HUP Fails Via Nonexistent Distributions 2: The Uniform Claim
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 17 Aug 2005 21:08:26 -0700
>>From Osher Doctorow mdoctorow@xxxxxxxxxxx
Max Jammer's The Philosophy of Quantum Mechanics, Wiley: N.Y. 1974,
page 6, points out that according to Born's probabilistic
interpretation the probability density of finding the system at
position q is:
1) /w(q)/^2
where w is the wave function. Also, the probability that q is in [L1,
L2) (I use L for his lambda) is:
2) P(L1 < = q < L2) = //(E_L2 - E_L1)w//^2 = I/w(q)/^2 dq
where I...dq is the integral from q1 to q2 with regard to q. The
expression E_L is called the "resolution of identity" such that:
3) E_L(w(q)) = w(q) for q < = L, E_L(w(q)) = 0 for q > L
Readers familiar with mathematical probability-statistics will
recognize E_L as the integral of the "indicator function" I_L:
4) I_L(q) = 1 if q < = L, I_L(q) = 0 if q > L
where the integral is from 0 (say) or -infinity to L.
The uniform probability distribution in mathematical
probability-statistics is the only one with these properties, which
means that Born assumed that the microscopic domain is uniformly
distributed! Since it is common to find Gaussian/normal distributions
in the quantum domain, not only is this experimentally unverified but
it is inconsistent with other results in quantum theory and quantum
practice. Moreover, if both position and momentum had uniform
distributions, then there is no sense to the claim that one is
"conjugate" to the other since both have definitely defined population
means/expectations and variances and standard deviations as well as
similarly defined probabilities. The only scenario that comes remotely
close to "conjugate" HUP is assigning one variable, say position, to
the uniform distribution on some finite interval, say (0, 1), and
assigning the other variable to some varying interval like (0, L), and
letting L --> infinity. Even if this were done, at no time and with no
scenario will either uniformly distributed random variable be "vague"
in any sense.
In fact, the graph of a uniform distribution is a horizontal line
segment. For example, the graph of a uniform probability density
function (pdf) on (a, b) or [a, b] is a horizontal line segment above
the x axis between a and b where a < b, and the rest of the graph has 0
vertical position, that is to say is the remainder of the x axis. The
vertical axis here represents the value of the pdf, usually between 0
and 1 except for example when distributions are extremely concentrated
around one point in which case it may exceed 1 for continuous
distributions. The height of the graph above the horizontal or x axis
is 1/(b - a). So if a = 0, the height would be 1/b, and if b = L -->
infinity, all that happens is that the height --> 0 without reaching 0
at any finite time. What this means "physically" is that whatever is
involved gets more and more "equally" or "uniformly" distributed along
more and more of the real line, with the limit being "equally
distributed" along the entire real line (but is never attained).
There is nothing "conjugate" about this scenario, but what is even more
remarkable, if two random variables were related in the above manner on
for example [0, 1] and [0, L], they would be influencing each other
rather than being causally unrelated! In other words, position x and
momentum p would themselves not be causally unrelated but one would
influence the other and presumably both influence each oher. Thus, the
HUP would imply either the Weak HUP (WHUP) or something very similar to
it.
It is true that as [0, L] gets bigger and bigger as an interval, the
variance increases since the latter is (1/12)(b - a)^2 or here
(1/12)L^2. However, for the uniform distribution all this reflects is
the largeness or large size of the entire real line, since "equally or
uniformly distributed" continues to hold for both variables
proportionately to the size of the domain (the line segment in which
the distribution has a support (is > 0)).
Osher Doctorow
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