"Craig Joyce" wrote in message
news:7e5f4e39.0211251916.32c3a682@posting.google.com...
> Hi! Can anyone explain the relation between Hilbert Space and the
> Schrodinger equation? Is the set of wave functions a vector space
> or something similar ? Please help. Thanks, 

Briefly and loosely, but not bereft of meaning, here it is:

The totality of the eigenfunctions of Schrodinger's equation for a
particular system are a "complete orthogonal set". That means that
*any* (reasonable) function can be expressed as a series expansion in
these eigenfunctions. In particular, any solution of Schrodinger's
equation can be so expressed. Fourier analysis is but one example of
this procedure.
Now, if you know the eigenfunctions in terms of which you expressed
the solution with which you are playing, you can write them in
sequence on an old envelope and stick it in your inside pocket. In
your actual calculations, all you need to do is to keep tabs on the
relative amplitudes of the terms in the expansion. (Rather like
expressing a complicated periodic electrical signal in terms of
sinusoidal harmonics, and thenceforth simply characterising the signal
in terms of the amplitudes of the harmonics) The "basis vectors" of
the Hilbert space you have chosen to work in are simply the actual set
of eigenfunctions. The components of the vector is the set of
amplitudes. The nice thing now is that the quantum mechanical
operators, which are the objects which enable you to study the
observables in your system, can be written in such a way that you only
need to operate on the vector in Hilbert space (i.e. the set of
amplitudes) rather than invoking the complete wave function.

A hardy theorist will probably lay hell out of me for this
explanation, but it is nevertheless a fairly close introduction to the
strict truth.

Franz Heymann