# Configuration Space And Hilbert Space For A Physicist Without Knowledge Of Analysis

I have passed calculus course, have basic knowledge of complex numbers and passed introductory linear algebra course. I am trying to study Griffith Quantum Mechanics book, but I am also checking some other books simultaneously, I see stuff about Hilbert Space, Configuration Space and other stuff I don't know about. I have not studied any analysis book, is there any appropriate book to get straightly to the intended topics? I mean a book which provides my needs?

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• Can you state your needs explicitly please? I don't quite get if you need references for the mathematics part or the physics part – Phoenix87 Dec 27 '14 at 10:29
• @Phoenix87 I need both references and advice about how to deal with it. – FreeMind Dec 27 '14 at 10:31
• A classic is Dirac's Principles, which is a good overview of the formalism used by physicists. For a more modern exposition there are books like Modern Quantum Mechanics by Sakurai, Quantum Mechanics by Messiah, or the volumes of Cohen-Tannoudji. For functional analysis, a good reference is Yosida, but there are many other resources (e.g. the book in here math.ucdavis.edu/~hunter/book/pdfbook.html) – Phoenix87 Dec 27 '14 at 10:39
• @FreeMind: if you would like a book on the issues that you ask, I can recommend "Quantum Theory: Concepts and Methods", by Asher Peres, (Kluwer Academic Publishers, 1993), chapter 4. It goes straightly to Hilbert spaces, and explains their properties. – Sofia Dec 27 '14 at 14:16
• @ACuriousMind : thanks, I recommended him Peres' book. – Sofia Dec 27 '14 at 14:18

The configuration space comprises all the pairs $(\vec r, \vec p)$, where $\vec r$ is the position and $\vec p$ the linear momentum. (In QM one cannot assume that for a given $\vec r$ the particle has to have a precise $\vec p$.) It is also named "phase space".

But in general the configuration space doesn't have to be positions and linear momenta, but some other variables. One can find in the literature the concepts of "generalized coordinates" and generalized momenta. In this case the Hilbert space comprises functions of these variables.

The Hilbert space is the collection of functions defined on some variables, and fulfilling some conditions as for instance integrability in absolute square (see a more rigorous definition in Wikipedia, http://en.wikipedia.org/wiki/Hilbert_space). A typical example of functions that can be elements of a Hilbert space (if the conditions are fulfilled) are the wave-functions defined on the positions of all the particles in a given system of particles (position representation). (Other representations are possible too, momentum representation if the variables in the function are momenta, or sometimes energy representation, and others). For belonging to a Hilbert space, a function has to fulfill some conditions, e.g. form a Hilbert space.

(Book recommended: "Quantum Theory: Concepts and Methods", by Asher Peres, (Kluwer Academic Publishers, 1993). Chapter 4 goes straightly to Hilbert spaces, and explains their properties.)

• While correct, OP is asking for references/resources to learn from, not explanations, so you are not really answering the question as it is posed. – ACuriousMind Dec 27 '14 at 13:45
• @FreeMind : the question is, does your library have that book of Peres? – Sofia Dec 27 '14 at 19:41
• @FreeMind : by the way, I see that many people refer to Griffith's QM book. Is it a good book? – Sofia Dec 27 '14 at 19:42
• @Sofia It it good, but sometimes it does not explain what I want to learn in depth. – FreeMind Dec 27 '14 at 22:19
• @Sofia I found the book :) – FreeMind Dec 27 '14 at 22:20