Since the energy scales with one over the square of n, high-n states all have energies close to zero. A very tenuously bound electron, with a very large n, can reach macroscopic size scales. While the ground state of hydrogen is the only truly stable state, in tenuous environments with low collision rates, atoms can get "stuck" in excited states that only have very rare radiative transitions available. These sorts of very excited atoms are called Rydberg atoms [2].
[1] http://info.phys.unm.edu/~ideutsch/Classes/Phys531F11/Proble...
Or does that mean that there is an infinitesimally small chance a electron could be found very far away from the nucleus because there is nothing else for it to interact with in empty deep space?
(The physics problem you linked to is far beyond my understanding)
It's not that the electron has an infinitesimally small chance to be observed far away from the nucleus - it's quite likely actually if it lives in a high-n state - it's that collisions rates are so low that the system is out of thermodynamic equilibrium. Section III of these notes [2] describe the physics of Rydberg aroms and the radio recombination lines we observe from them quite nicely.
[1] http://iopscience.iop.org/0004-637X/549/2/979/pdf/52126.web....
[2] http://www.ucolick.org/~krumholz/courses/spring10_ast230/not...
The images are of electrons scattered from hydrogen atoms, directed onto a plate by an electromagnet. The scattering almost certainly has a lens effect, similar to shining a torch at a mirror ball. From the article:
the team fired two lasers at hydrogen atoms inside a chamber, kicking off electrons at speeds and directions that depended on their underlying wave functions.
A strong electric field inside the chamber guided the electrons to positions on a planar detector that depended on their initial velocities rather than on their initial positions.
However, it's the imaging mechanism that produced a correspondingly large picture of an atom of hydrogen.
So then I felt normal again, perhaps a little more ashamed than when I started reading the article.
In the experiment you linked to the researchers managed to force an electron into a huge orbit, on the scale of millimetres. In the OP's article they are imaging normal Hydrogen atoms.
It's very interesting, but unrelated as far as I can tell.
This scale would have to be a size projected onto a detector. The scale of the source would then be determined by the details of the experiment.
Edit: These are probably higher energy states. e.g., the 6s orbital: http://en.wikipedia.org/wiki/File:HydrogenOrbitalsN6L0M0.png
edit 2: Previous comment was still not correct. More plots of electron density for different energy states: http://cronodon.com/Atomic/AtomTech4.html
Why would these be such high-energy states?