Before discussing specifics about the structure of the hydrogen atom, it is interesting to note what information about the hydrogen atom can be derived just from the Heisenberg uncertainty principle. A familiar form of the uncertainty principle looks like the following:
where 
and 
are the uncertainty in the x-component
of the position and momentum of a particle, respectively. Consider an
electron in a classical circular orbit in the xy-plane. It is then
reasonable to write
, where r is the radius of the orbit. Assuming a state
of minimum uncertainty, is
then known from the uncertainty principle, and it should be roughly
equal to the magnitude of the momentum for the circular orbit being
considered. That is,
Classically, the energy is simply
where m is the electron mass and e the electron charge. The last step results from the substitution of p from equation 2. The value of r is unknown, but one would expect it to have a value that minimizes the energy, as Nature likes to do. Differentiating equation 3 with respect to r and setting equal to zero gives
This yields
where 
is the Bohr radius. Substituting into equation 3
gives
The Bohr radius is exactly the radius of the circular orbit in the
ground state of the electron in Bohr theory, and it holds up as
representative of the extent of the orbit in Schrödinger theory.
The energy 
is the known ground state energy of the
hydrogen atom.
So, starting with only a very rough view of the structure of the atom and the
uncertainty principle, one can make some reasonable assumptions and
derive two extremely important fundamental results -- the ``size'' if the
hydrogen atom in its ground state and its ionization energy. Of course,
to get precisely the right results one needs to make the right assumptions,
and so this calculation is certainly not rigorously accurate. It merely
illustrates the relation of the fundamental physical structure of the
hydrogen atom to the uncertainty principle. The fact that these results
were derived assuming minimum uncertainty leads to a rather important
conclusion--the hydrogen atom in its ground state is essentially in a state
of minimum uncertainty. This explains why the electron in its ground state
cannot radiate, as
one expects classically, and get drawn in towards the nucleus -- to do so
would violate the uncertainty principle. If the electron were confined
closer to the nucleus, so that were much smaller, then
would be much larger and so it would not be possible to
consider the electron as necessarily bound to the nucleus.
