Author Archives: wdengquantum.me

In sufficiently symmetric geometries, the method of images provides a way to solve Poisson’s equation \(|\partial_{\textbf x}|^2\phi=-\rho/\varepsilon_0\) in a domain \(V\) subject to either Dirichlet or Neumann boundary conditions (required for the uniqueness theorem to hold) by strategically placing charges … Continue reading →

Problem: Distinguish between the terms “intrinsic semiconductor” and “extrinsic semiconductor“. Solution: An intrinsic semiconductor is pretty much what it sounds like, i.e. a “pure” semiconductor material like \(\text{Si}\) that is undoped with any impurity dopants. An extrinsic semiconductor is then … Continue reading →

Problem: A vibrating string with displacement profile \(y(x,t)\) has non-uniform mass per unit length \(\mu(x)\) and non-uniform tension \(T(x)\) experiences both an internal restoring force due to \(T(x)\) but also a linear “Hooke’s law” restoring force \(-k(x)y(x)\) everywhere so that … Continue reading →

Problem: Why is it important to learn how to learn? Solution: In a post-AGI world where each day feels like swallowing a fire hose of new (mis!)information (especially as a researcher), the most important arrow one can have in the … Continue reading →

Problem: What is an estimator \(\hat{\theta}\) for some parameter \(\theta\) of a random variable? Distinguish between point estimators and interval estimators. Solution: In the broadest sense, an estimator is any function of \(N\) i.i.d. draws of the random variable \(\hat{\theta}(X_1,…,X_N)\). … Continue reading →

Problem: Explain what the spacetime complexity of an algorithm on a problem of characteristic input size \(n\) is. Solution: Time complexity \(T(n)\), is the number of \(\Theta(1)\)-operations that an algorithm requires to produce a solution to a problem of input … Continue reading →

Problem: Given a linear operator \(H\) on some vector space, define the resolvent operator \(G_H(E)\) associated to \(H\). Solution: The resolvent \(G_H(E)\) of \(H\) is the operator-valued Mobius transformation of a complex variable \(E\in\textbf C\) defined by the inverse: \[G_H(E):=\frac{1}{E1-H}\] … Continue reading →

Although Feynman diagrams are often first encountered in statistical/quantum field theory contexts where they are employed in perturbative calculations of partition/correlation functions based on Wick’s theorem, there is a lot of “fluff” in these cases that obscures their underlying simplicity. … Continue reading →

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 →