From: John K Clark (johnkc@well.com)
Date: Sun Sep 28 1997 - 23:24:13 MDT
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On Sun, 28 Sep 1997 Geoff Smith <geoffs@unixg.ubc.ca> Wrote:
>Do you think an exact copy is impossible?
No.
>(and refresh my memory on the relevance of the Pauli Exlusion
>Principle to this question)
The idea is that the identity of indiscernibles must be true, electrons are
identical and position can't confer individually, otherwise the Pauli
Exclusion principle which says 2 atoms can't be in the same quantum state
would be untrue, but we know experimentally that it is.
The first philosopher to examine the principle of "The Identity Of
Indiscernibles" was Leibniz 300 years ago. He said that 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 the 20th century this idea had no observable consequences because
nobody could find two things exactly alike. Things changed dramatically when
it was discovered that atoms have no scratches on them to tell them apart.
Suppose you're looking at two electrons, you may think that you can find a
difference in them, because one is here and the other one is way over there,
but can you really? How do you know the particles are not changing positions?
Would the system be any different if they did? By asking these sort of
questions and using The Identity Of Indiscernibles we can derive The Pauli
Exclusion Principle, and that is the basis of the periodic table, and that
is the basis of chemistry, and that is the basis of life. We can also
discover the fact that there are two classes of particles, bosons like
photons and fermions like electrons.
Experimentally we can't measure the quantum wave function F(x) of a particle,
we can only measure the intensity of the wave function [F(x)]^2 because
that's probability and probability we can measure. P(x) =[F(x)]^2 is the
probability of finding two particles x distance apart. Now let's exchange
the position of the particles, the distance between them was x1 - x2 = x 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) . From this we see that [ F(x) ]^2 = [ F(-x)]^2
so the Quantum wave function can be an even function [ F(x) = +F(-x) ] or an
odd function [F(x) = -F(-x) ] , remember (-1)^2 = (+1)^2 =1.
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 it's own negative is zero so F(0) =0 . What this means is that the
wave function goes to zero and [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.
John K Clark johnkc@well.com
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