meekerdb wrote:
On 3/1/2015 2:18 PM, Bruce Kellett wrote:
The basis problem is even more severe. If observables are interpreted
as hermitian operators in Hilbert space, the eigenfunctions and
eigenvalues depend on the basis chosen for that space. The SWE
contains observables (operators) such as position, energy and momentum
and so on. What bases do we choose for these operators? The default,
that no one ever questions (to the extent that I doubt that many
people realize that it is an arbitrary choice) is that the eigenvalues
of the position operator are values along the real line and the
eigenfunctions are delta functions of reals. In any other basis,
position eigenvalues would be superpositions of these reals, and the
eigenfunctions would be quite different. Why do we unquestioningly
accept the basis commonly used?
In large part it's because we can't produce instruments that implement
other hermitian operators. The number of possible hermitian operators
for a system is enormous - but we only ever write down handful. Why?
Because those are only ones we know how to realize by a physical
instrument. Peres discusses this in Ch 11 in the context of classically
chaotic systems. The quantum version of those systems are not chaotic -
but they have eigenstates that are very complicated in the usual bases
we can measure. However, he notes that sometimes the experimentalist is
more clever than the we suppose and thinks of a way to measure something
theorists had not anticipated. He gives the example of spin-echo
measurements.
I have not read Peres' book, and looking at Amazon, it is not cheap!
However, I think you are right that it is a matter of the actual
measurements we can make. All this seems to point back to Bohr's
original insight that we can't begin with quantum mechanics without a
classical world as the background. The instruments we make to perform
the measurements we want are irreducibly classical. State preparation is
a classical operation for exactly the same reasons.
Bohm discusses this in the final chapter of his book on quantum theory.
He says: "This means that without an appeal to a classical level,
quantum theory would have no meaning. We conclude then that quantum
theory presupposes the classical level and the general correctness of
classical concepts in describing this level; /it does not deduce
classical concepts as limiting cases of quantum concepts./" (Emphasis in
the original.) He says later: "As we go from small scale to large scale
level, new (classical) properties then appear which cannot be deduced
from the quantum description in terms of the wave function alone, but
which must nevertheless be consistent with this quantum description."
"Quantum theory presupposes a classical level and the correctness of
classical concepts in describing this level. The classically definite
aspects of large-scale systems cannot be deduced from the
quantum-mechanical relationships of assumed small-scale elements."
This is essentially Bohr's insight into the relation between classical
and quantum concepts.
Bruce
I often challenge Bruno to come up with some testable prediction from
"comp". It seems that this kind of information theoretic question might
be one that mind-as-computation could address: Why is it we can only
think of the world in these limited, classical ways (if indeed that's
the case)? For example we do all our scattering calculations in
momentum-space; might there be sentient beings that conceptualize in
momentum space instead of position space? Maybe dolphins and bats do it
since they are more audio dependent.
Brent
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