Author Archives: wdengquantum.me

Problem #\(1\): What is the Landau-Ginzburg free energy functional \(F[m]\) for the Ising model? Solution #\(1\): It is defined implicitly through: \[e^{-\beta F[m]}=\sum_{\{\sigma_i\}\to_{\text{c.g.}}m(\textbf x)}e^{-\beta E_{\{\sigma_i\}}}\] where \(E_{\sigma_i}=-E_{\text{ext}}\sum_{i=1}^N\sigma_i-E_{\text{int}}\sum_{\langle i,j\rangle}\sigma_i\sigma_j\) is the energy of a given spin microstate \(\{\sigma_i\}\) and “\(\text{c.g.}\)” is … Continue reading →

Problem #\(1\): Define the Poincaré group. Solution #\(1\): In words, the Poincaré group is the isometry group of Minkowski spacetime \(\textbf R^{1,3}\). Mathematically, it is the semidirect product \(\textbf R^{1,3}⋊O(1,3)\) of the normal subgroup \(\textbf R^{1,3}\) of spacetime translations with … Continue reading →

Problem #\(1\): Write down a general \(\phi\)-dependent perturbation to the Klein-Gordon Lagrangian density \(\mathcal L\) for a real scalar field \(\phi\), and explain why in practice only the first \(2\) terms of such a perturbation need to be considered. Solution … Continue reading →

The purpose of this post is to explain what the toric code is, and its potential use as a fault-tolerant quantum error correcting stabilizer surface code for topological quantum computing. To begin, consider an \(N\times N\) square lattice \(\Lambda\) with … Continue reading →

The purpose of this post is to explain why, experimentally, one only observes \(4\) electric dipole transitions in the IR spectrum of buckminsterfullerene, also known as \(\text C_{60}\) or informally as the buckyball: Buckyball Basics The simplest conceptual way to … Continue reading →

The purpose of this post is to explain several techniques for laser cooling and laser trapping of atoms. In order to better emphasize key conceptual points, it will take the approach of posing problems, followed immediately by their solutions. Problem … Continue reading →

Problem: Given a point light source emitting with a Hermitian bilateral \(k\)-space irradiance spectrum \(I(k)\) (i.e. for \(k\geq 0\), the total irradiance contained in the interval \([k,k+dk]\) is \(2I(k)dk\) and furthermore \(I(-k):=I^{\dagger}(k)\)), show that the corresponding fringes \(I(x)\) measured by … Continue reading →

Problem: What does the phrase “spectral line shape” mean? Solution: A spectrum is typically a plot of the intensity \(I(\omega)\sim|E(\omega)|^2\) of some underlying time domain signal \(E(t)\). Problem: What are the \(2\) most prominent kinds of spectra one encounters? Solution: … Continue reading →

Problem: Draw a DNA molecule. Annotate the terms nucleotides, phosphate group, deoxyribose, phosphodiester linkage, adenine, thymine, cytosine, guanine, nitrogenous bases. Problem: Draw an RNA molecule, and make similar annotations as for DNA. Problem: What is the typical length scale \(x_{\text{cell}}\) … Continue reading →