Just a remark on:
Jan Skibinski begin to put finally this down:
> Here is our first attempt to model the abstract Dirac's
> formalism of Quantum Mechanics in Haskell.
> www.numeric-quest.com/haskell/QuantumVector.html
.....
> The base vectors are abstract: on one hand they are just
> used for identification purposes, on another -- they obey
> all the rules of a vector space. Any vector | x > can
> be represented as a linear combination of the base vectors
> and complex scalars. [..]
>
> We only require and impose the condition, that any two
> base vectors from the same basis are orthonormal, as in:
>
> < (i, j) | (p, q) > = d (i, j) (p, q)
>
> where the left hand side is a scalar product and on the
> right is a generalized definition of the classical Kronecker's
> delta.
=================
There is a lot of interesting things one can do in the domain of
Quantum Mechanics using the FP paradigms. The first thing to notice
is that the states in QM are elements of the Hilbert space, and in
order to do something with them at some abstract level (not just
symbolic manipulations like in M. Horbasch book) we *need* powerful,
higher-order functional system integrated with a decent mathematic
layer. That's why I insist that Haskell Num hierarchy should be
replaced or augmented by something more scholarly.
Having a possibility to work on quantum systems using functional
languages is not just a pastime for failed physicists. This is
a practical issue, a wonderful application domain. Moreover, FP
is - for the moment - a good starting point to construct a simulator
of a "quantum computer", which would be horribly slow, of course,
but at least it would permit many people to try to understand the
basic ideas. I know several computer scientists who would like to
know something more on that, but they have been repelled by the
fact that the examples are elaborated purely theoretically...
I hope that this work will progress. For the moment I would only
say, that the orthogonality requirement of Jan is a bit constraining,
limiting the applicability of the theory to discrete (even finite?)
spaces, while it would be interesting to work with, say |x>, with
x belonging to R3... Moreover, I assure you that some non-orthogonal
bases are of extreme importance in physics, a canonical example
being the coherent states in optics.
Jerzy Karczmarczuk
