joshuahhh

In section I.3 of QFT in a Nutshell, Zee examines the path integral for the Lagrangian \(\mathcal{L}(\phi) = \frac{1}{2}[(\partial \phi)^2-m^2\phi^2]\) (the “free”/”Gaussian” Lagrangian) with a source \(J\). This looks like $$\begin{align} Z(J)&=\int D\phi\,e^{i\int d^4x\,\{\frac{1}{2}[(\partial \phi)^2-m^2\phi^2]+J\phi\}}\\ &=\int D\phi\,e^{i\int d^4x\,\{-\frac{1}{2}\phi(\partial^2+m^2)\phi+J\phi\}}. \end{align}$$ (Integration by parts allows us to replace \((\partial\phi)^2\) with \(\phi(\partial^2\phi)\) in the exponent’s integrand.) The exponent [...]

On my last test, I had trouble with a couple questions because I wasn’t totally clueful about how angular momentum works in atoms. I should fix this. Orbital angular momentum should come naturally: Our standard way of understanding electron states in atoms is with orbitals, which are chosen to have well-defined angular momentum quantum number [...]

So the electron in the hydrogen atom is just a particle in a spherically-symmetric 1/r potential… you’ve got a ladder of energy eigenvalues indexed by a quantum number n. The nth eigenvalue has degeneracy n2, but that’s cool; picking an axis z, the total angular momentum operator L and the z-axis angular momentum operator Lz [...]