Subject: Re: CCL:Conservation of Difficulty

From: <[email-masked]>
Date: Wed, 6 Mar 1996 07:50:41 -0600

Equivalence of computational complexity is well known.  If you could solve
the traveling salesman problem, you could solve many other difficult computing
problems.  By gross extrapolation, not only have alot of smart people been
trying to find simple analytical solutions to the solutions to the many body 
Schroedinger equation without success, many more people in other fields have 
been working on problems that are computationally equivalent, also without 
success.  If there is a simple solution, finding it certainly is not easy!

That most of the problems we have solved tend to have simple solutions 
(Occam's
razor), does not imply that all problems have simple solutions.  Maybe we just
are not smart enough to solve the problems with really complicated solutions. 
 The conclusion from computer science is that there seem to be whole classes 
of problems that are just plain hard.

David

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From: [email-masked] (Konrad HINSEN)
Date: Wed, 6 Mar 1996 10:56:39 -0500

> right.  I firmly believe that the Schrodinger equation will one day
> be as easily dealt with as trignometric functions on a calculator.

Which means by numerical approximation ;-)

Seriously, I wonder what your confidence is based on. Of course it may
be possible to find further analytic solutions, but I don't see why
this should necessarily be true. Much less do I see how you can make
such a claim for *all* applications of the Schroedinger equation.

Of course all this depends on what you call a "solution". In the case
of trigonometry, all we have is a set of analytic relations between
various functions that often occur together. With the exception of a
few special cases, there is no "analytic" answer to a trigonometric
problem, in the sense that the result cannot be expressed in simpler
functions (i.e. sums, products, and powers). A comparative level of
"solution" of the Schroedinger equation would be a safe numerical
procedure that can find the solution to any problem to a specified
accuracy. This procedure could then be put into the theoretical
chemist's equivalent of a pocket calculator. If that's what you mean
by solution, I agree that it will probably one day be available.

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