Prof. J. C. Baird

A generalized space is defined based on the axiom of closure by which is meant that all points and all limit points are contained in the set defining the space. This idea of closure has the practical result that it creates a space containing all the points whether finite or infinite.

A space defined in this manner is called a topological space. In quantum mechanics, and elsewhere in physics and chemistry, a more restrictively defined space is needed. First, we like to measure distances between points in space in a well defined way. We may therefore define a metric space by defining the characteristics of distance. Such a space, that is one with closure and a metric, may be of any dimension. The N-dimensional Euclidian space is familiar and the infinitely-dimensional generalizaton is called a Hilbert space. The elements of the Hilbert space are the sequence of real numbers designated by x = {x1, x2, x3, ...} where the series and a distance is defined by with the further requirements that d(x,y) = d(y,x) and d(x,y) = 0 only when x = y. The triangle inequality also is assumed to hold d(x,y) + d(y,z) >= d(x,z). These spaces are usually defined by their components and we usually call them vector spaces. The idea of a vector, an element in the space decomposed into projections along the "axes" of the space causes us to call these vector spaces. It is also useful that there be an inner, or scalar product (u,v) between two "vectors" in this space. In the case of Hilbert space the basic vectors may be complex and the Hermitian scalar product is a complex function (u,v) = (v,u)* > 0 of the two vector variables.

Closure, Completness and Linear Independence?

Professor Tan's Notes on Hilbert Space

Interlude: Curvilinear Coordinates

The Basic Idea

In order to uniquely specify a physical theory and connect it to experimental measurement we need to establish a metric to the space as has been described above and a coordinate system within the space. The choice of coordinate system is ours to make, but the first step is the Cartesian Coordinate system. Any point in a three dimensional Euclidian space may be represented by three numbers which are referred to three mutually perpendicular axes. The distance between any two points is defined by .

Curvilinear Coordinates

Now if we have generalized coordinates, that is to say, some functional relation between the points in Cartesian space and some other set of elements, x(q1, q2, q3) for example then a specification of this transformation will define the new system in terms of our Cartesian space. We can do this for any general, well behaved functional relationship.

We may relate x, y, z to three new quantities by

Using the chain rule we may write

The square of the distance between two adjacent points is therefore

where

The line element, or distance between two points on a coordinate line, is

and the surface element and volume elements are

The cosine of the angle is

The in Curvilinear Coordinates

Orthogonal Coordinates

Orthogonal coordinates are defined by

Cylindrical Coordinates

Defining the relationship between Cartesian coordinates and cylindrical coordinates gives:

Spherical Polar Coordinates

Defining the relationship between Cartesian coordinates and Spherical Polar coordinates gives:

Oswald Veblen, Invariants of Quadratic Differential Forms, Cambridge University Press, 1952 L. Pontryagin, Topological Goups, Gordon and Breach, N.Y. 1966
H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1931
G. H. Hardy, Pure Mathematics, Cambridge University Press, 1951