Category Archives: Blog

Given a flow field \(\textbf v(\textbf x,t)\), the vorticity \(\boldsymbol{\omega}\) of \(\textbf v\) is defined by taking its curl \(\boldsymbol{\omega}:=\frac{\partial}{\partial\textbf x}\times\textbf v\). For a flow field rotating rigidly with angular velocity vector \(\boldsymbol{\omega}_0\) so that \(\textbf v=\boldsymbol{\omega}_0\times\textbf x\). The vorticity … Continue reading →

Often in biochemistry, if a single substrate \(\text S\) needs to be become a product \(\text P\) via a chemical reaction of the form \(\text S\to \text P\). Assuming this is a first-order elementary chemical reaction, it would merely have … Continue reading →

The free, dispersionless wave equation on \((ct,\textbf x)\in\textbf R\times \textbf R^d\) is: \[\frac{\partial^2\psi}{\partial (ct)^2}-\frac{\partial^2\psi}{\partial|\textbf x|^2}=0\] or more simply as \(☐^2\psi=0\), where the d’Alembert operator is defined by \(☐^2:=\frac{\partial^2}{\partial (ct)^2}-\frac{\partial^2}{\partial|\textbf x|^2}=\partial^{\mu}\partial_{\nu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\) with the usual metric \(\eta=\text{diag}(1,-1,-1,-1)\) defining the hyperbolic geometry of … Continue reading →

Problem #\(1\): Find the mistake in this derivation of the Green’s function for the inhomogeneous Helmholtz equation subject to the boundary condition of an outgoing wave: Solution #\(1\): First of all, the correct answer should be \(G(r)=\frac{e^{ikr}}{4\pi r}\), not this … Continue reading →

The purpose of this post is to review the classical theory of rigid body dynamics by working through a few illustrative problems in that regard. Problem #\(1\): What defines a rigid body? What is an immediate corollary of this? Solution … Continue reading →

A ChatGPT thesaurus of all the synonyms of “propagator” across different disciplines: Problem #\(1\): What is the retarded propagator of a quantum particle with Hamiltonian \(H\) in nonrelativistic quantum mechanics? Solution #\(1\): At its heart, the “retarded” part should make … Continue reading →

Newton’s second law in its most basic form states that for a single point mass, \(\dot{\textbf p}=\textbf F\) in any inertial frame. Combining this with Newton’s third law (antisymmetry of forces \(\textbf F_{i\to j}=-\textbf F_{j\to i}\), similar to many other … Continue reading →

The purpose of this post is to appreciate that the familiar idea of non-interacting particles from e.g. statistical mechanics manifests in the context of QFT as a free quantum field theory. Problem #\(1\): What defines a free classical field theory? … Continue reading →

Consider an arbitrary worldline \(\{(t,\textbf x)\}\) of a classical system of particles in configuration spacetime. In general, this worldline need not correspond to any physical/on-shell trajectory; it can be as wildly off-shell as one likes, the only caveat being that … Continue reading →

Problem: What is the defining property of the (Cauchy) stress tensor (field) \(\sigma(\textbf x,t)\) of a material. Solution: The idea is that if one wants to find the stress vector (also called traction) \(\boldsymbol{\sigma}\) acting on a plane with unit … Continue reading →