Help With Hilbert Space

Antimatter · 2007 February 17th, 02:16

Hello everyone,

We recently covered a small Quantum Mechanics section in our chemistry class which left me a little confused. I am no where close to having a complete understanding of QM. I'm having trouble with the concept of Hilbert Space. Wikipedia isn't exactly being helpful. I'd highly appreciate it if someone explained this concept. Sorry if I made some sort of mistake; I don't know much about QM (only 2 weeks worth of it, in fact:) ), so please excuse me if I type something stupid.

Thanks a bunch

Wilhelm · 2007 February 17th, 13:39

Quote:

Originally Posted by Antimatter

Hello everyone,

We recently covered a small Quantum Mechanics section in our chemistry class which left me a little confused. I am no where close to having a complete understanding of QM. I'm having trouble with the concept of Hilbert Space. Wikipedia isn't exactly being helpful. I'd highly appreciate it if someone explained this concept. Sorry if I made some sort of mistake; I don't know much about QM (only 2 weeks worth of it, in fact:) ), so please excuse me if I type something stupid.

Thanks a bunch

Don't worry about asking stupid questions around here. That's the whole point of having an anonymous forum for this sort of thing. Also, don't worry about not understanding QM completely, and don't believe someone who does tell you they understand it completely.

As for Hilbert space, it's something that's not too complicated to understand if you've had some background in linear algebra, but without that it just sounds like a lot of buzz words strung together. Basically, and "space" is just a set of "things" together with a set of operations on those "things". Any specific space is just a set of specific things with a set of specific operations. In Hilbert space the "things" are a set of vectors (or functions) defined over the complex numbers that can be combined by an operation called an inner product (or integral) that also has certain properties.

In terms of the interpretations of what all this means, the constraints on the wave functions in the Hilbert space force probability to be conserved (the vectors/functions are normalized), makes sure that the probabilities of things don't depend on the way YOU are looking at them (basis set invariance) etc. It’s not an easy thing (for me) to explain at an elementary level and I can’t think of a book that does it well, but there are a lot of people on here who know a lot more about this than I do who make be able to clarify this more and perhaps correct anything I’ve misspoken about. Best of luck.

Antimatter · 2007 February 17th, 14:04

Thanks a lot, Wilhelm. I think I have a better understanding of it now.