Category Archives: Blog

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 →

Problem: Describe how the classical Hall coefficient \(\rho^{-1}\) and explain why it’s “causally intuitive”. Solution: In the classical Hall effect, the “cause” is both an applied current density \(J\) together with an applied perpendicular magnetic field \(B\). The “effect” is … Continue reading →

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 →