"As to delocalization in particular, if you ask whether the electron
in a hydrogen atom is "a little bit here" *and* "a little bit there"
at any particular time, the answer is absolutely no.
This is an entirely nontrivial result, because it is directly
connected to the electrostatic self-energy. One part of the
"electron cloud" does not repel other parts. It's the same
electron after all, and it cannot repel itself."
I do not think this is a valid argument. That an electron in a
stationary state within an atom can be described as "smeared out"
over atomic space, does not imply the emergence of self-interaction.
I would say, it you bring in self-interaction for one configuration,
it will stay in all of them including the state described by an
eigenfunction of the position operator (infinitely sharply localized)
(See, e.g. Feynman's Lectures on Phys.)
Reversing the argument, - we know that the magnetic moments of the
1-electron quantum orbital states within an atom can be calculated as
the result of continuously distributed current (figuratively, spinning
of the electron probability cloud filling out the atomic space).
In this case the picture of the electron filling the atom gives exact
results (See, e.g., D. Blokhintsev, Quantum Mechanics).
"Also, if you pick any particular small region within the atom, and
ask whether the electron is in that region, the answer is very,
very likely to be no. If there are N such regions, N-1 of them
will be empty at any given time."
I absolutely disagree with this. If this were true, the electron
in a diffraction experiment would pass either one slit, or the other
with the result that we would not have any diffraction pattern at all.
Any claim that at any given time the electron occupies only one spot
leaving the others free is referral to hidden parameters, which were
shown to contadict the experiemnts (See, e.g., J. Bell,
Speakable and unspeakable in QM, and A. Aspect's experiements)
In another poster John says, that if we prepare an electron in a
localized state, we will find it later in this state. This is only
true in a special case when this state is stationary, that is we have
taken special care to put the electron in an appropriate very deep and
narrow potential well. Otherwise, the electron state will spread the
faster, the narrower its initial wave packet (in non-relativistic QM
- infintely fast for infinitely narrow initial state).