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Thread: Hilbert space

  1. #1 Hilbert space 
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    We seem to be encountering in several threads on quantum mechanics questions involving the nature of "wave functions" and Hilbert spaces.

    I will try to briefly sketch the salient points regarding the theory of Hilbert spaces. However, this will in no way be complete. It is a fairly large subject and there are several very good text books available. Some of the better references are:

    Introduction to Hilbert Space -- Paul Halmos

    A Hilbert Space Problem Book -- Paul Halmos

    Real and Complex Analysis -- Walter Rudin

    Functional Analysis -- Walter Rudin

    The most important examples of Hilbert spaces are the square summable sequences and more general square integrable functions on some set. The latter is the setting usually encountered in quantum mechanics, and requires the theory of the Lebesgue integral, which in turn requires the general theory of measure and integration. For that material good references are:

    Real and Complex Analysis -- Walter Rudin

    Measure Theory -- Paul Halmos

    Measure and Integration -- Leonard Richardson


    OK, here we go.

    1. A vector space is an algebraic structure consisting of:

    a) An abelian group of vectors (we use capital letters to denote vectors) with the operation of vector addition + and an identity for vector addition denoted by 0.

    b) A field of scalars (we use lower case letters to denote scalars), which we will always take to be either the real or complex numbers in all that follows. (In algebra one considers more general fields but in the theory of Hilbert spaces one considers only the real and complex numbers.

    c) An operation of scalars on vectors called scalar multiplication follows the expected algebraic relations:





    2. An inner product on a vector space is a function that takes pairs of vectors to scalars <X,Y> denoting the scalar associated with the pair of vectors X and Y such that:




    Such a bilinear form is called sesquilinear in the case of complex scalars, and simply bilinear in the case of real scalars in which the complex conjugate indicated is superfluous.

    With the notion of an inner product comes a natural notion of the magnitude of a vector:

    3.

    This gives us as well the notion of the "distance" between two vectors -- making an inner product space into a normed space which is thus a metric space in the sense of topology.


    Definition: A sequence of vectors is said to converge to if given there is a some such that whenever

    Definition A sequence of vectors is said to be Cauchy if given there is a some such that whenever

    Definition An inner product space (a vector spacewith an inner product) is complete if every Cauchy sequence converges.

    This definition of completeness extends rather naturally to the more general setting of a metric space (a topological space with a notion of distance) and is central to much of modern mathematical analysis. Completeness is the most important difference between the real numbers and the rational numbers. In simple terms the real number have no "holes" while the rational numbers have lots of "holes" (which are filled in by the irrational numbers when one constructs the real numbers. The complex numbers are also complete.

    Definition A Hilbert space is a complete inner product space.

    The most important Hilbert spaces are separable Hilbert spaces.

    Definition A Hilbert space is called separable if there exists in it a countable dense set. That is, there is some set of elements of the Hilbert space such that given any element in the Hilbert space and any real number then for some .

    If one has a separable Hilbert space then it can be proved that:

    1. There exists a countable set of orthonormal vectors that is complete, meaning that
    a) for all
    b) if
    c) the set of finite linear combinations of the is dense.

    2. Given any element in the Hilbert space then the series

    converges to as descrsibed above.

    Because of this a complete orthonormal set of vectors is called a Hilbert space basis. Note that is is a basis in the usual sense if the Hilbert space is finite-dimensional, but not if the Hilbert space is infinite-dimensional (a basis, also called a Hamel basis requires that every element in the vector space be represented as a finite linear combination of the basis vectors).

    Those who know the theory of Fourier series of periodic functions, will note that the expressions above bear a striking resemblance to the representation of functions in that theory. That is not an accident. The trigonometric functions that are periodic on some interval are a complete orthonormal set for the Hilbert space of functions defined on the interval that are square-integrable and the theory of Fourier series is really just the theory of that Hilbert space.

    Which brings us to the common examples of Hilbert spaces:

    1) The set of square summable sequences of complex numbers where wherein the inner product is given as



    2) The set of square integrable complex valued functions defined on some interval of the real line, the entire real line, some region of n-space or all of n-space with the inner product of two functions being


    3) The real Hilbert spaces defined analogously to 1) and 2) but using only the real number system.

    Caveat: You will note that I did not address the issue of completeness (convergence of Cauchy sequences) in the examples above. It can be shown that the Hilbert space of square-summable sequences is complete. It can also be shown that the Hilbert spaces of square integrable functions are complete, but this requires that one interpret the integral as the Lebesgue integral, which in turn requires elements of the theory of measure and integration. That is the reason for inclusion of texts on that topic in the list of references at the top of this piece. The Lebesgue integral is necessary because one can perform limiting operations with the Lebesgue integral that are not valid with the Riemann integral that one learns in ordinary calculus, and integrals exist for functions that are not Riemann integrable, so the Hilbert spaces include functions (really equivalence classes of functions) that are not integrable in the sense of Riemann. The good news is that if a function is Riemann integrable it is also Legesgue integrable and the two integrals are equal. So, if you are not familiar with the Lebesgue integral then take completeness as a given mystery and just pretend that the integrals are Riemann integrals.

    OK these are the very basics of the theory of Hilbert spaces. There is a great deal more to it -- the theory of operators on Hilbert space, spectral decompositions and the associated functional calculus, etc. These are much more advanced topics which one can find presented in the references given, but which require a great deal more background.

    The important thing to remember is that a Hilbert space is determined by the vector space structure and the inner product that is prescribed. One ought be overly concerned about a specific Hilbert space basis, or the representation of functions in terms of that basis. There are lots of possible choices for any given Hilbert space and one is always well advised to work in a coordinate-free manner until one simply must pick a basis and compute specific "components".
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  2. #2  
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    For those of you who have read the initial post and have had some time to think it about it a bit, it may have occurred to you that, in the case of separable Hilbert spaces, the existence of a countable Hilbert space basis gives a certain similarity between any such space and the example of square-summable sequences, often delnoted as .

    This impression is indeed correct. There is a natural isomorphism between any separable Hilbert space and which is obtained as follows:

    1) Let be a Hilbert space basis for the separable Hilbert space

    2) Send to the sequence where the occurs in the nth position.

    3) Extend linearly to the span of and then extend that by continuity to the whole of the Hilbert space

    One can then verify that this function is bijective, continuous and preserves the inner product of the Hilbert spaces; i.e. it is a Hilbert space isomorphism.

    Thus all separable Hilbert spaces are isomorphic. In fact one can show that Hilbert spaces are completely classified by the cardinality of their Hilbert space basis. In finite dimensions this is merely the fact that vector spaces are classified by dimension up to isomorphism, and the dot product is the canonical inner product in finite dimensions.

    Another thing that this tells you is that you may already know more about Hilbert spaces than you think that you do. If you know a bit about Fourier series for periodic functions on an interval, you already know a great deal about infinite dimensional separable Hilbert spaces, the most important example. For the theory of Fourier series of square integrable functions and the theory of separable infinite dimensional Hilbert spaces are the same theory -- the calculation of the Fourier components of a square-integrable function is precisely the correspondence between those functions and square summable sequences that was given above.
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