Author Archives: wdengquantum.me

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 →

The purpose of this post is to document the uses of several standard components used in optics experiments. Optical Fibers & APC Connectors An optical fiber is a waveguide for light waves. The idea is to use it to transmit … Continue reading →

In cold atom experiments, one very basic question one can ask is, given some atom cloud, what is the number of atoms \(N\) in the cloud? One way is to basically shine some light on the atom cloud and see … Continue reading →

The purpose of this post is to describe the relevant theory needed to understand the paper “High signal to noise absorption imaging of alkali atoms at moderatemagnetic fields” by Hans et al. In particular, a key paper which they cite … Continue reading →

Consider an non-interacting gas of identical fermions (e.g. electrons \(e^-\)); this is called an ideal Fermi gas. Because the Pauli exclusion principle prohibits identical fermions from occupying the same quantum state, the grand canonical partition function \(\mathcal Z\) for an … Continue reading →

The purpose of this post is to prove several general identities concerning the quantum statistical mechanics of an isolated, ideal Bose gas at equilibrium. Problem #\(1\): Specify the physics (i.e. write down the Hamiltonian \(H\) for an isolated, ideal Bose … Continue reading →

Problem: Consider an isolated atom with time-independent Hamiltonian \(H_0\). Such an atom will have many bound \(H_0\)-eigenstates, but for simplicity focus on just two such bound states (think of it as a qubit) \(|0\rangle\) and \(|1\rangle\) (called the ground state … Continue reading →

Problem #\(1\): Consider a simplistic classical model of an atom as a positively charged nucleus \(Q>0\) surrounded by a spherical, uniformly dense electron cloud \(-Q<0\) of radius \(a\). If this atom is subjected to a DC external electric field \(\textbf … Continue reading →

Problem: Explain how to make a Zener diode and the physics underlying its breakdown mechanism. Solution: A Zener diode is simply made by heavily doping a \(p\)-\(n\) junction, leading to a very thin depletion region. Then, upon applying a suitably … Continue reading →