

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 matrices and \(\hat{n}\) is a unit vector. Rather than using conventional eigenvalue methods, we demonstrate how to obtain the eigenvectors through a series of rotations in spin space. This approach not only yields the correct results but also provides deeper insights into why the Pauli matrices transform as vector quantities under rotations.

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 equation in one dimension and then move towards a more efficient solution for the scattering problem. Rather than following the conventional approach of solving from left to right and imposing continuity conditions at the boundaries, we introduce a faster method by assigning coefficients directly from the transmitted wave and building in the boundary conditions. We cover both scenarios where the particle’s energy is below and above the potential barrier, and we detail how to calculate the transmission and reflection coefficients in each case. To simplify the calculations, we introduce dimensionless parameters, which allow us to rewrite the transmission coefficient in a more intuitive form. This also helps us identify resonance wavelengths, which occur when integer multiples of half-wavelengths fit within the potential barrier.

quantum,scattering,tunneling,wave function