Right, but you can still have $\mathrm{SU}(2)$ 'phase factors' depending on your states, and this projective rep discussion looks like it could be adapted, but I'm not sure
Why would one expect that highly exicted harmonic oscillator eigenstates give the classical probability density of harmonic oscillator? Of course by correspondence principle, but the classical probability density is about finding a h.o. in a given position at a *random time*, while the statistical interpretation of QM has nothing to do with time
This is the first time I find a situation where it barely makes sense
I mean, the "classical probability density" is a trivial consequence of its oscillatory motion. Quantum probability density is about the statistical collapse of a wavefunction :/
classical limits are hard to get right, and they are not unique
you can equally well (perhaps even better) argue that the correct notion of "classical state" for the QHO is not "large $n$" but a coherent state with large $\alpha$
high $\alpha$ coherent states describe a localized system oscillating (what a classical h.o. actually is). The thing I wrote above seems to compare two conceptually different things
Just found a useful footnote on Griffiths. We can instead think of an ensemble of oscillators with different initial positions and we measure the ensemble positions all at the same time. This at leats makes it somehow closer to QM
@DanielSank I agree with ACM that most figures are just better without grid lines. Anecdotally it seems to me that they used to be more common a few decades ago and then we collectively moved away from that. Not sure if there's an actual reason or just due to default settings in software
