From: John K Clark (johnkc@well.com)
Date: Thu Nov 07 1996 - 21:46:53 MST
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On Wed, 06 Nov 1996 Michael Lorrey <retroman@tpk.net> Wrote:
>however, due to the Uncertainty principle, the hidden
>variable of each [atom] you can never know, so YOU could
>never know who was who.
We now know from experiments on Bell's inequality that if hidden variables
exist at all (most think they probably don't) they can not be the sort that
Einstein had in mind, they can't be local. Also, drinking a cup of coffee
will change you much more that quantum uncertainty ever will
On Wed, 06 Nov 1996 Chris Hind <chind@juno.com> Wrote:
>But two atoms of exactly the same kind *do* have
>individuality and that individuality is that they cannot
>occupy the same space at the same time.
Not all particles have this property (photons) but you are correct, atoms do.
It's called The Pauli Exclusion Principle.
>Their "individuality" is based on spacial location.
It's interesting that the Pauli Exclusion Principle can only be derived, in
fact it only makes sense, if we use an obscure idea in pure Philosophy, and
assume the exact opposite of what you say above. In 1690 the philosopher and
co-inventor of The Calculus, Leibniz, wrote about something he called
" The Identity Of Indiscernibles ". He said that things that you can measure
are what's important, and if there is no way to find a difference between two
things then they are identical and switching the position of the objects does
not change the physical state of the system. Until this century few thought
this idea was important, nobody thought it had observable consequences,
because nobody could find two things that were exactly alike. Things changed
dramatically when it was discovered that atoms have no scratches on them to
tell them apart.
The Schrodinger Wave Equation is one of the first and greatest discoveries in
Quantum Mechanics. It proved to be enormously useful in accurately predicting
the results of experiments, and as the name implies it's an equation
describing the movement of a wave, but embarrassingly it was not at all clear
what it was talking about. Exactly what was waving? Schrodinger thought it
was a matter wave, but that didn't seem right to Max Born. Born reasoned that
matter is not smeared around, only the probability of finding it is. Born was
correct, whenever an electron is detected it always acts like a particle, it
makes a dot when it hit's a phosphorus screen not a smudge, however the
probability of finding that electron does act like a wave so you can't be
certain exactly where that dot will be. Born showed that it's the square of
the wave equation that describes the probability, the wave equation itself is
sort of a useful mathematical fiction, like lines of longitude and latitude,
because experimentally we can't measure the quantum wave function F(x) of a
particle, we can only measure the intensity (square) of the wave function
[F(x)]^2 because that's a probability and probability we can measure.
Let's consider a very simple system with lots of space but only 2 particles
in it. P(x) is the probability of finding two particles x distance apart,
and we know that probability is the square of the wave function, so
P(x) =[F(x)]^2. Now let's exchange the position of the particles in the
system, the distance between them was x1 - x2 = x but is now x2 - x1 = -x.
The Identity Of Indiscernibles tells us that because the two particles are
the same, no measurable change has been made, no change in probability,
so P(x) = P(-x). Probability is just the square of the wave function so
[ F(x) ]^2 = [F(-x)]^2 . From this we can tell that the Quantum wave function
can be either an even function, F(x) = +F(-x), or an odd function,
F(x) = -F(-x). Either type of function would work in our probability equation
because the square of minus 1 is equal to the square of plus 1. It turns out
both solutions have physical significance, particles with integer spin,
Bosons, have even wave functions, particles with half integer spin, Fermions,
have odd wave functions.
If we put two Fermions like electrons in the same place then the distance
between them, x , is zero and because they must follow the laws of odd wave
functions, F(0) = -F(0), but the only number that is its own negative is
zero so F(0) =0 . What this means is that the wave function F(x) goes to
zero so of course [F(x)]^2 goes to zero, thus the probability of finding two
electrons in the same spot is zero, and that is The Pauli Exclusion Principle.
Two identical bosons, like photons of light, can sit on top of each other but
not so for fermions, The Pauli Exclusion Principle tells us that 2 identical
electrons can not be in the same orbit in an atom. If we didn't know that
then we wouldn't understand Chemistry, we wouldn't know why matter is rigid
and not infinitely compressible, and if we didn't know that atoms are
interchangeable we wouldn't understand any of that.
John K Clark johnkc@well.com
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