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Of course! What you're looking for is Noether's theorem; this tells us that for every (continuous) symmetry of a system one may construct a conserved quantity.
by Elpis1 4y ago
Of course! What you're looking for is Noether's theorem; this tells us that for every (continuous) symmetry of a system one may construct a conserved quantity. There are subtleties and exceptions, of course, but that's the gist of it. This is generally how we define things like angular momentum (Woit is referring to this song and dance when he says "Angular momentum is by definition the “infinitesimal generator” of the action of spatial rotations on the theory, both classically and quantum mechanically.")
As a quick example, a hydrogen atom has rotational symmetry, and this corresponds to conserved angular momentum. In turn, this leads to the structure behind the periodic table!
- the__alchemist 4y agoDoes it still have rotational symmetry if the elecron has n>1? Doesn't this lead to wavefunction shapes that aren't spherically symmetric? Thank you. (I'm coincidentally running into this conundrum while trying to build a chemistry visualizer. Have only attempted for n=1 with the potential being a single proton.) What about an electron in more complicated potentials, like the ones you'd see in real life vice textbook examples?
- Elpis1 4y agoGood question! What's important is that the Hamiltonian or Lagrangian has these symmetries. Particular solutions having rotational invariance may signify some nice properties of that solution, but it's quite irrelevant to Noether's theorem.
- wnoise 4y agoSymmetry of boundary conditions cashes out in symmetry of solutions in two ways. The obvious one is solutions that are themselves symmetrical. The less-obvious one is families of solutions, where the symmetry maps one solution to another. All of the "s" states are spherically symmetric. The "p" (and higher) states aren't spherically symmetric, but are instead a basis for an entire family of states that are related to each other through rotations.
- hpcjoe 4y agoAs I recall from grad-school-ish times (1/3 of a century ago ... geez), there was a nice discussion of this in Goldstein's Mechanics text.