Monthly Archives: August 2024

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 →