Author Archives: wdengquantum.me

Problem: Given a function \(f(x,p)\) defined on a single-particle classical phase space \((x,p)\) in \(1\) dimension, define the Weyl transform \(\hat f\) of \(f(x,p)\). Solution: \[\hat f=\int\frac{dx’dp’}{h}e^{i(p’X+x’P)/\hbar}\tilde f(x’, p’)\] where the phase-space Fourier transform \(\tilde f(x’,p’)\) of \(f(x,p)\) is given … Continue reading →

Problem: At a high level, what is the goal of classical hydrodynamics? Solution: The program of classical hydrodynamics seeks to bridge the physics of a many-body system at different length scales. The idea is to start from microscopics (i.e. Newton’s … Continue reading →

Problem: Prove the Sokhotski-Plemelj theorem on the real line \(\textbf R\): \[\frac{1}{\omega-\omega’\pm i0^+}=\mathcal P\frac{1}{\omega-\omega’}\mp i\pi\delta(\omega-\omega’)\] Solution: (in this solution, the function \(\chi(\omega)\) is assumed to be analytic on the integration interval in \(\textbf R\) to be able to apply the … Continue reading →

Problem: Distinguish between a supervised data set and unsupervised data set. Solution: Supervised data is of the form \(\{(\textbf x_1,y_1),…,(\textbf x_N,y_N)\}\). Unsupervised data consists of stripping away the target labels \(y_1,…,y_N\), leaving behind just the feature vectors \(\{\textbf x_1,…,\textbf x_N\}\). … Continue reading →

Problem: In a multilayer perceptron (MLP), how are layers conventionally counted? Solution: The input layer \(\textbf x\equiv\textbf a^{(0)}\) is also called “layer \(0\)”. However, if someone says that an MLP has e.g. \(7\) layers, what this means is that in … Continue reading →

Problem: Given a system of \(N\) identical bosons or fermions with Hamiltonian \(H\) in a mixed ensemble described by a density operator \(\rho\) (usually \(\rho=e^{-\beta H}/Z\) or \(\rho=e^{-\beta(H-\mu N)}/Z\) in equilibrium at temperature \(T=1/k_B\beta\) and chemical potential \(\mu\) though one … Continue reading →

Problem: What does it mean for \(N=2\) particles to be identical? Solution: \(N=2\) particles are identical iff their intrinsic properties are all identical; in classical mechanics this typically means mass \(m\), charge \(q\), etc. while in quantum mechanics this typically … Continue reading →

Problem: What is the meant by the phrase “elementary excitations” of an ideal Fermi gas? Solution: Basically “excitations” is a fancy word for “excited states”, in this case more precisely “many-body excited states”. One example is depicted in the diagram … Continue reading →

Problem: Somewhat bluntly, what is machine learning? Solution: Machine learning may be regarded (somewhat crudely) as just glorified “curve fitting”; an “architecture” is really just some “ansatz”/choice of fitting function that contains some number of unknown parameters, and machine learning … Continue reading →

Problem: Consider placing a fictitious open surface in an equilibrium ideal gas at temperature \(T\); although the net particle current density through such a surface would be \(\textbf J=\textbf 0\), if one only counts the particles that go through the … Continue reading →