Proton Electron Merger Can somebody explain what would happen if an electron & a proton, very close to each other are left to "fall" to each other in a straight line?
 A: I wish to point out that electrons are LEPTONS, and protons are HADRONS (forgive the SCREAMING). All protons are made up of 3 quarks (u u d). Neutrons have (u d d) quarks. Leptons have 0 quarks and do not participate in strong force interactions which are mediated by gluon exchanges between the hadron's component quarks. A lepton can only contribute energy (from its kinetic motion). While Coulomb attraction exists between the proton and incoming electron, you need a lot of energy to get a proton u quark to transition to d (a naive probability of ⅔, assuming you approach to within $10^{-17}$ meters), but to balance out all the quantum book-keeping, you also need an electron anti-neutrino!  A Feynman diagram would show this (see http://hst-archive.web.cern.ch/archiv/HST2002/feynman/examples.htm). As such, the electron would more probably lose its energy as Bremsstralung emissions.
A: They would form a hydrogen atom.
They would not merge, because merging would result in a neutron, which is heavier than a proton and an electron combined. However, if they are sufficiently forcibly "slammed" into each other, they might form a neutron (and an electron neutrino, which has an negligible mass). The neutron would later decay into proton, electron and electron anti-neutrino (it has a half-life of about 10 minutes).
A: If they are simply falling directly towards each other, they can't combine. To combine, they would need to form a neutron, but a neutron has slightly more mass. The extra mass would have to come from another particle, or source of energy - for example smashing them together forcefully enough.
So as they can't combine, they would remain as a proton and electron. They would be attracted to each other because of having an opposite charge, but when they got "too close", the nuclear interactions would become dominant (more powerful) and cause them to repel each other.
Another way of looking at the energy needed to merge, is its the energy needed to overcome that repulsion when they get very close.
So they would end up close but not too close. Attracted electrically, but unable to get very much closer or merge.
So it would remain as a hydrogen atom - a proton with a single bound electron.
A: 
Can somebody explain what would happen if an electron & a proton, very close to each other are left to "fall" to each other in a straight line?

One of the three solid evidences that classical electrodynamics and mechanics could not describe electrons, protons and atoms was exactly the fact that in classical electrodynamics the electron attracted by the proton charge would by acceleration fall on the proton neutralizing it, with a continuous electromagnetic radiation.
Instead there existed discrete  frequencies , the atomic spectra. Quantum mechanics was invented, leading to fitting the hydrogen spectra with quantized energy solutions.
The other two experimental no-goes of classical physics that quantum mechanics explained mathematically at the time were the photoelectric effect and black body radiation.
A: There is a probability that they will form a neutron, a hydrogen atom in some s state or an unbound electron proton system. Each of these possibilities can occur with a relative probability depending on the initial state. So it is not correct to say that a hydrogen atom must result, even without specifying in which state.
Quantum mechanics tells us that a very localised electron  centered at a proton corresponds to a superposition of hydrogen bound and ionised states. A very localised electron has very high kinetic energy which may exceed the potential energy. The quickest way to see this is to use Heisenberg's uncertainty principle for position and momentum. HUP tells you that a strongly localised electron wave function requires a superposition of very high momentum waves. Very high momentum means very high kinetic energy as well.
Note that if components with high enough kinetic energy are present to overcome the mass difference of neutron and proton and to create an electron neutrino of sufficient energy and momentum, also a neutron plus neutrino may be formed.
