Author Archives: wdengquantum.me

Given any Hausdorff, locally compact, abelian topological group \(G\), define a character \(\chi:G\to U(1)\) on \(G\) to be any continuous group homomorphism from \(G\) to \(U(1)\), that is for all \(g_1,g_2\in G\), \(\chi(g_1\cdot g_2)=\chi(g_1)\chi(g_2)\) (of course the notation here for … Continue reading →

Gross Structure of Hydrogenic Atoms In non-relativistic quantum mechanics, the gross structure Hamiltonian \(H_{\text{gross}}\) for a hydrogenic atom \(N^{Z+}\cup e^-\) consisting of a single electron \(e^-\) in a bound state with an atomic nucleus \(N^{Z+}\) of nuclear charge \(Z\in\textbf Z^+\) … Continue reading →

Consider the following \(2\)D dynamical system expressed in plane polar coordinates \((\rho,\phi)\): \[\dot{\rho}=\alpha\rho+\rho^3-\rho^5\] \[\dot{\phi}=\omega+\beta\rho^2\] with \(3\) real parameters \(\alpha,\omega,\beta\in\textbf R\). Since \(\rho\) is decoupled from \(\phi\) (but not vice versa) one can analyze \(\rho\) on its own. A simple calculation … Continue reading →

Problem: Write down the current density \(\mathbf J(\mathbf x,t)\) of an idealized Hertzian dipole. Hence, calculate the electric and magnetic fields \(\mathbf E(\mathbf x,t),\mathbf B(\mathbf x,t)\) of the Hertzian dipole. Solution: The current density for a Hertzian electric dipole \(\boldsymbol{\pi}(t)\) … Continue reading →

Problem: Let \(\psi(\mathbf x),\phi(\mathbf x)\) be arbitrary complex-valued maps on \(\mathbf x\in\mathbf R^d\) (in practice this could also just be time domain signals \(\psi(t),\phi(t)\), etc.). Define their convolution \((\psi*\phi)(\mathbf x)\) and their cross-correlation \((\psi\star\phi)(\mathbf x)\), and state how they are … Continue reading →

The goal of physics is to understand the what, how, and why of the universe. The twist is that sometimes one sometimes has to introduce auxiliary physical quantities as stepping stones towards such an understanding. An exemplar of this in … Continue reading →

Having been a member of Cambridge University Spaceflight (CUSF) during my first year as an undergraduate student, I thought it would be fun to document all the knowledge I’ve acquired with regards to the art of high-power rocketry. A good … Continue reading →

The purpose of this post is to explain the Born-Oppenheimer approximation. Ignoring relativistic fine/hyperfine structure effects, the gross structure molecular Hamiltonian \(H\) (i.e. the “theory of everything”) is: \[H=T_{\text n}+H_{\text e}\] where the nuclear kinetic energy is: \[T_{\text n}=\sum_{i}\frac{\textbf P_i^2}{2M_i}\] … Continue reading →

In this interview, well-known theoretical physicist Frank Wilczek commented that “still to this day I think the quantum theory of angular momentum is one of the absolute pinnacles of human achievement. Just beautiful”. The purpose of this post is to … Continue reading →

The purpose of this post will be to review a standard method for passing from the classical to the quantum world (or more poetically, turning \(\hbar=0\) on to \(\hbar=1\) in natural units). This procedure is called canonical quantization. It is … Continue reading →