Category Archives: Blog

Now suppose that first electron \(e^-\) naturally “burrows” its way down to the ground state \(\textbf n=\textbf k=\textbf 0\) in order to minimize its energy \(E=0\). Now put a second electron \(e^-\) into the box. In reality, the two electrons … Continue reading →

The purpose of this post is to discuss historically one of the first decision problems for which quantum computing was shown to provide an exponential advantage over classical computing. One of the initially striking discrepancies between classical logic gates such … Continue reading →

A qubit is any quantum system with a two-dimensional state space \(\mathcal H\cong\textbf C^2\). In particular because the state space is two-dimensional \(\dim\textbf C^2=2\), the Gram-Schmidt orthogonalization algorithm guarantees the existence of an orthonormal basis \(|0\rangle,|1\rangle\in\mathcal H\) of state vectors … Continue reading →

The purpose of this post is to quickly review some fundamentals of classical computation in order to better appreciate the distinctions between classical computing and quantum computing. Note that the word computation itself, whether classical or quantum, basically just means … Continue reading →

Problem: What does the phrase “\(N\) coupled harmonic oscillators” mean? Solution: Basically, just think of \(N\) masses \(m_1,m_2,…,m_N\) with some arbitrarily complicated network of springs (each of which could have different spring constants) connecting various pairs of masses together: Problem: … Continue reading →

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 →

Problem: Write down the Lagrangian \(L\) of a nonrelativistic classical point charge \(q\) of mass \(m\) moving in an electromagnetic field. Justify it. Solution: \(L=T-V\) as usual where the nonrelativistic kinetic energy \(T=m|\dot{\mathbf x}|^2/2\) (if it were relativistic then \(T=-mc^2/\gamma_{|\dot{\mathbf … 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 →