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 prove Bloch’s theorem from a physicist-oriented perspective. At the end, a few applications of Bloch’s theorem are given. Bloch’s Theorem: Given an arbitrary Bravais lattice \(\Lambda\) underlying some crystal with corresponding \(\Lambda\)-periodic potential … Continue reading →

Given a collection of \(N\) points \(\textbf x_1,\textbf x_2,…,\textbf x_N\in\textbf R^2\) in the Cartesian plane \(\textbf R^2\), one can enclose each such point \(\textbf x_i\) in a Voronoi cell consisting of the set of all points \(\textbf x\in\textbf R^2\) in … Continue reading →

The purpose of this post is to describe a theoretical model of solids based on the quantization of sound waves in solids into quasiparticles known as phonons due to Debye which culminates in a successful explanation for the experimentally observed … Continue reading →

The purpose of this post is to present a derivation of the Planck distribution which governs the phenomenon of so-called blackbody radiation. I feel a better (though longer) name would have been universal quantum thermal radiation (UQTR), because ultimately any … 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 →

In crystals, electrons roam nearly free,But weak potentials from atoms we see,Energy bands form, with gaps in between,Creating the structure of metals so keen. Near Brillouin zones, waves scatter and split,Forming conductors, insulators, bit by bit.Semiconductors bridge these worlds with … Continue reading →