Re: 10 questions on QM postulates

Hendrik van Hees wrote:

That's a very deep question, and I also asked it to my professor when he
introduced the concept. His answer was: "The operators, describing
observables, must have only real eigenvalues, and Hermitean operators
have only real eigenvalues."

In my opinion, that's a short but not really satisfactory answer. After
thinking about the foundations of quantum theory from time to time (I
apply quantum theory every day, but one does not think about its
foundations too often in this kind of work, so it's more a hobby of
mine to think about it from time to time, especially when I hear
something about modern quantum-optics experiments, which help a lot to
understand the foundations of quantum theory), I think the real
mathematical reason that the observables can be described without any
inconsistencies by Hermitean (or to be more precise by essentially
self-adjoint) operators is the spectral theorem, i.e., each quantum
state can be described by a superposition of a complete set of
generalized eigenvectors of an self-adjoint operator.

I do not claim that an operator must be necessarily self-adjoint to meet
this property. I'm not sure wheter it is or not, it might be sufficient
that an operator commutes with its adjoint, which is usually called a
"normal operator". Perhaps somebody more familiar with the mathematical
details may answer this question.


Normal operators A have real spectrum and a complete spectral
resolution of the identity
a) always in finite dimensions,
b) in infinite dimensions when they are bounded or when
(lambda-A)^-1 is bounded for some real lambda.




Also this is a delicate question. Admittedly, what I wrote above is a
mess, and most textbooks also write a mess about this point. That's
just to simplify the maths a little bit for us poor physicists. The
point is that continuum "states" are not true Hilbert-space states, but
"generalized" states. Mathematically they belong not to Hilbert space
but to a larger space, which is the dual space of a smaller subspace of
the Hilbert space, namely those vectors, where the self-adjoint
operator, describing the observable, is defined.


This is usually described in terms of 'rigged Hilbert space', or
'Gelfand triple' (two names for the same thing).

classical mechanics? In particular, how can we get from the
Schroedinger equation to the classical Hamiltonian (or vice versa)?


Look in a good textbook under "WKB approximation" or semi-classical
approximation. It's again a little bit difficult to squeeze the
interesting answer into a newsgroup posting, which becomes already
quite lengthy ;-)).


There is also a way to the classical limit via coherent states,
which is very neat though mathematically a little more demanding:
     LG Yaffe
     Large N limits as classical mechanics
     Reviews of Modern Physics 54 (1982 ), 407-435.


Arnold Neumaier

.

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