HilbertSpace

HilbertSpace - Physics 129a Hilbert Spaces 051014 Frank...

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Physics 129a Hilbert Spaces 051014 Frank Porter Revision 081009 1 Introduction It is a fundamental postulate of quantum mechanics that any physically allowed state of a system may be described as a vector in a separable Hilbert space of possible states. Hilbert spaces Fgure prominently in the theory of di±erential equations. Loosely, a Hilbert space adds the notion of a scalar product to the ingredients of linearity and continuity. These notes collect and remind you of several deFnitions connected with the notion of a Hilbert space. We also illustrate the relation of a Hilbert space to more general spaces from which it is derived. Two (of many, but I like these) references for further information are: 1. Guido ²ano, Mathematical Methods of Quantum Mechanics ,McGraw- Hill (1971). This is a text on the rigorous mathematical foundations of quantum mechanics. It is a very accessible read for the physics student. 2. John L. Kelley, General Topology , Van Nostrand Reinhold Company (1955). This is a textbook for mathematicians, but has lots of concise information for the impatient. This note may largely be summarized by ²ig. 1, showing the relationship between the di±erent abstract spaces discussed. 2 Some Preliminaries Defnition: A relation is a set of ordered pairs. Defnition: A ±unction is a relation such that no two distinct members have the same Frst coordinate. (To a mathematician, the following terms are synonymous: function, map, operator, transformation, cor- respondence). 1
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Linear Space Linearity Hausdorff Space Disjoint neighborhoods of distinct points Topological Linear Space Metric Space Distance between points Normed Space Length of a vector Banach Space Completeness Euclidean or pre-Hilbert Space Scalar product Hilbert Space Completeness Seperable Hilbert Space Denumerable dense set Topological Space Continuity Figure 1: The relationships between several abstract spaces. Structure is added from top to bottom, that is the lower boxes inherit the properties of the upper ones to which they are connected, with additional structure noted in the box. 2
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3C o n t i n u i t y The concept of a topological space embodies the notion of continuity in gen- eral. Defnition: A topological space T is a pair ( T ), where T is a non-empty set and τ is a family of subsets of T such that: 1. ∅∈ τ and T∈ τ , 2. the intersection of any two elements of τ is in τ ,and 3. the arbitrary union of elements of τ is in τ . The family τ is called a topology on T , and its elements are called the open sets of T . There are some special names for particular cases. For example, given any set T , the family τ = {∅ , T} de±nes a topology on T , called the indiscrete topology . Alternatively, the set of all subsets of T also de±nes a topology on T , called the discrete topology . A familiar topological space is the space of real numbers, with the usual topology generated by open intervals (also called the ordinary topology or the Euclidean topology ). With the topology understood, we often refer to T as a topological space.
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HilbertSpace - Physics 129a Hilbert Spaces 051014 Frank...

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