Author Archives: wdengquantum.me

Problem: What does it mean to say that a field \(\mathbf E(\mathbf x,t)\in\mathbf C^3\) is a plane wave with speed \(c\geq 0\) in direction \(\hat{\mathbf z}\in S^2\)? Show that a general such plane wave can be written as a Fourier … 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 →

Problem: Define an ideal Fermi gas. Solution: A non-interacting collection of identical fermions (e.g. electrons \(e^-\), neutrons \(n^0\), etc.). Mathematically, the “Fermi” part says that the state space is the antisymmetric submanifold \(\mathcal H=\bigwedge^NL^2(\mathbf R^3)\otimes\mathbf C^{2s+1}\) while the “ideal” part … 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 →

Problem #\(1\): Solution #\(1\): Problem: Solution: Problem #\(3\): Solution #\(3\): Problem #\(4\): Solution #\(4\): Problem #\(5\): Solution #\(5\): First, although this tight-binding model looks like a classical model, in fact it arises from the quantum Hamiltonian \(H=E_01-t_1\sum_n(|n+1\rangle\langle n|+|n-1\rangle\langle n|)\) together … Continue reading →

Problem #\(1\): Derive the Maxwell relation for a gas: \[\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial p}{\partial T}\right)_V\] And explain why Maxwell relations in general should be viewed as much more than just mathematical identities. Solution #\(1\): Here it is clear that one is … Continue reading →