Re: Fourier Derivation of Heisenberg Uncertainty is Nonsense
- From: Igor <thoovler@xxxxxxxxxx>
- Date: Sun, 23 Sep 2007 18:08:09 -0000
On Sep 22, 12:50 pm, OsherD <mdocto...@xxxxxxxxx> wrote:
From Osher Doctorow
According to Valery P. Dimitriyev dim...@xxxxxxxxxxxxx in sci.physics
and sci.math, Wed 05 Jan 7:52:24 GMT (Dimitriyev cites 2 of his own
publications in Stoch Mech 1990 and Galilean Electrodynamics 10(5)
95-99 1999), by taking the Fourier transform of the Gauss distribution
you get a Gaussian distribution function but with variance inversely
proportional to the variance of the original distribution so the
product is constant. This "idea" was recently ressurected (in the
last 2 or 3 days) in arXiv in an apparent attempt to defend the
Heisenberg Uncertainty Principle (HUP).
The derivation is nonsense. To base all of variance theory on the
Gaussian distribution in mathematical probability-statistics is like
basing physics on one example or even one object.
Quite a few physicists use the Gaussian distribution as an example,
mostly because sample mean distributions under a wide variety of
conditions converge to the Gaussian distribution as sample size
approaches infinity. It doesn't work for small samples (in fact, it's
often wildly wrong), it doesn't work in general for non-means (where
it's often wildly wrong), and under certain conditions it's wildly
wrong even for means.
This has caused a MAJOR trend away from the Gaussian/normal
distribution, which readers can easily discern from papers in arXiv
and especially Front for the Mathematics ArXiv under Mathematics
subcategory Statistics and also subcategory Probability, which extends
to physics and engineering (especially engineering Reliability
theory). This trend is especially pronounced in Large Deviations
Theory which is a major direction of current advanced research, in Fat-
tailed Distribution theory, in studies of maxima and minima and other
order statistics, etc.
Moreover, the Uniform and Exponential distributions are Maximum
Entropy for 1 or 2 unknown parameters among continuous distributions,
while Gaussian distributions are ONLY Maximum Entropy for 2 known
parameters among continuous distributions even using Shannon Entropy/
Information, not to mention Probable Causation/Influence (PI) where
other optimal distributions including Gamma (subtypes Exponential and
Chi Squared), F, beta, power, and other distributions are critical.
These other distributions are heavily used in Engineering Reliability
theory, one of the most practical applied theories around. In PI
optimality, Gaussian/Normal and other graph-symmetric real line
distributions are the least optimal of the 3 classes of optimal
distribution families.
Dimitriyev gives a second alternative derivation of the Heisenberg
Uncertainty Principle in rough outline using covariances of two random
variables, which he doesn't sufficiently clarify. It isn't related to
the above "derivation".
Osher Doctorow
Has it ever actually crossed your mind that the quantum wave packet is
a Gaussian? Or that the ground state of the simple harmonic
oscillator is a Gaussian? One usually needs to understand where
something is coming from before one can properly critique it.
.
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