Tetra Quark

Don’t drink and derive

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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

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

\(\text{Integral of the month: } \int \frac{x^\alpha dx}{x^2- 2 \beta x + 1}\)

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

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

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

\(\text{Integral of the month: } \int ds\,CRAZY(s)\)

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,complex integration,Integral of the month

\(\text{Integral of the month: } \int \frac{ dx}{x^n+ 1}\)

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

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.

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,reliability,redundancy

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 Mechanics,Second Quantization

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

\(2-\)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

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

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 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

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

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

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

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,LIGO,cavity

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,poisson

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.

exponential,poisson,gamma

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,poisson

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.

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

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

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,differential

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

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

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. We begin by deriving the Laplace operator in cylindrical coordinates, which is essential for understanding the behavior of physical systems. The focus is on defining the Green’s function as the solution to the corresponding differential equation, specifically in scenarios devoid of angular and axial dependencies, simplifying our analysis to radial dependence. The document explores the implications of the derived Green’s function in solving boundary value problems and its applications in various fields, including electrostatics, heat conduction, and fluid dynamics. Special attention is given to the careful handling of singularities and the application of Gauss’s theorem to validate the results.

Symmetry,Laplace

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

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

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.

DIY,painting,home projects

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 transformation

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

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

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

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.

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

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