Werner's course that you point out is geared toward Quantum Information
applications and so the emphasys on finite spectra is understandable from
context. My point is that, in conventional quantum mechanics intro courses,
you begin with problems like the harmonic oscillator or the infinite square
well or the delta potential where the spectrum of the Hamiltonian (or the
space of eigenfunctions) is not upwardly bound. These are quite
straighforwardly solved in the Schrodinger representation and
the infinity of the spectrum actually makes the solutions relatively
simpler than that of the simplest finite HS space you can formulate.
Granted the "peculiarities" of the quantum world do not depend on
the finiteness or infinity or countability of the HS!... The Hilbert
Space is not even a requirement for the formulation and solution
of quantum problems as the Path Integral technique demonstrates...
Tim May wrote:
> On Tuesday, December 31, 2002, at 07:02 AM, Joao Leao wrote:
> > I don't agree with Tim's suggestion that infinite-dimensional Hilbert
> > spaces
> > are somewhat "ancilliary" in QM and that all systems are calculable in
> > finite dimensional modes. In fact infinite sets of spaces are the rule
> > in
> > QM and
> > the finite dimensional subspaces only serve as toy systems.
> I said it is often done. Many of the details of the infinite case are
> just not needed. And QM is often taught this way, with no loss of
> rigor, provided any subtleties are pointed out to the student.
> For example, here are some fairly typical lecture notes for a course on
> "2.2. Hilbert Space
> Hilbert spaces are mentioned in most textbooks on quantum mechanics and
> functional analysis  . Therefore we will only mention some features,
> which are not found almost everywhere. We will also not have to go into
> the subtleties of topologies, continuous spectra, or unbounded
> operators, because throughout this course, we can assume that all
> Hilbert spaces are finite dimensional. Modifications in the infinite
> dimensional case will be mentioned in the notes. Our standard notation
> is <\phi ,\psi > for the scalar product of the vectors \phi ,\psi \in
> H, ||\phi ||=<\phi ,\phi >1/2 for the norm, and B(H) for the algebra of
> bounded linear operators on H. Of course, all linear operators on a
> finite dimensional space are bounded anyway, and the B is used mostly
> for conformity with the infinite dimensional case. "
> In nearly every area of physics, the issue of "infinity" is phrased in
> terms of sequences or structures approaching or growing towards the
> infinite or infinitesimal. For example, a test mass is assumed to be
> small enough not to perturb the curvature tensor. But actual infinite
> spaces are not needed, not even in thermodynamics.
> This dispenses with a lot of the mathematical cruft, alluded to above
> (continuous spectra, compactness, etc.). That cruft contains a lot of
> beautiful math, but physics just doesn't need it, at least not very
> I have no axe to grind on this. For those who want to study only the
> completely general, infinite-dimensional cases, cool. But a good
> understanding of finite-dimensional vector spaces (e.g., the Halmos
> book) provides the math one needs for QM, especially at the level we
> usually discuss it at here. (As many here perhaps already know, Halmos
> was Von Neumann's assistant, writing up his lectures, when he wrote his
> Provided the complex space is normed, and is complete, which all
> finite-dimensional vector spaces are, the math works. No infinities are
> needed, which is good.
> --Tim May
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