Let’s talk about the basic setup of quantum field theory. Start with a single quantum harmonic oscillator. Its energy eigenvalue equation is \[ H|\psi\rangle = E|\psi\rangle . \] It has a set of discrete energy eigenstates. For the harmonic oscillator, a more convenient set of eigenstates is the number states \(|n\rangle\) (\(n=0,1,2,\dots\)), satisfying \(H|n\rangle=E_n|n\rangle\). We introduce the annihilation operator \(a\) and the creation operator \(a^\dagger\), require them to satisfy \([a,a^\dagger]=1\), and define the number operator \(N=a^\dagger a\). The number states are also eigenstates of \(N\): ...

We want to classify different particles using wavefunctions. Consider a two-particle system: if we exchange the two particles, how does the state of the system change? In classical mechanics, even if two particles are completely identical in their physical properties, in principle we can still distinguish them by “labels” (such as “particle 1” and “particle 2”). Therefore, exchanging the positions of two classical particles produces a new, distinguishable microscopic configuration. ...

Today, on a whim, I wanted to study various courses in English, like biology and geography. There are quite a few The Great Courses offerings on Bilibili, but based on my previous viewing experience, these courses often use a lot of terminology; just listening—even with English subtitles—I still can’t fully understand them. So I figured I’d add bilingual Chinese–English subtitles to these course videos and then watch them. The technical plan is straightforward: use yutto to download videos from Bilibili, ffmpeg to extract audio from the videos, whisper.cpp to transcribe the audio into SRT subtitles (by the way, I’m using Apple M3 Silicon), and finally use the DeepSeek API to translate the SRT subtitles. ...

This morning at work I wanted to listen to an audiobook, so I searched online and found this 【“Madness and Civilization” Michel Foucault【Audiobook | Synchronized Text】】 I think what he’s done is very appropriate. For sociology- and philosophy-related books, just reading often feels dull and leads to giving up; just listening often means forgetting the context, zoning out, and not knowing what’s being said. These kinds of books are best suited for audiobooks with synchronized text. So I started thinking: how did he make an audiobook like this? What AI tools did he use? How did he align the text and audio precisely? I dug through r/localllama for quite a while. Open-source solutions are usually very complicated: first convert the ebook into audio and timestamped subtitles, then synchronize the subtitles with the audio to produce a video. ...

For the one-dimensional harmonic oscillator problem, its Hamiltonian can be written as \(H = \frac{p^2}{2} + \frac{1}{2}\omega^2 x^2\). Our goal is to solve its energy eigenvalue equation \(H\varphi = E\varphi\). To solve it using a more concise algebraic method, we introduce a pair of ladder operators defined as: \[ a^{\pm} = \frac{p \pm i\omega x}{\sqrt{2\omega}} \] Here \(a^+\) is usually called the creation operator \(a^\dagger\), and \(a^-\) is called the annihilation operator \(a\). With these operators, we can express the Hamiltonian in a more compact form. We define the number operator as \(N = a^\dagger a\). After some derivation, we find that the relation between the Hamiltonian and the number operator is: ...

To solve for the state of a particle in a system with a central potential, such as an electron in an atom, we describe its state using a wavefunction \( \psi(r, \theta, \phi) \) in spherical coordinates. A key mathematical technique here is the separation of variables. This method is applicable because the system’s potential energy is spherically symmetric, meaning it only depends on the distance \(r\) from the center, not on the angles \( \theta \) or \( \phi \). This symmetry allows us to decompose the wavefunction into a product of a radial part \( R(r) \) and an angular part \( Y(\theta, \phi) \). By expressing \( \psi(r, \theta, \phi) = R(r)Y(\theta, \phi) \), we can transform the single complex Schrödinger equation into a set of simpler, one-dimensional ordinary differential equations, which can be solved separately. ...

A symmetry can be understood as an operator that, when applied to a system, leaves its fundamental characteristics unchanged. For instance, a crystal lattice exhibits translational symmetry; shifting its position by a lattice vector does not alter its structure. In quantum mechanics, this concept is deeply tied to the degeneracy of energy levels. Degeneracy means that different quantum states can share the same energy. While symmetry sometimes implies degeneracy, but degeneracy in energy levels is always a sign of an underlying symmetry in the system. ...

In quantum mechanics, the state of a physical system is described by a state vector, denoted using Dirac’s bra-ket notation as \(|\psi\rangle\). These state vectors are elements of a complex vector space called a Hilbert space. Given two state vectors \(|\psi\rangle\) and \(|\phi\rangle\), their inner product is a complex number denoted by \(\langle\phi|\psi\rangle\). If the inner product of two different states is zero, i.e., \(\langle\phi|\psi\rangle = 0\), the states are said to be orthogonal, representing completely independent physical situations. ...

Following Susskind’s Statistical Mechanics course, lecture 9. The energy of a single spin in the Ising model is given by: \[E = -J\sigma\]where: \(E\) is the energy of the spin \(J\) is the coupling constant (strength of interaction) \(\sigma\) is the spin value, which can be either +1 (spin up) or -1 (spin down) Focus on a single spin. Its partition function \(Z\), summing over the two configurations (\(\sigma = +1, \sigma = -1\)), is: ...