From: John K Clark (johnkclark@fastmail.fm)
Date: Mon Feb 09 2009 - 10:32:08 MST
On Mon, 9 Feb 2009 "Stuart Armstrong"
<dragondreaming@googlemail.com> said:
> > On the
> > Extropian List I used nothing but
> > The Identity of Indiscernibles and high school algebra to derive The
> > Pauli Exclusion Principle; and that is the root of chemistry, and that
> > is the root of biology. By the way I'm rather pleased with that
> > derivation.
>
> Sounds rather impressive - do you still have the derivation? And what
> assumptions do you put into it - basically how did you distinguish
> bosons (which don't obey the exclusion principle) from fermions?
> Indiscernibles wouldn't be enough.
=========================================
>From my post to the Extropian list:
In about 1690 the philosopher Leibniz discovered something 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.
Leibniz's idea turned out to be very practical, although until the 20th
century nobody realized it, before that his idea had no 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.
By using The Identity Of Indiscernibles you can deduce one of the
foundations of modern physics the fact that there must be two classes of
particles, bosons like photons and fermions like electrons, and from
there
you can deduce The Pauli Exclusion Principle, and that is the basis
of the periodic table of elements, and that is the basis of chemistry,
and
that is the basis of life. If The Identity Of Indiscernibles is wrong
then
this entire chain breaks down and you can throw Science into the trash
can.
The Schrödinger Wave Equation 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? Schrödinger 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.
We MUST assume that atoms are interchangeable and have no individuality
or
modern Physics becomes incomprehensible. 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 it's 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.
Bottom line: Atoms have no individuality and If they can't even give
themselves this property they can't give it to us.
John K Clark
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