Re: Uncertainty principle, fourier transform

Bob Cain wrote:

I know that I'm stepping in over my head here but one of my aborted
attempts to go deeper into QM left me with this understanding. The pair
of non-commuting observables may only be considered as measurement
distributions, i.e. ensembles. Both distributions are normal, Gaussian,
and the HUP says that that, for any particular setup that measures both
of them, the deviations of the distributions measured for each will be
inversely proportional to twice h-bar. I believe that the Fourier basis
is the only basis for transforming domains which has the property of
mapping Gaussian distributions to Gaussian distributions with deviations
that are inversely proportional.

That sounds nice enough to be true.

As Mati Meron mentions, the significance of the Gaussian distributions
would be a limiting case: any other distribution gives greater than the
minimum product of distributions -- or so I'm told. This, presumably,
under Fourier transformations, or some suitable generalization of them.

As to whether there is anything "special" about Fourier transformations
-- on the one hand I'm technically uncertain (no pun) just what special
properties they may buy us (like the story you just told), but I think
the question is likely semantic: there may be some standard examples
of uncertainty relations which fill this bill, and they may be
understood as members of a larger class of relations which don't quite
fill this bill. The more general relation involving the commutator
may be of the class described as "sufficiently general to be useless".


With regard to the results involving measurments on ensembles of
identically prepared system, that would be a correct understanding, in
my book. We can also assign the result to properties of the
wavefunction -- which are just those which predict the ensemble
statistics you describe. We could also tell stories about one
measurement affecting the next, or the photon changing the electrons
momentum when we localize the electron. These points of view seem to
be partially ... but not completely ... out of fashion. Certainly
nature must conspire _somehow_ to prevent us from making repeatable
measurements of uncertainty-related-observables on the same system to
arbitrary precision: otherwise we could use such measurments to produce
and ensemble of systems violating an uncertainty principle for future
measurments. Can the position/momentum uncertainty relation, say, then
be understood as a quantitative predictions about the product of the
experimental uncertanties remaing after simultaneous or sequential
measurements on a _single_ system, as opposed to an ensemble?

Apparently the answer is "Yes": for a correctly calculated
experimental uncertainty following a measurement must be interpretable
as the uncertainty attaching to future, indefinitely precise,
remeasurements of the same property -- which precisely takes us back to
the ensemble interpretation.

Once again I learn something unknown to both myself and the sender
prior to transmission, demonsrating the spontaneous creation of
information in a transmitter receiver couple, or
"Aufbaugebbenscaboosnessheit".

Then, again, I may have misunderstood or misremembered what I read and I
certainly don't remember how the fact that the FT is complex factors
into this. Is the FT of a real Gaussian distribution also real?

It wouldn't matter, since the observables are real. Both
representations of the wavefunction (position and momentum, say) will
in general be complex, but the standard deviation associated with each
is real in any case: the complex numbers are magnituded out.

.

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