To answer your questions one by one:
First:
There are multiple lines of evidence supporting the exclusion
principle, the principle that there cannot be more than one fermion
particle in any given quantum state.
1. It obviously works empirically in explaining why the electrons
in atoms occupy the states that they do, rather than all piling up into
low-energy states.
2. There is a deep theorem, called the spin-statistics theorem,
which shows that in any quantum theory consistent with special
relativity the exclusion principle has to hold for all particles whose
spin is a half-integral number of units. Of course, the fermions are
precisely these particles, so the formal theorem matches perfectly with
observed behaviors.
3. There are countless other experimental and theoretical
developments linking quantum mechanics, special relativity, and the
behavior of all sorts of particles. Breaking the exclusion principle
would require destroying all of this intricate and highly successful
structure.
However, even with the most compelling principles, there is always
the possibility that in some future physics the whole framework will
have changed and the current principle will be seen merely as an
approximation. Usually it's best to keep that possibility far to the
back of your mind.
Second:
Again, Heisenberg's uncertainty principle is a direct consequence
of the whole structure of quantum mechanics. Quantum mechanics has been
a spectacularly successful way of predicting the behavior of everything
on a small to medium scale. We don't have any alternative theory that
is even close to being able to describe the microscopic behavior of
things. At any rate, the most troubling aspects of the uncertainty
relation are confirmed by experiments showing that nature violates the
Bell inequalities, which would be obeyed by any processes which have
definite values for measurable quantities. So even if we someday
transcend our current quantum mechanics, it seems that something like
the uncertainty relations will have to be a permanent part of physics.
Third:
Heisenberg tried to write a formal mathematical description of
atomic behavior, and the uncertainty relation popped right out of the
math. Schroedinger wrote a very different looking wave description of
atomic phenomena, and the same relations emerged from it. The two
descriptions turned out to be mathematically equivalent.
Pauli initially was trying to account both for atomic spectra and
for the effects of magnetic fields on these spectra. Later, he went
back to confirm his picture by proving the spin-statistics theorem.
Fourth:
There are many forms of energy, of which the classical kinetic term
you mention is just one. In quantum mechanics, energy is really another
word for frequency, and the basic rules for how things change in time
do not allow for it to change. The fundamental theorem here, due to
Emmy Noether, is that so long as the same laws of physics apply at all
times, the variable (frequency or energy) describing how things change
in time must itself be conserved. As for how the classical kinetic
energy term arises, it pops up as an approximation for the relativistic
expression for the net energy. The fact that the same laws of physics
apply in all different parts of space gives, via Noether's theorem, the
conservation of momentum.
Mike W.
p.s. We're still working on your deep question about why electron clouds don't self-repel. It's hard.
(published on 10/22/2007)
