Author Archives: wdengquantum.me

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 →

Problem: Explain what is meant by a classical field. Solution: A classical field is any function \(\phi(x)\) (denoted as \(\phi\) because “phi” sounds like “field”) on spacetime \(x:=(ct,\mathbf x)\in\mathbf R^{1,3}\) which is a \(c\)-number, that is to say \(\phi(x)\in\mathbf R\) … Continue reading →

Consider an atomic two-level system with ground state \(|0\rangle\) and excited state \(|1\rangle\). Recall that in the interaction picture, after making the rotating wave approximation and boosting into a steady-state rotating frame, one had the resultant time-independent steady-state Hamiltonian: \[H_{\infty}=\frac{\hbar}{2}\tilde{\boldsymbol{\Omega}}\cdot\boldsymbol{\sigma}\] … Continue reading →

The purpose of this post is to explain the \(2\) key models of classical optics, namely geometrical optics (also known as ray optics) and physical optics (also known as wave optics). Although historically geometrical optics came before physical optics, and … Continue reading →

Problem #\(1\): When someone comes up to you on the street and just says “Ising model”, what should be the first thing you think of? Solution #\(1\): The classical Hamiltonian: \[H=-E_{\text{int}}\sum_{\langle i,j\rangle}\sigma_i\sigma_j-E_{\text{ext}}\sum_i\sigma_i\] (keeping in mind though that there many variants … Continue reading →

The purpose of this post is to study the universal properties of fully developed turbulence \(\text{Re}\gg\text{Re}^*\sim 10^3\). Thanks to direct numerical simulation (DNS), there is strong evidence to suggest that the nonlinear advective term \(\left(\textbf v\cdot\frac{\partial}{\partial\textbf x}\right)\textbf v\) in the … Continue reading →