Author Archives: wdengquantum.me

The purpose of this post is to lay out the basic theory of ensembles in statistical mechanics in addition to some examples. At the end, the goal will be to convincingly demonstrate ensemble equivalence. Any system (e.g. a gas, a … Continue reading →

Problem: What is the state space \(\mathcal H_{\text{TB}}\) of the tight-binding model on a lattice \(\Lambda\) in \(\mathbf R^d\), ignoring spin degrees of freedom and assuming a single state/”orbital” per lattice point in \(\Lambda\). Solution: The tight-binding model assumes a … Continue reading →

Problem: Consider a classical particle subject to Newton’s second law \(\dot{\textbf p}=\textbf F\). If one takes the dot product of both sides with the particle’s velocity \(\dot{\textbf x}\) so that \(\dot{\textbf p}\cdot\dot{\textbf x}=\textbf F\cdot\dot{\textbf x}\), the left hand side is … Continue reading →

The purpose of this post is to contrast the classical treatment of Rutherford scattering with its quantum mechanical treatment. At the end, the surprise will be that for certain important quantities such as the differential cross-section, the classical and quantum … Continue reading →

Problem: Mobius transformations (also called fractional linear transformations) are maps \(\mathcal M:\textbf C\cup\{\infty\}\to\textbf C\cup\{\infty\}\) on the Riemann sphere \(\textbf C\cup\{\infty\}\) of the form \(\mathcal M(z):=\frac{az+b}{cz+d}\) for \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}\neq 0\). The purpose of this problem is to gain a deeper appreciation for … Continue reading →

The purpose of this post is to prove a simple but somewhat unintuitive result in nonrelativistic \(1\)D scattering of quantum particles. Specifically, consider a potential energy landscape \(V(x)\) that decays asymptotically to \(\lim_{|x|\to\infty}V(x)=0\) at \(|x|\to\infty\) which may be asymmetric so … Continue reading →

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 →