Category Archives: Blog

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 →

Problem: To what kinds of waves does the concept of “wave impedance” \(Z\) apply to? Solution: (Transverse/longitudinal/dispersive/non-dispersive/plane/non-planar) travelling waves Problem: Why does it make more sense conceptually to consider the reciprocal of the impedance \(Y:=1/Z\) (called the admittance)? Solution: In … Continue reading →

Suppose you know that \(p\) is the derivative of some function with respect to \(v\). A natural question is whether or not the roles of \(v\) and \(p\) can be reversed, that is, can \(v\) also be viewed as the … Continue reading →