[seqfan] Re: Quantum Pascal's Pyramid

[Brad Klee]

Hi Xavier,

I think the notebook, and references therein, are a good place to start
once you have a basic idea of the quantum harmonic oscillator (
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator ). This topic is
discussed in every textbook on quantum mechanics. You need to do the one
dimensional case and then generalize to two or more dimensions. Then you
can start to think about motions of crystals and molecules.

The two-dimensional case is essentially the same as the one-dimensional
case except that it becomes possible to transform from Cartesian
coordinates to Polar coordinates. This is one way to introduce what I call
the "Pseudotop analogy": Quantum objects fixed in space can appear to
rotate. This is a fundamental quantum principle, which is the corner-stone
of many investigations including *Crystal Field Splitting*. It is also a
serious headache once you try to formulate the concept of total angular
momentum.

Coordinate transformation is what we are studying in the context of this
number pyramid.

I figured out a better way to formulate the matrix elements and have
started writing a draft:

https://oeis.org/draft/A268533

The draft already contains some nice examples of how the coordinate
transformations work, but it's important to remember that the space of
irreducible representations is infinite dimensional. Taking the N^th block
of the pyramid, we can transform any set of polynomials:

{ ( x + d/dx)^n * (y+d/dy )^m * 1 : m+n = N },

or alternatively, any representation of angular momentum operators ( J_1,
J_2, J_3 ).

Thanks,

Brad







On Tue, Feb 23, 2016 at 2:14 AM, Xavier Combelle <xavier.combelle at gmail.com>
wrote:

> 2016-02-21 21:22 GMT+01:00 Brad Klee <bradklee at gmail.com>:
>
> > Hi Seqfans,
> >
> > Many of you are probably familiar with the su(2) algebra, as it has been
> > important in both math and physics for nearly 100 years.
> >
>
> I'm just a beginner in math, is there any easy introduction to it on the
> web ?
> the wikipedia page (https://en.wikipedia.org/wiki/Special_unitary_group)
> is
> too much synthetic
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>