In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 $$ and then he computes the following $$ \frac{\partial\mathcal{L}}{\partial(\partia...

I'm confusing myself. Let's represent some state in the eigenbasis for Hydrogen: $$|\psi\rangle = \sum_{n,l,m}|n,l,m\rangle\langle n,l,m|\psi\rangle.$$ Now denote the initial state by $\psi(t=0)\equiv\psi_o$, and hit this thing with time evolution: $$U|\psi\rangle = \sum_{n,l,m}e^{-iE_nt/\hbar...

In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily checked that $$\vec\nabla\times\vec{v}=0, \qquad \rho\neq 0.\tag{2} $$ Is it possible to write down a...

Dry ice sublimates at -78.5°C, while the lowest temperature registered is about -89.2°C around the Vostok station in Antarctica. Imagine a person living on that station on that day (or night, since it was measured during the polar night). If that person goes outside, would they find any dry ice ...