# A question regarding state spaces

1. Jan 15, 2004

### vacuum

I am trying to get oriented in all those QM spaces.

I would appreciate if somenone could point out some source of information (preferably online, though paper is also fine :-) ) which explains the matter in form understandable to humans.

More specifically I need explanation of generalised functions basises (Latin plural of basis would be 'basii'?) and representation of state functions in them as well as how does the Dirac notation transcend to these new objects (As far as I know it was defined with the aid of space - dual space product, which is somewhat more complicated when talking about rigged Hilbert spaces etc.)

2. Jan 15, 2004

### lethe

plural in latin and in english (check your dictionary next time you need to know a spelling, or if you don t have one, use dictionary.com) of the word "basis" is "bases"

3. Jan 16, 2004

### vacuum

Thank you very much for resolving this issue Lethe.

However those other questions still remain. Perhaps I should post this in some other section? Any hints?

4. Jan 16, 2004

### lethe

i m not quite sure what your question is, which is why i didn t say anything. if you want to learn about hilbert spaces, you can probably start with a beginning quantum mechanics book. or if you are more serious, a functional analysis book.

5. Jan 16, 2004

### vacuum

Yes, when I re-read my original post I found it rather uncomprehensible. I wanted to put too much questions in too little space.

Let me try to explain what I need more precisely:

I had a course of mathematical physics in which (for example) a Dirac notation was introduced and this course was based finite dimension spaces.

Now, when I got to Hilbert spaces there was a leap without much mathematical rigour from finite spaces to Hilbert spaces using the same mathematical formalism! I know now how to employ those mathematic instruments to solve problems, but the justification for using them was left out.

Since I didn't find any sort of strict mathematical transition from one case to another in literature I have on disposal, I was wondering if anyone here can recommend me something (more specific than 'functional analysis book' :)

6. Jan 16, 2004

### lethe

are there any specific issues you would like to ask questions about?

anyway, a standard textbook for functional analysis is Reed + Simon, Functional Analysis, part of his series on mathematical physics. also try Quantum Mathematical Physics by Thirring, and Quantum Field Theory by Ticciati. those books also deal with the mathematics of Hilbert spaces rather rigorously.

i will retract my earlier advice that you check a beginning quantum mechanics book. those books tend to gloss over at best, or ignore entirely at worst, the mathematical problems that arise when you move from finite dimensional to infinite dimensional Hilbert spaces.

Last edited: Jan 16, 2004
7. Jan 17, 2004

### Arcon

I was refreshing myself this past summer on this and thought it would be nice to make a web page to have a clear straightforward description of all this. Here is the page I created --
http://www.geocities.com/physics_world/qm/state_space.htm

Hope it helps

8. Jan 19, 2004

### vacuum

Thanks people. This was very helpful for me!

9. Jan 27, 2004

### venkat

hi!
there is a good functional analysis book by Erwin Kreyzig...which has hilbert spaces and banach spaces and stuff!