Finally, we consider a system of Hydrogen atoms which have been left
alone to radiate all of their excess energy. Classically, we would
eventually (after about 
) expect to find that all of the
electrons have fallen into the nuclei at R=0. According to the
uncertainty principle, however, there must remain an uncertainty in
the position 
(measured from the proton) and momentum 
of the electrons. Again, we expect 
. But
that there will be some spread 
and 
in the
distributions.
In a hydrogen atom the energy of the electron is given by
where M and e are the charge and mass of the electron respectively. The average energy we expect for our system of hydrogen atoms is then
Here, as with the box, we are again being sloppy with precise values and are just making an order of magnitude estimate in order to gain qualitative understanding of the physics of the hydrogen atom.
We now apply the uncertainty principle, which tells us that 
so that
Once again, we find a lower bound on <E> by finding the minimum of
. Setting the derivative equal to zero,
The value we find for the width of the distribution of the electron
about the proton is precisely the experimental size of the Hydrogen atom! The
combination of constants which given this distance, 
, appears so often in atomic physics that is given
a special name, the Bohr radius.
Finally, we have our estimate of the average energy of our systems in their ground state,
which happens to be precisely the ionization energy of Hydrogen, the energy it takes to remove an electron from a hydrogen atom when it is in its ground state!
The exact agreement is accidental because this is only a semiquantitative argument. The fact that we got the correct order of magnitude is not an accident. It is a consequence of the fact that our argument contains all of the correct physics in this situations.
Note also the from this argument alone, we cannot understand why it is that all hydrogen atoms in their ground state have precisely the same ionization energy. This argument merely puts a limit on the lower bound the average ionization energy may have. There is no problem if all hydrogen atoms require the same energy for ionization, so long at that energy does not violate (4). The equality of the ionization energies for all hydrogen atoms in the ground state is an indication that there are strong correlations in the distributions for R and P. (See the notes on statistics for a discussion of correlation).
Finally, our results are in accord with the classical limit. As
we find 
and 
; the electron ``falls'' into the infinite
well of the proton as expected classically.
