...although apparently there are exactly two readers/writers
of this thread on this list. Oh, well, it is as boring as any
other subject.
Jan Skibinski comments my observation:
> > ... I assure you that some non-orthogonal
> > bases are of extreme importance in physics, a canonical example
> > being the coherent states in optics.
> I think Jerzy is talking about cases conceptually sketched below.
>
> / e2 |e2'
> / |
> / |
> /_______ | contravariant basis
> e1 \
> covariant basis \
> \ e1'
>
> e1 * e2' = 0 e1*e1' = 1
> e2 * e1' = 0 e2*e2' = 1
==
Not really. This is not a contra/co geometric problem. In fact, this
begins to be interesting in infinite-dimensional spaces. The coherent
states (eigenstates of the annihilation operator) describe laser beams,
superfluidity, currents in the Josephson junction, and (quasi)
"classical"
distribution within the quantum formalism. They are truly
non-orthogonal,
and redundant (it is an over-complete basis; there are *analytic*
relations
between various basis vectors, which are labeled by complex numbers.
That's why they are called "coherent").
Now, what has all that to do with Haskell?
For most of you probably nothing.
I confess that I became interested in Haskell *because* of its possible
applications to scientific computing, and *in particular* to quantum
physics. (And some statistical physics; the underlying math is very
similar, and this is not accidental).
Mind you, this is a domain where you see immediately the necessity of
computing using higher-order functions!
Your states are functions. Your mathematical objects are functions. Your
physical quantities (observables) are functions acting on states.
Most problems in QM cannot be solved without using perturbation methods.
The perturbation formulae are usually very tedious to implement, unless
one dares to use some lazy coding.
Then you can economize a few days of pencil work, and you can spend this
time rolling on the ground and laughing at the people who claim that
Haskell is useless for practical computations, because they don't know
how to implement some middle-Chinese chess in it.
Jerzy Karczmarczuk
Caen, France
