Category Archives: Blog

Given a mixed ensemble \(\{(|\psi_n\rangle, p_n)\}\) of pure quantum states \(|\psi_n\rangle\in\mathcal H\) with statistical probabilities \(p_n\in[0,1]\), the Hermitian, positive semi-definite, unit trace density operator \(\rho_{\{(|\psi_n\rangle, p_n)\}}:\mathcal H\to\mathcal H\) of that mixed ensemble is defined by the formula: \[\rho_{\{(|\psi_n\rangle, p_n)\}}:=\sum_np_n|\psi_n\rangle\langle\psi_n|\] In … Continue reading →

The purpose of this post is to flesh out some of the subtleties of the Stern-Gerlach experiment which historically was a demonstration of angular momentum quantization in quantum mechanics. Problem #\(1\): What should the charge \(Q\) of the particles used … Continue reading →

Parity Classically, if one takes a trajectory \(\textbf x(t)\) and reflects it about the origin to obtain the reflected trajectory \(\textbf x'(t)=-\textbf x(t)\), then the momentum of the particle \(\textbf p=m\dot{\textbf x}\) is correspondingly reflected \(\textbf p’=m\dot{\textbf x’}=-m\dot{\textbf x}=-\textbf p\). … Continue reading →

In number theory, the fundamental theorem of arithmetic clarifies why prime numbers are so important, namely that they form a “multiplicative basis” with which one can uniquely factorize any positive integer \(n\in \textbf Z^+\). In the same spirit, Schur’s lemma … Continue reading →

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 →

As a \(2024\) summer intern in the University of Toronto’s Summer Undergraduate Research Program, I will be working with Dr. Nina Gusinskaia and Dr. Rik van Lieshout (both postdoctoral researchers working jointly at the Canadian Institute for Theoretical Astrophysics (CITA) … Continue reading →

Given two real-valued analog signals \(f(t), g(t)\), there are two very similar but distinct bilinear signal processing operations that are commonly performed in the field of analog signal processing on \(f\) and \(g\) to obtain a new function. The first … 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 →