Tuesday 19 December 2006

Conclusions from functional analysis

Actually I would like to talk about how I got up at 1:30 pm at a day when it already starts becoming dark at 3:30. And how it is to celebrate finishing a lab before I am quite done with all the lab reports. But this would not fit to what my blog's name promises.

What I am talking about today is not really chemistry either but related to it. Maybe it is not interesting to non-mathematicians but I have to post it because I like the conclusion.

I found this book "Lectures in Functional Analysis and Operator Theory"[1] at my grandpa's house. Since functional analysis seems to be a really big thing in theoretical chemistry I took a look into it. What I found out is that a functional analyst is not "sqeamish about using Zorn's lemma", that he "definitely relishes the use of topology", and that "he doesn't stand in the way of the internal algebraic impulses of the subject".

I liked this even though I don't know what topology is and I don't have any idea about the algebraic impulses. But I do know what Zorn's lemma is. We talked about it (squeamishly) in a linear algebra course I took.

Zorn's lemma is equivalent to the axiom of choice. It is used to deal with infinite sets. One interesting conclusion is that it can be used to prove that it is possible to take a solid ball in three dimensional space, cut it up into finitely many pieces, and rearrange the pieces to get two solid balls of the same size (Banach-Tarski paradox).

Does this pertain to a chemist? Do we need such abstract laws with strange conclusions? Well we do. Zorn's lemma is needed to proof that even an infinite vector space has a basis. You need this combined with the spectral theorem to proof the variation principle. And that is something most chemists probably have heard about. It is the basis for pretty much every orbital approximation.

In other words: Next time you look at an orbital, think of the fact that it was derived using the same mathematical principle that states that you can cut up a solid ball into pieces, rearrange the pieces and make two solid balls of the same size out of it.

By the way, I haven't really gotten farther in the book than the text I've quoted. Maybe eventually I will do some extra math and then I will be able to perform some nice Kohn-Sham equations.

[1] Berberian, SK. Lectures in Functional Analysis and Operator Theory. 1974. Springer.


Ψ*Ψ said...

Yay DFT!
This is strangely exciting. Maybe I am a closeted spectroscopist?

Felix said...

Your name kind of suggests that :), especially if the star means "complex conjugate" and not just "times"

Ψ*Ψ said...

but of course it's the complex conjugate!
all i need is an excuse to start a side project involving lasers. alas, i have no such excuse.