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The first job of QM is to construct a NEUTRAL
atom/molecule/object by keeping equal and
opposite charges APART! This is the first defiance
of Earnshaw.
A home-brew proof of Earnshaw's Theorem (actually a
corollary thereof): If there were a point in empty space
(no charge there) where a positive test charge would be
in stable equilibrium under only electrostatic forces, the
electric field would have to be zero at that point, and at
every point on the surface of a neighboring sphere
(centered on that point) the field would have to point
inward toward that point. (outward if you use a negative
test charge). Since E lines are continuous and only start
and terminate on charges (Div E(r) = 0 in charge free
space), this is impossible. QED