Blog

Johnson Noise

This paper explores Johnson-Nyquist noise, a fundamental electronic phenomenon arising from the thermal agitation of charge carriers in conductors. We examine the…

Adding two Weibull random numbers

We analyze the probability distribution of the sum of two Weibull random variables by examining two redundant system configurations: sequential and interleaved operation. We…

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…

Second Quantization

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…

Adding images to Plotly

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…

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…

\(2-\)D Quantum Oscillator

This article explores the two-dimensional quantum harmonic oscillator through interactive visualizations. We consider a perturbation potential \(V(x,y)\propto xy\) and compute…

A refresher on statistical mechanics

This article provides a comprehensive refresher on fundamental concepts in statistical mechanics, drawing inspiration from Leonard Susskind’s lectures. Beginning with…

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…

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…

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…

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…

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…

Red balls, Green Balls

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…

Fabry-Perot Cavity

This article explores the physics and applications of Fabry-Perot cavities, with a particular focus on their role in LIGO (Laser Interferometer Gravitational-Wave…

Renewal Processes

This article explores renewal processes and their relationship to Poisson processes in probability theory. We examine the fundamental concepts of interarrival times…

Musings on the Gamma Distribution

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…

Musings on the Exponential Distribution

This article presents a rigorous mathematical proof demonstrating why exponential distributions naturally emerge in memoryless physical processes. We show that the…

Dirac delta with a Function inside

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…

Diffraction

This post explores the physics and mathematics of diffraction, a fundamental wave phenomenon that limits the resolution of optical instruments like telescopes. Starting from…

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

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…

Coil calculations

Electromagnetism

This article provides a comprehensive analysis of electromagnetic coil calculations, from basic principles to advanced mathematical treatments. We begin with simplified…

Separation of variables in spherical coordinates

This article presents a comprehensive derivation of the separation of variables technique applied to partial differential equations in spherical coordinates. We examine the…

Potentials of a split cylindrical shell

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…

An introduction to conformal maps

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…

Radial Green’s Function in Cylindrical Coordinates

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…

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…

Difference of two linearly distributed random numbers

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…

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…

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…

The math of low pass filtered PWM

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…

Interactive Timeline of JPL Missions

This interactive visualization presents a comprehensive timeline of Jet Propulsion Laboratory (JPL) missions throughout history. The timeline features an intuitive interface…

Eigenvectors of \(\hat{n}\cdot\vec{\sigma}\)

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…

Quantum scattering in one dimension

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…