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MathGroup Archive 2001

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Re: Any quantum chemists / physicists?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30260] Re: Any quantum chemists / physicists?
  • From: "Urs Schreiber" <Urs.Schreiber at uni-essen.de>
  • Date: Sat, 4 Aug 2001 01:14:27 -0400 (EDT)
  • Organization: University of Essen, Germany
  • References: <9k5r6g$hc1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"Gustavo Seabra" <gseabra at swbell.net> wrote in article
news:9k5r6g$hc1$1 at smc.vnet.net...

>     Are there any quantum chemists / physicists around using Mathematica?
I
> was wondering if there is any package that handles things like
commutators,
> operators algebra, second quantization, superoperators, etc...


Hello everybody,

I am new to this list and pretty new to Mathematica. I am currently working
on my diploma thesis in theoretical physics on supersymmetric quantum
cosmologies. I purchased Mathematica a couple of weeks ago and started
coding right away in kind of a hurry, because I was in bad need for some
computations that were just too tedious to do by hand. But I am constantly
worrying that I am reinventing wheels, therefore I got interested in this
discussion here. I'd be grateful for hints on what of the following is
covered by publicly available packages:

Supersymmetric quantum mechanics is intimately related to differential
geometry, so the first thing I needed was a differential geometry package. I
have written code that takes a metric and spits out all the important
tensors (connection, curvature, etc.) as well as the exterior derivatives
and the Hodge Laplacian. For the latter I had to implement some exterior
algebra with anticommutation relations and such. This works fine now, but
since I all made it up as I needed it I am somewhat unsatisfied with the
generality of the code. (But at least on tensor analysis there are surely
vast amounts of code out there?)
Since the question arose on this thread: Second quantization is covered in
(very) small part by this code: Since Fermi operators on curved manifolds
are isomorphic to differential forms, implementing one means implementing
the other. The exterior derivative can be regarded as a combination of a
Bose and a Fermi operator (then usually called a supercharge). I told
Mathematica how to convert my exterior operators into matrices of
differential operators on the space of forms (i.e. the Hilbert space of
states of the susy sigma model), that is: generalized Dirac operators. I
need to find the states annihilated by these operators.
Now that Mathematica generates the equations (systems of pde's) that I need
I want it to solve them, or at least help out a little in solving them. But
this turns out to be unexpectedly tough. Am I correct in observing that only
a very restricted amount of types of pde's is supported by Mathematica? So
currently I am trying to simplify the questions submitted to the kernel and
that's probably fine the way it is, since I should do something myself :-)
(and this is the only way to find deeper structures)  But if anyone knows
where to find packages that, for example, know how to compute closed,
coclosed and harmonic differential forms, I'd appreciate references.
Alternatively, or even better, I'd like to know about literature on the
theory of explicitly *finding* harmonic forms computationally (I know the
theory on the *existence* of closed form, i.e. Hodge's theorem and the like,
but even that does not help much in my case, since I am concerned with
manifolds not compact).

Regards,
Urs Schreiber

--
Urs.Schreiber at uni-essen.de





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