Monthly Archives: September 2024

The purpose of this post is to answer some questions posed to me by ChatGPT regarding my understanding of the \(z\)-transform in digital signal processing and mathematics more broadly. My inquiry to it was simply: “Ask me some questions to … Continue reading →

The purpose of this post is to describe the Aharanov-Bohm phase and its relevance to Dirac’s classic \(1931\) argument for the quantization of magnetic charge (if magnetic monopoles were to exist) using the principles of quantum mechanics. Finally, a few … Continue reading →

The purpose of this post is to explain how the Landau levels of a nonrelativistic charged quantum particle moving in a magnetic field arise. Classically, charged particles spiral along \(\textbf B\)-field lines at the cyclotron frequency \(\omega_B=\frac{qB}{m}\), and indeed this … Continue reading →

The purpose of this post is to provide some intuition for Bloch’s theorem, a result which might be more descriptively called the “periodic potential lemma” or even the “fundamental theorem of condensed matter physics“. Problem #\(1\): Solve the \(1\)st order … Continue reading →

The purpose of this post is to serve as a reference on standard properties of the driven, damped harmonic oscillator: \[\ddot{x}+\Delta\omega\dot{x}+\omega_0^2x=f(t)\] where the damping coefficient \(\Delta\omega>0\) describes the resonant bandwidth of the system’s frequency response (reciprocal to the oscillator lifetime … Continue reading →

Problem #\(1\): Write down a formula for the total energy \(E\) stored in the transverse and longitudinal acoustic phonons of an insulator at temperature \(T\). Solution #\(1\): This should feel completely natural: \[E=\int_0^{k_D}dk 3\frac{V}{(2\pi)^3}4\pi k^2\times\hbar\omega_k\times\frac{1}{e^{\beta\hbar\omega_k}-1}\] where the Debye wavenumber \(k_D\sim(N/V)^{1/3}\) … Continue reading →

Problem: What is the position space wavefunction \(\psi(\textbf x)\) of a photon in a box of volume \(V=L^3\) whose opposite faces are subject to periodic boundary conditions? What about reflecting boundary conditions? Solution: With periodic boundary conditions, the wavefunctions are … Continue reading →

The purpose of this post is to explore the rich physics encoded in the Van der Waals equation of state: \[\left(p+\frac{aN^2}{V^2}\right)(V-Nb)=NkT\] Essentially, the Van der Waals equation of state dispenses with two assumptions implicit in the ideal gas law: First, … Continue reading →

The purpose of this post is to demonstrate how the scattering operator \(S:\mathcal H_{\text{incident}}^{\infty}\to\mathcal H_{\text{scattered}}^{\infty}\), also called the \(S\)-operator for short, despite being defined to enact the scattering \(|\psi’_{\infty}\rangle=S|\psi_{\infty}\rangle\) of asymptotic incident waves \(|\psi_{\infty}\rangle\) off a potential \(V\) into asymptotic … Continue reading →

Problem #\(1\): Derive the band structure of the \(1\)D nearly free electron model. Solution #\(1\): In the \(1\)D extended zone scheme, the band structure (a butchering of the free particle parabolic dispersion) looks like: or in the \(1\)D reduced zone … Continue reading →