The h Bar

Main thing of interest in QM is that $D_\nu(z)$ is basically what you get for the harmonic oscillator wavefunction if you didn't worry about boundary conditions

Emilio Pisanty

9:31 PM

But

Semiclassical

when $\nu$ is a nonnegative integer you get that form of hermite polynomial * Gaussian

Emilio Pisanty

Uh

But

You do care about bcs

Semiclassical

sure. But suppose you want to do WKB on some other potential

Emilio Pisanty

@Semiclassical that does make more sense though

Semiclassical

if you want to do WKB in a way that's better behaved than the usual "linear at turning points" approach, you do quadratic WKB stuff instead of linear

and then you very much care about the parabolic cylinder functions when $\nu$ is almost but not quite integer

Emilio Pisanty

9:33 PM

@Semiclassical ugh

Semiclassical

lol

it shouldn't be too surprising, really. doing linear wkb means you don't worry too much about what happens at the bottom of a well; you just care about the turning points

but for that reason the usual linear WKB stuff doesn't work so well for the ground state

doing quadratic WKB fixes that

though tbh I don't really care about all that

for me it's more of a test case of "given an integral representation, how do asymptotic expansions work"

and I'm using it as an excuse to learn Borel summation stuff

Emilio Pisanty

@Semiclassical oh, no, it sounds perfectly reasonable

It just sounds super boring

Semiclassical

lol

Emilio Pisanty

@Semiclassical that does sound interesting enough though

Semiclassical

Some stuff in that vein is in those notes I linked, which I still think you should browse through

plus these slides: icts.res.in/sites/default/files/…

(they talk about the Airy function and its stokes lines, which I think we've touched on before?)

lots of fun stuff

Emilio Pisanty

9:41 PM

1

A: Unitary Matrix for Block Diagonalization

I suspect if you posted this in the math SE you'd get excessively systematic and pithy answers, and aim at generality. Here, I'll just remind you what your basic linear algebra text almost certainly covers. The point is you are meant to immediately observe $C_3$ (is the cyclic shift matrix and ...

Lolz

@Semiclassical I'll take a look soonish

I'm pretty frazzled atm

Semiclassical

Fair enough

bolbteppa

10:01 PM

"The Poincare group has rank two, so we expect to find two Casimir operators. I’ll look for a good mathematical argument for this and add it here if I find it. The way I think of it is that the translation part doesn’t matter for the rank since it’s abelian, so we’re left with SO(1,3), which is basically SO(4), which (as we will discuss later) is SU(2) × SU(2). Now SU(2) has rank 1 so SU(2) ×SU(2) has rank 2 so that’s the rank of the full group" hmm

"There is a way to "understand" why the number of Casimirs of the Poincaré group is 2. The Poincaré group is a Wigner-İnönü contraction of the de-Sitter group SO(4,1) which is semisimple and of rank 2. The Casimirs of the Poincaré group can be obtained from the Casimirs of SO(4,1) explicitly in the contraction process. This is not a full proof because group contractions are singular limits, but at least it is a way to understand the case of the Poincaré group"

Two hand-waving arguments for this, that I guess you have to take seriously :(

peterh

@Secret It may be so in many cases. However, the behavior of the so-named "autistic children", particularly of the easier cases, seem to me quite similar to the behavior of children who are simply revolting against the care of their parents - but they are not capable to explain their problems on a more verbal way.

Jaime Gallego

10:47 PM

@BernardoMeurer This actually overlaps with an experiment I've thought of doing.

It basically is developing an esoteric digital radio mode that encodes data through melodies. The challenge is to algorithmically generate songs we can consider "harmonious" from the data, not just a simple mapping to avoid repetition.

It looks feasible to me (given that mathematical music theory exists!) but that kind of fields merger between music and signals sure looks complicated.

dmckee

11:22 PM

There is an old pseudo-proof that there are no un-interesting numbers.

Semiclassical

"once you choose one, it's interesting"

dmckee

If only some numbers are interesting then there is a set of uninteresting numbers. But one of those must be the lowest uninteresting number which would be an interesting thing about it. By induction the set of uninteresting numbers must be empty.